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In Chapter 3 I talked a lot about variables, how they’re assigned and some of the things you can do with them, but there’s a lot of additional complexities. That’s not a surprise of course. However, some of those issues are worth drawing your attention to now. So that’s the goal of this section; to cover a few extra topics. As a consequence, this section is basically a bunch of things that I want to briefly mention, but don’t really fit in anywhere else. In short, I’ll talk about several different issues in this section, which are only loosely connected to one another.
Special values
The first thing I want to mention are some of the “special” values that you might see R produce. Most likely you’ll see them in situations where you were expecting a number, but there are quite a few other ways you can encounter them. These values are `Inf`, `NaN`, `NA` and `NULL`. These values can crop up in various different places, and so it’s important to understand what they mean.
• Infinity (`Inf`). The easiest of the special values to explain is `Inf`, since it corresponds to a value that is infinitely large. You can also have `-Inf`. The easiest way to get `Inf` is to divide a positive number by 0:
````1 / 0`
```
``## [1] Inf``
In most real world data analysis situations, if you’re ending up with infinite numbers in your data, then something has gone awry. Hopefully you’ll never have to see them.
• Not a Number (`NaN`). The special value of `NaN` is short for “not a number”, and it’s basically a reserved keyword that means “there isn’t a mathematically defined number for this”. If you can remember your high school maths, remember that it is conventional to say that 0/0 doesn’t have a proper answer: mathematicians would say that 0/0 is undefined. R says that it’s not a number:
`` 0 / 0``
````## [1] NaN`
```
Nevertheless, it’s still treated as a “numeric” value. To oversimplify, `NaN` corresponds to cases where you asked a proper numerical question that genuinely has no meaningful answer.
• Not available (`NA`). `NA` indicates that the value that is “supposed” to be stored here is missing. To understand what this means, it helps to recognise that the `NA` value is something that you’re most likely to see when analysing data from real world experiments. Sometimes you get equipment failures, or you lose some of the data, or whatever. The point is that some of the information that you were “expecting” to get from your study is just plain missing. Note the difference between `NA` and `NaN`. For `NaN`, we really do know what’s supposed to be stored; it’s just that it happens to correspond to something like 0/0 that doesn’t make any sense at all. In contrast, `NA` indicates that we actually don’t know what was supposed to be there. The information is missing.
• No value (`NULL`). The `NULL` value takes this “absence” concept even further. It basically asserts that the variable genuinely has no value whatsoever. This is quite different to both `NaN` and `NA`. For `NaN` we actually know what the value is, because it’s something insane like 0/0. For `NA`, we believe that there is supposed to be a value “out there”, but a dog ate our homework and so we don’t quite know what it is. But for `NULL` we strongly believe that there is no value at all.
Assigning names to vector elements
One thing that is sometimes a little unsatisfying about the way that R prints out a vector is that the elements come out unlabelled. Here’s what I mean. Suppose I’ve got data reporting the quarterly profits for some company. If I just create a no-frills vector, I have to rely on memory to know which element corresponds to which event. That is:
``````profit <- c( 3.1, 0.1, -1.4, 1.1 )
profit``````
``## [1] 3.1 0.1 -1.4 1.1``
You can probably guess that the first element corresponds to the first quarter, the second element to the second quarter, and so on, but that’s only because I’ve told you the back story and because this happens to be a very simple example. In general, it can be quite difficult. This is where it can be helpful to assign `names` to each of the elements. Here’s how you do it:
``````names(profit) <- c("Q1","Q2","Q3","Q4")
profit``````
``````## Q1 Q2 Q3 Q4
## 3.1 0.1 -1.4 1.1``````
This is a slightly odd looking command, admittedly, but it’s not too difficult to follow. All we’re doing is assigning a vector of labels (character strings) to `names(profit)`. You can always delete the names again by using the command `names(profit) <- NULL`. It’s also worth noting that you don’t have to do this as a two stage process. You can get the same result with this command:
``````profit <- c( "Q1" = 3.1, "Q2" = 0.1, "Q3" = -1.4, "Q4" = 1.1 )
profit``````
``````## Q1 Q2 Q3 Q4
## 3.1 0.1 -1.4 1.1``````
The important things to notice are that (a) this does make things much easier to read, but (b) the names at the top aren’t the “real” data. The value of `profit[1]` is still `3.1`; all I’ve done is added a name to `profit[1]` as well. Nevertheless, names aren’t purely cosmetic, since R allows you to pull out particular elements of the vector by referring to their names:
``profit["Q1"]``
``````## Q1
## 3.1``````
And if I ever need to pull out the names themselves, then I just type `names(profit)`.
Variable classes
As we’ve seen, R allows you to store different kinds of data. In particular, the variables we’ve defined so far have either been character data (text), numeric data, or logical data.56 It’s important that we remember what kind of information each variable stores (and even more important that R remembers) since different kinds of variables allow you to do different things to them. For instance, if your variables have numerical information in them, then it’s okay to multiply them together:
``````x <- 5 # x is numeric
y <- 4 # y is numeric
x * y ``````
``## [1] 20``
But if they contain character data, multiplication makes no sense whatsoever, and R will complain if you try to do it:
``````x <- "apples" # x is character
y <- "oranges" # y is character
x * y ``````
````## Error in x * y: non-numeric argument to binary operator`
```
Even R is smart enough to know you can’t multiply `"apples"` by `"oranges"`. It knows this because the quote marks are indicators that the variable is supposed to be treated as text, not as a number.
This is quite useful, but notice that it means that R makes a big distinction between `5` and `"5"`. Without quote marks, R treats `5` as the number five, and will allow you to do calculations with it. With the quote marks, R treats `"5"` as the textual character five, and doesn’t recognise it as a number any more than it recognises `"p"` or `"five"` as numbers. As a consequence, there’s a big difference between typing `x <- 5` and typing `x <- "5"`. In the former, we’re storing the number `5`; in the latter, we’re storing the character `"5"`. Thus, if we try to do multiplication with the character versions, R gets stroppy:
``````x <- "5" # x is character
y <- "4" # y is character
x * y ``````
``## Error in x * y: non-numeric argument to binary operator``
Okay, let’s suppose that I’ve forgotten what kind of data I stored in the variable `x` (which happens depressingly often). R provides a function that will let us find out. Or, more precisely, it provides three functions: `class()`, `mode()` and `typeof()`. Why the heck does it provide three functions, you might be wondering? Basically, because R actually keeps track of three different kinds of information about a variable:
1. The class of a variable is a “high level” classification, and it captures psychologically (or statistically) meaningful distinctions. For instance `"2011-09-12"` and `"my birthday"` are both text strings, but there’s an important difference between the two: one of them is a date. So it would be nice if we could get R to recognise that `"2011-09-12"` is a date, and allow us to do things like add or subtract from it. The class of a variable is what R uses to keep track of things like that. Because the class of a variable is critical for determining what R can or can’t do with it, the `class()` function is very handy.
2. The mode of a variable refers to the format of the information that the variable stores. It tells you whether R has stored text data or numeric data, for instance, which is kind of useful, but it only makes these “simple” distinctions. It can be useful to know about, but it’s not the main thing we care about. So I’m not going to use the `mode()` function very much.57
3. The type of a variable is a very low level classification. We won’t use it in this book, but (for those of you that care about these details) this is where you can see the distinction between integer data, double precision numeric, etc. Almost none of you actually will care about this, so I’m not even going to bother demonstrating the `typeof()` function.
For purposes, it’s the `class()` of the variable that we care most about. Later on, I’ll talk a bit about how you can convince R to “coerce” a variable to change from one class to another (Section 7.10). That’s a useful skill for real world data analysis, but it’s not something that we need right now. In the meantime, the following examples illustrate the use of the `class()` function:
``````x <- "hello world" # x is text
class(x)``` ```
``## [1] "character"``
``````x <- TRUE # x is logical
class(x)```
```
``## [1] "logical"``
``````x <- 100 # x is a number
class(x)```
```
``## [1] "numeric"``
Exciting, no? | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/04%3A_Additional_R_Concepts/4.06%3A_Useful_Things_to_Know_about_Variables.txt |
Okay, it’s time to start introducing some of the data types that are somewhat more specific to statistics. If you remember back to Chapter 2, when we assign numbers to possible outcomes, these numbers can mean quite different things depending on what kind of variable we are attempting to measure. In particular, we commonly make the distinction between nominal, ordinal, interval and ratio scale data. How do we capture this distinction in R? Currently, we only seem to have a single numeric data type. That’s probably not going to be enough, is it?
A little thought suggests that the numeric variable class in R is perfectly suited for capturing ratio scale data. For instance, if I were to measure response time (RT) for five different events, I could store the data in R like this:
``RT <- c(342, 401, 590, 391, 554)``
where the data here are measured in milliseconds, as is conventional in the psychological literature. It’s perfectly sensible to talk about “twice the response time”, 2×RT, or the “response time plus 1 second”, RT+1000, and so both of the following are perfectly reasonable things for R to do:
````2 * RT`
```
``## [1] 684 802 1180 782 1108``
````RT + 1000`
```
``## [1] 1342 1401 1590 1391 1554``
And to a lesser extent, the “numeric” class is okay for interval scale data, as long as we remember that multiplication and division aren’t terribly interesting for these sorts of variables. That is, if my IQ score is 110 and yours is 120, it’s perfectly okay to say that you’re 10 IQ points smarter than me58, but it’s not okay to say that I’m only 92% as smart as you are, because intelligence doesn’t have a natural zero.59 We might even be willing to tolerate the use of numeric variables to represent ordinal scale variables, such as those that you typically get when you ask people to rank order items (e.g., like we do in Australian elections), though as we will see R actually has a built in tool for representing ordinal data (see Section 7.11.2) However, when it comes to nominal scale data, it becomes completely unacceptable, because almost all of the “usual” rules for what you’re allowed to do with numbers don’t apply to nominal scale data. It is for this reason that R has factors.
Introducing factors
Suppose, I was doing a study in which people could belong to one of three different treatment conditions. Each group of people were asked to complete the same task, but each group received different instructions. Not surprisingly, I might want to have a variable that keeps track of what group people were in. So I could type in something like this
``group <- c(1,1,1,2,2,2,3,3,3)``
so that `group[i]` contains the group membership of the `i`-th person in my study. Clearly, this is numeric data, but equally obviously this is a nominal scale variable. There’s no sense in which “group 1” plus “group 2” equals “group 3”, but nevertheless if I try to do that, R won’t stop me because it doesn’t know any better:
````group + 2`
```
``## [1] 3 3 3 4 4 4 5 5 5``
Apparently R seems to think that it’s allowed to invent “group 4” and “group 5”, even though they didn’t actually exist. Unfortunately, R is too stupid to know any better: it thinks that `3` is an ordinary number in this context, so it sees no problem in calculating `3 + 2`. But since we’re not that stupid, we’d like to stop R from doing this. We can do so by instructing R to treat `group` as a factor. This is easy to do using the `as.factor()` function.60
``````group <- as.factor(group)
group```
```
``````## [1] 1 1 1 2 2 2 3 3 3
## Levels: 1 2 3``````
It looks more or less the same as before (though it’s not immediately obvious what all that `Levels` rubbish is about), but if we ask R to tell us what the class of the `group` variable is now, it’s clear that it has done what we asked:
````class(group)`
```
``## [1] "factor"``
Neat. Better yet, now that I’ve converted `group` to a factor, look what happens when I try to add 2 to it:
````group + 2`
```
``## Warning in Ops.factor(group, 2): '+' not meaningful for factors``
``## [1] NA NA NA NA NA NA NA NA NA``
This time even R is smart enough to know that I’m being an idiot, so it tells me off and then produces a vector of missing values. (i.e., `NA`: see Section 4.6.1).
Labelling the factor levels
I have a confession to make. My memory is not infinite in capacity; and it seems to be getting worse as I get older. So it kind of annoys me when I get data sets where there’s a nominal scale variable called `gender`, with two levels corresponding to males and females. But when I go to print out the variable I get something like this:
``gender``
``````## [1] 1 1 1 1 1 2 2 2 2
## Levels: 1 2``````
Okaaaay. That’s not helpful at all, and it makes me very sad. Which number corresponds to the males and which one corresponds to the females? Wouldn’t it be nice if R could actually keep track of this? It’s way too hard to remember which number corresponds to which gender. And besides, the problem that this causes is much more serious than a single sad nerd… because R has no way of knowing that the `1`s in the `group` variable are a very different kind of thing to the `1`s in the `gender` variable. So if I try to ask which elements of the `group` variable are equal to the corresponding elements in `gender`, R thinks this is totally kosher, and gives me this:
``group == gender``
````## Error in Ops.factor(group, gender): level sets of factors are different`
```
Well, that’s … especially stupid.61 The problem here is that R is very literal minded. Even though you’ve declared both `group` and `gender` to be factors, it still assumes that a `1` is a `1` no matter which variable it appears in.
To fix both of these problems (my memory problem, and R’s infuriating literal interpretations), what we need to do is assign meaningful labels to the different levels of each factor. We can do that like this:
``````levels(group) <- c("group 1", "group 2", "group 3")
print(group)``````
``````## [1] group 1 group 1 group 1 group 2 group 2 group 2 group 3 group 3 group 3
## Levels: group 1 group 2 group 3``````
``````levels(gender) <- c("male", "female")
print(gender)```
```
``````## [1] male male male male male female female female female
## Levels: male female``````
That’s much easier on the eye, and better yet, R is smart enough to know that `"female"` is not equal to `"group 2"`, so now when I try to ask which group memberships are “equal to” the gender of the corresponding person,
``group == gender``
``## Error in Ops.factor(group, gender): level sets of factors are different``
R correctly tells me that I’m an idiot.
Moving on…
Factors are very useful things, and we’ll use them a lot in this book: they’re the main way to represent a nominal scale variable. And there are lots of nominal scale variables out there. I’ll talk more about factors in 7.11.2, but for now you know enough to be able to get started. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/04%3A_Additional_R_Concepts/4.07%3A_Factors.txt |
It’s now time to go back and deal with the somewhat confusing thing that happened in Section ?? when we tried to open up a CSV file. Apparently we succeeded in loading the data, but it came to us in a very odd looking format. At the time, I told you that this was a data frame. Now I’d better explain what that means.
Introducing data frames
In order to understand why R has created this funny thing called a data frame, it helps to try to see what problem it solves. So let’s go back to the little scenario that I used when introducing factors in Section 4.7. In that section I recorded the `group` and `gender` for all 9 participants in my study. Let’s also suppose I recorded their ages and their `score` on “Dan’s Terribly Exciting Psychological Test”:
``````age <- c(17, 19, 21, 37, 18, 19, 47, 18, 19)
score <- c(12, 10, 11, 15, 16, 14, 25, 21, 29)``````
Assuming no other variables are in the workspace, if I type `who()` I get this:
````who()`
```
``````## -- Name -- -- Class -- -- Size --
## age numeric 9
## gender factor 9
## group factor 9
## score numeric 9``````
So there are four variables in the workspace, `age`, `gender`, `group` and `score`. And it just so happens that all four of them are the same size (i.e., they’re all vectors with 9 elements). Aaaand it just so happens that `age[1]` corresponds to the age of the first person, and `gender[1]` is the gender of that very same person, etc. In other words, you and I both know that all four of these variables correspond to the same data set, and all four of them are organised in exactly the same way.
However, R doesn’t know this! As far as it’s concerned, there’s no reason why the `age` variable has to be the same length as the `gender` variable; and there’s no particular reason to think that `age[1]` has any special relationship to `gender[1]` any more than it has a special relationship to `gender[4]`. In other words, when we store everything in separate variables like this, R doesn’t know anything about the relationships between things. It doesn’t even really know that these variables actually refer to a proper data set. The data frame fixes this: if we store our variables inside a data frame, we’re telling R to treat these variables as a single, fairly coherent data set.
To see how they do this, let’s create one. So how do we create a data frame? One way we’ve already seen: if we import our data from a CSV file, R will store it as a data frame. A second way is to create it directly from some existing variables using the `data.frame()` function. All you have to do is type a list of variables that you want to include in the data frame. The output of a `data.frame()` command is, well, a data frame. So, if I want to store all four variables from my experiment in a data frame called `expt` I can do so like this:
``````expt <- data.frame ( age, gender, group, score )
expt ```
```
``````## age gender group score
## 1 17 male group 1 12
## 2 19 male group 1 10
## 3 21 male group 1 11
## 4 37 male group 2 15
## 5 18 male group 2 16
## 6 19 female group 2 14
## 7 47 female group 3 25
## 8 18 female group 3 21
## 9 19 female group 3 29``````
Note that `expt` is a completely self-contained variable. Once you’ve created it, it no longer depends on the original variables from which it was constructed. That is, if we make changes to the original `age` variable, it will not lead to any changes to the age data stored in `expt`.
At this point, our workspace contains only the one variable, a data frame called `expt`. But as we can see when we told R to print the variable out, this data frame contains 4 variables, each of which has 9 observations. So how do we get this information out again? After all, there’s no point in storing information if you don’t use it, and there’s no way to use information if you can’t access it. So let’s talk a bit about how to pull information out of a data frame.
The first thing we might want to do is pull out one of our stored variables, let’s say `score`. One thing you might try to do is ignore the fact that `score` is locked up inside the `expt` data frame. For instance, you might try to print it out like this:
``score``
````## Error in eval(expr, envir, enclos): object 'score' not found`
```
This doesn’t work, because R doesn’t go “peeking” inside the data frame unless you explicitly tell it to do so. There’s actually a very good reason for this, which I’ll explain in a moment, but for now let’s just assume R knows what it’s doing. How do we tell R to look inside the data frame? As is always the case with R there are several ways. The simplest way is to use the `\$` operator to extract the variable you’re interested in, like this:
``expt\$score``
``## [1] 12 10 11 15 16 14 25 21 29``
Getting information about a data frame
One problem that sometimes comes up in practice is that you forget what you called all your variables. Normally you might try to type `objects()` or `who()`, but neither of those commands will tell you what the names are for those variables inside a data frame! One way is to ask R to tell you what the names of all the variables stored in the data frame are, which you can do using the `names()` function:
``names(expt)``
``## [1] "age" "gender" "group" "score"``
An alternative method is to use the `who()` function, as long as you tell it to look at the variables inside data frames. If you set `expand = TRUE` then it will not only list the variables in the workspace, but it will “expand” any data frames that you’ve got in the workspace, so that you can see what they look like. That is:
``who(expand = TRUE)``
``````## -- Name -- -- Class -- -- Size --
## expt data.frame 9 x 4
## \$age numeric 9
## \$gender factor 9
## \$group factor 9
## \$score numeric 9``````
or, since `expand` is the first argument in the `who()` function you can just type `who(TRUE)`. I’ll do that a lot in this book.
Looking for more on data frames?
There’s a lot more that can be said about data frames: they’re fairly complicated beasts, and the longer you use R the more important it is to make sure you really understand them. We’ll talk a lot more about them in Chapter 7. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/04%3A_Additional_R_Concepts/4.08%3A_Data_frames.txt |
The next kind of data I want to mention are lists. Lists are an extremely fundamental data structure in R, and as you start making the transition from a novice to a savvy R user you will use lists all the time. I don’t use lists very often in this book – not directly – but most of the advanced data structures in R are built from lists (e.g., data frames are actually a specific type of list). Because lists are so important to how R stores things, it’s useful to have a basic understanding of them. Okay, so what is a list, exactly? Like data frames, lists are just “collections of variables.” However, unlike data frames – which are basically supposed to look like a nice “rectangular” table of data – there are no constraints on what kinds of variables we include, and no requirement that the variables have any particular relationship to one another. In order to understand what this actually means, the best thing to do is create a list, which we can do using the `list()` function. If I type this as my command:
``````Dan <- list( age = 34,
nerd = TRUE,
parents = c("Joe","Liz")
)``````
R creates a new list variable called `Dan`, which is a bundle of three different variables: `age`, `nerd` and `parents`. Notice, that the `parents` variable is longer than the others. This is perfectly acceptable for a list, but it wouldn’t be for a data frame. If we now print out the variable, you can see the way that R stores the list:
````print( Dan )`
```
``````## \$age
## [1] 34
##
## \$nerd
## [1] TRUE
##
## \$parents
## [1] "Joe" "Liz"``````
As you might have guessed from those `\$` symbols everywhere, the variables are stored in exactly the same way that they are for a data frame (again, this is not surprising: data frames are a type of list). So you will (I hope) be entirely unsurprised and probably quite bored when I tell you that you can extract the variables from the list using the `\$` operator, like so:
````Dan\$nerd`
```
``## [1] TRUE``
If you need to add new entries to the list, the easiest way to do so is to again use `\$`, as the following example illustrates. If I type a command like this
``Dan\$children <- "Alex"``
then R creates a new entry to the end of the list called `children`, and assigns it a value of `"Alex"`. If I were now to `print()` this list out, you’d see a new entry at the bottom of the printout. Finally, it’s actually possible for lists to contain other lists, so it’s quite possible that I would end up using a command like `Dan\$children\$age` to find out how old my son is. Or I could try to remember it myself I suppose.
4.10: Formulas
The last kind of variable that I want to introduce before finally being able to start talking about statistics is the formula. Formulas were originally introduced into R as a convenient way to specify a particular type of statistical model (see Chapter15) but they’re such handy things that they’ve spread. Formulas are now used in a lot of different contexts, so it makes sense to introduce them early.
Stated simply, a formula object is a variable, but it’s a special type of variable that specifies a relationship between other variables. A formula is specified using the “tilde operator” #~#. A very simple example of a formula is shown below:62
``````formula1 <- out ~ pred
formula1``````
``## out ~ pred``
The precise meaning of this formula depends on exactly what you want to do with it, but in broad terms it means “the `out` (outcome) variable, analysed in terms of the `pred` (predictor) variable”. That said, although the simplest and most common form of a formula uses the “one variable on the left, one variable on the right” format, there are others. For instance, the following examples are all reasonably common
``````formula2 <- out ~ pred1 + pred2 # more than one variable on the right
formula3 <- out ~ pred1 * pred2 # different relationship between predictors
formula4 <- ~ var1 + var2 # a 'one-sided' formula``````
and there are many more variants besides. Formulas are pretty flexible things, and so different functions will make use of different formats, depending on what the function is intended to do. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/04%3A_Additional_R_Concepts/4.09%3A_Lists.txt |
There’s one really important thing that I omitted when I discussed functions earlier on in Section 3.5, and that’s the concept of a generic function. The two most notable examples that you’ll see in the next few chapters are `summary()` and `plot()`, although you’ve already seen an example of one working behind the scenes, and that’s the `print()` function. The thing that makes generics different from the other functions is that their behaviour changes, often quite dramatically, depending on the `class()` of the input you give it. The easiest way to explain the concept is with an example. With that in mind, lets take a closer look at what the `print()` function actually does. I’ll do this by creating a formula, and printing it out in a few different ways. First, let’s stick with what we know:
``````my.formula <- blah ~ blah.blah # create a variable of class "formula"
print( my.formula ) # print it out using the generic print() function``````
``## blah ~ blah.blah``
So far, there’s nothing very surprising here. But there’s actually a lot going on behind the scenes here. When I type `print( my.formula )`, what actually happens is the `print()` function checks the class of the `my.formula` variable. When the function discovers that the variable it’s been given is a formula, it goes looking for a function called `print.formula()`, and then delegates the whole business of printing out the variable to the `print.formula()` function.63 For what it’s worth, the name for a “dedicated” function like `print.formula()` that exists only to be a special case of a generic function like `print()` is a method, and the name for the process in which the generic function passes off all the hard work onto a method is called method dispatch. You won’t need to understand the details at all for this book, but you do need to know the gist of it; if only because a lot of the functions we’ll use are actually generics. Anyway, to help expose a little more of the workings to you, let’s bypass the `print()` function entirely and call the formula method directly:
``````print.formula( my.formula ) # print it out using the print.formula() method
## Appears to be deprecated``````
There’s no difference in the output at all. But this shouldn’t surprise you because it was actually the `print.formula()` method that was doing all the hard work in the first place. The `print()` function itself is a lazy bastard that doesn’t do anything other than select which of the methods is going to do the actual printing.
Okay, fair enough, but you might be wondering what would have happened if `print.formula()` didn’t exist? That is, what happens if there isn’t a specific method defined for the class of variable that you’re using? In that case, the generic function passes off the hard work to a “default” method, whose name in this case would be `print.default()`. Let’s see what happens if we bypass the `print()` formula, and try to print out `my.formula` using the `print.default()` function:
``print.default( my.formula ) # print it out using the print.default() method``
``````## blah ~ blah.blah
## attr(,"class")
## [1] "formula"
## attr(,".Environment")
## <environment: R_GlobalEnv>```
```
Hm. You can kind of see that it is trying to print out the same formula, but there’s a bunch of ugly low-level details that have also turned up on screen. This is because the `print.default()` method doesn’t know anything about formulas, and doesn’t know that it’s supposed to be hiding the obnoxious internal gibberish that R produces sometimes.
At this stage, this is about as much as we need to know about generic functions and their methods. In fact, you can get through the entire book without learning any more about them than this, so it’s probably a good idea to end this discussion here. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/04%3A_Additional_R_Concepts/4.11%3A_Generic_Functions.txt |
The very last topic I want to mention in this chapter is where to go to find help. Obviously, I’ve tried to make this book as helpful as possible, but it’s not even close to being a comprehensive guide, and there’s thousands of things it doesn’t cover. So where should you go for help?
Other resources
• The Rseek website (www.rseek.org). One thing that I really find annoying about the R help documentation is that it’s hard to search properly. When coupled with the fact that the documentation is dense and highly technical, it’s often a better idea to search or ask online for answers to your questions. With that in mind, the Rseek website is great: it’s an R specific search engine. I find it really useful, and it’s almost always my first port of call when I’m looking around.
• The R-help mailing list (see http://www.r-project.org/mail.html for details). This is the official R help mailing list. It can be very helpful, but it’s very important that you do your homework before posting a question. The list gets a lot of traffic. While the people on the list try as hard as they can to answer questions, they do so for free, and you really don’t want to know how much money they could charge on an hourly rate if they wanted to apply market rates. In short, they are doing you a favour, so be polite. Don’t waste their time asking questions that can be easily answered by a quick search on Rseek (it’s rude), make sure your question is clear, and all of the relevant information is included. In short, read the posting guidelines carefully (http://www.r-project.org/posting-guide.html), and make use of the `help.request()` function that R provides to check that you’re actually doing what you’re expected.
Taken together, Chapters 3 and 4 provide enough of a background that you can finally get started doing some statistics! Yes, there’s a lot more R concepts that you ought to know (and we’ll talk about some of them in Chapters 7 and 8), but I think that we’ve talked quite enough about programming for the moment. It’s time to see how your experience with programming can be used to do some data analysis…
References
Fox, J., and S. Weisberg. 2011. An R Companion to Applied Regression. 2nd ed. Los Angeles: Sage.
1. Notice that I used `print(keeper)` rather than just typing `keeper`. Later on in the text I’ll sometimes use the `print()` function to display things because I think it helps make clear what I’m doing, but in practice people rarely do this.
2. More precisely, there are 5000 or so packages on CRAN, the Comprehensive R Archive Network.
3. Basically, the reason is that there are 5000 packages, and probably about 4000 authors of packages, and no-one really knows what all of them do. Keeping the installation separate from the loading minimizes the chances that two packages will interact with each other in a nasty way.
4. If you’re using the command line, you can get the same information by typing `library()` at the command line.
5. The logit function a simple mathematical function that happens not to have been included in the basic R distribution.
6. Tip for advanced users. You can get R to use the one from the `car` package by using `car::logit()` as your command rather than `logit()`, since the `car::` part tells R explicitly which package to use. See also `:::` if you’re especially keen to force R to use functions it otherwise wouldn’t, but take care, since `:::` can be dangerous.
7. It is not very difficult.
8. This would be especially annoying if you’re reading an electronic copy of the book because the text displayed by the `who()` function is searchable, whereas text shown in a screen shot isn’t!
9. Mind you, all that means is that it’s been removed from the workspace. If you’ve got the data saved to file somewhere, then that file is perfectly safe.
10. Well, the partition, technically.
11. One additional thing worth calling your attention to is the `file.choose()` function. Suppose you want to load a file and you don’t quite remember where it is, but would like to browse for it. Typing `file.choose()` at the command line will open a window in which you can browse to find the file; when you click on the file you want, R will print out the full path to that file. This is kind of handy.
12. Notably those with .rda, .Rd, .Rhistory, .rdb and .rdx extensions
13. In a lot of books you’ll see the `read.table()` function used for this purpose instead of `read.csv()`. They’re more or less identical functions, with the same arguments and everything. They differ only in the default values.
14. Note that I didn’t to this in my earlier example when loading the .Rdata
15. A word of warning: what you don’t want to do is use the “File” menu. If you look in the “File” menu you will see “Save” and “Save As…” options, but they don’t save the workspace. Those options are used for dealing with scripts, and so they’ll produce `.R` files. We won’t get to those until Chapter 8.
16. Or functions. But let’s ignore functions for the moment.
17. Actually, I don’t think I ever use this in practice. I don’t know why I bother to talk about it in the book anymore.
18. Taking all the usual caveats that attach to IQ measurement as a given, of course.
19. Or, more precisely, we don’t know how to measure it. Arguably, a rock has zero intelligence. But it doesn’t make sense to say that the IQ of a rock is 0 in the same way that we can say that the average human has an IQ of 100. And without knowing what the IQ value is that corresponds to a literal absence of any capacity to think, reason or learn, then we really can’t multiply or divide IQ scores and expect a meaningful answer.
20. Once again, this is an example of coercing a variable from one class to another. I’ll talk about coercion in more detail in Section 7.10.
21. Some users might wonder why R even allows the `==` operator for factors. The reason is that sometimes you really do have different factors that have the same levels. For instance, if I was analysing data associated with football games, I might have a factor called `home.team`, and another factor called `winning.team`. In that situation I really should be able to ask if `home.team == winning.team`.
22. Note that, when I write out the formula, R doesn’t check to see if the `out` and `pred` variables actually exist: it’s only later on when you try to use the formula for something that this happens.
23. For readers with a programming background: what I’m describing is the very basics of how S3 methods work. However, you should be aware that R has two entirely distinct systems for doing object oriented programming, known as S3 and S4. Of the two, S3 is simpler and more informal, whereas S4 supports all the stuff that you might expect of a fully object oriented language. Most of the generics we’ll run into in this book use the S3 system, which is convenient for me because I’m still trying to figure out S4.
4.13: Summary
This chapter continued where Chapter 3 left off. The focus was still primarily on introducing basic R concepts, but this time at least you can see how those concepts are related to data analysis:
• Installing, loading and updating packages. Knowing how to extend the functionality of R by installing and using packages is critical to becoming an effective R user (Section 4.2)
• Getting around. Section 4.3 talked about how to manage your workspace and how to keep it tidy. Similarly, Section 4.4 talked about how to get R to interact with the rest of the file system.
• Loading and saving data. Finally, we encountered actual data files. Loading and saving data is obviously a crucial skill, one we discussed in Section 4.5.
• Useful things to know about variables. In particular, we talked about special values, element names and classes (Section 4.6).
• More complex types of variables. R has a number of important variable types that will be useful when analysing real data. I talked about factors in Section 4.7, data frames in Section 4.8, lists in Section 4.9 and formulas in Section 4.10.
• Generic functions. How is it that some function seem to be able to do lots of different things? Section 4.11 tells you how.
• Getting help. Assuming that you’re not looking for counselling, Section 4.12 covers several possibilities. If you are looking for counselling, well, this book really can’t help you there. Sorry. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/04%3A_Additional_R_Concepts/4.12%3A_Getting_help.txt |
Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what descriptive statistics (as opposed to inferential statistics) is all about. In fact, to many people the term “statistics” is synonymous with descriptive statistics. It is this topic that we’ll consider in this chapter, but before going into any details, let’s take a moment to get a sense of why we need descriptive statistics. To do this, let’s load the `aflsmall.Rdata` file, and use the `who()` function in the `lsr` package to see what variables are stored in the file:
``````load( "./data/aflsmall.Rdata" )
library(lsr)``````
``## Warning: package 'lsr' was built under R version 3.5.2``
``who()``
``````## -- Name -- -- Class -- -- Size --
## afl.finalists factor 400
## afl.margins numeric 176``````
There are two variables here, `afl.finalists` and `afl.margins`. We’ll focus a bit on these two variables in this chapter, so I’d better tell you what they are. Unlike most of data sets in this book, these are actually real data, relating to the Australian Football League (AFL)64 The `afl.margins` variable contains the winning margin (number of points) for all 176 home and away games played during the 2010 season. The `afl.finalists` variable contains the names of all 400 teams that played in all 200 finals matches played during the period 1987 to 2010. Let’s have a look at the `afl.margins` variable:
``print(afl.margins)``
``````## [1] 56 31 56 8 32 14 36 56 19 1 3 104 43 44 72 9 28
## [18] 25 27 55 20 16 16 7 23 40 48 64 22 55 95 15 49 52
## [35] 50 10 65 12 39 36 3 26 23 20 43 108 53 38 4 8 3
## [52] 13 66 67 50 61 36 38 29 9 81 3 26 12 36 37 70 1
## [69] 35 12 50 35 9 54 47 8 47 2 29 61 38 41 23 24 1
## [86] 9 11 10 29 47 71 38 49 65 18 0 16 9 19 36 60 24
## [103] 25 44 55 3 57 83 84 35 4 35 26 22 2 14 19 30 19
## [120] 68 11 75 48 32 36 39 50 11 0 63 82 26 3 82 73 19
## [137] 33 48 8 10 53 20 71 75 76 54 44 5 22 94 29 8 98
## [154] 9 89 1 101 7 21 52 42 21 116 3 44 29 27 16 6 44
## [171] 3 28 38 29 10 10``````
This output doesn’t make it easy to get a sense of what the data are actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data. In order to get some idea about what’s going on, we need to calculate some descriptive statistics (this chapter) and draw some nice pictures (Chapter 6. Since the descriptive statistics are the easier of the two topics, I’ll start with those, but nevertheless I’ll show you a histogram of the `afl.margins` data, since it should help you get a sense of what the data we’re trying to describe actually look like. But for what it’s worth, this histogram – which is shown in Figure 5.1 – was generated using the `hist()` function. We’ll talk a lot more about how to draw histograms in Section 6.3. For now, it’s enough to look at the histogram and note that it provides a fairly interpretable representation of the `afl.margins` data.
05: Descriptive Statistics
Drawing pictures of the data, as I did in Figure 5.1 is an excellent way to convey the “gist” of what the data is trying to tell you, it’s often extremely useful to try to condense the data into a few simple “summary” statistics. In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you’d like to know something about the “average” or “middle” of your data lies. The two most commonly used measures are the mean, median and mode; occasionally people will also report a trimmed mean. I’ll explain each of these in turn, and then discuss when each of them is useful.
mean
The mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values. The first five AFL margins were 56, 31, 56, 8 and 32, so the mean of these observations is just:
$\dfrac{56+31+56+8+32}{5}=\dfrac{183}{5}=36.60 \nonumber$
Of course, this definition of the mean isn’t news to anyone: averages (i.e., means) are used so often in everyday life that this is pretty familiar stuff. However, since the concept of a mean is something that everyone already understands, I’ll use this as an excuse to start introducing some of the mathematical notation that statisticians use to describe this calculation, and talk about how the calculations would be done in R.
The first piece of notation to introduce is N, which we’ll use to refer to the number of observations that we’re averaging (in this case N=5). Next, we need to attach a label to the observations themselves. It’s traditional to use X for this, and to use subscripts to indicate which observation we’re actually talking about. That is, we’ll use X1 to refer to the first observation, X2 to refer to the second observation, and so on, all the way up to XN for the last one. Or, to say the same thing in a slightly more abstract way, we use Xi to refer to the i-th observation. Just to make sure we’re clear on the notation, the following table lists the 5 observations in the afl.margins variable, along with the mathematical symbol used to refer to it, and the actual value that the observation corresponds to:
The Observation Its Symbol The Observed Value
winning margin, game 1 X1 56 points
winning margin, game 2 X2 31 points
winning margin, game 3 X3 56 points
winning margin, game 4 X4 8 points
winning margin, game 5 X5 32 points
Okay, now let’s try to write a formula for the mean. By tradition, we use x̄ as the notation for the mean. So the calculation for the mean could be expressed using the following formula:
$\bar{X}=\dfrac{X_{1}+X_{2}+\ldots+X_{N-1}+X_{N}}{N} \nonumber$
This formula is entirely correct, but it’s terribly long, so we make use of the summation symbol ∑ to shorten it.65 If I want to add up the first five observations, I could write out the sum the long way, X1+X2+X3+X4+X5 or I could use the summation symbol to shorten it to this:
$\sum_{i=1}^{5} X_{i} \nonumber$
Taken literally, this could be read as “the sum, taken over all i values from 1 to 5, of the value Xi”. But basically, what it means is “add up the first five observations”. In any case, we can use this notation to write out the formula for the mean, which looks like this:
$\bar{X}=\dfrac{1}{N} \sum_{i=1}^{N} X_{i} \nonumber$
In all honesty, I can’t imagine that all this mathematical notation helps clarify the concept of the mean at all. In fact, it’s really just a fancy way of writing out the same thing I said in words: add all the values up, and then divide by the total number of items. However, that’s not really the reason I went into all that detail. My goal was to try to make sure that everyone reading this book is clear on the notation that we’ll be using throughout the book: $\bar{X}$ for the mean, ∑ for the idea of summation, Xi for the ith observation, and N for the total number of observations. We’re going to be re-using these symbols a fair bit, so it’s important that you understand them well enough to be able to “read” the equations, and to be able to see that it’s just saying “add up lots of things and then divide by another thing”.
Calculating the mean in R
Okay that’s the maths, how do we get the magic computing box to do the work for us? If you really wanted to, you could do this calculation directly in R. For the first 5 AFL scores, do this just by typing it in as if R were a calculator…
(56 + 31 + 56 + 8 + 32) / 5
## [1] 36.6
… in which case R outputs the answer 36.6, just as if it were a calculator. However, that’s not the only way to do the calculations, and when the number of observations starts to become large, it’s easily the most tedious. Besides, in almost every real world scenario, you’ve already got the actual numbers stored in a variable of some kind, just like we have with the afl.margins variable. Under those circumstances, what you want is a function that will just add up all the values stored in a numeric vector. That’s what the sum() function does. If we want to add up all 176 winning margins in the data set, we can do so using the following command:66
sum( afl.margins )
## [1] 6213
If we only want the sum of the first five observations, then we can use square brackets to pull out only the first five elements of the vector. So the command would now be:
sum( afl.margins[1:5] )
## [1] 183
To calculate the mean, we now tell R to divide the output of this summation by five, so the command that we need to type now becomes the following:
sum( afl.margins[1:5] ) / 5
## [1] 36.6
Although it’s pretty easy to calculate the mean using the sum() function, we can do it in an even easier way, since R also provides us with the mean() function. To calculate the mean for all 176 games, we would use the following command:
mean( x = afl.margins )
## [1] 35.30114
However, since x is the first argument to the function, I could have omitted the argument name. In any case, just to show you that there’s nothing funny going on, here’s what we would do to calculate the mean for the first five observations:
mean( afl.margins[1:5] )
## [1] 36.6
As you can see, this gives exactly the same answers as the previous calculations.
median
The second measure of central tendency that people use a lot is the median, and it’s even easier to describe than the mean. The median of a set of observations is just the middle value. As before let’s imagine we were interested only in the first 5 AFL winning margins: 56, 31, 56, 8 and 32. To figure out the median, we sort these numbers into ascending order:
8,31,32,56,56
From inspection, it’s obvious that the median value of these 5 observations is 32, since that’s the middle one in the sorted list (I’ve put it in bold to make it even more obvious). Easy stuff. But what should we do if we were interested in the first 6 games rather than the first 5? Since the sixth game in the season had a winning margin of 14 points, our sorted list is now
8,14,31,32,56,56
and there are two middle numbers, 31 and 32. The median is defined as the average of those two numbers, which is of course 31.5. As before, it’s very tedious to do this by hand when you’ve got lots of numbers. To illustrate this, here’s what happens when you use R to sort all 176 winning margins. First, I’ll use the sort() function (discussed in Chapter 7) to display the winning margins in increasing numerical order:
sort( x = afl.margins )
## [1] 0 0 1 1 1 1 2 2 3 3 3 3 3 3 3 3 4
## [18] 4 5 6 7 7 8 8 8 8 8 9 9 9 9 9 9 10
## [35] 10 10 10 10 11 11 11 12 12 12 13 14 14 15 16 16 16
## [52] 16 18 19 19 19 19 19 20 20 20 21 21 22 22 22 23 23
## [69] 23 24 24 25 25 26 26 26 26 27 27 28 28 29 29 29 29
## [86] 29 29 30 31 32 32 33 35 35 35 35 36 36 36 36 36 36
## [103] 37 38 38 38 38 38 39 39 40 41 42 43 43 44 44 44 44
## [120] 44 47 47 47 48 48 48 49 49 50 50 50 50 52 52 53 53
## [137] 54 54 55 55 55 56 56 56 57 60 61 61 63 64 65 65 66
## [154] 67 68 70 71 71 72 73 75 75 76 81 82 82 83 84 89 94
## [171] 95 98 101 104 108 116
The middle values are 30 and 31, so the median winning margin for 2010 was 30.5 points. In real life, of course, no-one actually calculates the median by sorting the data and then looking for the middle value. In real life, we use the median command:
median( x = afl.margins )
## [1] 30.5
which outputs the median value of 30.5.
Mean or median? What’s the difference?
Knowing how to calculate means and medians is only a part of the story. You also need to understand what each one is saying about the data, and what that implies for when you should use each one. This is illustrated in Figure 5.2 the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data. What this implies, as far as which one you should use, depends a little on what type of data you’ve got and what you’re trying to achieve. As a rough guide:
• If your data are nominal scale, you probably shouldn’t be using either the mean or the median. Both the mean and the median rely on the idea that the numbers assigned to values are meaningful. If the numbering scheme is arbitrary, then it’s probably best to use the mode (Section 5.1.7) instead.
• If your data are ordinal scale, you’re more likely to want to use the median than the mean. The median only makes use of the order information in your data (i.e., which numbers are bigger), but doesn’t depend on the precise numbers involved. That’s exactly the situation that applies when your data are ordinal scale. The mean, on the other hand, makes use of the precise numeric values assigned to the observations, so it’s not really appropriate for ordinal data.
• For interval and ratio scale data, either one is generally acceptable. Which one you pick depends a bit on what you’re trying to achieve. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values, as we’ll see in Section 5.1.6.
Let’s expand on that last part a little. One consequence is that there’s systematic differences between the mean and the median when the histogram is asymmetric (skewed; see Section 5.3). This is illustrated in Figure 5.2 notice that the median (right hand side) is located closer to the “body” of the histogram, whereas the mean (left hand side) gets dragged towards the “tail” (where the extreme values are). To give a concrete example, suppose Bob (income $50,000), Kate (income$60,000) and Jane (income $65,000) are sitting at a table: the average income at the table is$58,333 and the median income is $60,000. Then Bill sits down with them (income$100,000,000). The average income has now jumped to $25,043,750 but the median rises only to$62,500. If you’re interested in looking at the overall income at the table, the mean might be the right answer; but if you’re interested in what counts as a typical income at the table, the median would be a better choice here.
real life example
To try to get a sense of why you need to pay attention to the differences between the mean and the median, let’s consider a real life example. Since I tend to mock journalists for their poor scientific and statistical knowledge, I should give credit where credit is due. This is from an excellent article on the ABC news website67 24 September, 2010:
Senior Commonwealth Bank executives have travelled the world in the past couple of weeks with a presentation showing how Australian house prices, and the key price to income ratios, compare favourably with similar countries. “Housing affordability has actually been going sideways for the last five to six years,” said Craig James, the chief economist of the bank’s trading arm, CommSec.
This probably comes as a huge surprise to anyone with a mortgage, or who wants a mortgage, or pays rent, or isn’t completely oblivious to what’s been going on in the Australian housing market over the last several years. Back to the article:
CBA has waged its war against what it believes are housing doomsayers with graphs, numbers and international comparisons. In its presentation, the bank rejects arguments that Australia’s housing is relatively expensive compared to incomes. It says Australia’s house price to household income ratio of 5.6 in the major cities, and 4.3 nationwide, is comparable to many other developed nations. It says San Francisco and New York have ratios of 7, Auckland’s is 6.7, and Vancouver comes in at 9.3.
More excellent news! Except, the article goes on to make the observation that…
Many analysts say that has led the bank to use misleading figures and comparisons. If you go to page four of CBA’s presentation and read the source information at the bottom of the graph and table, you would notice there is an additional source on the international comparison – Demographia. However, if the Commonwealth Bank had also used Demographia’s analysis of Australia’s house price to income ratio, it would have come up with a figure closer to 9 rather than 5.6 or 4.3
That’s, um, a rather serious discrepancy. One group of people say 9, another says 4-5. Should we just split the difference, and say the truth lies somewhere in between? Absolutely not: this is a situation where there is a right answer and a wrong answer. Demographia are correct, and the Commonwealth Bank is incorrect. As the article points out
[An] obvious problem with the Commonwealth Bank’s domestic price to income figures is they compare average incomes with median house prices (unlike the Demographia figures that compare median incomes to median prices). The median is the mid-point, effectively cutting out the highs and lows, and that means the average is generally higher when it comes to incomes and asset prices, because it includes the earnings of Australia’s wealthiest people. To put it another way: the Commonwealth Bank’s figures count Ralph Norris’ multi-million dollar pay packet on the income side, but not his (no doubt) very expensive house in the property price figures, thus understating the house price to income ratio for middle-income Australians.
Couldn’t have put it better myself. The way that Demographia calculated the ratio is the right thing to do. The way that the Bank did it is incorrect. As for why an extremely quantitatively sophisticated organisation such as a major bank made such an elementary mistake, well… I can’t say for sure, since I have no special insight into their thinking, but the article itself does happen to mention the following facts, which may or may not be relevant:
[As] Australia’s largest home lender, the Commonwealth Bank has one of the biggest vested interests in house prices rising. It effectively owns a massive swathe of Australian housing as security for its home loans as well as many small business loans.
My, my.
Trimmed mean
One of the fundamental rules of applied statistics is that the data are messy. Real life is never simple, and so the data sets that you obtain are never as straightforward as the statistical theory says.68 This can have awkward consequences. To illustrate, consider this rather strange looking data set:
−100,2,3,4,5,6,7,8,9,10
If you were to observe this in a real life data set, you’d probably suspect that something funny was going on with the −100 value. It’s probably an outlier, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely, and in this particular case I’d probably agree with that course of action. In real life, however, you don’t always get such cut-and-dried examples. For instance, you might get this instead:
−15,2,3,4,5,6,7,8,9,12
The −15 looks a bit suspicious, but not anywhere near as much as that −100 did. In this case, it’s a little trickier. It might be a legitimate observation, it might not.
When faced with a situation where some of the most extreme-valued observations might not be quite trustworthy, the mean is not necessarily a good measure of central tendency. It is highly sensitive to one or two extreme values, and is thus not considered to be a robust measure. One remedy that we’ve seen is to use the median. A more general solution is to use a “trimmed mean”. To calculate a trimmed mean, what you do is “discard” the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. The goal is to preserve the best characteristics of the mean and the median: just like a median, you aren’t highly influenced by extreme outliers, but like the mean, you “use” more than one of the observations. Generally, we describe a trimmed mean in terms of the percentage of observation on either side that are discarded. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. Not surprisingly, the 0% trimmed mean is just the regular mean, and the 50% trimmed mean is the median. In that sense, trimmed means provide a whole family of central tendency measures that span the range from the mean to the median.
For our toy example above, we have 10 observations, and so a 10% trimmed mean is calculated by ignoring the largest value (i.e., 12) and the smallest value (i.e., -15) and taking the mean of the remaining values. First, let’s enter the data
dataset <- c( -15,2,3,4,5,6,7,8,9,12 )
Next, let’s calculate means and medians:
mean( x = dataset )
## [1] 4.1
median( x = dataset )
## [1] 5.5
That’s a fairly substantial difference, but I’m tempted to think that the mean is being influenced a bit too much by the extreme values at either end of the data set, especially the −15 one. So let’s just try trimming the mean a bit. If I take a 10% trimmed mean, we’ll drop the extreme values on either side, and take the mean of the rest:
mean( x = dataset, trim = .1)
## [1] 5.5
which in this case gives exactly the same answer as the median. Note that, to get a 10% trimmed mean you write trim = .1, not trim = 10. In any case, let’s finish up by calculating the 5% trimmed mean for the afl.margins data,
mean( x = afl.margins, trim = .05)
## [1] 33.75
Mode
The mode of a sample is very simple: it is the value that occurs most frequently. To illustrate the mode using the AFL data, let’s examine a different aspect to the data set. Who has played in the most finals? The afl.finalists variable is a factor that contains the name of every team that played in any AFL final from 1987-2010, so let’s have a look at it. To do this we will use the head() command. head() is useful when you’re working with a data.frame with a lot of rows since you can use it to tell you how many rows to return. There have been a lot of finals in this period so printing afl.finalists using print(afl.finalists) will just fill us the screen. The command below tells R we just want the first 25 rows of the data.frame.
head(afl.finalists, 25)
## [1] Hawthorn Melbourne Carlton Melbourne Hawthorn
## [6] Carlton Melbourne Carlton Hawthorn Melbourne
## [11] Melbourne Hawthorn Melbourne Essendon Hawthorn
## [16] Geelong Geelong Hawthorn Collingwood Melbourne
## [21] Collingwood West Coast Collingwood Essendon Collingwood
## 17 Levels: Adelaide Brisbane Carlton Collingwood Essendon ... Western Bulldogs
There are actually 400 entries (aren’t you glad we didn’t print them all?). We could read through all 400, and count the number of occasions on which each team name appears in our list of finalists, thereby producing a frequency table. However, that would be mindless and boring: exactly the sort of task that computers are great at. So let’s use the table() function (discussed in more detail in Section 7.1) to do this task for us:
table( afl.finalists )
## afl.finalists
## Adelaide Brisbane Carlton Collingwood
## 26 25 26 28
## Essendon Fitzroy Fremantle Geelong
## 32 0 6 39
## Hawthorn Melbourne North Melbourne Port Adelaide
## 27 28 28 17
## Richmond St Kilda Sydney West Coast
## 6 24 26 38
## Western Bulldogs
## 24
Now that we have our frequency table, we can just look at it and see that, over the 24 years for which we have data, Geelong has played in more finals than any other team. Thus, the mode of the finalists data is "Geelong". The core packages in R don’t have a function for calculating the mode69. However, I’ve included a function in the lsr package that does this. The function is called modeOf(), and here’s how you use it:
modeOf( x = afl.finalists )
## [1] "Geelong"
There’s also a function called maxFreq() that tells you what the modal frequency is. If we apply this function to our finalists data, we obtain the following:
maxFreq( x = afl.finalists )
## [1] 39
Taken together, we observe that Geelong (39 finals) played in more finals than any other team during the 1987-2010 period.
One last point to make with respect to the mode. While it’s generally true that the mode is most often calculated when you have nominal scale data (because means and medians are useless for those sorts of variables), there are some situations in which you really do want to know the mode of an ordinal, interval or ratio scale variable. For instance, let’s go back to thinking about our afl.margins variable. This variable is clearly ratio scale (if it’s not clear to you, it may help to re-read Section 2.2), and so in most situations the mean or the median is the measure of central tendency that you want. But consider this scenario… a friend of yours is offering a bet. They pick a football game at random, and (without knowing who is playing) you have to guess the exact margin. If you guess correctly, you win $50. If you don’t, you lose$1. There are no consolation prizes for “almost” getting the right answer. You have to guess exactly the right margin70 For this bet, the mean and the median are completely useless to you. It is the mode that you should bet on. So, we calculate this modal value
modeOf( x = afl.margins )
## [1] 3
maxFreq( x = afl.margins )
## [1] 8
So the 2010 data suggest you should bet on a 3 point margin, and since this was observed in 8 of the 176 game (4.5% of games) the odds are firmly in your favour. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.01%3A_Measures_of_Central_Tendency.txt |
The statistics that we’ve discussed so far all relate to central tendency. That is, they all talk about which values are “in the middle” or “popular” in the data. However, central tendency is not the only type of summary statistic that we want to calculate. The second thing that we really want is a measure of the variability of the data. That is, how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be? For now, let’s assume that the data are interval or ratio scale, so we’ll continue to use the afl.margins data. We’ll use this data to discuss several different measures of spread, each with different strengths and weaknesses.
Range
The range of a variable is very simple: it’s the biggest value minus the smallest value. For the AFL winning margins data, the maximum value is 116, and the minimum value is 0. We can calculate these values in R using the max() and min() functions:
max( afl.margins )
## [1] 116
min( afl.margins )
## [1] 0
where I’ve omitted the output because it’s not interesting. The other possibility is to use the range() function; which outputs both the minimum value and the maximum value in a vector, like this:
range( afl.margins )
## [1] 0 116
Although the range is the simplest way to quantify the notion of “variability”, it’s one of the worst. Recall from our discussion of the mean that we want our summary measure to be robust. If the data set has one or two extremely bad values in it, we’d like our statistics not to be unduly influenced by these cases. If we look once again at our toy example of a data set containing very extreme outliers…
−100,2,3,4,5,6,7,8,9,10
… it is clear that the range is not robust, since this has a range of 110, but if the outlier were removed we would have a range of only 8.
Interquartile range
The interquartile range (IQR) is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. Probably you already know what a quantile is (they’re more commonly called percentiles), but if not: the 10th percentile of a data set is the smallest number x such that 10% of the data is less than x. In fact, we’ve already come across the idea: the median of a data set is its 50th quantile / percentile! R actually provides you with a way of calculating quantiles, using the (surprise, surprise) quantile() function. Let’s use it to calculate the median AFL winning margin:
quantile( x = afl.margins, probs = .5)
## 50%
## 30.5
And not surprisingly, this agrees with the answer that we saw earlier with the median() function. Now, we can actually input lots of quantiles at once, by specifying a vector for the probs argument. So lets do that, and get the 25th and 75th percentile:
quantile( x = afl.margins, probs = c(.25,.75) )
## 25% 75%
## 12.75 50.50
And, by noting that 50.5−12.75=37.75, we can see that the interquartile range for the 2010 AFL winning margins data is 37.75. Of course, that seems like too much work to do all that typing, so R has a built in function called IQR() that we can use:
IQR( x = afl.margins )
## [1] 37.75
While it’s obvious how to interpret the range, it’s a little less obvious how to interpret the IQR. The simplest way to think about it is like this: the interquartile range is the range spanned by the “middle half” of the data. That is, one quarter of the data falls below the 25th percentile, one quarter of the data is above the 75th percentile, leaving the “middle half” of the data lying in between the two. And the IQR is the range covered by that middle half.
Mean absolute deviation
The two measures we’ve looked at so far, the range and the interquartile range, both rely on the idea that we can measure the spread of the data by looking at the quantiles of the data. However, this isn’t the only way to think about the problem. A different approach is to select a meaningful reference point (usually the mean or the median) and then report the “typical” deviations from that reference point. What do we mean by “typical” deviation? Usually, the mean or median value of these deviations! In practice, this leads to two different measures, the “mean absolute deviation (from the mean)” and the “median absolute deviation (from the median)”. From what I’ve read, the measure based on the median seems to be used in statistics, and does seem to be the better of the two, but to be honest I don’t think I’ve seen it used much in psychology. The measure based on the mean does occasionally show up in psychology though. In this section I’ll talk about the first one, and I’ll come back to talk about the second one later.
Since the previous paragraph might sound a little abstract, let’s go through the mean absolute deviation from the mean a little more slowly. One useful thing about this measure is that the name actually tells you exactly how to calculate it. Let’s think about our AFL winning margins data, and once again we’ll start by pretending that there’s only 5 games in total, with winning margins of 56, 31, 56, 8 and 32. Since our calculations rely on an examination of the deviation from some reference point (in this case the mean), the first thing we need to calculate is the mean, $\bar{X}$. For these five observations, our mean is $\bar{X}$=36.6. The next step is to convert each of our observations Xi into a deviation score. We do this by calculating the difference between the observation Xi and the mean $\bar{X}$. That is, the deviation score is defined to be Xi−$\bar{X}$. For the first observation in our sample, this is equal to 56−36.6=19.4. Okay, that’s simple enough. The next step in the process is to convert these deviations to absolute deviations. As we discussed earlier when talking about the abs() function in R (Section 3.5), we do this by converting any negative values to positive ones. Mathematically, we would denote the absolute value of −3 as |−3|, and so we say that |−3|=3. We use the absolute value function here because we don’t really care whether the value is higher than the mean or lower than the mean, we’re just interested in how close it is to the mean. To help make this process as obvious as possible, the table below shows these calculations for all five observations:
the observation its symbol the observed value
winning margin, game 2 X2 31 points
winning margin, game 5 X5 32 points
winning margin, game 1 X1 56 points
winning margin, game 3 X3 56 points
winning margin, game 4 X4 8 points
Now that we have calculated the absolute deviation score for every observation in the data set, all that we have to do to calculate the mean of these scores. Let’s do that:
$\dfrac{19.4+5.6+19.4+28.6+4.6}{5}=15.52 \nonumber$
And we’re done. The mean absolute deviation for these five scores is 15.52.
However, while our calculations for this little example are at an end, we do have a couple of things left to talk about. Firstly, we should really try to write down a proper mathematical formula. But in order do to this I need some mathematical notation to refer to the mean absolute deviation. Irritatingly, “mean absolute deviation” and “median absolute deviation” have the same acronym (MAD), which leads to a certain amount of ambiguity, and since R tends to use MAD to refer to the median absolute deviation, I’d better come up with something different for the mean absolute deviation. Sigh. What I’ll do is use AAD instead, short for average absolute deviation. Now that we have some unambiguous notation, here’s the formula that describes what we just calculated:
$(X)=\dfrac{1}{N} \sum_{i=1}^{N}\left|X_{i}-\bar{X}\right| \nonumber$
The last thing we need to talk about is how to calculate AAD in R. One possibility would be to do everything using low level commands, laboriously following the same steps that I used when describing the calculations above. However, that’s pretty tedious. You’d end up with a series of commands that might look like this:
X <- c(56, 31,56,8,32) # enter the data
X.bar <- mean( X ) # step 1. the mean of the data
AD <- abs( X - X.bar ) # step 2. the absolute deviations from the mean
AAD <- mean( AD ) # step 3. the mean absolute deviations
print( AAD ) # print the results
## [1] 15.52
Each of those commands is pretty simple, but there’s just too many of them. And because I find that to be too much typing, the lsr package has a very simple function called aad() that does the calculations for you. If we apply the aad() function to our data, we get this:
library(lsr)
aad( X )
## [1] 15.52
No suprises there.
Variance
Although the mean absolute deviation measure has its uses, it’s not the best measure of variability to use. From a purely mathematical perspective, there are some solid reasons to prefer squared deviations rather than absolute deviations. If we do that, we obtain a measure is called the variance, which has a lot of really nice statistical properties that I’m going to ignore,71(X)\$ and Var(Y) respectively. Now imagine I want to define a new variable Z that is the sum of the two, Z=X+Y. As it turns out, the variance of Z is equal to Var(X)+Var(Y). This is a very useful property, but it’s not true of the other measures that I talk about in this section.] and one massive psychological flaw that I’m going to make a big deal out of in a moment. The variance of a data set X is sometimes written as Var(X), but it’s more commonly denoted s2 (the reason for this will become clearer shortly). The formula that we use to calculate the variance of a set of observations is as follows:
\begin{aligned} &\operatorname{Var}(X)=\dfrac{1}{N} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}\ &\operatorname{Var}(X)=\dfrac{\sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}}{N} \end{aligned} \nonumber
As you can see, it’s basically the same formula that we used to calculate the mean absolute deviation, except that instead of using “absolute deviations” we use “squared deviations”. It is for this reason that the variance is sometimes referred to as the “mean square deviation”.
Now that we’ve got the basic idea, let’s have a look at a concrete example. Once again, let’s use the first five AFL games as our data. If we follow the same approach that we took last time, we end up with the following table:
Table 5.1: Basic arithmetic operations in R. These five operators are used very frequently throughout the text, so it’s important to be familiar with them at the outset.
Notation [English] i [which game] Xi [value] Xi−$\bar{X}$ [deviation from mean] (Xi−$\bar{X}$)2 [absolute deviation]
5 32 -4.6 21.16
2 31 -5.6 31.36
1 56 19.4 376.36
3 56 19.4 376.36
4 8 -28.6 817.96
That last column contains all of our squared deviations, so all we have to do is average them. If we do that by typing all the numbers into R by hand…
( 376.36 + 31.36 + 376.36 + 817.96 + 21.16 ) / 5
## [1] 324.64
… we end up with a variance of 324.64. Exciting, isn’t it? For the moment, let’s ignore the burning question that you’re all probably thinking (i.e., what the heck does a variance of 324.64 actually mean?) and instead talk a bit more about how to do the calculations in R, because this will reveal something very weird.
As always, we want to avoid having to type in a whole lot of numbers ourselves. And as it happens, we have the vector X lying around, which we created in the previous section. With this in mind, we can calculate the variance of X by using the following command,
mean( (X - mean(X) )^2)
## [1] 324.64
and as usual we get the same answer as the one that we got when we did everything by hand. However, I still think that this is too much typing. Fortunately, R has a built in function called var() which does calculate variances. So we could also do this…
var(X)
## [1] 405.8
and you get the same… no, wait… you get a completely different answer. That’s just weird. Is R broken? Is this a typo? Is Dan an idiot?
As it happens, the answer is no.72 It’s not a typo, and R is not making a mistake. To get a feel for what’s happening, let’s stop using the tiny data set containing only 5 data points, and switch to the full set of 176 games that we’ve got stored in our afl.margins vector. First, let’s calculate the variance by using the formula that I described above:
mean( (afl.margins - mean(afl.margins) )^2)
## [1] 675.9718
Now let’s use the var() function:
var( afl.margins )
## [1] 679.8345
Hm. These two numbers are very similar this time. That seems like too much of a coincidence to be a mistake. And of course it isn’t a mistake. In fact, it’s very simple to explain what R is doing here, but slightly trickier to explain why R is doing it. So let’s start with the “what”. What R is doing is evaluating a slightly different formula to the one I showed you above. Instead of averaging the squared deviations, which requires you to divide by the number of data points N, R has chosen to divide by N−1. In other words, the formula that R is using is this one
$\dfrac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2} \nonumber It’s easy enough to verify that this is what’s happening, as the following command illustrates: sum( (X-mean(X))^2 ) / 4 ## [1] 405.8 This is the same answer that R gave us originally when we calculated var(X) originally. So that’s the what. The real question is why R is dividing by N−1 and not by N. After all, the variance is supposed to be the mean squared deviation, right? So shouldn’t we be dividing by N, the actual number of observations in the sample? Well, yes, we should. However, as we’ll discuss in Chapter 10, there’s a subtle distinction between “describing a sample” and “making guesses about the population from which the sample came”. Up to this point, it’s been a distinction without a difference. Regardless of whether you’re describing a sample or drawing inferences about the population, the mean is calculated exactly the same way. Not so for the variance, or the standard deviation, or for many other measures besides. What I outlined to you initially (i.e., take the actual average, and thus divide by N) assumes that you literally intend to calculate the variance of the sample. Most of the time, however, you’re not terribly interested in the sample in and of itself. Rather, the sample exists to tell you something about the world. If so, you’re actually starting to move away from calculating a “sample statistic”, and towards the idea of estimating a “population parameter”. However, I’m getting ahead of myself. For now, let’s just take it on faith that R knows what it’s doing, and we’ll revisit the question later on when we talk about estimation in Chapter 10. Okay, one last thing. This section so far has read a bit like a mystery novel. I’ve shown you how to calculate the variance, described the weird “N−1” thing that R does and hinted at the reason why it’s there, but I haven’t mentioned the single most important thing… how do you interpret the variance? Descriptive statistics are supposed to describe things, after all, and right now the variance is really just a gibberish number. Unfortunately, the reason why I haven’t given you the human-friendly interpretation of the variance is that there really isn’t one. This is the most serious problem with the variance. Although it has some elegant mathematical properties that suggest that it really is a fundamental quantity for expressing variation, it’s completely useless if you want to communicate with an actual human… variances are completely uninterpretable in terms of the original variable! All the numbers have been squared, and they don’t mean anything anymore. This is a huge issue. For instance, according to the table I presented earlier, the margin in game 1 was “376.36 points-squared higher than the average margin”. This is exactly as stupid as it sounds; and so when we calculate a variance of 324.64, we’re in the same situation. I’ve watched a lot of footy games, and never has anyone referred to “points squared”. It’s not a real unit of measurement, and since the variance is expressed in terms of this gibberish unit, it is totally meaningless to a human. Standard deviation Okay, suppose that you like the idea of using the variance because of those nice mathematical properties that I haven’t talked about, but – since you’re a human and not a robot – you’d like to have a measure that is expressed in the same units as the data itself (i.e., points, not points-squared). What should you do? The solution to the problem is obvious: take the square root of the variance, known as the standard deviation, also called the “root mean squared deviation”, or RMSD. This solves out problem fairly neatly: while nobody has a clue what “a variance of 324.68 points-squared” really means, it’s much easier to understand “a standard deviation of 18.01 points”, since it’s expressed in the original units. It is traditional to refer to the standard deviation of a sample of data as s, though “sd” and “std dev.” are also used at times. Because the standard deviation is equal to the square root of the variance, you probably won’t be surprised to see that the formula is: \[ s=\sqrt{\dfrac{1}{N} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}} \nonumber$
and the R function that we use to calculate it is sd(). However, as you might have guessed from our discussion of the variance, what R actually calculates is slightly different to the formula given above. Just like the we saw with the variance, what R calculates is a version that divides by N−1 rather than N. For reasons that will make sense when we return to this topic in Chapter@refch:estimation I’ll refer to this new quantity as $\hat{\sigma}$ (read as: “sigma hat”), and the formula for this is
$\hat{\sigma}=\sqrt{\dfrac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}} \nonumber$
With that in mind, calculating standard deviations in R is simple:
sd( afl.margins )
## [1] 26.07364
Interpreting standard deviations is slightly more complex. Because the standard deviation is derived from the variance, and the variance is a quantity that has little to no meaning that makes sense to us humans, the standard deviation doesn’t have a simple interpretation. As a consequence, most of us just rely on a simple rule of thumb: in general, you should expect 68% of the data to fall within 1 standard deviation of the mean, 95% of the data to fall within 2 standard deviation of the mean, and 99.7% of the data to fall within 3 standard deviations of the mean. This rule tends to work pretty well most of the time, but it’s not exact: it’s actually calculated based on an assumption that the histogram is symmetric and “bell shaped.”73 As you can tell from looking at the AFL winning margins histogram in Figure 5.1, this isn’t exactly true of our data! Even so, the rule is approximately correct. As it turns out, 65.3% of the AFL margins data fall within one standard deviation of the mean. This is shown visually in Figure 5.3.
Median absolute deviation
The last measure of variability that I want to talk about is the median absolute deviation (MAD). The basic idea behind MAD is very simple, and is pretty much identical to the idea behind the mean absolute deviation (Section 5.2.3). The difference is that you use the median everywhere. If we were to frame this idea as a pair of R commands, they would look like this:
# mean absolute deviation from the mean:
mean( abs(afl.margins - mean(afl.margins)) )
## [1] 21.10124
# *median* absolute deviation from the *median*:
median( abs(afl.margins - median(afl.margins)) )
## [1] 19.5
This has a straightforward interpretation: every observation in the data set lies some distance away from the typical value (the median). So the MAD is an attempt to describe a typical deviation from a typical value in the data set. It wouldn’t be unreasonable to interpret the MAD value of 19.5 for our AFL data by saying something like this:
The median winning margin in 2010 was 30.5, indicating that a typical game involved a winning margin of about 30 points. However, there was a fair amount of variation from game to game: the MAD value was 19.5, indicating that a typical winning margin would differ from this median value by about 19-20 points.
As you’d expect, R has a built in function for calculating MAD, and you will be shocked no doubt to hear that it’s called mad(). However, it’s a little bit more complicated than the functions that we’ve been using previously. If you want to use it to calculate MAD in the exact same way that I have described it above, the command that you need to use specifies two arguments: the data set itself x, and a constant that I’ll explain in a moment. For our purposes, the constant is 1, so our command becomes
mad( x = afl.margins, constant = 1 )
## [1] 19.5
Apart from the weirdness of having to type that constant = 1 part, this is pretty straightforward.
Okay, so what exactly is this constant = 1 argument? I won’t go into all the details here, but here’s the gist. Although the “raw” MAD value that I’ve described above is completely interpretable on its own terms, that’s not actually how it’s used in a lot of real world contexts. Instead, what happens a lot is that the researcher actually wants to calculate the standard deviation. However, in the same way that the mean is very sensitive to extreme values, the standard deviation is vulnerable to the exact same issue. So, in much the same way that people sometimes use the median as a “robust” way of calculating “something that is like the mean”, it’s not uncommon to use MAD as a method for calculating “something that is like the standard deviation”. Unfortunately, the raw MAD value doesn’t do this. Our raw MAD value is 19.5, and our standard deviation was 26.07. However, what some clever person has shown is that, under certain assumptions74, you can multiply the raw MAD value by 1.4826 and obtain a number that is directly comparable to the standard deviation. As a consequence, the default value of constant is 1.4826, and so when you use the mad() command without manually setting a value, here’s what you get:
mad( afl.margins )
## [1] 28.9107
I should point out, though, that if you want to use this “corrected” MAD value as a robust version of the standard deviation, you really are relying on the assumption that the data are (or at least, are “supposed to be” in some sense) symmetric and basically shaped like a bell curve. That’s really not true for our afl.margins data, so in this case I wouldn’t try to use the MAD value this way.
Which measure to use?
We’ve discussed quite a few measures of spread (range, IQR, MAD, variance and standard deviation), and hinted at their strengths and weaknesses. Here’s a quick summary:
• Range. Gives you the full spread of the data. It’s very vulnerable to outliers, and as a consequence it isn’t often used unless you have good reasons to care about the extremes in the data.
• Interquartile range. Tells you where the “middle half” of the data sits. It’s pretty robust, and complements the median nicely. This is used a lot.
• Mean absolute deviation. Tells you how far “on average” the observations are from the mean. It’s very interpretable, but has a few minor issues (not discussed here) that make it less attractive to statisticians than the standard deviation. Used sometimes, but not often.
• Variance. Tells you the average squared deviation from the mean. It’s mathematically elegant, and is probably the “right” way to describe variation around the mean, but it’s completely uninterpretable because it doesn’t use the same units as the data. Almost never used except as a mathematical tool; but it’s buried “under the hood” of a very large number of statistical tools.
• Standard deviation. This is the square root of the variance. It’s fairly elegant mathematically, and it’s expressed in the same units as the data so it can be interpreted pretty well. In situations where the mean is the measure of central tendency, this is the default. This is by far the most popular measure of variation.
• Median absolute deviation. The typical (i.e., median) deviation from the median value. In the raw form it’s simple and interpretable; in the corrected form it’s a robust way to estimate the standard deviation, for some kinds of data sets. Not used very often, but it does get reported sometimes.
In short, the IQR and the standard deviation are easily the two most common measures used to report the variability of the data; but there are situations in which the others are used. I’ve described all of them in this book because there’s a fair chance you’ll run into most of these somewhere. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.02%3A_Measures_of_Variability.txt |
There are two more descriptive statistics that you will sometimes see reported in the psychological literature, known as skew and kurtosis. In practice, neither one is used anywhere near as frequently as the measures of central tendency and variability that we’ve been talking about. Skew is pretty important, so you do see it mentioned a fair bit; but I’ve actually never seen kurtosis reported in a scientific article to date.
## [1] -0.9174977
## [1] 0.009023979
## [1] 0.9250898
Since it’s the more interesting of the two, let’s start by talking about the skewness. Skewness is basically a measure of asymmetry, and the easiest way to explain it is by drawing some pictures. As Figure 5.4 illustrates, if the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel), then we say that the data are negatively skewed. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are positively skewed. That’s the qualitative idea behind skewness. The actual formula for the skewness of a data set is as follows
$\text { skewness }(X)=\dfrac{1}{N \hat{\sigma}\ ^{3}} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{3} \label{skew}$
where N is the number of observations, $\bar{X}$ is the sample mean, and $\hat{\sigma}$ is the standard deviation (the “divide by N−1” version, that is). Perhaps more helpfully, it might be useful to point out that the psych package contains a skew() function that you can use to calculate skewness. So if we wanted to use this function to calculate the skewness of the afl.margins data, we’d first need to load the package
library( psych )
which now makes it possible to use the following command:
skew( x = afl.margins )
## [1] 0.7671555
Not surprisingly, it turns out that the AFL winning margins data is fairly skewed.
The final measure that is sometimes referred to is the kurtosis of a data set. Put simply, kurtosis is a measure of the “tailedness”, or outlier character, of the data. Historically, it was thought that this statistic measures “pointiness” or “flatness” of a distribution, but this has been shown to be an error of interpretation. See Figure 5.5.
## [1] -0.9631805
## [1] 0.02226287
## [1] 1.994329
By mathematical calculations, the “normal curve” (black lines) has zero kurtosis, so the outlier character of a data set is assessed relative to this curve. In this Figure, the data on the left are less outlier-prone, so the kurtosis is negative and we call the data platykurtic. The data on the right are more outlier-prone, so the kurtosis is positive and we say that the data is leptokurtic. But the data in the middle are similar in their outlier character, so we say that it is mesokurtic and has kurtosis zero. This is summarised in the table below:
informal term technical name kurtosis value
just pointy enough mesokurtic zero
too pointy leptokurtic positive
too flat platykurtic negative
The equation for kurtosis is pretty similar in spirit to the formulas we’ve seen already for the variance and the skewness (Equation \ref{skew}); except that where the variance involved squared deviations and the skewness involved cubed deviations, the kurtosis involves raising the deviations to the fourth power:75
$\text { kurtosis }(X)=\dfrac{1}{N \hat{\sigma}\ ^{4}} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{4}-3$
The psych package has a function called kurtosi() that you can use to calculate the kurtosis of your data. For instance, if we were to do this for the AFL margins,
kurtosi( x = afl.margins )
## [1] 0.02962633
we discover that the AFL winning margins data are just pointy enough.
Contributors
• Danielle Navarro (Associate Professor (Psychology) at University of New South Wales)
• Peter H. Westfall (Paul Whitfield Horn Professor and James and Marguerite Niver Professor, Texas Tech University) | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.03%3A_Skew_and_Kurtosis.txt |
Up to this point in the chapter I’ve explained several different summary statistics that are commonly used when analysing data, along with specific functions that you can use in R to calculate each one. However, it’s kind of annoying to have to separately calculate means, medians, standard deviations, skews etc. Wouldn’t it be nice if R had some helpful functions that would do all these tedious calculations at once? Something like `summary()` or `describe()`, perhaps? Why yes, yes it would. So much so that both of these functions exist. The `summary()` function is in the `base` package, so it comes with every installation of R. The `describe()` function is part of the `psych` package, which we loaded earlier in the chapter.
“Summarising” a variable
The `summary()` function is an easy thing to use, but a tricky thing to understand in full, since it’s a generic function (see Section 4.11. The basic idea behind the `summary()` function is that it prints out some useful information about whatever object (i.e., variable, as far as we’re concerned) you specify as the `object` argument. As a consequence, the behaviour of the `summary()` function differs quite dramatically depending on the class of the object that you give it. Let’s start by giving it a numeric object:
``summary( object = afl.margins ) ``
``````## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.00 12.75 30.50 35.30 50.50 116.00``` ```
For numeric variables, we get a whole bunch of useful descriptive statistics. It gives us the minimum and maximum values (i.e., the range), the first and third quartiles (25th and 75th percentiles; i.e., the IQR), the mean and the median. In other words, it gives us a pretty good collection of descriptive statistics related to the central tendency and the spread of the data.
Okay, what about if we feed it a logical vector instead? Let’s say I want to know something about how many “blowouts” there were in the 2010 AFL season. I operationalise the concept of a blowout (see Chapter 2) as a game in which the winning margin exceeds 50 points. Let’s create a logical variable `blowouts` in which the i-th element is `TRUE` if that game was a blowout according to my definition,
``````blowouts <- afl.margins > 50
blowouts``````
``````## [1] TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [12] TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [23] FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE FALSE FALSE
## [34] TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [45] FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE
## [56] TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [67] TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [78] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [89] FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [100] FALSE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE
## [111] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [122] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE
## [133] FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE
## [144] TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE
## [155] TRUE FALSE TRUE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE
## [166] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE``````
So that’s what the `blowouts` variable looks like. Now let’s ask R for a `summary()`
``summary( object = blowouts )``
``````## Mode FALSE TRUE
## logical 132 44``````
In this context, the `summary()` function gives us a count of the number of `TRUE` values, the number of `FALSE` values, and the number of missing values (i.e., the `NA`s). Pretty reasonable behaviour.
Next, let’s try to give it a factor. If you recall, I’ve defined the `afl.finalists` vector as a factor, so let’s use that:
````summary( object = afl.finalists )`
```
``````## Adelaide Brisbane Carlton Collingwood
## 26 25 26 28
## Essendon Fitzroy Fremantle Geelong
## 32 0 6 39
## Hawthorn Melbourne North Melbourne Port Adelaide
## 27 28 28 17
## Richmond St Kilda Sydney West Coast
## 6 24 26 38
## Western Bulldogs
## 24``````
For factors, we get a frequency table, just like we got when we used the `table()` function. Interestingly, however, if we convert this to a character vector using the `as.character()` function (see Section 7.10, we don’t get the same results:
``````f2 <- as.character( afl.finalists )
summary( object = f2 )``````
``````## Length Class Mode
## 400 character character``````
This is one of those situations I was referring to in Section 4.7, in which it is helpful to declare your nominal scale variable as a factor rather than a character vector. Because I’ve defined `afl.finalists` as a factor, R knows that it should treat it as a nominal scale variable, and so it gives you a much more detailed (and helpful) summary than it would have if I’d left it as a character vector.
“Summarising” a data frame
Okay what about data frames? When you pass a data frame to the `summary()` function, it produces a slightly condensed summary of each variable inside the data frame. To give you a sense of how this can be useful, let’s try this for a new data set, one that you’ve never seen before. The data is stored in the `clinicaltrial.Rdata` file, and we’ll use it a lot in Chapter 14 (you can find a complete description of the data at the start of that chapter). Let’s load it, and see what we’ve got:
``````load( "./data/clinicaltrial.Rdata" )
who(TRUE)``````
``````## -- Name -- -- Class -- -- Size --
## clin.trial data.frame 18 x 3
## \$drug factor 18
## \$therapy factor 18
## \$mood.gain numeric 18``````
There’s a single data frame called `clin.trial` which contains three variables, `drug`, `therapy` and `mood.gain`. Presumably then, this data is from a clinical trial of some kind, in which people were administered different drugs; and the researchers looked to see what the drugs did to their mood. Let’s see if the `summary()` function sheds a little more light on this situation:
``summary( clin.trial )``
``````## drug therapy mood.gain
## placebo :6 no.therapy:9 Min. :0.1000
## anxifree:6 CBT :9 1st Qu.:0.4250
## joyzepam:6 Median :0.8500
## Mean :0.8833
## 3rd Qu.:1.3000
## Max. :1.8000``````
Evidently there were three drugs: a placebo, something called “anxifree” and something called “joyzepam”; and there were 6 people administered each drug. There were 9 people treated using cognitive behavioural therapy (CBT) and 9 people who received no psychological treatment. And we can see from looking at the summary of the `mood.gain` variable that most people did show a mood gain (mean =.88), though without knowing what the scale is here it’s hard to say much more than that. Still, that’s not too bad. Overall, I feel that I learned something from that.
“Describing” a data frame
The `describe()` function (in the `psych` package) is a little different, and it’s really only intended to be useful when your data are interval or ratio scale. Unlike the `summary()` function, it calculates the same descriptive statistics for any type of variable you give it. By default, these are:
• `var`. This is just an index: 1 for the first variable, 2 for the second variable, and so on.
• `n`. This is the sample size: more precisely, it’s the number of non-missing values.
• `mean`. This is the sample mean (Section 5.1.1).
• `sd`. This is the (bias corrected) standard deviation (Section 5.2.5).
• `median`. The median (Section 5.1.3).
• `trimmed`. This is trimmed mean. By default it’s the 10% trimmed mean (Section 5.1.6).
• `mad`. The median absolute deviation (Section 5.2.6).
• `min`. The minimum value.
• `max`. The maximum value.
• `range`. The range spanned by the data (Section 5.2.1).
• `skew`. The skewness (Section 5.3).
• `kurtosis`. The kurtosis (Section 5.3).
• `se`. The standard error of the mean (Chapter 10).
Notice that these descriptive statistics generally only make sense for data that are interval or ratio scale (usually encoded as numeric vectors). For nominal or ordinal variables (usually encoded as factors), most of these descriptive statistics are not all that useful. What the `describe()` function does is convert factors and logical variables to numeric vectors in order to do the calculations. These variables are marked with `*` and most of the time, the descriptive statistics for those variables won’t make much sense. If you try to feed it a data frame that includes a character vector as a variable, it produces an error.
With those caveats in mind, let’s use the `describe()` function to have a look at the `clin.trial` data frame. Here’s what we get:
````describe( x = clin.trial )`
```
``````## vars n mean sd median trimmed mad min max range skew
## drug* 1 18 2.00 0.84 2.00 2.00 1.48 1.0 3.0 2.0 0.00
## therapy* 2 18 1.50 0.51 1.50 1.50 0.74 1.0 2.0 1.0 0.00
## mood.gain 3 18 0.88 0.53 0.85 0.88 0.67 0.1 1.8 1.7 0.13
## kurtosis se
## drug* -1.66 0.20
## therapy* -2.11 0.12
## mood.gain -1.44 0.13``````
As you can see, the output for the asterisked variables is pretty meaningless, and should be ignored. However, for the `mood.gain` variable, there’s a lot of useful information. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.04%3A_Getting_an_Overall_Summary_of_a_Variable.txt |
It is very commonly the case that you find yourself needing to look at descriptive statistics, broken down by some grouping variable. This is pretty easy to do in R, and there are three functions in particular that are worth knowing about: `by()`, `describeBy()` and `aggregate()`. Let’s start with the `describeBy()` function, which is part of the `psych` package. The `describeBy()` function is very similar to the `describe()` function, except that it has an additional argument called `group` which specifies a grouping variable. For instance, let’s say, I want to look at the descriptive statistics for the `clin.trial` data, broken down separately by `therapy` type. The command I would use here is:
``describeBy( x=clin.trial, group=clin.trial\$therapy )``
``````##
## Descriptive statistics by group
## group: no.therapy
## vars n mean sd median trimmed mad min max range skew kurtosis
## drug* 1 9 2.00 0.87 2.0 2.00 1.48 1.0 3.0 2.0 0.00 -1.81
## therapy* 2 9 1.00 0.00 1.0 1.00 0.00 1.0 1.0 0.0 NaN NaN
## mood.gain 3 9 0.72 0.59 0.5 0.72 0.44 0.1 1.7 1.6 0.51 -1.59
## se
## drug* 0.29
## therapy* 0.00
## mood.gain 0.20
## --------------------------------------------------------
## group: CBT
## vars n mean sd median trimmed mad min max range skew
## drug* 1 9 2.00 0.87 2.0 2.00 1.48 1.0 3.0 2.0 0.00
## therapy* 2 9 2.00 0.00 2.0 2.00 0.00 2.0 2.0 0.0 NaN
## mood.gain 3 9 1.04 0.45 1.1 1.04 0.44 0.3 1.8 1.5 -0.03
## kurtosis se
## drug* -1.81 0.29
## therapy* NaN 0.00
## mood.gain -1.12 0.15``````
As you can see, the output is essentially identical to the output that the `describe()` function produce, except that the output now gives you means, standard deviations etc separately for the `CBT` group and the `no.therapy` group. Notice that, as before, the output displays asterisks for factor variables, in order to draw your attention to the fact that the descriptive statistics that it has calculated won’t be very meaningful for those variables. Nevertheless, this command has given us some really useful descriptive statistics `mood.gain` variable, broken down as a function of `therapy`.
A somewhat more general solution is offered by the `by()` function. There are three arguments that you need to specify when using this function: the `data` argument specifies the data set, the `INDICES` argument specifies the grouping variable, and the `FUN` argument specifies the name of a function that you want to apply separately to each group. To give a sense of how powerful this is, you can reproduce the `describeBy()` function by using a command like this:
``by( data=clin.trial, INDICES=clin.trial\$therapy, FUN=describe )``
``````## clin.trial\$therapy: no.therapy
## vars n mean sd median trimmed mad min max range skew kurtosis
## drug* 1 9 2.00 0.87 2.0 2.00 1.48 1.0 3.0 2.0 0.00 -1.81
## therapy* 2 9 1.00 0.00 1.0 1.00 0.00 1.0 1.0 0.0 NaN NaN
## mood.gain 3 9 0.72 0.59 0.5 0.72 0.44 0.1 1.7 1.6 0.51 -1.59
## se
## drug* 0.29
## therapy* 0.00
## mood.gain 0.20
## --------------------------------------------------------
## clin.trial\$therapy: CBT
## vars n mean sd median trimmed mad min max range skew
## drug* 1 9 2.00 0.87 2.0 2.00 1.48 1.0 3.0 2.0 0.00
## therapy* 2 9 2.00 0.00 2.0 2.00 0.00 2.0 2.0 0.0 NaN
## mood.gain 3 9 1.04 0.45 1.1 1.04 0.44 0.3 1.8 1.5 -0.03
## kurtosis se
## drug* -1.81 0.29
## therapy* NaN 0.00
## mood.gain -1.12 0.15``````
This will produce the exact same output as the command shown earlier. However, there’s nothing special about the `describe()` function. You could just as easily use the `by()` function in conjunction with the `summary()` function. For example:
``by( data=clin.trial, INDICES=clin.trial\$therapy, FUN=summary )` `
``````## clin.trial\$therapy: no.therapy
## drug therapy mood.gain
## placebo :3 no.therapy:9 Min. :0.1000
## anxifree:3 CBT :0 1st Qu.:0.3000
## joyzepam:3 Median :0.5000
## Mean :0.7222
## 3rd Qu.:1.3000
## Max. :1.7000
## --------------------------------------------------------
## clin.trial\$therapy: CBT
## drug therapy mood.gain
## placebo :3 no.therapy:0 Min. :0.300
## anxifree:3 CBT :9 1st Qu.:0.800
## joyzepam:3 Median :1.100
## Mean :1.044
## 3rd Qu.:1.300
## Max. :1.800``````
Again, this output is pretty easy to interpret. It’s the output of the `summary()` function, applied separately to `CBT` group and the `no.therapy` group. For the two factors (`drug` and `therapy`) it prints out a frequency table, whereas for the numeric variable (`mood.gain`) it prints out the range, interquartile range, mean and median.
What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy. It is actually possible to do this using the `by()` and `describeBy()` functions, but I usually find it more convenient to use the `aggregate()` function in this situation. There are again three arguments that you need to specify. The `formula` argument is used to indicate which variable you want to analyse, and which variables are used to specify the groups. For instance, if you want to look at `mood.gain` separately for each possible combination of `drug` and `therapy`, the formula you want is `mood.gain ~ drug + therapy`. The `data` argument is used to specify the data frame containing all the data, and the `FUN` argument is used to indicate what function you want to calculate for each group (e.g., the `mean`). So, to obtain group means, use this command:
``````aggregate( formula = mood.gain ~ drug + therapy, # mood.gain by drug/therapy combination
data = clin.trial, # data is in the clin.trial data frame
FUN = mean # print out group means
)```
```
``````## drug therapy mood.gain
## 1 placebo no.therapy 0.300000
## 2 anxifree no.therapy 0.400000
## 3 joyzepam no.therapy 1.466667
## 4 placebo CBT 0.600000
## 5 anxifree CBT 1.033333
## 6 joyzepam CBT 1.500000``````
or, alternatively, if you want to calculate the standard deviations for each group, you would use the following command (argument names omitted this time):
````aggregate( mood.gain ~ drug + therapy, clin.trial, sd )`
```
``````## drug therapy mood.gain
## 1 placebo no.therapy 0.2000000
## 2 anxifree no.therapy 0.2000000
## 3 joyzepam no.therapy 0.2081666
## 4 placebo CBT 0.3000000
## 5 anxifree CBT 0.2081666
## 6 joyzepam CBT 0.2645751`````` | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.05%3A_Descriptive_Statistics_Separately_for_each_Group.txt |
Suppose my friend is putting together a new questionnaire intended to measure “grumpiness”. The survey has 50 questions, which you can answer in a grumpy way or not. Across a big sample (hypothetically, let’s imagine a million people or so!) the data are fairly normally distributed, with the mean grumpiness score being 17 out of 50 questions answered in a grumpy way, and the standard deviation is 5. In contrast, when I take the questionnaire, I answer 35 out of 50 questions in a grumpy way. So, how grumpy am I? One way to think about would be to say that I have grumpiness of 35/50, so you might say that I’m 70% grumpy. But that’s a bit weird, when you think about it. If my friend had phrased her questions a bit differently, people might have answered them in a different way, so the overall distribution of answers could easily move up or down depending on the precise way in which the questions were asked. So, I’m only 70% grumpy with respect to this set of survey questions. Even if it’s a very good questionnaire, this isn’t very a informative statement.
A simpler way around this is to describe my grumpiness by comparing me to other people. Shockingly, out of my friend’s sample of 1,000,000 people, only 159 people were as grumpy as me (that’s not at all unrealistic, frankly), suggesting that I’m in the top 0.016% of people for grumpiness. This makes much more sense than trying to interpret the raw data. This idea – that we should describe my grumpiness in terms of the overall distribution of the grumpiness of humans – is the qualitative idea that standardisation attempts to get at. One way to do this is to do exactly what I just did, and describe everything in terms of percentiles. However, the problem with doing this is that “it’s lonely at the top”. Suppose that my friend had only collected a sample of 1000 people (still a pretty big sample for the purposes of testing a new questionnaire, I’d like to add), and this time gotten a mean of 16 out of 50 with a standard deviation of 5, let’s say. The problem is that almost certainly, not a single person in that sample would be as grumpy as me.
However, all is not lost. A different approach is to convert my grumpiness score into a standard score, also referred to as a z-score. The standard score is defined as the number of standard deviations above the mean that my grumpiness score lies. To phrase it in “pseudo-maths” the standard score is calculated like this:
standard score $=\dfrac{\text { raw score }-\text { mean }}{\text { standard deviation }}$
In actual maths, the equation for the z-score is
$z_{i}=\dfrac{X_{i}-\bar{X}}{\hat{\sigma}}$
So, going back to the grumpiness data, we can now transform Dan’s raw grumpiness into a standardised grumpiness score.76 If the mean is 17 and the standard deviation is 5 then my standardised grumpiness score would be77
$z=\dfrac{35-17}{5}=3.6 \nonumber$
To interpret this value, recall the rough heuristic that I provided in Section 5.2.5, in which I noted that 99.7% of values are expected to lie within 3 standard deviations of the mean. So the fact that my grumpiness corresponds to a z score of 3.6 indicates that I’m very grumpy indeed. Later on, in Section 9.5, I’ll introduce a function called pnorm() that allows us to be a bit more precise than this. Specifically, it allows us to calculate a theoretical percentile rank for my grumpiness, as follows:
pnorm( 3.6 )
## [1] 0.9998409
At this stage, this command doesn’t make too much sense, but don’t worry too much about it. It’s not important for now. But the output is fairly straightforward: it suggests that I’m grumpier than 99.98% of people. Sounds about right.
In addition to allowing you to interpret a raw score in relation to a larger population (and thereby allowing you to make sense of variables that lie on arbitrary scales), standard scores serve a second useful function. Standard scores can be compared to one another in situations where the raw scores can’t. Suppose, for instance, my friend also had another questionnaire that measured extraversion using a 24 items questionnaire. The overall mean for this measure turns out to be 13 with standard deviation 4; and I scored a 2. As you can imagine, it doesn’t make a lot of sense to try to compare my raw score of 2 on the extraversion questionnaire to my raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are “about” fundamentally different things, so this would be like comparing apples to oranges.
What about the standard scores? Well, this is a little different. If we calculate the standard scores, we get z=(35−17)/5=3.6 for grumpiness and z=(2−13)/4=−2.75 for extraversion. These two numbers can be compared to each other.78 I’m much less extraverted than most people (z=−2.75) and much grumpier than most people (z=3.6): but the extent of my unusualness is much more extreme for grumpiness (since 3.6 is a bigger number than 2.75). Because each standardised score is a statement about where an observation falls relative to its own population, it is possible to compare standardised scores across completely different variables. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.06%3A_Standard_Scores.txt |
Up to this point we have focused entirely on how to construct descriptive statistics for a single variable. What we haven’t done is talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables. But first, we need some data.
data
After spending so much time looking at the AFL data, I’m starting to get bored with sports. Instead, let’s turn to a topic close to every parent’s heart: sleep. The following data set is fictitious, but based on real events. Suppose I’m curious to find out how much my infant son’s sleeping habits affect my mood. Let’s say that I can rate my grumpiness very precisely, on a scale from 0 (not at all grumpy) to 100 (grumpy as a very, very grumpy old man). And, lets also assume that I’ve been measuring my grumpiness, my sleeping patterns and my son’s sleeping patterns for quite some time now. Let’s say, for 100 days. And, being a nerd, I’ve saved the data as a file called parenthood.Rdata. If we load the data…
load( "./data/parenthood.Rdata" )
who(TRUE)
## -- Name -- -- Class -- -- Size --
## parenthood data.frame 100 x 4
## $dan.sleep numeric 100 ##$baby.sleep numeric 100
## $dan.grump numeric 100 ##$day integer 100
… we see that the file contains a single data frame called parenthood, which contains four variables dan.sleep, baby.sleep, dan.grump and day. If we peek at the data using head() out the data, here’s what we get:
head(parenthood,10)
## dan.sleep baby.sleep dan.grump day
## 1 7.59 10.18 56 1
## 2 7.91 11.66 60 2
## 3 5.14 7.92 82 3
## 4 7.71 9.61 55 4
## 5 6.68 9.75 67 5
## 6 5.99 5.04 72 6
## 7 8.19 10.45 53 7
## 8 7.19 8.27 60 8
## 9 7.40 6.06 60 9
## 10 6.58 7.09 71 10
Next, I’ll calculate some basic descriptive statistics:
describe( parenthood )
## vars n mean sd median trimmed mad min max range
## dan.sleep 1 100 6.97 1.02 7.03 7.00 1.09 4.84 9.00 4.16
## baby.sleep 2 100 8.05 2.07 7.95 8.05 2.33 3.25 12.07 8.82
## dan.grump 3 100 63.71 10.05 62.00 63.16 9.64 41.00 91.00 50.00
## day 4 100 50.50 29.01 50.50 50.50 37.06 1.00 100.00 99.00
## skew kurtosis se
## dan.sleep -0.29 -0.72 0.10
## baby.sleep -0.02 -0.69 0.21
## dan.grump 0.43 -0.16 1.00
## day 0.00 -1.24 2.90
Finally, to give a graphical depiction of what each of the three interesting variables looks like, Figure 5.6 plots histograms.
One thing to note: just because R can calculate dozens of different statistics doesn’t mean you should report all of them. If I were writing this up for a report, I’d probably pick out those statistics that are of most interest to me (and to my readership), and then put them into a nice, simple table like the one in Table ??.79 Notice that when I put it into a table, I gave everything “human readable” names. This is always good practice. Notice also that I’m not getting enough sleep. This isn’t good practice, but other parents tell me that it’s standard practice.
Table 5.2: Descriptive statistics for the parenthood data.
variable min max mean median std. dev IQR
Dan’s grumpiness 41 91 63.71 62 10.05 14
Dan’s hours slept 4.84 9 6.97 7.03 1.02 1.45
Dan’s son’s hours slept 3.25 12.07 8.05 7.95 2.07 3.21
strength and direction of a relationship
We can draw scatterplots to give us a general sense of how closely related two variables are. Ideally though, we might want to say a bit more about it than that. For instance, let’s compare the relationship between dan.sleep and dan.grump (Figure 5.7 with that between baby.sleep and dan.grump (Figure 5.8. When looking at these two plots side by side, it’s clear that the relationship is qualitatively the same in both cases: more sleep equals less grump! However, it’s also pretty obvious that the relationship between dan.sleep and dan.grump is stronger than the relationship between baby.sleep and dan.grump. The plot on the left is “neater” than the one on the right. What it feels like is that if you want to predict what my mood is, it’d help you a little bit to know how many hours my son slept, but it’d be more helpful to know how many hours I slept.
In contrast, let’s consider Figure 5.8 vs. Figure 5.9. If we compare the scatterplot of “baby.sleep v dan.grump” to the scatterplot of “baby.sleep v dan.sleep”, the overall strength of the relationship is the same, but the direction is different. That is, if my son sleeps more, I get more sleep (positive relationship, but if he sleeps more then I get less grumpy (negative relationship).
correlation coefficient
We can make these ideas a bit more explicit by introducing the idea of a correlation coefficient (or, more specifically, Pearson’s correlation coefficient), which is traditionally denoted by r. The correlation coefficient between two variables X and Y (sometimes denoted rXY), which we’ll define more precisely in the next section, is a measure that varies from −1 to 1. When r=−1 it means that we have a perfect negative relationship, and when r=1 it means we have a perfect positive relationship. When r=0, there’s no relationship at all. If you look at Figure 5.10, you can see several plots showing what different correlations look like.
The formula for the Pearson’s correlation coefficient can be written in several different ways. I think the simplest way to write down the formula is to break it into two steps. Firstly, let’s introduce the idea of a covariance. The covariance between two variables X and Y is a generalisation of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables that isn’t terribly informative to humans:
$\operatorname{Cov}(X, Y)=\dfrac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right) \nonumber$
Because we’re multiplying (i.e., taking the “product” of) a quantity that depends on X by a quantity that depends on Y and then averaging80, you can think of the formula for the covariance as an “average cross product” between X and Y. The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If the relationship between them is positive (in the sense shown in Figure@reffig:corr) then the covariance is also positive; and if the relationship is negative then the covariance is also negative. In other words, the covariance captures the basic qualitative idea of correlation. Unfortunately, the raw magnitude of the covariance isn’t easy to interpret: it depends on the units in which X and Y are expressed, and worse yet, the actual units that the covariance itself is expressed in are really weird. For instance, if X refers to the dan.sleep variable (units: hours) and Y refers to the dan.grump variable (units: grumps), then the units for their covariance are “hours × grumps”. And I have no freaking idea what that would even mean.
The Pearson correlation coefficient r fixes this interpretation problem by standardising the covariance, in pretty much the exact same way that the z-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations.81 In other words, the correlation between X and Y can be written as follows:
$r_{X Y}=\dfrac{\operatorname{Cov}(X, Y)}{\hat{\sigma}_{X} \hat{\sigma}_{Y}} \nonumber$
By doing this standardisation, not only do we keep all of the nice properties of the covariance discussed earlier, but the actual values of r are on a meaningful scale: r=1 implies a perfect positive relationship, and r=−1 implies a perfect negative relationship. I’ll expand a little more on this point later, in Section@refsec:interpretingcorrelations. But before I do, let’s look at how to calculate correlations in R.
Calculating correlations in R
Calculating correlations in R can be done using the cor() command. The simplest way to use the command is to specify two input arguments x and y, each one corresponding to one of the variables. The following extract illustrates the basic usage of the function:82
cor( x = parenthood$dan.sleep, y = parenthood$dan.grump )
## [1] -0.903384
However, the cor() function is a bit more powerful than this simple example suggests. For example, you can also calculate a complete “correlation matrix”, between all pairs of variables in the data frame:83
# correlate all pairs of variables in "parenthood":
cor( x = parenthood )
## dan.sleep baby.sleep dan.grump day
## dan.sleep 1.00000000 0.62794934 -0.90338404 -0.09840768
## baby.sleep 0.62794934 1.00000000 -0.56596373 -0.01043394
## dan.grump -0.90338404 -0.56596373 1.00000000 0.07647926
## day -0.09840768 -0.01043394 0.07647926 1.00000000
Interpreting a correlation
Naturally, in real life you don’t see many correlations of 1. So how should you interpret a correlation of, say r=.4? The honest answer is that it really depends on what you want to use the data for, and on how strong the correlations in your field tend to be. A friend of mine in engineering once argued that any correlation less than .95 is completely useless (I think he was exaggerating, even for engineering). On the other hand there are real cases – even in psychology – where you should really expect correlations that strong. For instance, one of the benchmark data sets used to test theories of how people judge similarities is so clean that any theory that can’t achieve a correlation of at least .9 really isn’t deemed to be successful. However, when looking for (say) elementary correlates of intelligence (e.g., inspection time, response time), if you get a correlation above .3 you’re doing very very well. In short, the interpretation of a correlation depends a lot on the context. That said, the rough guide in Table ?? is pretty typical.
knitr::kable(
rbind(
c("-1.0 to -0.9" ,"Very strong", "Negative"),
c("-0.9 to -0.7", "Strong", "Negative") ,
c("-0.7 to -0.4", "Moderate", "Negative") ,
c("-0.4 to -0.2", "Weak", "Negative"),
c("-0.2 to 0","Negligible", "Negative") ,
c("0 to 0.2","Negligible", "Positive"),
c("0.2 to 0.4", "Weak", "Positive"),
c("0.4 to 0.7", "Moderate", "Positive"),
c("0.7 to 0.9", "Strong", "Positive"),
c("0.9 to 1.0", "Very strong", "Positive")), col.names=c("Correlation", "Strength", "Direction"),
booktabs = TRUE)
Correlation Strength Direction
-1.0 to -0.9 Very strong Negative
-0.9 to -0.7 Strong Negative
-0.7 to -0.4 Moderate Negative
-0.4 to -0.2 Weak Negative
-0.2 to 0 Negligible Negative
0 to 0.2 Negligible Positive
0.2 to 0.4 Weak Positive
0.4 to 0.7 Moderate Positive
0.7 to 0.9 Strong Positive
0.9 to 1.0 Very strong Positive
However, something that can never be stressed enough is that you should always look at the scatterplot before attaching any interpretation to the data. A correlation might not mean what you think it means. The classic illustration of this is “Anscombe’s Quartet” (??? Anscombe1973), which is a collection of four data sets. Each data set has two variables, an X and a Y. For all four data sets the mean value for X is 9 and the mean for Y is 7.5. The, standard deviations for all X variables are almost identical, as are those for the the Y variables. And in each case the correlation between X and Y is r=0.816. You can verify this yourself, since the dataset comes distributed with R. The commands would be:
cor( anscombe$x1, anscombe$y1 )
## [1] 0.8164205
cor( anscombe$x2, anscombe$y2 )
## [1] 0.8162365
and so on.
You’d think that these four data setswould look pretty similar to one another. They do not. If we draw scatterplots of X against Y for all four variables, as shown in Figure 5.11 we see that all four of these are spectacularly different to each other.
The lesson here, which so very many people seem to forget in real life is “always graph your raw data”. This will be the focus of Chapter 6.
Spearman’s rank correlations
The Pearson correlation coefficient is useful for a lot of things, but it does have shortcomings. One issue in particular stands out: what it actually measures is the strength of the linear relationship between two variables. In other words, what it gives you is a measure of the extent to which the data all tend to fall on a single, perfectly straight line. Often, this is a pretty good approximation to what we mean when we say “relationship”, and so the Pearson correlation is a good thing to calculation. Sometimes, it isn’t.
One very common situation where the Pearson correlation isn’t quite the right thing to use arises when an increase in one variable X really is reflected in an increase in another variable Y, but the nature of the relationship isn’t necessarily linear. An example of this might be the relationship between effort and reward when studying for an exam. If you put in zero effort (X) into learning a subject, then you should expect a grade of 0% (Y). However, a little bit of effort will cause a massive improvement: just turning up to lectures means that you learn a fair bit, and if you just turn up to classes, and scribble a few things down so your grade might rise to 35%, all without a lot of effort. However, you just don’t get the same effect at the other end of the scale. As everyone knows, it takes a lot more effort to get a grade of 90% than it takes to get a grade of 55%. What this means is that, if I’ve got data looking at study effort and grades, there’s a pretty good chance that Pearson correlations will be misleading.
To illustrate, consider the data plotted in Figure 5.12, showing the relationship between hours worked and grade received for 10 students taking some class. The curious thing about this – highly fictitious – data set is that increasing your effort always increases your grade. It might be by a lot or it might be by a little, but increasing effort will never decrease your grade. The data are stored in effort.Rdata:
> load( "effort.Rdata" )
> who(TRUE)
-- Name -- -- Class -- -- Size --
effort data.frame 10 x 2
$hours numeric 10$grade numeric 10
The raw data look like this:
> effort
hours grade
1 2 13
2 76 91
3 40 79
4 6 14
5 16 21
6 28 74
7 27 47
8 59 85
9 46 84
10 68 88
If we run a standard Pearson correlation, it shows a strong relationship between hours worked and grade received,
> cor( effort$hours, effort$grade )
[1] 0.909402
but this doesn’t actually capture the observation that increasing hours worked always increases the grade. There’s a sense here in which we want to be able to say that the correlation is perfect but for a somewhat different notion of what a “relationship” is. What we’re looking for is something that captures the fact that there is a perfect ordinal relationship here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade. That’s not what a correlation of r=.91 says at all.
How should we address this? Actually, it’s really easy: if we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, lets rank all 10 of our students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2). Notice that I’m using “rank =1” to mean “low rank”. Sometimes in everyday language we talk about “rank = 1” to mean “top rank” rather than “bottom rank”. So be careful: you can rank “from smallest value to largest value” (i.e., small equals rank 1) or you can rank “from largest value to smallest value” (i.e., large equals rank 1). In this case, I’m ranking from smallest to largest, because that’s the default way that R does it. But in real life, it’s really easy to forget which way you set things up, so you have to put a bit of effort into remembering!
Okay, so let’s have a look at our students when we rank them from worst to best in terms of effort and reward:
rank (hours worked) rank (grade received)
student 1 1
student 2 10
student 3 6
student 4 2
student 5 3
student 6 5
student 7 4
student 8 8
student 9 7
student 10 9
Hm. These are identical. The student who put in the most effort got the best grade, the student with the least effort got the worst grade, etc. We can get R to construct these rankings using the rank() function, like this:
> hours.rank <- rank( effort$hours ) # rank students by hours worked > grade.rank <- rank( effort$grade ) # rank students by grade received
As the table above shows, these two rankings are identical, so if we now correlate them we get a perfect relationship:
> cor( hours.rank, grade.rank )
[1] 1
What we’ve just re-invented is Spearman’s rank order correlation, usually denoted ρ to distinguish it from the Pearson correlation r. We can calculate Spearman’s ρ using R in two different ways. Firstly we could do it the way I just showed, using the rank() function to construct the rankings, and then calculate the Pearson correlation on these ranks. However, that’s way too much effort to do every time. It’s much easier to just specify the method argument of the cor() function.
> cor( effort$hours, effort$grade, method = "spearman")
[1] 1
The default value of the method argument is "pearson", which is why we didn’t have to specify it earlier on when we were doing Pearson correlations.
correlate() function
As we’ve seen, the cor() function works pretty well, and handles many of the situations that you might be interested in. One thing that many beginners find frustrating, however, is the fact that it’s not built to handle non-numeric variables. From a statistical perspective, this is perfectly sensible: Pearson and Spearman correlations are only designed to work for numeric variables, so the cor() function spits out an error.
Here’s what I mean. Suppose you were keeping track of how many hours you worked in any given day, and counted how many tasks you completed. If you were doing the tasks for money, you might also want to keep track of how much pay you got for each job. It would also be sensible to keep track of the weekday on which you actually did the work: most of us don’t work as much on Saturdays or Sundays. If you did this for 7 weeks, you might end up with a data set that looks like this one:
> load("work.Rdata")
> who(TRUE)
-- Name -- -- Class -- -- Size --
work data.frame 49 x 7
$hours numeric 49$tasks numeric 49
$pay numeric 49$day integer 49
$weekday factor 49$week numeric 49
$day.type factor 49 > head(work) hours tasks pay day weekday week day.type 1 7.2 14 41 1 Tuesday 1 weekday 2 7.4 11 39 2 Wednesday 1 weekday 3 6.6 14 13 3 Thursday 1 weekday 4 6.5 22 47 4 Friday 1 weekday 5 3.1 5 4 5 Saturday 1 weekend 6 3.0 7 12 6 Sunday 1 weekend Obviously, I’d like to know something about how all these variables correlate with one another. I could correlate hours with pay quite using cor(), like so: > cor(work$hours,work\$pay)
[1] 0.7604283
But what if I wanted a quick and easy way to calculate all pairwise correlations between the numeric variables? I can’t just input the work data frame, because it contains two factor variables, weekday and day.type. If I try this, I get an error:
> cor(work)
Error in cor(work) : 'x' must be numeric
It order to get the correlations that I want using the cor() function, is create a new data frame that doesn’t contain the factor variables, and then feed that new data frame into the cor() function. It’s not actually very hard to do that, and I’ll talk about how to do it properly in Section@refsec:subsetdataframe. But it would be nice to have some function that is smart enough to just ignore the factor variables. That’s where the correlate() function in the lsr package can be handy. If you feed it a data frame that contains factors, it knows to ignore them, and returns the pairwise correlations only between the numeric variables:
> correlate(work)
CORRELATIONS
============
- correlation type: pearson
- correlations shown only when both variables are numeric
hours tasks pay day weekday week day.type
hours . 0.800 0.760 -0.049 . 0.018 .
tasks 0.800 . 0.720 -0.072 . -0.013 .
pay 0.760 0.720 . 0.137 . 0.196 .
day -0.049 -0.072 0.137 . . 0.990 .
weekday . . . . . . .
week 0.018 -0.013 0.196 0.990 . . .
day.type . . . . . . .
The output here shows a . whenever one of the variables is non-numeric. It also shows a . whenever a variable is correlated with itself (it’s not a meaningful thing to do). The correlate() function can also do Spearman correlations, by specifying the corr.method to use:
> correlate( work, corr.method="spearman" )
CORRELATIONS
============
- correlation type: spearman
- correlations shown only when both variables are numeric
hours tasks pay day weekday week day.type
hours . 0.805 0.745 -0.047 . 0.010 .
tasks 0.805 . 0.730 -0.068 . -0.008 .
pay 0.745 0.730 . 0.094 . 0.154 .
day -0.047 -0.068 0.094 . . 0.990 .
weekday . . . . . . .
week 0.010 -0.008 0.154 0.990 . . .
day.type . . . . . . .
Obviously, there’s no new functionality in the correlate() function, and any advanced R user would be perfectly capable of using the cor() function to get these numbers out. But if you’re not yet comfortable with extracting a subset of a data frame, the correlate()` function is for you. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.07%3A_Correlations.txt |
There’s one last topic that I want to discuss briefly in this chapter, and that’s the issue of missing data. Real data sets very frequently turn out to have missing values: perhaps someone forgot to fill in a particular survey question, for instance. Missing data can be the source of a lot of tricky issues, most of which I’m going to gloss over. However, at a minimum, you need to understand the basics of handling missing data in R.
single variable case
Let’s start with the simplest case, in which you’re trying to calculate descriptive statistics for a single variable which has missing data. In R, this means that there will be `NA` values in your data vector. Let’s create a variable like that:
``> partial <- c(10, 20, NA, 30)``
Let’s assume that you want to calculate the mean of this variable. By default, R assumes that you want to calculate the mean using all four elements of this vector, which is probably the safest thing for a dumb automaton to do, but it’s rarely what you actually want. Why not? Well, remember that the basic interpretation of `NA` is “I don’t know what this number is”. This means that `1 + NA = NA`: if I add 1 to some number that I don’t know (i.e., the `NA`) then the answer is also a number that I don’t know. As a consequence, if you don’t explicitly tell R to ignore the `NA` values, and the data set does have missing values, then the output will itself be a missing value. If I try to calculate the mean of the `partial` vector, without doing anything about the missing value, here’s what happens:
``````> mean( x = partial )
[1] NA``````
Technically correct, but deeply unhelpful.
To fix this, all of the descriptive statistics functions that I’ve discussed in this chapter (with the exception of `cor()` which is a special case I’ll discuss below) have an optional argument called `na.rm`, which is shorthand for “remove NA values”. By default, `na.rm = FALSE`, so R does nothing about the missing data problem. Let’s try setting `na.rm = TRUE` and see what happens:
When calculating sums and means when missing data are present (i.e., when there are `NA` values) there’s actually an additional argument to the function that you should be aware of. This argument is called `na.rm`, and is a logical value indicating whether R should ignore (or “remove”) the missing data for the purposes of doing the calculations. By default, R assumes that you want to keep the missing values, so unless you say otherwise it will set `na.rm = FALSE`. However, R assumes that `1 + NA = NA`: if I add 1 to some number that I don’t know (i.e., the `NA`) then the answer is also a number that I don’t know. As a consequence, if you don’t explicitly tell R to ignore the `NA` values, and the data set does have missing values, then the output will itself be a missing value. This is illustrated in the following extract:
``````> mean( x = partial, na.rm = TRUE )
[1] 20``````
Notice that the mean is `20` (i.e., `60 / 3`) and not `15`. When R ignores a `NA` value, it genuinely ignores it. In effect, the calculation above is identical to what you’d get if you asked for the mean of the three-element vector `c(10, 20, 30)`.
As indicated above, this isn’t unique to the `mean()` function. Pretty much all of the other functions that I’ve talked about in this chapter have an `na.rm` argument that indicates whether it should ignore missing values. However, its behaviour is the same for all these functions, so I won’t waste everyone’s time by demonstrating it separately for each one.
Missing values in pairwise calculations
I mentioned earlier that the `cor()` function is a special case. It doesn’t have an `na.rm` argument, because the story becomes a lot more complicated when more than one variable is involved. What it does have is an argument called `use` which does roughly the same thing, but you need to think little more carefully about what you want this time. To illustrate the issues, let’s open up a data set that has missing values, `parenthood2.Rdata`. This file contains the same data as the original parenthood data, but with some values deleted. It contains a single data frame, `parenthood2`:
``````> load( "parenthood2.Rdata" )
> print( parenthood2 )
dan.sleep baby.sleep dan.grump day
1 7.59 NA 56 1
2 7.91 11.66 60 2
3 5.14 7.92 82 3
4 7.71 9.61 55 4
5 6.68 9.75 NA 5
6 5.99 5.04 72 6
BLAH BLAH BLAH``````
If I calculate my descriptive statistics using the `describe()` function
``````> describe( parenthood2 )
var n mean sd median trimmed mad min max BLAH
dan.sleep 1 91 6.98 1.02 7.03 7.02 1.13 4.84 9.00 BLAH
baby.sleep 2 89 8.11 2.05 8.20 8.13 2.28 3.25 12.07 BLAH
dan.grump 3 92 63.15 9.85 61.00 62.66 10.38 41.00 89.00 BLAH
day 4 100 50.50 29.01 50.50 50.50 37.06 1.00 100.00 BLAH``````
we can see from the `n` column that there are 9 missing values for `dan.sleep`, 11 missing values for `baby.sleep` and 8 missing values for `dan.grump`.84 Suppose what I would like is a correlation matrix. And let’s also suppose that I don’t bother to tell R how to handle those missing values. Here’s what happens:
``````> cor( parenthood2 )
dan.sleep baby.sleep dan.grump day
dan.sleep 1 NA NA NA
baby.sleep NA 1 NA NA
dan.grump NA NA 1 NA
day NA NA NA 1``````
Annoying, but it kind of makes sense. If I don’t know what some of the values of `dan.sleep` and `baby.sleep` actually are, then I can’t possibly know what the correlation between these two variables is either, since the formula for the correlation coefficient makes use of every single observation in the data set. Once again, it makes sense: it’s just not particularly helpful.
To make R behave more sensibly in this situation, you need to specify the `use` argument to the `cor()` function. There are several different values that you can specify for this, but the two that we care most about in practice tend to be `"complete.obs"` and `"pairwise.complete.obs"`. If we specify `use = "complete.obs"`, R will completely ignore all cases (i.e., all rows in our `parenthood2` data frame) that have any missing values at all. So, for instance, if you look back at the extract earlier when I used the `head()` function, notice that observation 1 (i.e., day 1) of the `parenthood2` data set is missing the value for `baby.sleep`, but is otherwise complete? Well, if you choose `use = "complete.obs"` R will ignore that row completely: that is, even when it’s trying to calculate the correlation between `dan.sleep` and `dan.grump`, observation 1 will be ignored, because the value of `baby.sleep` is missing for that observation. Here’s what we get:
``````> cor(parenthood2, use = "complete.obs")
dan.sleep baby.sleep dan.grump day
dan.sleep 1.00000000 0.6394985 -0.89951468 0.06132891
baby.sleep 0.63949845 1.0000000 -0.58656066 0.14555814
dan.grump -0.89951468 -0.5865607 1.00000000 -0.06816586
day 0.06132891 0.1455581 -0.06816586 1.00000000``````
The other possibility that we care about, and the one that tends to get used more often in practice, is to set `use = "pairwise.complete.obs"`. When we do that, R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of `parenthood2` is for `baby.sleep` R will only drop observation 1 when `baby.sleep` is one of the variables involved: and so R keeps observation 1 when trying to correlate `dan.sleep` and `dan.grump`. When we do it this way, here’s what we get:
``````> cor(parenthood2, use = "pairwise.complete.obs")
dan.sleep baby.sleep dan.grump day
dan.sleep 1.00000000 0.61472303 -0.903442442 -0.076796665
baby.sleep 0.61472303 1.00000000 -0.567802669 0.058309485
dan.grump -0.90344244 -0.56780267 1.000000000 0.005833399
day -0.07679667 0.05830949 0.005833399 1.000000000``````
Similar, but not quite the same. It’s also worth noting that the `correlate()` function (in the `lsr` package) automatically uses the “pairwise complete” method:
``````> correlate(parenthood2)
CORRELATIONS
============
- correlation type: pearson
- correlations shown only when both variables are numeric
dan.sleep baby.sleep dan.grump day
dan.sleep . 0.615 -0.903 -0.077
baby.sleep 0.615 . -0.568 0.058
dan.grump -0.903 -0.568 . 0.006
day -0.077 0.058 0.006 .``````
The two approaches have different strengths and weaknesses. The “pairwise complete” approach has the advantage that it keeps more observations, so you’re making use of more of your data and (as we’ll discuss in tedious detail in Chapter 10 and it improves the reliability of your estimated correlation. On the other hand, it means that every correlation in your correlation matrix is being computed from a slightly different set of observations, which can be awkward when you want to compare the different correlations that you’ve got.
So which method should you use? It depends a lot on why you think your values are missing, and probably depends a little on how paranoid you are. For instance, if you think that the missing values were “chosen” completely randomly85 then you’ll probably want to use the pairwise method. If you think that missing data are a cue to thinking that the whole observation might be rubbish (e.g., someone just selecting arbitrary responses in your questionnaire), but that there’s no pattern to which observations are “rubbish” then it’s probably safer to keep only those observations that are complete. If you think there’s something systematic going on, in that some observations are more likely to be missing than others, then you have a much trickier problem to solve, and one that is beyond the scope of this book. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.08%3A_Handling_Missing_Values.txt |
Calculating some basic descriptive statistics is one of the very first things you do when analysing real data, and descriptive statistics are much simpler to understand than inferential statistics, so like every other statistics textbook I’ve started with descriptives. In this chapter, we talked about the following topics:
• Measures of central tendency. Broadly speaking, central tendency measures tell you where the data are. There’s three measures that are typically reported in the literature: the mean, median and mode. (Section 5.1)
• Measures of variability. In contrast, measures of variability tell you about how “spread out” the data are. The key measures are: range, standard deviation, interquartile reange (Section 5.2)
• Getting summaries of variables in R. Since this book focuses on doing data analysis in R, we spent a bit of time talking about how descriptive statistics are computed in R. (Section 2.8 and 5.5)
• Standard scores. The z-score is a slightly unusual beast. It’s not quite a descriptive statistic, and not quite an inference. We talked about it in Section 5.6. Make sure you understand that section: it’ll come up again later.
• Correlations. Want to know how strong the relationship is between two variables? Calculate a correlation. (Section 5.7)
• Missing data. Dealing with missing data is one of those frustrating things that data analysts really wish the didn’t have to think about. In real life it can be hard to do well. For the purpose of this book, we only touched on the basics in Section 5.8
In the next section we’ll move on to a discussion of how to draw pictures! Everyone loves a pretty picture, right? But before we do, I want to end on an important point. A traditional first course in statistics spends only a small proportion of the class on descriptive statistics, maybe one or two lectures at most. The vast majority of the lecturer’s time is spent on inferential statistics, because that’s where all the hard stuff is. That makes sense, but it hides the practical everyday importance of choosing good descriptives. With that in mind…
5.10: Epilogue- Good Descriptive Statistics Are Descriptive
The death of one man is a tragedy. The death of millions is a statistic.
– Josef Stalin, Potsdam 1945
950,000 – 1,200,000
– Estimate of Soviet repression deaths, 1937-1938 (Ellman 2002)
Stalin’s infamous quote about the statistical character death of millions is worth giving some thought. The clear intent of his statement is that the death of an individual touches us personally and its force cannot be denied, but that the deaths of a multitude are incomprehensible, and as a consequence mere statistics, more easily ignored. I’d argue that Stalin was half right. A statistic is an abstraction, a description of events beyond our personal experience, and so hard to visualise. Few if any of us can imagine what the deaths of millions is “really” like, but we can imagine one death, and this gives the lone death its feeling of immediate tragedy, a feeling that is missing from Ellman’s cold statistical description.
Yet it is not so simple: without numbers, without counts, without a description of what happened, we have no chance of understanding what really happened, no opportunity event to try to summon the missing feeling. And in truth, as I write this, sitting in comfort on a Saturday morning, half a world and a whole lifetime away from the Gulags, when I put the Ellman estimate next to the Stalin quote a dull dread settles in my stomach and a chill settles over me. The Stalinist repression is something truly beyond my experience, but with a combination of statistical data and those recorded personal histories that have come down to us, it is not entirely beyond my comprehension. Because what Ellman’s numbers tell us is this: over a two year period, Stalinist repression wiped out the equivalent of every man, woman and child currently alive in the city where I live. Each one of those deaths had it’s own story, was it’s own tragedy, and only some of those are known to us now. Even so, with a few carefully chosen statistics, the scale of the atrocity starts to come into focus.
Thus it is no small thing to say that the first task of the statistician and the scientist is to summarise the data, to find some collection of numbers that can convey to an audience a sense of what has happened. This is the job of descriptive statistics, but it’s not a job that can be told solely using the numbers. You are a data analyst, not a statistical software package. Part of your job is to take these statistics and turn them into a description. When you analyse data, it is not sufficient to list off a collection of numbers. Always remember that what you’re really trying to do is communicate with a human audience. The numbers are important, but they need to be put together into a meaningful story that your audience can interpret. That means you need to think about framing. You need to think about context. And you need to think about the individual events that your statistics are summarising.
References
Ellman, Michael. 2002. “Soviet Repression Statistics: Some Comments.” Europe-Asia Studies 54 (7). Taylor & Francis: 1151–72.
1. Note for non-Australians: the AFL is an Australian rules football competition. You don’t need to know anything about Australian rules in order to follow this section.
2. The choice to use Σ to denote summation isn’t arbitrary: it’s the Greek upper case letter sigma, which is the analogue of the letter S in that alphabet. Similarly, there’s an equivalent symbol used to denote the multiplication of lots of numbers: because multiplications are also called “products”, we use the Π symbol for this; the Greek upper case pi, which is the analogue of the letter P.
3. Note that, just as we saw with the combine function `c()` and the remove function `rm()`, the `sum()` function has unnamed arguments. I’ll talk about unnamed arguments later in Section 8.4.1, but for now let’s just ignore this detail.
4. www.abc.net.au/news/stories/2010/09/24/3021480.htm
5. Or at least, the basic statistical theory – these days there is a whole subfield of statistics called robust statistics that tries to grapple with the messiness of real data and develop theory that can cope with it.
6. As we saw earlier, it does have a function called `mode()`, but it does something completely different.
7. This is called a “0-1 loss function”, meaning that you either win (1) or you lose (0), with no middle ground.
8. Well, I will very briefly mention the one that I think is coolest, for a very particular definition of “cool”, that is. Variances are additive. Here’s what that means: suppose I have two variables X and Y, whose variances are \$
9. With the possible exception of the third question.
10. Strictly, the assumption is that the data are normally distributed, which is an important concept that we’ll discuss more in Chapter 9, and will turn up over and over again later in the book.
11. The assumption again being that the data are normally-distributed!
12. The “−3” part is something that statisticians tack on to ensure that the normal curve has kurtosis zero. It looks a bit stupid, just sticking a “-3” at the end of the formula, but there are good mathematical reasons for doing this.
13. I haven’t discussed how to compute z-scores, explicitly, but you can probably guess. For a variable `X`, the simplest way is to use a command like `(X - mean(X)) / sd(X)`. There’s also a fancier function called `scale()` that you can use, but it relies on somewhat more complicated R concepts that I haven’t explained yet.
14. Technically, because I’m calculating means and standard deviations from a sample of data, but want to talk about my grumpiness relative to a population, what I’m actually doing is estimating a z score. However, since we haven’t talked about estimation yet (see Chapter 10) I think it’s best to ignore this subtlety, especially as it makes very little difference to our calculations.
15. Though some caution is usually warranted. It’s not always the case that one standard deviation on variable A corresponds to the same “kind” of thing as one standard deviation on variable B. Use common sense when trying to determine whether or not the z scores of two variables can be meaningfully compared.
16. Actually, even that table is more than I’d bother with. In practice most people pick one measure of central tendency, and one measure of variability only.
17. Just like we saw with the variance and the standard deviation, in practice we divide by N−1 rather than N.
18. This is an oversimplification, but it’ll do for our purposes.
19. If you are reading this after having already completed Chapter 11 you might be wondering about hypothesis tests for correlations. R has a function called `cor.test()` that runs a hypothesis test for a single correlation, and the `psych` package contains a version called `corr.test()` that can run tests for every correlation in a correlation matrix; hypothesis tests for correlations are discussed in more detail in Section 15.6.
20. An alternative usage of `cor()` is to correlate one set of variables with another subset of variables. If `X` and `Y` are both data frames with the same number of rows, then `cor(x = X, y = Y)` will produce a correlation matrix that correlates all variables in `X` with all variables in `Y`.
21. It’s worth noting that, even though we have missing data for each of these variables, the output doesn’t contain any `NA` values. This is because, while `describe()` also has an `na.rm` argument, the default value for this function is `na.rm = TRUE`.
22. The technical term here is “missing completely at random” (often written MCAR for short). Makes sense, I suppose, but it does sound ungrammatical to me. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/05%3A_Descriptive_Statistics/5.09%3A_Summary.txt |
Above all else show the data.
–Edward Tufte86
Visualising data is one of the most important tasks facing the data analyst. It’s important for two distinct but closely related reasons. Firstly, there’s the matter of drawing “presentation graphics”: displaying your data in a clean, visually appealing fashion makes it easier for your reader to understand what you’re trying to tell them. Equally important, perhaps even more important, is the fact that drawing graphs helps you to understand the data. To that end, it’s important to draw “exploratory graphics” that help you learn about the data as you go about analysing it. These points might seem pretty obvious, but I cannot count the number of times I’ve seen people forget them.
``## Warning: package 'HistData' was built under R version 3.5.2``
To give a sense of the importance of this chapter, I want to start with a classic illustration of just how powerful a good graph can be. To that end, Figure 6.1 shows a redrawing of one of the most famous data visualisations of all time: John Snow’s 1854 map of cholera deaths. The map is elegant in its simplicity. In the background we have a street map, which helps orient the viewer. Over the top, we see a large number of small dots, each one representing the location of a cholera case. The larger symbols show the location of water pumps, labelled by name. Even the most casual inspection of the graph makes it very clear that the source of the outbreak is almost certainly the Broad Street pump. Upon viewing this graph, Dr Snow arranged to have the handle removed from the pump, ending the outbreak that had killed over 500 people. Such is the power of a good data visualisation.
The goals in this chapter are twofold: firstly, to discuss several fairly standard graphs that we use a lot when analysing and presenting data, and secondly, to show you how to create these graphs in R. The graphs themselves tend to be pretty straightforward, so in that respect this chapter is pretty simple. Where people usually struggle is learning how to produce graphs, and especially, learning how to produce good graphs.87 Fortunately, learning how to draw graphs in R is reasonably simple, as long as you’re not too picky about what your graph looks like. What I mean when I say this is that R has a lot of very good graphing functions, and most of the time you can produce a clean, high-quality graphic without having to learn very much about the low-level details of how R handles graphics. Unfortunately, on those occasions when you do want to do something non-standard, or if you need to make highly specific changes to the figure, you actually do need to learn a fair bit about the these details; and those details are both complicated and boring. With that in mind, the structure of this chapter is as follows: I’ll start out by giving you a very quick overview of how graphics work in R. I’ll then discuss several different kinds of graph and how to draw them, as well as showing the basics of how to customise these plots. I’ll then talk in more detail about R graphics, discussing some of those complicated and boring issues. In a future version of this book, I intend to finish this chapter off by talking about what makes a good or a bad graph, but I haven’t yet had the time to write that section.
06: Drawing Graphs
Reduced to its simplest form, you can think of an R graphic as being much like a painting. You start out with an empty canvas. Every time you use a graphics function, it paints some new things onto your canvas. Later on, you can paint more things over the top if you want; but just like painting, you can’t “undo” your strokes. If you make a mistake, you have to throw away your painting and start over. Fortunately, this is way more easy to do when using R than it is when painting a picture in real life: you delete the plot and then type a new set of commands.88 This way of thinking about drawing graphs is referred to as the painter’s model. So far, this probably doesn’t sound particularly complicated, and for the vast majority of graphs you’ll want to draw it’s exactly as simple as it sounds. Much like painting in real life, the headaches usually start when we dig into details. To see why, I’ll expand this “painting metaphor” a bit further just to show you the basics of what’s going on under the hood, but before I do I want to stress that you really don’t need to understand all these complexities in order to draw graphs. I’d been using R for years before I even realised that most of these issues existed! However, I don’t want you to go through the same pain I went through every time I inadvertently discovered one of these things, so here’s a quick overview.
Firstly, if you want to paint a picture, you need to paint it on something. In real life, you can paint on lots of different things. Painting onto canvas isn’t the same as painting onto paper, and neither one is the same as painting on a wall. In R, the thing that you paint your graphic onto is called a device. For most applications that we’ll look at in this book, this “device” will be a window on your computer. If you’re using Windows as your operating system, then the name for this device is `windows`; on a Mac it’s called `quartz` because that’s the name of the software that the Mac OS uses to draw pretty pictures; and on Linux/Unix, you’re probably using `X11`. On the other hand, if you’re using Rstudio (regardless of which operating system you’re on), there’s a separate device called `RStudioGD` that forces R to paint inside the “plots” panel in Rstudio. However, from the computers perspective there’s nothing terribly special about drawing pictures on screen: and so R is quite happy to paint pictures directly into a file. R can paint several different types of image files: `jpeg`, `png`, `pdf`, `postscript`, `tiff` and `bmp` files are all among the options that you have available to you. For the most part, these different devices all behave the same way, so you don’t really need to know much about the differences between them when learning how to draw pictures. But, just like real life painting, sometimes the specifics do matter. Unless stated otherwise, you can assume that I’m drawing a picture on screen, using the appropriate device (i.e., `windows`, `quartz`, `X11` or `RStudioGD`). One the rare occasions where these behave differently from one another, I’ll try to point it out in the text.
Secondly, when you paint a picture you need to paint it with something. Maybe you want to do an oil painting, but maybe you want to use watercolour. And, generally speaking, you pretty much have to pick one or the other. The analog to this in R is a “graphics system”. A graphics system defines a collection of very low-level graphics commands about what to draw and where to draw it. Something that surprises most new R users is the discovery that R actually has two completely independent graphics systems, known as traditional graphics (in the `graphics` package) and grid graphics (in the `grid` package).89 Not surprisingly, the traditional graphics system is the older of the two: in fact, it’s actually older than R since it has it’s origins in S, the system from which R is descended. Grid graphics are newer, and in some respects more powerful, so many of the more recent, fancier graphical tools in R make use of grid graphics. However, grid graphics are somewhat more complicated beasts, so most people start out by learning the traditional graphics system. Nevertheless, as long as you don’t want to use any low-level commands yourself, then you don’t really need to care about whether you’re using traditional graphics or grid graphics. However, the moment you do want to tweak your figure by using some low-level commands you do need to care. Because these two different systems are pretty much incompatible with each other, there’s a pretty big divide in R graphics universe. Unless stated otherwise, you can assume that everything I’m saying pertains to traditional graphics.
Thirdly, a painting is usually done in a particular style. Maybe it’s a still life, maybe it’s an impressionist piece, or maybe you’re trying to annoy me by pretending that cubism is a legitimate artistic style. Regardless, each artistic style imposes some overarching aesthetic and perhaps even constraints on what can (or should) be painted using that style. In the same vein, R has quite a number of different packages, each of which provide a collection of high-level graphics commands. A single high-level command is capable of drawing an entire graph, complete with a range of customisation options. Most but not all of the high-level commands that I’ll talk about in this book come from the `graphics` package itself, and so belong to the world of traditional graphics. These commands all tend to share a common visual style, although there are a few graphics that I’ll use that come from other packages that differ in style somewhat. On the other side of the great divide, the grid universe relies heavily on two different packages – `lattice` and `ggplots2` – each of which provides a quite different visual style. As you’ve probably guessed, there’s a whole separate bunch of functions that you’d need to learn if you want to use `lattice` graphics or make use of the `ggplots2`. However, for the purposes of this book I’ll restrict myself to talking about the basic `graphics` tools.
At this point, I think we’ve covered more than enough background material. The point that I’m trying to make by providing this discussion isn’t to scare you with all these horrible details, but rather to try to convey to you the fact that R doesn’t really provide a single coherent graphics system. Instead, R itself provides a platform, and different people have built different graphical tools using that platform. As a consequence of this fact, there’s two different universes of graphics, and a great multitude of packages that live in them. At this stage you don’t need to understand these complexities, but it’s useful to know that they’re there. But for now, I think we can be happy with a simpler view of things: we’ll draw pictures on screen using the traditional graphics system, and as much as possible we’ll stick to high level commands only.
So let’s start painting. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.01%3A_An_Overview_of_R_Graphics.txt |
Before I discuss any specialised graphics, let’s start by drawing a few very simple graphs just to get a feel for what it’s like to draw pictures using R. To that end, let’s create a small vector `Fibonacci` that contains a few numbers we’d like R to draw for us. Then, we’ll ask R to `plot()` those numbers:
``````> Fibonacci <- c( 1,1,2,3,5,8,13 )
> plot( Fibonacci )``````
The result is Figure 6.2.
As you can see, what R has done is plot the values stored in the `Fibonacci` variable on the vertical axis (y-axis) and the corresponding index on the horizontal axis (x-axis). In other words, since the 4th element of the vector has a value of 3, we get a dot plotted at the location (4,3). That’s pretty straightforward, and the image in Figure 6.2 is probably pretty close to what you would have had in mind when I suggested that we plot the `Fibonacci` data. However, there’s quite a lot of customisation options available to you, so we should probably spend a bit of time looking at some of those options. So, be warned: this ends up being a fairly long section, because there’s so many possibilities open to you. Don’t let it overwhelm you though… while all of the options discussed here are handy to know about, you can get by just fine only knowing a few of them. The only reason I’ve included all this stuff right at the beginning is that it ends up making the rest of the chapter a lot more readable!
tedious digression
Before we go into any discussion of customising plots, we need a little more background. The important thing to note when using the `plot()` function, is that it’s another example of a generic function (Section 4.11, much like `print()` and `summary()`, and so its behaviour changes depending on what kind of input you give it. However, the `plot()` function is somewhat fancier than the other two, and its behaviour depends on two arguments, `x` (the first input, which is required) and `y` (which is optional). This makes it (a) extremely powerful once you get the hang of it, and (b) hilariously unpredictable, when you’re not sure what you’re doing. As much as possible, I’ll try to make clear what type of inputs produce what kinds of outputs. For now, however, it’s enough to note that I’m only doing very basic plotting, and as a consequence all of the work is being done by the `plot.default()` function.
What kinds of customisations might we be interested in? If you look at the help documentation for the default plotting method (i.e., type `?plot.default` or `help("plot.default")`) you’ll see a very long list of arguments that you can specify to customise your plot. I’ll talk about several of them in a moment, but first I want to point out something that might seem quite wacky. When you look at all the different options that the help file talks about, you’ll notice that some of the options that it refers to are “proper” arguments to the `plot.default()` function, but it also goes on to mention a bunch of things that look like they’re supposed to be arguments, but they’re not listed in the “Usage” section of the file, and the documentation calls them graphical parameters instead. Even so, it’s usually possible to treat them as if they were arguments of the plotting function. Very odd. In order to stop my readers trying to find a brick and look up my home address, I’d better explain what’s going on; or at least give the basic gist behind it.
What exactly is a graphical parameter? Basically, the idea is that there are some characteristics of a plot which are pretty universal: for instance, regardless of what kind of graph you’re drawing, you probably need to specify what colour to use for the plot, right? So you’d expect there to be something like a `col` argument to every single graphics function in R? Well, sort of. In order to avoid having hundreds of arguments for every single function, what R does is refer to a bunch of these “graphical parameters” which are pretty general purpose. Graphical parameters can be changed directly by using the low-level `par()` function, which I discuss briefly in Section ?? though not in a lot of detail. If you look at the help files for graphical parameters (i.e., type `?par`) you’ll see that there’s lots of them. Fortunately, (a) the default settings are generally pretty good so you can ignore the majority of the parameters, and (b) as you’ll see as we go through this chapter, you very rarely need to use `par()` directly, because you can “pretend” that graphical parameters are just additional arguments to your high-level function (e.g. `plot.default()`). In short… yes, R does have these wacky “graphical parameters” which can be quite confusing. But in most basic uses of the plotting functions, you can act as if they were just undocumented additional arguments to your function.
Customising the title and the axis labels
One of the first things that you’ll find yourself wanting to do when customising your plot is to label it better. You might want to specify more appropriate axis labels, add a title or add a subtitle. The arguments that you need to specify to make this happen are:
• `main`. A character string containing the title.
• `sub`. A character string containing the subtitle.
• `xlab`. A character string containing the x-axis label.
• `ylab`. A character string containing the y-axis label.
These aren’t graphical parameters, they’re arguments to the high-level function. However, because the high-level functions all rely on the same low-level function to do the drawing90 the names of these arguments are identical for pretty much every high-level function I’ve come across. Let’s have a look at what happens when we make use of all these arguments. Here’s the command…
``````> plot( x = Fibonacci,
+ main = "You specify title using the 'main' argument",
+ sub = "The subtitle appears here! (Use the 'sub' argument for this)",
+ xlab = "The x-axis label is 'xlab'",
+ ylab = "The y-axis label is 'ylab'"
+ )``````
The picture that this draws is shown in Figure 6.3.
It’s more or less as you’d expect. The plot itself is identical to the one we drew in Figure 6.2, except for the fact that we’ve changed the axis labels, and added a title and a subtitle. Even so, there’s a couple of interesting features worth calling your attention to. Firstly, notice that the subtitle is drawn below the plot, which I personally find annoying; as a consequence I almost never use subtitles. You may have a different opinion, of course, but the important thing is that you remember where the subtitle actually goes. Secondly, notice that R has decided to use boldface text and a larger font size for the title. This is one of my most hated default settings in R graphics, since I feel that it draws too much attention to the title. Generally, while I do want my reader to look at the title, I find that the R defaults are a bit overpowering, so I often like to change the settings. To that end, there are a bunch of graphical parameters that you can use to customise the font style:
• Font styles: `font.main`, `font.sub`, `font.lab`, `font.axis`. These four parameters control the font style used for the plot title (`font.main`), the subtitle (`font.sub`), the axis labels (`font.lab`: note that you can’t specify separate styles for the x-axis and y-axis without using low level commands), and the numbers next to the tick marks on the axis (`font.axis`). Somewhat irritatingly, these arguments are numbers instead of meaningful names: a value of 1 corresponds to plain text, 2 means boldface, 3 means italic and 4 means bold italic.
• Font colours: `col.main`, `col.sub`, `col.lab`, `col.axis`. These parameters do pretty much what the name says: each one specifies a colour in which to type each of the different bits of text. Conveniently, R has a very large number of named colours (type `colours()` to see a list of over 650 colour names that R knows), so you can use the English language name of the colour to select it.91 Thus, the parameter value here string like `"red"`, `"gray25"` or `"springgreen4"` (yes, R really does recognise four different shades of “spring green”).
• Font size: `cex.main`, `cex.sub`, `cex.lab`, `cex.axis`. Font size is handled in a slightly curious way in R. The “cex” part here is short for “character expansion”, and it’s essentially a magnification value. By default, all of these are set to a value of 1, except for the font title: `cex.main` has a default magnification of 1.2, which is why the title font is 20% bigger than the others.
• Font family: `family`. This argument specifies a font family to use: the simplest way to use it is to set it to `"sans"`, `"serif"`, or `"mono"`, corresponding to a san serif font, a serif font, or a monospaced font. If you want to, you can give the name of a specific font, but keep in mind that different operating systems use different fonts, so it’s probably safest to keep it simple. Better yet, unless you have some deep objections to the R defaults, just ignore this parameter entirely. That’s what I usually do.
To give you a sense of how you can use these parameters to customise your titles, the following command can be used to draw Figure 6.4:
``````> plot( x = Fibonacci, # the data to plot
+ main = "The first 7 Fibonacci numbers", # the title
+ xlab = "Position in the sequence", # x-axis label
+ ylab = "The Fibonacci number", # y-axis label
+ font.main = 1, # plain text for title
+ cex.main = 1, # normal size for title
+ font.axis = 2, # bold text for numbering
+ col.lab = "gray50" # grey colour for labels
+ )``````
Although this command is quite long, it’s not complicated: all it does is override a bunch of the default parameter values. The only difficult aspect to this is that you have to remember what each of these parameters is called, and what all the different values are. And in practice I never remember: I have to look up the help documentation every time, or else look it up in this book.
Changing the plot type
Adding and customising the titles associated with the plot is one way in which you can play around with what your picture looks like. Another thing that you’ll want to do is customise the appearance of the actual plot! To start with, let’s look at the single most important options that the `plot()` function (or, recalling that we’re dealing with a generic function, in this case the `plot.default()` function, since that’s the one doing all the work) provides for you to use, which is the `type` argument. The type argument specifies the visual style of the plot. The possible values for this are:
• `type = "p"`. Draw the points only.
• `type = "l"`. Draw a line through the points.
• `type = "o"`. Draw the line over the top of the points.
• `type = "b"`. Draw both points and lines, but don’t overplot.
• `type = "h"`. Draw “histogram-like” vertical bars.
• `type = "s"`. Draw a staircase, going horizontally then vertically.
• `type = "S"`. Draw a Staircase, going vertically then horizontally.
• `type = "c"`. Draw only the connecting lines from the “b” version.
• `type = "n"`. Draw nothing. (Apparently this is useful sometimes?)
The simplest way to illustrate what each of these really looks like is just to draw them. To that end, Figure 6.5 shows the same Fibonacci data, drawn using six different `types` of plot. As you can see, by altering the type argument you can get a qualitatively different appearance to your plot. In other words, as far as R is concerned, the only difference between a scatterplot (like the ones we drew in Section 5.7 and a line plot is that you draw a scatterplot by setting `type = "p"` and you draw a line plot by setting `type = "l"`. However, that doesn’t imply that you should think of them as begin equivalent to each other. As you can see by looking at Figure 6.5, a line plot implies that there is some notion of continuity from one point to the next, whereas a scatterplot does not.
Changing other features of the plot
In Section ?? we talked about a group of graphical parameters that are related to the formatting of titles, axis labels etc. The second group of parameters I want to discuss are those related to the formatting of the plot itself:
• Colour of the plot: `col`. As we saw with the previous colour-related parameters, the simplest way to specify this parameter is using a character string: e.g., `col = "blue"`. It’s a pretty straightforward parameter to specify: the only real subtlety is that every high-level function tends to draw a different “thing” as it’s output, and so this parameter gets interpreted a little differently by different functions. However, for the `plot.default()` function it’s pretty simple: the `col` argument refers to the colour of the points and/or lines that get drawn!
• Character used to plot points: `pch`. The plot character parameter is a number, usually between 1 and 25. What it does is tell R what symbol to use to draw the points that it plots. The simplest way to illustrate what the different values do is with a picture. Figure 6.6 a shows the first 25 plotting characters. The default plotting character is a hollow circle (i.e., `pch = 1`).
• Plot size: `cex`. This parameter describes a character expansion factor (i.e., magnification) for the plotted characters. By default `cex=1`, but if you want bigger symbols in your graph you should specify a larger value.
• Line type: `lty`. The line type parameter describes the kind of line that R draws. It has seven values which you can specify using a number between `0` and `7`, or using a meaningful character string: `"blank"`, `"solid"`, `"dashed"`, `"dotted"`, `"dotdash"`, `"longdash"`, or `"twodash"`. Note that the “blank” version (value 0) just means that R doesn’t draw the lines at all. The other six versions are shown in Figure 6.6 b.
• Line width: `lwd`. The last graphical parameter in this category that I want to mention is the line width parameter, which is just a number specifying the width of the line. The default value is 1. Not surprisingly, larger values produce thicker lines and smaller values produce thinner lines. Try playing around with different values of `lwd` to see what happens.
To illustrate what you can do by altering these parameters, let’s try the following command:
``````> plot( x = Fibonacci, # the data set
+ type = "b", # plot both points and lines
+ col = "blue", # change the plot colour to blue
+ pch = 19, # plotting character is a solid circle
+ cex = 5, # plot it at 5x the usual size
+ lty = 2, # change line type to dashed
+ lwd = 4 # change line width to 4x the usual
+ )``````
The output is shown in Figure 6.7.
``````plot( x = Fibonacci,
type = "b",
col = "blue",
pch = 19,
cex=5,
lty=2,
lwd=4)``````
Changing the appearance of the axes
There are several other possibilities worth discussing. Ignoring graphical parameters for the moment, there’s a few other arguments to the `plot.default()` function that you might want to use. As before, many of these are standard arguments that are used by a lot of high level graphics functions:
• Changing the axis scales: `xlim`, `ylim`. Generally R does a pretty good job of figuring out where to set the edges of the plot. However, you can override its choices by setting the `xlim` and `ylim` arguments. For instance, if I decide I want the vertical scale of the plot to run from 0 to 100, then I’d set `ylim = c(0, 100)`.
• Suppress labelling: `ann`. This is a logical-valued argument that you can use if you don’t want R to include any text for a title, subtitle or axis label. To do so, set `ann = FALSE`. This will stop R from including any text that would normally appear in those places. Note that this will override any of your manual titles. For example, if you try to add a title using the `main` argument, but you also specify `ann = FALSE`, no title will appear.
• Suppress axis drawing: `axes`. Again, this is a logical valued argument. Suppose you don’t want R to draw any axes at all. To suppress the axes, all you have to do is add `axes = FALSE`. This will remove the axes and the numbering, but not the axis labels (i.e. the `xlab` and `ylab` text). Note that you can get finer grain control over this by specifying the `xaxt` and `yaxt` graphical parameters instead (see below).
• Include a framing box: `frame.plot`. Suppose you’ve removed the axes by setting `axes = FALSE`, but you still want to have a simple box drawn around the plot; that is, you only wanted to get rid of the numbering and the tick marks, but you want to keep the box. To do that, you set `frame.plot = TRUE`.
Note that this list isn’t exhaustive. There are a few other arguments to the `plot.default` function that you can play with if you want to, but those are the ones you are probably most likely to want to use. As always, however, if these aren’t enough options for you, there’s also a number of other graphical parameters that you might want to play with as well. That’s the focus of the next section. In the meantime, here’s a command that makes use of all these different options:
``````> plot( x = Fibonacci, # the data
+ xlim = c(0, 15), # expand the x-scale
+ ylim = c(0, 15), # expand the y-scale
+ ann = FALSE, # delete all annotations
+ axes = FALSE, # delete the axes
+ frame.plot = TRUE # but include a framing box
+ )``````
The output is shown in Figure 6.8, and it’s pretty much exactly as you’d expect. The axis scales on both the horizontal and vertical dimensions have been expanded, the axes have been suppressed as have the annotations, but I’ve kept a box around the plot.
``````plot( x = Fibonacci,
xlim = c(0, 15),
ylim = c(0, 15),
ann = FALSE,
axes = FALSE,
frame.plot = TRUE)``````
Before moving on, I should point out that there are several graphical parameters relating to the axes, the box, and the general appearance of the plot which allow finer grain control over the appearance of the axes and the annotations.
• Suppressing the axes individually: `xaxt`, `yaxt`. These graphical parameters are basically just fancier versions of the `axes` argument we discussed earlier. If you want to stop R from drawing the vertical axis but you’d like it to keep the horizontal axis, set `yaxt = "n"`. I trust that you can figure out how to keep the vertical axis and suppress the horizontal one!
• Box type: `bty`. In the same way that `xaxt`, `yaxt` are just fancy versions of `axes`, the box type parameter is really just a fancier version of the `frame.plot` argument, allowing you to specify exactly which out of the four borders you want to keep. The way we specify this parameter is a bit stupid, in my opinion: the possible values are `"o"` (the default), `"l"`, `"7"`, `"c"`, `"u"`, or `"]"`, each of which will draw only those edges that the corresponding character suggests. That is, the letter `"c"` has a top, a bottom and a left, but is blank on the right hand side, whereas `"7"` has a top and a right, but is blank on the left and the bottom. Alternatively a value of `"n"` means that no box will be drawn.
• Orientation of the axis labels `las`. I presume that the name of this parameter is an acronym of label style or something along those lines; but what it actually does is govern the orientation of the text used to label the individual tick marks (i.e., the numbering, not the `xlab` and `ylab` axis labels). There are four possible values for `las`: A value of 0 means that the labels of both axes are printed parallel to the axis itself (the default). A value of 1 means that the text is always horizontal. A value of 2 means that the labelling text is printed at right angles to the axis. Finally, a value of 3 means that the text is always vertical.
Again, these aren’t the only possibilities. There are a few other graphical parameters that I haven’t mentioned that you could use to customise the appearance of the axes,92 but that’s probably enough (or more than enough) for now. To give a sense of how you could use these parameters, let’s try the following command:
``````> plot( x = Fibonacci, # the data
+ xaxt = "n", # don't draw the x-axis
+ bty = "]", # keep bottom, right and top of box only
+ las = 1 # rotate the text
+ )``````
The output is shown in Figure 6.9. As you can see, this isn’t a very useful plot at all. However, it does illustrate the graphical parameters we’re talking about, so I suppose it serves its purpose.
``````plot( x = Fibonacci,
xaxt = "n",
bty = "]",
las = 1 )``````
Don’t panic
At this point, a lot of readers will be probably be thinking something along the lines of, “if there’s this much detail just for drawing a simple plot, how horrible is it going to get when we start looking at more complicated things?” Perhaps, contrary to my earlier pleas for mercy, you’ve found a brick to hurl and are right now leafing through an Adelaide phone book trying to find my address. Well, fear not! And please, put the brick down. In a lot of ways, we’ve gone through the hardest part: we’ve already covered vast majority of the plot customisations that you might want to do. As you’ll see, each of the other high level plotting commands we’ll talk about will only have a smallish number of additional options. Better yet, even though I’ve told you about a billion different ways of tweaking your plot, you don’t usually need them. So in practice, now that you’ve read over it once to get the gist, the majority of the content of this section is stuff you can safely forget: just remember to come back to this section later on when you want to tweak your plot. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.02%3A_An_Introduction_to_Plotting.txt |
Now that we’ve tamed (or possibly fled from) the beast that is R graphical parameters, let’s talk more seriously about some real life graphics that you’ll want to draw. We begin with the humble histogram. Histograms are one of the simplest and most useful ways of visualising data. They make most sense when you have an interval or ratio scale (e.g., the `afl.margins` data from Chapter 5 and what you want to do is get an overall impression of the data. Most of you probably know how histograms work, since they’re so widely used, but for the sake of completeness I’ll describe them. All you do is divide up the possible values into bins, and then count the number of observations that fall within each bin. This count is referred to as the frequency of the bin, and is displayed as a bar: in the AFL winning margins data, there are 33 games in which the winning margin was less than 10 points, and it is this fact that is represented by the height of the leftmost bar in Figure 6.10. Drawing this histogram in R is pretty straightforward. The function you need to use is called `hist()`, and it has pretty reasonable default settings. In fact, Figure 6.10 is exactly what you get if you just type this:
``> hist( afl.margins ) # panel a``
``````load("./rbook-master/data/aflsmall.Rdata")
hist(afl.margins) # panel a``````
Although this image would need a lot of cleaning up in order to make a good presentation graphic (i.e., one you’d include in a report), it nevertheless does a pretty good job of describing the data. In fact, the big strength of a histogram is that (properly used) it does show the entire spread of the data, so you can get a pretty good sense about what it looks like. The downside to histograms is that they aren’t very compact: unlike some of the other plots I’ll talk about it’s hard to cram 20-30 histograms into a single image without overwhelming the viewer. And of course, if your data are nominal scale (e.g., the `afl.finalists` data) then histograms are useless.
The main subtlety that you need to be aware of when drawing histograms is determining where the `breaks` that separate bins should be located, and (relatedly) how many breaks there should be. In Figure 6.10, you can see that R has made pretty sensible choices all by itself: the breaks are located at 0, 10, 20, … 120, which is exactly what I would have done had I been forced to make a choice myself. On the other hand, consider the two histograms in Figure 6.11 and 6.12, which I produced using the following two commands:
``hist( x = afl.margins, breaks = 3 ) # panel b``
``hist( x = afl.margins, breaks = 0:116 ) # panel c``
In Figure 6.12, the bins are only 1 point wide. As a result, although the plot is very informative (it displays the entire data set with no loss of information at all!) the plot is very hard to interpret, and feels quite cluttered. On the other hand, the plot in Figure 6.11 has a bin width of 50 points, and has the opposite problem: it’s very easy to “read” this plot, but it doesn’t convey a lot of information. One gets the sense that this histogram is hiding too much. In short, the way in which you specify the breaks has a big effect on what the histogram looks like, so it’s important to make sure you choose the breaks sensibly. In general R does a pretty good job of selecting the breaks on its own, since it makes use of some quite clever tricks that statisticians have devised for automatically selecting the right bins for a histogram, but nevertheless it’s usually a good idea to play around with the breaks a bit to see what happens.
There is one fairly important thing to add regarding how the `breaks` argument works. There are two different ways you can specify the breaks. You can either specify how many breaks you want (which is what I did for panel b when I typed `breaks = 3`) and let R figure out where they should go, or you can provide a vector that tells R exactly where the breaks should be placed (which is what I did for panel c when I typed `breaks = 0:116`). The behaviour of the `hist()` function is slightly different depending on which version you use. If all you do is tell it how many breaks you want, R treats it as a “suggestion” not as a demand. It assumes you want “approximately 3” breaks, but if it doesn’t think that this would look very pretty on screen, it picks a different (but similar) number. It does this for a sensible reason – it tries to make sure that the breaks are located at sensible values (like 10) rather than stupid ones (like 7.224414). And most of the time R is right: usually, when a human researcher says “give me 3 breaks”, he or she really does mean “give me approximately 3 breaks, and don’t put them in stupid places”. However, sometimes R is dead wrong. Sometimes you really do mean “exactly 3 breaks”, and you know precisely where you want them to go. So you need to invoke “real person privilege”, and order R to do what it’s bloody well told. In order to do that, you have to input the full vector that tells R exactly where you want the breaks. If you do that, R will go back to behaving like the nice little obedient calculator that it’s supposed to be.
Visual style of your histogram
Okay, so at this point we can draw a basic histogram, and we can alter the number and even the location of the `breaks`. However, the visual style of the histograms shown in Figure @ref(fig:hist1a; hist1b; hist1c) could stand to be improved. We can fix this by making use of some of the other arguments to the `hist()` function. Most of the things you might want to try doing have already been covered in Section 6.2, but there’s a few new things:
• Shading lines: `density`, `angle`. You can add diagonal lines to shade the bars: the `density` value is a number indicating how many lines per inch R should draw (the default value of `NULL` means no lines), and the `angle` is a number indicating how many degrees from horizontal the lines should be drawn at (default is `angle = 45` degrees).
• Specifics regarding colours: `col`, `border`. You can also change the colours: in this instance the `col` parameter sets the colour of the shading (either the shading lines if there are any, or else the colour of the interior of the bars if there are not), and the `border` argument sets the colour of the edges of the bars.
• Labelling the bars: `labels`. You can also attach labels to each of the bars using the `labels` argument. The simplest way to do this is to set `labels = TRUE`, in which case R will add a number just above each bar, that number being the exact number of observations in the bin. Alternatively, you can choose the labels yourself, by inputting a vector of strings, e.g., `labels = c("label 1","label 2","etc")`
Not surprisingly, this doesn’t exhaust the possibilities. If you type `help("hist")` or `?hist` and have a look at the help documentation for histograms, you’ll see a few more options. A histogram that makes use of the histogram-specific customisations as well as several of the options we discussed in Section ?? is shown in Figure ??. The R command that I used to draw it is this:
``````hist( x = afl.margins,
main = "2010 AFL margins", # title of the plot
xlab = "Margin", # set the x-axis label
density = 10, # draw shading lines: 10 per inch
angle = 40, # set the angle of the shading lines is 40 degrees
border = "gray20", # set the colour of the borders of the bars
col = "gray80", # set the colour of the shading lines
labels = TRUE, # add frequency labels to each bar
ylim = c(0,40) # change the scale of the y-axis
)``````
Overall, this is a much nicer histogram than the default ones. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.03%3A_Histograms.txt |
Histograms are one of the most widely used methods for displaying the observed values for a variable. They’re simple, pretty, and very informative. However, they do take a little bit of effort to draw. Sometimes it can be quite useful to make use of simpler, if less visually appealing, options. One such alternative is the stem and leaf plot. To a first approximation you can think of a stem and leaf plot as a kind of text-based histogram. Stem and leaf plots aren’t used as widely these days as they were 30 years ago, since it’s now just as easy to draw a histogram as it is to draw a stem and leaf plot. Not only that, they don’t work very well for larger data sets. As a consequence you probably won’t have as much of a need to use them yourself, though you may run into them in older publications. These days, the only real world situation where I use them is if I have a small data set with 20-30 data points and I don’t have a computer handy, because it’s pretty easy to quickly sketch a stem and leaf plot by hand.
With all that as background, lets have a look at stem and leaf plots. The AFL margins data contains 176 observations, which is at the upper end for what you can realistically plot this way. The function in R for drawing stem and leaf plots is called `stem()` and if we ask for a stem and leaf plot of the `afl.margins` data, here’s what we get:
``stem( afl.margins )``
``````##
## The decimal point is 1 digit(s) to the right of the |
##
## 0 | 001111223333333344567788888999999
## 1 | 0000011122234456666899999
## 2 | 00011222333445566667788999999
## 3 | 01223555566666678888899
## 4 | 012334444477788899
## 5 | 00002233445556667
## 6 | 0113455678
## 7 | 01123556
## 8 | 122349
## 9 | 458
## 10 | 148
## 11 | 6``````
The values to the left of the `|` are called stems and the values to the right are called leaves. If you just look at the shape that the leaves make, you can see something that looks a lot like a histogram made out of numbers, just rotated by 90 degrees. But if you know how to read the plot, there’s quite a lot of additional information here. In fact, it’s also giving you the actual values of all of the observations in the data set. To illustrate, let’s have a look at the last line in the stem and leaf plot, namely `11 | 6`. Specifically, let’s compare this to the largest values of the `afl.margins` data set:
``````> max( afl.margins )
[1] 116``````
Hm… `11 | 6` versus `116`. Obviously the stem and leaf plot is trying to tell us that the largest value in the data set is 116. Similarly, when we look at the line that reads `10 | 148`, the way we interpret it to note that the stem and leaf plot is telling us that the data set contains observations with values 101, 104 and 108. Finally, when we see something like `5 | 00002233445556667` the four `0`s in the the stem and leaf plot are telling us that there are four observations with value 50.
I won’t talk about them in a lot of detail, but I should point out that some customisation options are available for stem and leaf plots in R. The two arguments that you can use to do this are:
• `scale`. Changing the `scale` of the plot (default value is 1), which is analogous to changing the number of breaks in a histogram. Reducing the scale causes R to reduce the number of stem values (i.e., the number of breaks, if this were a histogram) that the plot uses.
• `width`. The second way that to can customise a stem and leaf plot is to alter the `width` (default value is 80). Changing the width alters the maximum number of leaf values that can be displayed for any given stem.
However, since stem and leaf plots aren’t as important as they used to be, I’ll leave it to the interested reader to investigate these options. Try the following two commands to see what happens:
``````> stem( x = afl.margins, scale = .25 )
> stem( x = afl.margins, width = 20 )``````
The only other thing to note about stem and leaf plots is the line in which R tells you where the decimal point is. If our data set had included only the numbers .11, .15, .23, .35 and .59 and we’d drawn a stem and leaf plot of these data, then R would move the decimal point: the stem values would be 1,2,3,4 and 5, but R would tell you that the decimal point has moved to the left of the `|` symbol. If you want to see this in action, try the following command:
``> stem( x = afl.margins / 1000 )``
The stem and leaf plot itself will look identical to the original one we drew, except for the fact that R will tell you that the decimal point has moved. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.04%3A_Stem_and_Leaf_Plots.txt |
Another alternative to histograms is a boxplot, sometimes called a “box and whiskers” plot. Like histograms, they’re most suited to interval or ratio scale data. The idea behind a boxplot is to provide a simple visual depiction of the median, the interquartile range, and the range of the data. And because they do so in a fairly compact way, boxplots have become a very popular statistical graphic, especially during the exploratory stage of data analysis when you’re trying to understand the data yourself. Let’s have a look at how they work, again using the `afl.margins` data as our example. Firstly, let’s actually calculate these numbers ourselves using the `summary()` function:93
``````> summary( afl.margins )
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.00 12.75 30.50 35.30 50.50 116.00 ``````
So how does a boxplot capture these numbers? The easiest way to describe what a boxplot looks like is just to draw one. The function for doing this in R is (surprise, surprise) `boxplot()`. As always there’s a lot of optional arguments that you can specify if you want, but for the most part you can just let R choose the defaults for you. That said, I’m going to override one of the defaults to start with by specifying the `range` option, but for the most part you won’t want to do this (I’ll explain why in a minute). With that as preamble, let’s try the following command:
``boxplot( x = afl.margins, range = 100 )``
What R draws is shown in Figure ??, the most basic boxplot possible. When you look at this plot, this is how you should interpret it: the thick line in the middle of the box is the median; the box itself spans the range from the 25th percentile to the 75th percentile; and the “whiskers” cover the full range from the minimum value to the maximum value. This is summarised in the annotated plot in Figure ??.
In practice, this isn’t quite how boxplots usually work. In most applications, the “whiskers” don’t cover the full range from minimum to maximum. Instead, they actually go out to the most extreme data point that doesn’t exceed a certain bound. By default, this value is 1.5 times the interquartile range, corresponding to a `range` value of 1.5. Any observation whose value falls outside this range is plotted as a circle instead of being covered by the whiskers, and is commonly referred to as an outlier. For our AFL margins data, there is one observation (a game with a margin of 116 points) that falls outside this range. As a consequence, the upper whisker is pulled back to the next largest observation (a value of 108), and the observation at 116 is plotted as a circle. This is illustrated in Figure @ref(fig:boxplot2a. Since the default value is `range = 1.5` we can draw this plot using the simple command
``boxplot( afl.margins )``
Visual style of your boxplot
I’ll talk a little more about the relationship between boxplots and outliers in the Section ??, but before I do let’s take the time to clean this figure up. Boxplots in R are extremely customisable. In addition to the usual range of graphical parameters that you can tweak to make the plot look nice, you can also exercise nearly complete control over every element to the plot. Consider the boxplot in Figure ??: in this version of the plot, not only have I added labels (`xlab`, `ylab`) and removed the stupid border (`frame.plot`), I’ve also dimmed all of the graphical elements of the boxplot except the central bar that plots the median (`border`) so as to draw more attention to the median rather than the rest of the boxplot.
You’ve seen all these options in previous sections in this chapter, so hopefully those customisations won’t need any further explanation. However, I’ve done two new things as well: I’ve deleted the cross-bars at the top and bottom of the whiskers (known as the “staples” of the plot), and converted the whiskers themselves to solid lines. The arguments that I used to do this are called by the ridiculous names of `staplewex` and `whisklty`,94 and I’ll explain these in a moment.
But first, here’s the actual command I used to draw this figure:
``````> boxplot( x = afl.margins, # the data
+ xlab = "AFL games, 2010", # x-axis label
+ ylab = "Winning Margin", # y-axis label
+ border = "grey50", # dim the border of the box
+ frame.plot = FALSE, # don't draw a frame
+ staplewex = 0, # don't draw staples
+ whisklty = 1 # solid line for whisker
+ )``````
Overall, I think the resulting boxplot is a huge improvement in visual design over the default version. In my opinion at least, there’s a fairly minimalist aesthetic that governs good statistical graphics. Ideally, every visual element that you add to a plot should convey part of the message. If your plot includes things that don’t actually help the reader learn anything new, you should consider removing them. Personally, I can’t see the point of the cross-bars on a standard boxplot, so I’ve deleted them.
Okay, what commands can we use to customise the boxplot? If you type `?boxplot` and flick through the help documentation, you’ll notice that it does mention `staplewex` as an argument, but there’s no mention of `whisklty`. The reason for this is that the function that handles the drawing is called `bxp()`, so if you type `?bxp` all the gory details appear. Here’s the short summary. In order to understand why these arguments have such stupid names, you need to recognise that they’re put together from two components. The first part of the argument name specifies one part of the box plot: `staple` refers to the staples of the plot (i.e., the cross-bars), and `whisk` refers to the whiskers. The second part of the name specifies a graphical parameter: `wex` is a width parameter, and `lty` is a line type parameter. The parts of the plot you can customise are:
• `box`. The box that covers the interquartile range.
• `med`. The line used to show the median.
• `whisk`. The vertical lines used to draw the whiskers.
• `staple`. The cross bars at the ends of the whiskers.
• `out`. The points used to show the outliers.
The actual graphical parameters that you might want to specify are slightly different for each visual element, just because they’re different shapes from each other. As a consequence, the following options are available:
• Width expansion: `boxwex, staplewex, outwex`. These are scaling factors that govern the width of various parts of the plot. The default scaling factor is (usually) 0.8 for the box, and 0.5 for the other two. Note that in the case of the outliers this parameter is meaningless unless you decide to draw lines plotting the outliers rather than use points.
• Line type: `boxlty, medlty, whisklty, staplelty, outlty`. These govern the line type for the relevant elements. The values for this are exactly the same as those used for the regular `lty` parameter, with two exceptions. There’s an additional option where you can set `medlty = "blank"` to suppress the median line completely (useful if you want to draw a point for the median rather than plot a line). Similarly, by default the outlier line type is set to `outlty = "blank"`, because the default behaviour is to draw outliers as points instead of lines.
• Line width: `boxlwd, medlwd, whisklwd, staplelwd, outlwd`. These govern the line widths for the relevant elements, and behave the same way as the regular `lwd` parameter. The only thing to note is that the default value for `medlwd` value is three times the value of the others.
• Line colour: `boxcol, medcol, whiskcol, staplecol, outcol`. These govern the colour of the lines used to draw the relevant elements. Specify a colour in the same way that you usually do.
• Fill colour: `boxfill`. What colour should we use to fill the box?
• Point character: `medpch, outpch`. These behave like the regular `pch` parameter used to select the plot character. Note that you can set `outpch = NA` to stop R from plotting the outliers at all, and you can also set `medpch = NA` to stop it from drawing a character for the median (this is the default!)
• Point expansion: `medcex, outcex`. Size parameters for the points used to plot medians and outliers. These are only meaningful if the corresponding points are actually plotted. So for the default boxplot, which includes outlier points but uses a line rather than a point to draw the median, only the `outcex` parameter is meaningful.
• Background colours: `medbg, outbg`. Again, the background colours are only meaningful if the points are actually plotted.
Taken as a group, these parameters allow you almost complete freedom to select the graphical style for your boxplot that you feel is most appropriate to the data set you’re trying to describe. That said, when you’re first starting out there’s no shame in using the default settings! But if you want to master the art of designing beautiful figures, it helps to try playing around with these parameters to see what works and what doesn’t. Finally, I should mention a few other arguments that you might want to make use of:
• `horizontal`. Set this to `TRUE` to display the plot horizontally rather than vertically.
• `varwidth`. Set this to `TRUE` to get R to scale the width of each box so that the areas are proportional to the number of observations that contribute to the boxplot. This is only useful if you’re drawing multiple boxplots at once (see Section 6.5.3.
• `show.names`. Set this to `TRUE` to get R to attach labels to the boxplots.
• `notch`. If you set `notch = TRUE`, R will draw little notches in the sides of each box. If the notches of two boxplots don’t overlap, then there is a “statistically significant” difference between the corresponding medians. If you haven’t read Chapter 11, ignore this argument – we haven’t discussed statistical significance, so this doesn’t mean much to you. I’m mentioning it only because you might want to come back to the topic later on. (see also the `notch.frac` option when you type `?bxp`).
Using box plots to detect outliers
Because the boxplot automatically (unless you change the `range` argument) separates out those observations that lie within a certain range, people often use them as an informal method for detecting outliers: observations that are “suspiciously” distant from the rest of the data. Here’s an example. Suppose that I’d drawn the boxplot for the AFL margins data, and it came up looking like Figure 6.15.
It’s pretty clear that something funny is going on with one of the observations. Apparently, there was one game in which the margin was over 300 points! That doesn’t sound right to me. Now that I’ve become suspicious, it’s time to look a bit more closely at the data. One function that can be handy for this is the `which()` function; it takes as input a vector of logicals, and outputs the indices of the `TRUE` cases. This is particularly useful in the current context because it lets me do this:
``````> suspicious.cases <- afl.margins > 300
> which( suspicious.cases )
[1] 137``````
although in real life I probably wouldn’t bother creating the `suspicious.cases` variable: I’d just cut out the middle man and use a command like `which( afl.margins > 300 )`. In any case, what this has done is shown me that the outlier corresponds to game 137. Then, I find the recorded margin for that game:
``````> afl.margins[137]
[1] 333``````
Hm. That definitely doesn’t sound right. So then I go back to the original data source (the internet!) and I discover that the actual margin of that game was 33 points. Now it’s pretty clear what happened. Someone must have typed in the wrong number. Easily fixed, just by typing `afl.margins[137] <- 33`. While this might seem like a silly example, I should stress that this kind of thing actually happens a lot. Real world data sets are often riddled with stupid errors, especially when someone had to type something into a computer at some point. In fact, there’s actually a name for this phase of data analysis, since in practice it can waste a huge chunk of our time: data cleaning. It involves searching for typos, missing data and all sorts of other obnoxious errors in raw data files.95
What about the real data? Does the value of 116 constitute a funny observation not? Possibly. As it turns out the game in question was Fremantle v Hawthorn, and was played in round 21 (the second last home and away round of the season). Fremantle had already qualified for the final series and for them the outcome of the game was irrelevant; and the team decided to rest several of their star players. As a consequence, Fremantle went into the game severely underpowered. In contrast, Hawthorn had started the season very poorly but had ended on a massive winning streak, and for them a win could secure a place in the finals. With the game played on Hawthorn’s home turf96 and with so many unusual factors at play, it is perhaps no surprise that Hawthorn annihilated Fremantle by 116 points. Two weeks later, however, the two teams met again in an elimination final on Fremantle’s home ground, and Fremantle won comfortably by 30 points.97
So, should we exclude the game from subsequent analyses? If this were a psychology experiment rather than an AFL season, I’d be quite tempted to exclude it because there’s pretty strong evidence that Fremantle weren’t really trying very hard: and to the extent that my research question is based on an assumption that participants are genuinely trying to do the task. On the other hand, in a lot of studies we’re actually interested in seeing the full range of possible behaviour, and that includes situations where people decide not to try very hard: so excluding that observation would be a bad idea. In the context of the AFL data, a similar distinction applies. If I’d been trying to make tips about who would perform well in the finals, I would have (and in fact did) disregard the Round 21 massacre, because it’s way too misleading. On the other hand, if my interest is solely in the home and away season itself, I think it would be a shame to throw away information pertaining to one of the most distinctive (if boring) games of the year. In other words, the decision about whether to include outliers or exclude them depends heavily on why you think the data look they way they do, and what you want to use the data for. Statistical tools can provide an automatic method for suggesting candidates for deletion, but you really need to exercise good judgment here. As I’ve said before, R is a mindless automaton. It doesn’t watch the footy, so it lacks the broader context to make an informed decision. You are not a mindless automaton, so you should exercise judgment: if the outlier looks legitimate to you, then keep it. In any case, I’ll return to the topic again in Section 15.9, so let’s return to our discussion of how to draw boxplots.
Drawing multiple boxplots
One last thing. What if you want to draw multiple boxplots at once? Suppose, for instance, I wanted separate boxplots showing the AFL margins not just for 2010, but for every year between 1987 and 2010. To do that, the first thing we’ll have to do is find the data. These are stored in the `aflsmall2.Rdata` file. So let’s load it and take a quick peek at what’s inside:
``````> load( "aflsmall2.Rdata" )
> who( TRUE )
-- Name -- -- Class -- -- Size --
afl2 data.frame 4296 x 2
\$margin numeric 4296
\$year numeric 4296 ``````
Notice that `afl2` data frame is pretty big. It contains 4296 games, which is far more than I want to see printed out on my computer screen. To that end, R provides you with a few useful functions to print out only a few of the row in the data frame. The first of these is `head()` which prints out the first 6 rows, of the data frame, like this:
``````> head( afl2 )
margin year
1 33 1987
2 59 1987
3 45 1987
4 91 1987
5 39 1987
6 1 1987``````
You can also use the `tail()` function to print out the last 6 rows. The `car` package also provides a handy little function called `some()` which prints out a random subset of the rows.
In any case, the important thing is that we have the `afl2` data frame which contains the variables that we’re interested in. What we want to do is have R draw boxplots for the `margin` variable, plotted separately for each separate `year`. The way to do this using the `boxplot()` function is to input a `formula` rather than a variable as the input. In this case, the formula we want is `margin ~ year`. So our boxplot command now looks like this:
The result is shown in Figure 6.16.98 Even this, the default version of the plot, gives a sense of why it’s sometimes useful to choose boxplots instead of histograms. Even before taking the time to turn this basic output into something more readable, it’s possible to get a good sense of what the data look like from year to year without getting overwhelmed with too much detail. Now imagine what would have happened if I’d tried to cram 24 histograms into this space: no chance at all that the reader is going to learn anything useful.
That being said, the default boxplot leaves a great deal to be desired in terms of visual clarity. The outliers are too visually prominent, the dotted lines look messy, and the interesting content (i.e., the behaviour of the median and the interquartile range across years) gets a little obscured. Fortunately, this is easy to fix, since we’ve already covered a lot of tools you can use to customise your output. After playing around with several different versions of the plot, the one I settled on is shown in Figure 6.17. The command I used to produce it is long, but not complicated:
``````> boxplot( formula = margin ~ year, # the formula
+ data = afl2, # the data set
+ xlab = "AFL season", # x axis label
+ ylab = "Winning Margin", # y axis label
+ frame.plot = FALSE, # don't draw a frame
+ staplewex = 0, # don't draw staples
+ staplecol = "white", # (fixes a tiny display issue)
+ boxwex = .75, # narrow the boxes slightly
+ boxfill = "grey80", # lightly shade the boxes
+ whisklty = 1, # solid line for whiskers
+ whiskcol = "grey70", # dim the whiskers
+ boxcol = "grey70", # dim the box borders
+ outcol = "grey70", # dim the outliers
+ outpch = 20, # outliers as solid dots
+ outcex = .5, # shrink the outliers
+ medlty = "blank", # no line for the medians
+ medpch = 20, # instead, draw solid dots
+ medlwd = 1.5 # make them larger
+ )``````
Of course, given that the command is that long, you might have guessed that I didn’t spend ages typing all that rubbish in over and over again. Instead, I wrote a script, which I kept tweaking until it produced the figure that I wanted. We’ll talk about scripts later in Section 8.1, but given the length of the command I thought I’d remind you that there’s an easier way of trying out different commands than typing them all in over and over. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.05%3A_Boxplots.txt |
Scatterplots are a simple but effective tool for visualising data. We’ve already seen scatterplots in this chapter, when using the `plot()` function to draw the `Fibonacci` variable as a collection of dots (Section 6.2. However, for the purposes of this section I have a slightly different notion in mind. Instead of just plotting one variable, what I want to do with my scatterplot is display the relationship between two variables, like we saw with the figures in the section on correlation (Section 5.7. It’s this latter application that we usually have in mind when we use the term “scatterplot”. In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don’t really have a clear opinions about what the causal relationship is (e.g., does A cause B, or does B cause A, or does some other variable C control both A and B). If that’s the case, it doesn’t really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it’s conventional to plot the cause variable on the x-axis, and the effect variable on the y-axis. With that in mind, let’s look at how to draw scatterplots in R, using the same `parenthood` data set (i.e. `parenthood.Rdata`) that I used when introducing the idea of correlations.
Suppose my goal is to draw a scatterplot displaying the relationship between the amount of sleep that I get (`dan.sleep`) and how grumpy I am the next day (`dan.grump`). As you might expect given our earlier use of `plot()` to display the `Fibonacci` data, the function that we use is the `plot()` function, but because it’s a generic function all the hard work is still being done by the `plot.default()` function. In any case, there are two different ways in which we can get the plot that we’re after. The first way is to specify the name of the variable to be plotted on the `x` axis and the variable to be plotted on the `y` axis. When we do it this way, the command looks like this:
``````plot( x = parenthood\$dan.sleep, # data on the x-axis
y = parenthood\$dan.grump # data on the y-axis
) ``````
The second way do to it is to use a “formula and data frame” format, but I’m going to avoid using it.99 For now, let’s just stick with the `x` and `y` version. If we do this, the result is the very basic scatterplot shown in Figure 6.19. This serves fairly well, but there’s a few customisations that we probably want to make in order to have this work properly. As usual, we want to add some labels, but there’s a few other things we might want to do as well. Firstly, it’s sometimes useful to rescale the plots. In Figure 6.19 R has selected the scales so that the data fall neatly in the middle. But, in this case, we happen to know that the grumpiness measure falls on a scale from 0 to 100, and the hours slept falls on a natural scale between 0 hours and about 12 or so hours (the longest I can sleep in real life). So the command I might use to draw this is:
``````plot( x = parenthood\$dan.sleep, # data on the x-axis
y = parenthood\$dan.grump, # data on the y-axis
xlab = "My sleep (hours)", # x-axis label
ylab = "My grumpiness (0-100)", # y-axis label
xlim = c(0,12), # scale the x-axis
ylim = c(0,100), # scale the y-axis
pch = 20, # change the plot type
col = "gray50", # dim the dots slightly
frame.plot = FALSE # don't draw a box
)``````
This command produces the scatterplot in Figure ??, or at least very nearly. What it doesn’t do is draw the line through the middle of the points. Sometimes it can be very useful to do this, and I can do so using `lines()`, which is a low level plotting function. Better yet, the arguments that I need to specify are pretty much the exact same ones that I use when calling the `plot()` function. That is, suppose that I want to draw a line that goes from the point (4,93) to the point (9.5,37). Then the `x` locations can be specified by the vector `c(4,9.5)` and the `y` locations correspond to the vector `c(93,37)`. In other words, I use this command:
``````plot( x = parenthood\$dan.sleep, # data on the x-axis
y = parenthood\$dan.grump, # data on the y-axis
xlab = "My sleep (hours)", # x-axis label
ylab = "My grumpiness (0-100)", # y-axis label
xlim = c(0,12), # scale the x-axis
ylim = c(0,100), # scale the y-axis
pch = 20, # change the plot type
col = "gray50", # dim the dots slightly
frame.plot = FALSE # don't draw a box
)
lines( x = c(4,9.5), # the horizontal locations
y = c(93,37), # the vertical locations
lwd = 2 # line width
)``````
And when I do so, R plots the line over the top of the plot that I drew using the previous command. In most realistic data analysis situations you absolutely don’t want to just guess where the line through the points goes, since there’s about a billion different ways in which you can get R to do a better job. However, it does at least illustrate the basic idea.
One possibility, if you do want to get R to draw nice clean lines through the data for you, is to use the `scatterplot()` function in the `car` package. Before we can use `scatterplot()` we need to load the package:
``> library( car )``
Having done so, we can now use the function. The command we need is this one:
``````> scatterplot( dan.grump ~ dan.sleep,
+ data = parenthood,
+ smooth = FALSE
+ )```
```
``## Loading required package: carData``
The first two arguments should be familiar: the first input is a formula `dan.grump ~ dan.sleep` telling R what variables to plot,100 and the second specifies a `data` frame. The third argument `smooth` I’ve set to `FALSE` to stop the `scatterplot()` function from drawing a fancy “smoothed” trendline (since it’s a bit confusing to beginners). The scatterplot itself is shown in Figure 6.20. As you can see, it’s not only drawn the scatterplot, but its also drawn boxplots for each of the two variables, as well as a simple line of best fit showing the relationship between the two variables.
More elaborate options
Often you find yourself wanting to look at the relationships between several variables at once. One useful tool for doing so is to produce a scatterplot matrix, analogous to the correlation matrix.
``````> cor( x = parenthood ) # calculate correlation matrix
dan.sleep baby.sleep dan.grump day
dan.sleep 1.00000000 0.62794934 -0.90338404 -0.09840768
baby.sleep 0.62794934 1.00000000 -0.56596373 -0.01043394
dan.grump -0.90338404 -0.56596373 1.00000000 0.07647926
day -0.09840768 -0.01043394 0.07647926 1.00000000``````
We can get a the corresponding scatterplot matrix by using the `pairs()` function:101
``pairs( x = parenthood ) # draw corresponding scatterplot matrix ``
The output of the `pairs()` command is shown in Figure ??. An alternative way of calling the `pairs()` function, which can be useful in some situations, is to specify the variables to include using a one-sided formula. For instance, this
``````> pairs( formula = ~ dan.sleep + baby.sleep + dan.grump,
+ data = parenthood
+ )``````
would produce a 3×3 scatterplot matrix that only compare `dan.sleep`, `dan.grump` and `baby.sleep`. Obviously, the first version is much easier, but there are cases where you really only want to look at a few of the variables, so it’s nice to use the formula interface. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.06%3A_Scatterplots.txt |
Another form of graph that you often want to plot is the bar graph. The main function that you can use in R to draw them is the `barplot()` function.102 And to illustrate the use of the function, I’ll use the `finalists` variable that I introduced in Section 5.1.7. What I want to do is draw a bar graph that displays the number of finals that each team has played in over the time spanned by the `afl` data set. So, let’s start by creating a vector that contains this information. I’ll use the `tabulate()` function to do this (which will be discussed properly in Section ??, since it creates a simple numeric vector:
``````> freq <- tabulate( afl.finalists )
> print( freq )
[1] 26 25 26 28 32 0 6 39 27 28 28 17 6 24 26 39 24``````
This isn’t exactly the prettiest of frequency tables, of course. I’m only doing it this way so that you can see the `barplot()` function in it’s “purest” form: when the input is just an ordinary numeric vector. That being said, I’m obviously going to need the team names to create some labels, so let’s create a variable with those. I’ll do this using the `levels()` function, which outputs the names of all the levels of a factor (see Section 4.7:
``````> teams <- levels( afl.finalists )
> print( teams )
[1] "Adelaide" "Brisbane" "Carlton" "Collingwood"
[5] "Essendon" "Fitzroy" "Fremantle" "Geelong"
[9] "Hawthorn" "Melbourne" "North Melbourne" "Port Adelaide"
[13] "Richmond" "St Kilda" "Sydney" "West Coast"
[17] "Western Bulldogs"``````
Okay, so now that we have the information we need, let’s draw our bar graph. The main argument that you need to specify for a bar graph is the `height` of the bars, which in our case correspond to the values stored in the `freq` variable:
``````> barplot( height = freq ) # specifying the argument name (panel a)
> barplot( freq ) # the lazier version (panel a)``````
Either of these two commands will produce the simple bar graph shown in Figure 6.21.
As you can see, R has drawn a pretty minimal plot. It doesn’t have any labels, obviously, because we didn’t actually tell the `barplot()` function what the labels are! To do this, we need to specify the `names.arg` argument. The `names.arg` argument needs to be a vector of character strings containing the text that needs to be used as the label for each of the items. In this case, the `teams` vector is exactly what we need, so the command we’re looking for is:
``barplot( height = freq, names.arg = teams ) ``
This is an improvement, but not much of an improvement. R has only included a few of the labels, because it can’t fit them in the plot. This is the same behaviour we saw earlier with the multiple-boxplot graph in Figure 6.16. However, in Figure 6.16 it wasn’t an issue: it’s pretty obvious from inspection that the two unlabelled plots in between 1987 and 1990 must correspond to the data from 1988 and 1989. However, the fact that `barplot()` has omitted the names of every team in between Adelaide and Fitzroy is a lot more problematic.
The simplest way to fix this is to rotate the labels, so that the text runs vertically not horizontally. To do this, we need to alter set the `las` parameter, which I discussed briefly in Section ??. What I’ll do is tell R to rotate the text so that it’s always perpendicular to the axes (i.e., I’ll set `las = 2`). When I do that, as per the following command…
`````` barplot(height = freq, # the frequencies
names.arg = teams, # the label
las = 2) # rotate the labels``````
… the result is the bar graph shown in Figure 6.23. We’ve fixed the problem, but we’ve created a new one: the axis labels don’t quite fit anymore. To fix this, we have to be a bit cleverer again. A simple fix would be to use shorter names rather than the full name of all teams, and in many situations that’s probably the right thing to do. However, at other times you really do need to create a bit more space to add your labels, so I’ll show you how to do that.
Changing global settings using `par()`
Altering the margins to the plot is actually a somewhat more complicated exercise than you might think. In principle it’s a very simple thing to do: the size of the margins is governed by a graphical parameter called `mar`, so all we need to do is alter this parameter. First, let’s look at what the `mar` argument specifies. The `mar` argument is a vector containing four numbers: specifying the amount of space at the bottom, the left, the top and then the right. The units are “number of `lines'". The default value ``for`mar`is`c(5.1, 4.1, 4.1, 2.1)`, meaning that R leaves 5.1”lines" empty at the bottom, 4.1 lines on the left and the bottom, and only 2.1 lines on the right. In order to make more room at the bottom, what I need to do is change the first of these numbers. A value of 10.1 should do the trick.
So far this doesn’t seem any different to the other graphical parameters that we’ve talked about. However, because of the way that the traditional graphics system in R works, you need to specify what the margins will be before calling your high-level plotting function. Unlike the other cases we’ve see, you can’t treat `mar` as if it were just another argument in your plotting function. Instead, you have to use the `par()` function to change the graphical parameters beforehand, and only then try to draw your figure. In other words, the first thing I would do is this:
``> par( mar = c( 10.1, 4.1, 4.1, 2.1) )``
There’s no visible output here, but behind the scenes R has changed the graphical parameters associated with the current device (remember, in R terminology all graphics are drawn onto a “device”). Now that this is done, we could use the exact same command as before, but this time you’d see that the labels all fit, because R now leaves twice as much room for the labels at the bottom. However, since I’ve now figured out how to get the labels to display properly, I might as well play around with some of the other options, all of which are things you’ve seen before:
``````barplot( height = freq,
names.arg = teams,
las=2,
ylab = "Number of Finals",
main = "Finals Played, 1987-2010",
density = 10,
angle = 20)``````
However, one thing to remember about the `par()` function is that it doesn’t just change the graphical parameters for the current plot. Rather, the changes pertain to any subsequent plot that you draw onto the same device. This might be exactly what you want, in which case there’s no problem. But if not, you need to reset the graphical parameters to their original settings. To do this, you can either close the device (e.g., close the window, or click the “Clear All” button in the Plots panel in Rstudio) or you can reset the graphical parameters to their original values, using a command like this:
````> par( mar = c(5.1, 4.1, 4.1, 2.1) )`
``` | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.07%3A_Bar_Graphs.txt |
Hold on, you might be thinking. What’s the good of being able to draw pretty pictures in R if I can’t save them and send them to friends to brag about how awesome my data is? How do I save the picture? This is another one of those situations where the easiest thing to do is to use the RStudio tools.
If you’re running R through Rstudio, then the easiest way to save your image is to click on the “Export” button in the Plot panel (i.e., the area in Rstudio where all the plots have been appearing). When you do that you’ll see a menu that contains the options “Save Plot as PDF” and “Save Plot as Image”. Either version works. Both will bring up dialog boxes that give you a few options that you can play with, but besides that it’s pretty simple.
This works pretty nicely for most situations. So, unless you’re filled with a burning desire to learn the low level details, feel free to skip the rest of this section.
ugly details (advanced)
As I say, the menu-based options should be good enough for most people most of the time. However, one day you might want to be a bit more sophisticated, and make use of R’s image writing capabilities at a lower level. In this section I’ll give you a very basic introduction to this. In all honesty, this barely scratches the surface, but it will help a little bit in getting you started if you want to learn the details.
Okay, as I hinted earlier, whenever you’re drawing pictures in R you’re deemed to be drawing to a device of some kind. There are devices that correspond to a figure drawn on screen, and there are devices that correspond to graphics files that R will produce for you. For the purposes of this section I’ll assume that you’re using the default application in either Windows or Mac OS, not Rstudio. The reason for this is that my experience with the graphical device provided by Rstudio has led me to suspect that it still has a bunch on non-standard (or possibly just undocumented) features, and so I don’t quite trust that it always does what I expect. I’ve no doubt they’ll smooth it out later, but I can honestly say that I don’t quite get what’s going on with the `RStudioGD` device. In any case, we can ask R to list all of the graphics devices that currently exist, simply by using the command `dev.list()`. If there are no figure windows open, then you’ll see this:
``````> dev.list()
NULL``````
which just means that R doesn’t have any graphics devices open. However, suppose if you’ve just drawn a histogram and you type the same command, R will now give you a different answer. For instance, if you’re using Windows:
``````> hist( afl.margins )
> dev.list()
windows
2``````
What this means is that there is one graphics device (device 2) that is currently open, and it’s a figure window. If you did the same thing on a Mac, you get basically the same answer, except that the name of the device would be `quartz` rather than `windows`. If you had several graphics windows open (which, incidentally, you can do by using the `dev.new()` command) then you’d see something like this:
``````> dev.list()
windows windows windows
2 3 4 ``````
Okay, so that’s the basic idea behind graphics devices. The key idea here is that graphics files (like JPEG images etc) are also graphics devices as far as R is concerned. So what you want to do is to copy the contents of one graphics device to another one. There’s a command called `dev.copy()` that does this, but what I’ll explain to you is a simpler one called `dev.print()`. It’s pretty simple:
``````> dev.print( device = jpeg, # what are we printing to?
+ filename = "thisfile.jpg", # name of the image file
+ width = 480, # how many pixels wide should it be
+ height = 300 # how many pixels high should it be
+ )``````
This takes the “active” figure window, copies it to a jpeg file (which R treats as a device) and then closes that device. The `filename = "thisfile.jpg"` part tells R what to name the graphics file, and the `width = 480` and `height = 300` arguments tell R to draw an image that is 300 pixels high and 480 pixels wide. If you want a different kind of file, just change the device argument from `jpeg` to something else. R has devices for `png`, `tiff` and `bmp` that all work in exactly the same way as the `jpeg` command, but produce different kinds of files. Actually, for simple cartoonish graphics like this histogram, you’d be better advised to use PNG or TIFF over JPEG. The JPEG format is very good for natural images, but is wasteful for simple line drawings. The information above probably covers most things you might want to. However, if you want more information about what kinds of options you can specify using R, have a look at the help documentation by typing `?jpeg` or `?tiff` or whatever. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.08%3A_Saving_Image_Files_Using_R_and_Rstudio.txt |
Perhaps I’m a simple minded person, but I love pictures. Every time I write a new scientific paper, one of the first things I do is sit down and think about what the pictures will be. In my head, an article is really just a sequence of pictures, linked together by a story. All the rest of it is just window dressing. What I’m really trying to say here is that the human visual system is a very powerful data analysis tool. Give it the right kind of information and it will supply a human reader with a massive amount of knowledge very quickly. Not for nothing do we have the saying “a picture is worth a thousand words”. With that in mind, I think that this is one of the most important chapters in the book. The topics covered were:
• Basic overview to R graphics. In Section 6.1 we talked about how graphics in R are organised, and then moved on to the basics of how they’re drawn in Section 6.2.
• Common plots. Much of the chapter was focused on standard graphs that statisticians like to produce: histograms (Section 6.3, stem and leaf plots (Section 6.4, boxplots (Section 6.5, scatterplots (Section 6.6 and bar graphs (Section 6.7.
• Saving image files. The last part of the chapter talked about how to export your pictures (Section 6.8
One final thing to point out. At the start of the chapter I mentioned that R has several completely distinct systems for drawing figures. In this chapter I’ve focused on the traditional graphics system. It’s the easiest one to get started with: you can draw a histogram with a command as simple as `hist(x)`. However, it’s not the most powerful tool for the job, and after a while most R users start looking to shift to fancier systems. One of the most popular graphics systems is provided by the `ggplot2` package (see ), which is loosely based on “The grammar of graphics” @[Wilkinson2006]. It’s not for novices: you need to have a pretty good grasp of R before you can start using it, and even then it takes a while to really get the hang of it. But when you’re finally at that stage, it’s worth taking the time to teach yourself, because it’s a much cleaner system.
1. The origin of this quote is Tufte’s lovely book The Visual Display of Quantitative Information.
2. I should add that this isn’t unique to R. Like everything in R there’s a pretty steep learning curve to learning how to draw graphs, and like always there’s a massive payoff at the end in terms of the quality of what you can produce. But to be honest, I’ve seen the same problems show up regardless of what system people use. I suspect that the hardest thing to do is to force yourself to take the time to think deeply about what your graphs are doing. I say that in full knowledge that only about half of my graphs turn out as well as they ought to. Understanding what makes a good graph is easy: actually designing a good graph is hard.
3. Or, since you can always use the up and down keys to scroll through your recent command history, you can just pull up your most recent commands and edit them to fix your mistake. It becomes even easier once you start using scripts (Section 8.1, since all you have to do is edit your script and then run it again.
4. Of course, even that is a slightly misleading description, since some R graphics tools make use of external graphical rendering systems like OpenGL (e.g., the `rgl` package). I absolutely will not be talking about OpenGL or the like in this book, but as it happens there is one graph in this book that relies on them: Figure 15.6.
5. The low-level function that does this is called `title()` in case you ever need to know, and you can type `?title` to find out a bit more detail about what these arguments do.
6. On the off chance that this isn’t enough freedom for you, you can select a colour directly as a “red, green, blue” specification using the `rgb()` function, or as a “hue, saturation, value” specification using the `hsv()` function.
7. Also, there’s a low level function called `axis()` that allows a lot more control over the appearance of the axes.
8. R being what it is, it’s no great surprise that there’s also a `fivenum()` function that does much the same thing.
9. I realise there’s a kind of logic to the way R names are constructed, but they still sound dumb. When I typed this sentence, all I could think was that it sounded like the name of a kids movie if it had been written by Lewis Carroll: “The frabjous gambolles of Staplewex and Whisklty” or something along those lines.
10. Sometimes it’s convenient to have the boxplot automatically label the outliers for you. The original `boxplot()` function doesn’t allow you to do this; however, the `Boxplot()` function in the `car` package does. The design of the `Boxplot()` function is very similar to `boxplot()`. It just adds a few new arguments that allow you to tweak the labelling scheme. I’ll leave it to the reader to check this out.
11. Sort of. The game was played in Launceston, which is a de facto home away from home for Hawthorn.
12. Contrast this situation with the next largest winning margin in the data set, which was Geelong’s 108 point demolition of Richmond in round 6 at their home ground, Kardinia Park. Geelong have been one of the most dominant teams over the last several years, a period during which they strung together an incredible 29-game winning streak at Kardinia Park. Richmond have been useless for several years. This is in no meaningful sense an outlier. Geelong have been winning by these margins (and Richmond losing by them) for quite some time. Frankly I’m surprised that the result wasn’t more lopsided: as happened to Melbourne in 2011 when Geelong won by a modest 186 points.
13. Actually, there’s other ways to do this. If the input argument `x` is a list object (see Section 4.9, the `boxplot()` function will draw a separate boxplot for each variable in that list. Relatedly, since the `plot()` function – which we’ll discuss shortly – is a generic (see Section 4.11, you might not be surprised to learn that one of its special cases is a boxplot: specifically, if you use `plot()` where the first argument `x` is a factor and the second argument `y` is numeric, then the result will be a boxplot, showing the values in `y`, with a separate boxplot for each level. For instance, something like `plot(x = afl2\$year, y = afl2\$margin)` would work.
14. The reason is that there’s an annoying design flaw in the way the `plot()` function handles this situation. The problem is that the `plot.formula()` function uses different names to for the arguments than the `plot()` function expects. As a consequence, you can’t specify the formula argument by name. If you just specify a formula as the first argument without using the name it works fine, because the `plot()` function thinks the formula corresponds to the `x` argument, and the `plot.formula()` function thinks it corresponds to the `formula` argument; and surprisingly, everything works nicely. But the moment that you, the user, tries to be unambiguous about the name, one of those two functions is going to cry.
15. You might be wondering why I haven’t specified the argument name for the formula. The reason is that there’s a bug in how the `scatterplot()` function is written: under the hood there’s one function that expects the argument to be named `x` and another one that expects it to be called `formula`. I don’t know why the function was written this way, but it’s not an isolated problem: this particular kind of bug repeats itself in a couple of other functions (you’ll see it again in Chapter 13. The solution in such cases is to omit the argument name: that way, one function “thinks” that you’ve specified `x` and the other one “thinks” you’ve specified `formula` and everything works the way it’s supposed to. It’s not a great state of affairs, I’ll admit, but it sort of works.
16. Yet again, we could have produced this output using the `plot()` function: when the `x` argument is a data frame containing numeric variables only, then the output is a scatterplot matrix. So, once again, what I could have done is just type `plot( parenthood )`.
17. Once again, it’s worth noting the link to the generic `plot()` function. If the `x` argument to `plot()` is a factor (and no `y` argument is given), the result is a bar graph. So you could use `plot( afl.finalists )` and get the same output as `barplot( afl.finalists )`. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/06%3A_Drawing_Graphs/6.09%3A__Summary.txt |
The garden of life never seems to confine itself to the plots philosophers have laid out for its convenience. Maybe a few more tractors would do the trick.
–Roger Zelazny103
This is a somewhat strange chapter, even by my standards. My goal in this chapter is to talk a bit more honestly about the realities of working with data than you’ll see anywhere else in the book. The problem with real world data sets is that they are messy. Very often the data file that you start out with doesn’t have the variables stored in the right format for the analysis you want to do. Sometimes might be a lot of missing values in your data set. Sometimes you only want to analyse a subset of the data. Et cetera. In other words, there’s a lot of data manipulation that you need to do, just to get all your data set into the format that you need it. The purpose of this chapter is to provide a basic introduction to all these pragmatic topics. Although the chapter is motivated by the kinds of practical issues that arise when manipulating real data, I’ll stick with the practice that I’ve adopted through most of the book and rely on very small, toy data sets that illustrate the underlying issue. Because this chapter is essentially a collection of “tricks” and doesn’t tell a single coherent story, it may be useful to start with a list of topics:
• Section 7.1. Tabulating data.
• Section 7.2. Transforming or recoding a variable.
• Section 7.3. Some useful mathematical functions.
• Section 7.4. Extracting a subset of a vector.
• Section 7.5. Extracting a subset of a data frame.
• Section 7.6. Sorting, flipping or merging data sets.
• Section 7.7. Reshaping a data frame.
• Section 7.8. Manipulating text.
• Section 7.9. Opening data from different file types.
• Section 7.10. Coercing data from one type to another.
• Section 7.11. Other important data types.
• Section 7.12. Miscellaneous topics.
As you can see, the list of topics that the chapter covers is pretty broad, and there’s a lot of content there. Even though this is one of the longest and hardest chapters in the book, I’m really only scratching the surface of several fairly different and important topics. My advice, as usual, is to read through the chapter once and try to follow as much of it as you can. Don’t worry too much if you can’t grasp it all at once, especially the later sections. The rest of the book is only lightly reliant on this chapter, so you can get away with just understanding the basics. However, what you’ll probably find is that later on you’ll need to flick back to this chapter in order to understand some of the concepts that I refer to here.
07: Pragmatic Matters
A very common task when analysing data is the construction of frequency tables, or cross-tabulation of one variable against another. There are several functions that you can use in R for that purpose. In this section I’ll illustrate the use of three functions – `table()`, `xtabs()` and `tabulate()` – though there are other options (e.g., `ftable()`) available.
Creating tables from vectors
Let’s start with a simple example. As the father of a small child, I naturally spend a lot of time watching TV shows like In the Night Garden. In the `nightgarden.Rdata` file, I’ve transcribed a short section of the dialogue. The file contains two variables, `speaker` and `utterance`, and when we take a look at the data, it becomes very clear what happened to my sanity.
``````library(lsr)
load("./rbook-master/data/nightgarden.Rdata" )
who()``````
``````## -- Name -- -- Class -- -- Size --
## speaker character 10
## utterance character 10``````
````print( speaker )`
```
``````## [1] "upsy-daisy" "upsy-daisy" "upsy-daisy" "upsy-daisy" "tombliboo"
## [6] "tombliboo" "makka-pakka" "makka-pakka" "makka-pakka" "makka-pakka"``````
``print( utterance )``
``## [1] "pip" "pip" "onk" "onk" "ee" "oo" "pip" "pip" "onk" "onk"``
With these as my data, one task I might find myself needing to do is construct a frequency count of the number of words each character speaks during the show. The `table()` function provides a simple way do to this. The basic usage of the `table()` function is as follows:
``table(speaker)``
``````## speaker
## makka-pakka tombliboo upsy-daisy
## 4 2 4``````
The output here tells us on the first line that what we’re looking at is a tabulation of the `speaker` variable. On the second line it lists all the different speakers that exist in the data, and on the third line it tells you how many times that speaker appears in the data. In other words, it’s a frequency table104 Notice that in the command above I didn’t name the argument, since `table()` is another function that makes use of unnamed arguments. You just type in a list of the variables that you want R to tabulate, and it tabulates them. For instance, if I type in the name of two variables, what I get as the output is a cross-tabulation:
``table(speaker, utterance)``
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0 2 0 2
## tombliboo 1 0 1 0
## upsy-daisy 0 2 0 2``````
When interpreting this table, remember that these are counts: so the fact that the first row and second column corresponds to a value of 2 indicates that Makka-Pakka (row 1) says “onk” (column 2) twice in this data set. As you’d expect, you can produce three way or higher order cross tabulations just by adding more objects to the list of inputs. However, I won’t discuss that in this section.
Creating tables from data frames
Most of the time your data are stored in a data frame, not kept as separate variables in the workspace. Let’s create one:
``````itng <- data.frame( speaker, utterance )
itng``````
``````## speaker utterance
## 1 upsy-daisy pip
## 2 upsy-daisy pip
## 3 upsy-daisy onk
## 4 upsy-daisy onk
## 5 tombliboo ee
## 6 tombliboo oo
## 7 makka-pakka pip
## 8 makka-pakka pip
## 9 makka-pakka onk
## 10 makka-pakka onk``````
There’s a couple of options under these circumstances. Firstly, if you just want to cross-tabulate all of the variables in the data frame, then it’s really easy:
````table(itng)`
```
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0 2 0 2
## tombliboo 1 0 1 0
## upsy-daisy 0 2 0 2``````
However, it’s often the case that you want to select particular variables from the data frame to tabulate. This is where the `xtabs()` function is useful. In this function, you input a one sided `formula` in order to list all the variables you want to cross-tabulate, and the name of the `data` frame that stores the data:
``xtabs( formula = ~ speaker + utterance, data = itng )``
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0 2 0 2
## tombliboo 1 0 1 0
## upsy-daisy 0 2 0 2``````
Clearly, this is a totally unnecessary command in the context of the `itng` data frame, but in most situations when you’re analysing real data this is actually extremely useful, since your data set will almost certainly contain lots of variables and you’ll only want to tabulate a few of them at a time.
Converting a table of counts to a table of proportions
The tabulation commands discussed so far all construct a table of raw frequencies: that is, a count of the total number of cases that satisfy certain conditions. However, often you want your data to be organised in terms of proportions rather than counts. This is where the `prop.table()` function comes in handy. It has two arguments:
• `x`. The frequency table that you want to convert.
• `margin`. Which “dimension” do you want to calculate proportions for. By default, R assumes you want the proportion to be expressed as a fraction of all possible events. See examples for details.
To see how this works:
``````itng.table <- table(itng) # create the table, and assign it to a variable
itng.table # display the table again, as a reminder``````
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0 2 0 2
## tombliboo 1 0 1 0
## upsy-daisy 0 2 0 2``` ```
``prop.table( x = itng.table ) # express as proportion:``
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0.0 0.2 0.0 0.2
## tombliboo 0.1 0.0 0.1 0.0
## upsy-daisy 0.0 0.2 0.0 0.2``````
Notice that there were 10 observations in our original data set, so all that R has done here is divide all our raw frequencies by 10. That’s a sensible default, but more often you actually want to calculate the proportions separately by row (`margin = 1`) or by column (`margin = 2`). Again, this is most clearly seen by looking at examples:
``prop.table( x = itng.table, margin = 1)``
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0.0 0.5 0.0 0.5
## tombliboo 0.5 0.0 0.5 0.0
## upsy-daisy 0.0 0.5 0.0 0.5``````
Notice that each row now sums to 1, but that’s not true for each column. What we’re looking at here is the proportions of utterances made by each character. In other words, 50% of Makka-Pakka’s utterances are “pip”, and the other 50% are “onk”. Let’s contrast this with the following command:
``prop.table( x = itng.table, margin = 2)``
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0.0 0.5 0.0 0.5
## tombliboo 1.0 0.0 1.0 0.0
## upsy-daisy 0.0 0.5 0.0 0.5``````
Now the columns all sum to 1 but the rows don’t. In this version, what we’re seeing is the proportion of characters associated with each utterance. For instance, whenever the utterance “ee” is made (in this data set), 100% of the time it’s a Tombliboo saying it.
level tabulation
One final function I want to mention is the `tabulate()` function, since this is actually the low-level function that does most of the hard work. It takes a numeric vector as input, and outputs frequencies as outputs:
``````some.data <- c(1,2,3,1,1,3,1,1,2,8,3,1,2,4,2,3,5,2)
tabulate(some.data)``````
``## [1] 6 5 4 1 1 0 0 1`` | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.01%3A_Tabulating_and_Cross-tabulating_Data.txt |
It’s not uncommon in real world data analysis to find that one of your variables isn’t quite equivalent to the variable that you really want. For instance, it’s often convenient to take a continuous-valued variable (e.g., age) and break it up into a smallish number of categories (e.g., younger, middle, older). At other times, you may need to convert a numeric variable into a different numeric variable (e.g., you may want to analyse at the absolute value of the original variable). In this section I’ll describe a few key tricks that you can make use of to do this.
Creating a transformed variable
The first trick to discuss is the idea of transforming a variable. Taken literally, anything you do to a variable is a transformation, but in practice what it usually means is that you apply a relatively simple mathematical function to the original variable, in order to create new variable that either (a) provides a better way of describing the thing you’re actually interested in or (b) is more closely in agreement with the assumptions of the statistical tests you want to do. Since – at this stage – I haven’t talked about statistical tests or their assumptions, I’ll show you an example based on the first case.
To keep the explanation simple, the variable we’ll try to transform (`likert.raw`) isn’t inside a data frame, though in real life it almost certainly would be. However, I think it’s useful to start with an example that doesn’t use data frames because it illustrates the fact that you already know how to do variable transformations. To see this, let’s go through an example. Suppose I’ve run a short study in which I ask 10 people a single question:
On a scale of 1 (strongly disagree) to 7 (strongly agree), to what extent do you agree with the proposition that “Dinosaurs are awesome”?
Now let’s load and look at the data. The data file `likert.Rdata` contains a single variable that contains the raw Likert-scale responses:
``````load("./rbook-master/data/likert.Rdata")
likert.raw``````
``## [1] 1 7 3 4 4 4 2 6 5 5``
However, if you think about it, this isn’t the best way to represent these responses. Because of the fairly symmetric way that we set up the response scale, there’s a sense in which the midpoint of the scale should have been coded as 0 (no opinion), and the two endpoints should be +3 (strong agree) and −3 (strong disagree). By recoding the data in this way, it’s a bit more reflective of how we really think about the responses. The recoding here is trivially easy: we just subtract 4 from the raw scores:
``````likert.centred <- likert.raw - 4
likert.centred``````
``## [1] -3 3 -1 0 0 0 -2 2 1 1``
One reason why it might be useful to have the data in this format is that there are a lot of situations where you might prefer to analyse the strength of the opinion separately from the direction of the opinion. We can do two different transformations on this `likert.centred` variable in order to distinguish between these two different concepts. Firstly, to compute an `opinion.strength` variable, we want to take the absolute value of the centred data (using the `abs()` function that we’ve seen previously), like so:
``````opinion.strength <- abs( likert.centred )
opinion.strength``````
``## [1] 3 3 1 0 0 0 2 2 1 1``
Secondly, to compute a variable that contains only the direction of the opinion and ignores the strength, we can use the `sign()` function to do this. If you type `?sign` you’ll see that this function is really simple: all negative numbers are converted to −1, all positive numbers are converted to 1 and zero stays as 0. So, when we apply the `sign()` function we obtain the following:
``````opinion.dir <- sign( likert.centred )
opinion.dir``````
``## [1] -1 1 -1 0 0 0 -1 1 1 1``
And we’re done. We now have three shiny new variables, all of which are useful transformations of the original `likert.raw` data. All of this should seem pretty familiar to you. The tools that you use to do regular calculations in R (e.g., Chapters 3 and 4) are very much the same ones that you use to transform your variables! To that end, in Section 7.3 I’ll revisit the topic of doing calculations in R because there’s a lot of other functions and operations that are worth knowing about.
Before moving on, you might be curious to see what these calculations look like if the data had started out in a data frame. To that end, it may help to note that the following example does all of the calculations using variables inside a data frame, and stores the variables created inside it:
``````df <- data.frame( likert.raw ) # create data frame
df\$likert.centred <- df\$likert.raw - 4 # create centred data
df\$opinion.strength <- abs( df\$likert.centred ) # create strength variable
df\$opinion.dir <- sign( df\$likert.centred ) # create direction variable
df ``````
``````## likert.raw likert.centred opinion.strength opinion.dir
## 1 1 -3 3 -1
## 2 7 3 3 1
## 3 3 -1 1 -1
## 4 4 0 0 0
## 5 4 0 0 0
## 6 4 0 0 0
## 7 2 -2 2 -1
## 8 6 2 2 1
## 9 5 1 1 1
## 10 5 1 1 1``````
In other words, the commands you use are basically ones as before: it’s just that every time you want to read a variable from the data frame or write to the data frame, you use the `\$` operator. That’s the easiest way to do it, though I should make note of the fact that people sometimes make use of the `within()` function to do the same thing. However, since (a) I don’t use the `within()` function anywhere else in this book, and (b) the `\$` operator works just fine, I won’t discuss it any further.
Cutting a numeric variable into categories
One pragmatic task that arises more often than you’d think is the problem of cutting a numeric variable up into discrete categories. For instance, suppose I’m interested in looking at the age distribution of people at a social gathering:
``age <- c( 60,58,24,26,34,42,31,30,33,2,9 )``
In some situations it can be quite helpful to group these into a smallish number of categories. For example, we could group the data into three broad categories: young (0-20), adult (21-40) and older (41-60). This is a quite coarse-grained classification, and the labels that I’ve attached only make sense in the context of this data set (e.g., viewed more generally, a 42 year old wouldn’t consider themselves as “older”). We can slice this variable up quite easily using the `cut()` function.105 To make things a little cleaner, I’ll start by creating a variable that defines the boundaries for the categories:
``````age.breaks <- seq( from = 0, to = 60, by = 20 )
age.breaks``````
``## [1] 0 20 40 60``
and another one for the labels:
``````age.labels <- c( "young", "adult", "older" )
age.labels``````
``## [1] "young" "adult" "older"``
Note that there are four numbers in the `age.breaks` variable, but only three labels in the `age.labels` variable; I’ve done this because the `cut()` function requires that you specify the edges of the categories rather than the mid-points. In any case, now that we’ve done this, we can use the `cut()` function to assign each observation to one of these three categories. There are several arguments to the `cut()` function, but the three that we need to care about are:
• `x`. The variable that needs to be categorised.
• `breaks`. This is either a vector containing the locations of the breaks separating the categories, or a number indicating how many categories you want.
• `labels`. The labels attached to the categories. This is optional: if you don’t specify this R will attach a boring label showing the range associated with each category.
Since we’ve already created variables corresponding to the breaks and the labels, the command we need is just:
``````age.group <- cut( x = age, # the variable to be categorised
breaks = age.breaks, # the edges of the categories
labels = age.labels ) # the labels for the categories``````
Note that the output variable here is a factor. In order to see what this command has actually done, we could just print out the `age.group` variable, but I think it’s actually more helpful to create a data frame that includes both the original variable and the categorised one, so that you can see the two side by side:
``data.frame(age, age.group)``
``````## age age.group
## 1 60 older
## 2 58 older
## 3 24 adult
## 4 26 adult
## 5 34 adult
## 6 42 older
## 7 31 adult
## 8 30 adult
## 9 33 adult
## 10 2 young
## 11 9 young``````
It can also be useful to tabulate the output, just to see if you’ve got a nice even division of the sample:
``table( age.group )``
``````## age.group
## young adult older
## 2 6 3``````
In the example above, I made all the decisions myself. Much like the `hist()` function that we saw in Chapter 6, if you want to you can delegate a lot of the choices to R. For instance, if you want you can specify the number of categories you want, rather than giving explicit ranges for them, and you can allow R to come up with some labels for the categories. To give you a sense of how this works, have a look at the following example:
``age.group2 <- cut( x = age, breaks = 3 )``
With this command, I’ve asked for three categories, but let R make the choices for where the boundaries should be. I won’t bother to print out the `age.group2` variable, because it’s not terribly pretty or very interesting. Instead, all of the important information can be extracted by looking at the tabulated data:
````table( age.group2 )`
```
``````## age.group2
## (1.94,21.3] (21.3,40.7] (40.7,60.1]
## 2 6 3``````
This output takes a little bit of interpretation, but it’s not complicated. What R has done is determined that the lowest age category should run from 1.94 years up to 21.3 years, the second category should run from 21.3 years to 40.7 years, and so on. The formatting on those labels might look a bit funny to those of you who haven’t studied a lot of maths, but it’s pretty simple. When R describes the first category as corresponding to the range (1.94,21.3] what it’s saying is that the range consists of those numbers that are larger than 1.94 but less than or equal to 21.3. In other words, the weird asymmetric brackets is R s way of telling you that if there happens to be a value that is exactly equal to 21.3, then it belongs to the first category, not the second one. Obviously, this isn’t actually possible since I’ve only specified the ages to the nearest whole number, but R doesn’t know this and so it’s trying to be precise just in case. This notation is actually pretty standard, but I suspect not everyone reading the book will have seen it before. In any case, those labels are pretty ugly, so it’s usually a good idea to specify your own, meaningful labels to the categories.
Before moving on, I should take a moment to talk a little about the mechanics of the `cut()` function. Notice that R has tried to divide the `age` variable into three roughly equal sized bins. Unless you specify the particular breaks you want, that’s what it will do. But suppose you want to divide the `age` variable into three categories of different size, but with approximately identical numbers of people. How would you do that? Well, if that’s the case, then what you want to do is have the breaks correspond to the 0th, 33rd, 66th and 100th percentiles of the data. One way to do this would be to calculate those values using the `quantiles()` function and then use those quantiles as input to the `cut()` function. That’s pretty easy to do, but it does take a couple of lines to type. So instead, the `lsr` package has a function called `quantileCut()` that does exactly this:
``````age.group3 <- quantileCut( x = age, n = 3 )
table( age.group3 )```
```
``````## age.group3
## (1.94,27.3] (27.3,33.7] (33.7,60.1]
## 4 3 4``````
Notice the difference in the boundaries that the `quantileCut()` function selects. The first and third categories now span an age range of about 25 years each, whereas the middle category has shrunk to a span of only 6 years. There are some situations where this is genuinely what you want (that’s why I wrote the function!), but in general you should be careful. Usually the numeric variable that you’re trying to cut into categories is already expressed in meaningful units (i.e., it’s interval scale), but if you cut it into unequal bin sizes then it’s often very difficult to attach meaningful interpretations to the resulting categories.
More generally, regardless of whether you’re using the original `cut()` function or the `quantileCut()` version, it’s important to take the time to figure out whether or not the resulting categories make any sense at all in terms of your research project. If they don’t make any sense to you as meaningful categories, then any data analysis that uses those categories is likely to be just as meaningless. More generally, in practice I’ve noticed that people have a very strong desire to carve their (continuous and messy) data into a few (discrete and simple) categories; and then run analysis using the categorised data instead of the original one.106 I wouldn’t go so far as to say that this is an inherently bad idea, but it does have some fairly serious drawbacks at times, so I would advise some caution if you are thinking about doing it. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.02%3A_Transforming_and_Recoding_a_Variable.txt |
In Section 7.2 I discussed the ideas behind variable transformations, and showed that a lot of the transformations that you might want to apply to your data are based on fairly simple mathematical functions and operations, of the kind that we discussed in Chapter 3. In this section I want to return to that discussion, and mention several other mathematical functions and arithmetic operations that I didn’t bother to mention when introducing you to R, but are actually quite useful for a lot of real world data analysis. Table 7.1 gives a brief overview of the various mathematical functions I want to talk about (and some that I already have talked about). Obviously this doesn’t even come close to cataloging the range of possibilities available in R, but it does cover a very wide range of functions that are used in day to day data analysis.
Table 7.1: Some of the mathematical functions available in R.
mathematical.function R.function example.input answer
square root sqrt() sqrt(25) 5
absolute value abs() abs(-23) 23
logarithm (base 10) log10() log10(1000) 3
logarithm (base e) log() log(1000) 6.908
exponentiation exp() exp(6.908) 1000.245
rounding to nearest round() round(1.32) 1
rounding down floor() floor(1.32) 1
rounding up ceiling() ceiling(1.32) 2
Rounding a number
One very simple transformation that crops up surprisingly often is the need to round a number to the nearest whole number, or to a certain number of significant digits. To start with, let’s assume that we want to round to a whole number. To that end, there are three useful functions in R you want to know about: `round()`, `floor()` and `ceiling()`. The `round()` function just rounds to the nearest whole number. So if you round the number 4.3, it “rounds down” to `4`, like so:
``round( x = 4.3 )``
``## [1] 4``
In contrast, if we want to round the number 4.7, we would round upwards to 5. In everyday life, when someone talks about “rounding”, they usually mean “round to nearest”, so this is the function we use most of the time. However sometimes you have reasons to want to always round up or always round down. If you want to always round down, use the `floor()` function instead; and if you want to force R to round up, then use `ceiling()`. That’s the only difference between the three functions. What if you want to round to a certain number of digits? Let’s suppose you want to round to a fixed number of decimal places, say 2 decimal places. If so, what you need to do is specify the `digits` argument to the `round()` function. That’s pretty straightforward:
``round( x = 0.0123, digits = 2 )``
``## [1] 0.01``
The only subtlety that you need to keep in mind is that sometimes what you want to do is round to 2 significant digits and not to two decimal places. The difference is that, when determining the number of significant digits, zeros don’t count. To see this, let’s apply the `signif()` function instead of the `round()` function:
``signif( x = 0.0123, digits = 2 )``
``## [1] 0.012``
This time around, we get an answer of 0.012 because the zeros don’t count as significant digits. Quite often scientific journals will ask you to report numbers to two or three significant digits, so it’s useful to remember the distinction.
Modulus and integer division
Table 7.2: Two more arithmetic operations that sometimes come in handy
operation operator example.input answer
integer division %/% 42 %/% 10 4
modulus %% 42 %% 10 2
Since we’re on the topic of simple calculations, there are two other arithmetic operations that I should mention, since they can come in handy when working with real data. These operations are calculating a modulus and doing integer division. They don’t come up anywhere else in this book, but they are worth knowing about. First, let’s consider integer division. Suppose I have \$42 in my wallet, and want to buy some sandwiches, which are selling for \$10 each. How many sandwiches can I afford107 to buy? The answer is of course 4. Note that it’s not 4.2, since no shop will sell me one-fifth of a sandwich. That’s integer division. In R we perform integer division by using the `%/%` operator:
``42 %/% 10``
``## [1] 4``
Okay, that’s easy enough. What about the modulus? Basically, a modulus is the remainder after integer division, and it’s calculated using the `%%` operator. For the sake of argument, let’s suppose I buy four overpriced \$10 sandwiches. If I started out with \$42, how much money do I have left? The answer, as both R and common sense tells us, is \$2:
````42 %% 10`
```
``## [1] 2``
So that’s also pretty easy. There is, however, one subtlety that I need to mention, and this relates to how negative numbers are handled. Firstly, what would happen if I tried to do integer division with a negative number? Let’s have a look:
``-42 %/% 10``
``## [1] -5``
This might strike you as counterintuitive: why does `42 %/% 10` produce an answer of `4`, but `-42 %/% 10` gives us an answer of `-5`? Intuitively you might think that the answer to the second one should be `-4`. The way to think about it is like this. Suppose I owe the sandwich shop \$42, but I don’t have any money. How many sandwiches would I have to give them in order to stop them from calling security? The answer108 here is 5, not 4. If I handed them 4 sandwiches, I’d still owe them \$2, right? So I actually have to give them 5 sandwiches. And since it’s me giving them the sandwiches, the answer to `-42 %/% 10` is `-5`. As you might expect, the behaviour of the modulus operator has a similar pattern. If I’ve handed 5 sandwiches over to the shop in order to pay off my debt of \$42, then they now owe me \$8. So the modulus is now:
``-42 %% 10``
``## [1] 8``
Logarithms and exponentials
As I’ve mentioned earlier, R has an incredible range of mathematical functions built into it, and there really wouldn’t be much point in trying to describe or even list all of them. For the most part, I’ve focused only on those functions that are strictly necessary for this book. However I do want to make an exception for logarithms and exponentials. Although they aren’t needed anywhere else in this book, they are everywhere in statistics more broadly, and not only that, there are a lot of situations in which it is convenient to analyse the logarithm of a variable (i.e., to take a “log-transform” of the variable). I suspect that many (maybe most) readers of this book will have encountered logarithms and exponentials before, but from past experience I know that there’s a substantial proportion of students who take a social science statistics class who haven’t touched logarithms since high school, and would appreciate a bit of a refresher.
In order to understand logarithms and exponentials, the easiest thing to do is to actually calculate them and see how they relate to other simple calculations. There are three R functions in particular that I want to talk about, namely `log()`, `log10()` and `exp()`. To start with, let’s consider `log10()`, which is known as the “logarithm in base 10”. The trick to understanding a logarithm is to understand that it’s basically the “opposite” of taking a power. Specifically, the logarithm in base 10 is closely related to the powers of 10. So let’s start by noting that 10-cubed is 1000. Mathematically, we would write this:
103=1000
and in R we’d calculate it by using the command `10^3`. The trick to understanding a logarithm is to recognise that the statement that “10 to the power of 3 is equal to 1000” is equivalent to the statement that “the logarithm (in base 10) of 1000 is equal to 3”. Mathematically, we write this as follows,
log10(1000)=3
and if we wanted to do the calculation in R we would type this:
``log10( 1000 )``
``## [1] 3``
Obviously, since you already know that 103=1000 there’s really no point in getting R to tell you that the base-10 logarithm of 1000 is 3. However, most of the time you probably don’t know what right answer is. For instance, I can honestly say that I didn’t know that 102.69897=500, so it’s rather convenient for me that I can use R to calculate the base-10 logarithm of 500:
``log10( 500 )``
``## [1] 2.69897``
Or at least it would be convenient if I had a pressing need to know the base-10 logarithm of 500.
Okay, since the `log10()` function is related to the powers of 10, you might expect that there are other logarithms (in bases other than 10) that are related to other powers too. And of course that’s true: there’s not really anything mathematically special about the number 10. You and I happen to find it useful because decimal numbers are built around the number 10, but the big bad world of mathematics scoffs at our decimal numbers. Sadly, the universe doesn’t actually care how we write down numbers. Anyway, the consequence of this cosmic indifference is that there’s nothing particularly special about calculating logarithms in base 10. You could, for instance, calculate your logarithms in base 2, and in fact R does provide a function for doing that, which is (not surprisingly) called `log2()`. Since we know that 23=2×2×2=8, it’s not surprise to see that
``log2( 8 )``
``## [1] 3``
Alternatively, a third type of logarithm – and one we see a lot more of in statistics than either base 10 or base 2 – is called the natural logarithm, and corresponds to the logarithm in base e. Since you might one day run into it, I’d better explain what e is. The number e, known as Euler’s number, is one of those annoying “irrational” numbers whose decimal expansion is infinitely long, and is considered one of the most important numbers in mathematics. The first few digits of e are:
e=2.718282
There are quite a few situation in statistics that require us to calculate powers of e, though none of them appear in this book. Raising e to the power x is called the exponential of x, and so it’s very common to see ex written as exp(x). And so it’s no surprise that R has a function that calculate exponentials, called `exp()`. For instance, suppose I wanted to calculate e3. I could try typing in the value of e manually, like this:
``2.718282 ^ 3``
``## [1] 20.08554``
but it’s much easier to do the same thing using the `exp()` function:
``exp( 3 )``
``## [1] 20.08554``
Anyway, because the number e crops up so often in statistics, the natural logarithm (i.e., logarithm in base e) also tends to turn up. Mathematicians often write it as loge(x) or ln(x), or sometimes even just log(x). In fact, R works the same way: the `log()` function corresponds to the natural logarithm109 Anyway, as a quick check, let’s calculate the natural logarithm of 20.08554 using R:
``log( 20.08554 )``
``## [1] 3``
And with that, I think we’ve had quite enough exponentials and logarithms for this book! | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.03%3A_A_few_More_Mathematical_Functions_and_Operations.txt |
One very important kind of data handling is being able to extract a particular subset of the data. For instance, you might be interested only in analysing the data from one experimental condition, or you may want to look closely at the data from people over 50 years in age. To do this, the first step is getting R to extract the subset of the data corresponding to the observations that you’re interested in. In this section I’ll talk about subsetting as it applies to vectors, extending the discussion from Chapters 3 and 4. In Section 7.5 I’ll go on to talk about how this discussion extends to data frames.
Refresher
This section returns to the `nightgarden.Rdata` data set. If you’re reading this whole chapter in one sitting, then you should already have this data set loaded. If not, don’t forget to use the `load("nightgarden.Rdata")` command. For this section, let’s ignore the `itng` data frame that we created earlier, and focus instead on the two vectors `speaker` and `utterance` (see Section 7.1 if you’ve forgotten what those vectors look like). Suppose that what I want to do is pull out only those utterances that were made by Makka-Pakka. To that end, I could first use the equality operator to have R tell me which cases correspond to Makka-Pakka speaking:
``````is.MP.speaking <- speaker == "makka-pakka"
is.MP.speaking``````
``## [1] FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE``
and then use logical indexing to get R to print out those elements of `utterance` for which `is.MP.speaking` is true, like so:
``utterance[ is.MP.speaking ]``
``## [1] "pip" "pip" "onk" "onk"``
Or, since I’m lazy, I could collapse it to a single command like so:
``utterance[ speaker == "makka-pakka" ]``
``## [1] "pip" "pip" "onk" "onk"``
Using `%in%`match multiple cases
A second useful trick to be aware of is the `%in%` operator110. It’s actually very similar to the `==` operator, except that you can supply a collection of acceptable values. For instance, suppose I wanted to keep only those cases when the utterance is either “pip” or “oo”. One simple way do to this is:
``utterance %in% c("pip","oo") ``
``## [1] TRUE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE``
What this does if return `TRUE` for those elements of `utterance` that are either `"pip"` or `"oo"` and returns `FALSE` for all the others. What that means is that if I want a list of all those instances of characters speaking either of these two words, I could do this:
``speaker[ utterance %in% c("pip","oo") ]``
``## [1] "upsy-daisy" "upsy-daisy" "tombliboo" "makka-pakka" "makka-pakka"``
Using negative indices to drop elements
Before moving onto data frames, there’s a couple of other tricks worth mentioning. The first of these is to use negative values as indices. Recall from Section 3.10 that we can use a vector of numbers to extract a set of elements that we would like to keep. For instance, suppose I want to keep only elements 2 and 3 from `utterance`. I could do so like this:
``utterance[2:3]``
``## [1] "pip" "onk"``
But suppose, on the other hand, that I have discovered that observations 2 and 3 are untrustworthy, and I want to keep everything except those two elements. To that end, R lets you use negative numbers to remove specific values, like so:
``utterance [ -(2:3) ]``
``## [1] "pip" "onk" "ee" "oo" "pip" "pip" "onk" "onk"``
The output here corresponds to element 1 of the original vector, followed by elements 4, 5, and so on. When all you want to do is remove a few cases, this is a very handy convention.
Splitting a vector by group
One particular example of subsetting that is especially common is the problem of splitting one one variable up into several different variables, one corresponding to each group. For instance, in our In the Night Garden example, I might want to create subsets of the `utterance` variable for every character. One way to do this would be to just repeat the exercise that I went through earlier separately for each character, but that quickly gets annoying. A faster way do it is to use the `split()` function. The arguments are:
• `x`. The variable that needs to be split into groups.
• `f`. The grouping variable.
What this function does is output a list (Section 4.9), containing one variable for each group. For instance, I could split up the `utterance` variable by `speaker` using the following command:
``````speech.by.char <- split( x = utterance, f = speaker )
speech.by.char``````
``````## \$`makka-pakka`
## [1] "pip" "pip" "onk" "onk"
##
## \$tombliboo
## [1] "ee" "oo"
##
## \$`upsy-daisy`
## [1] "pip" "pip" "onk" "onk"``````
Once you’re starting to become comfortable working with lists and data frames, this output is all you need, since you can work with this list in much the same way that you would work with a data frame. For instance, if you want the first utterance made by Makka-Pakka, all you need to do is type this:
``speech.by.char\$`makka-pakka`[1]``
``## [1] "pip"``
Just remember that R does need you to add the quoting characters (i.e. `'`). Otherwise, there’s nothing particularly new or difficult here.
However, sometimes – especially when you’re just starting out – it can be convenient to pull these variables out of the list, and into the workspace. This isn’t too difficult to do, though it can be a little daunting to novices. To that end, I’ve included a function called `importList()` in the `lsr` package that does this.111 First, here’s what you’d have if you had wiped the workspace before the start of this section:
``who()``
``````## -- Name -- -- Class -- -- Size --
## age numeric 11
## age.breaks numeric 4
## age.group factor 11
## age.group2 factor 11
## age.group3 factor 11
## age.labels character 3
## df data.frame 10 x 4
## is.MP.speaking logical 10
## itng data.frame 10 x 2
## itng.table table 3 x 4
## likert.centred numeric 10
## likert.raw numeric 10
## opinion.dir numeric 10
## opinion.strength numeric 10
## some.data numeric 18
## speaker character 10
## speech.by.char list 3
## utterance character 10``````
Now we use the `importList()` function to copy all of the variables within the `speech.by.char` list:
``importList( speech.by.char, ask = FALSE)``
Because the `importList()` function is attempting to create new variables based on the names of the elements of the list, it pauses to check that you’re okay with the variable names. The reason it does this is that, if one of the to-be-created variables has the same name as a variable that you already have in your workspace, that variable will end up being overwritten, so it’s a good idea to check. Assuming that you type `y`, it will go on to create the variables. Nothing appears to have happened, but if we look at our workspace now:
````who()`
```
``````## -- Name -- -- Class -- -- Size --
## age numeric 11
## age.breaks numeric 4
## age.group factor 11
## age.group2 factor 11
## age.group3 factor 11
## age.labels character 3
## df data.frame 10 x 4
## is.MP.speaking logical 10
## itng data.frame 10 x 2
## itng.table table 3 x 4
## likert.centred numeric 10
## likert.raw numeric 10
## makka.pakka character 4
## opinion.dir numeric 10
## opinion.strength numeric 10
## some.data numeric 18
## speaker character 10
## speech.by.char list 3
## tombliboo character 2
## upsy.daisy character 4
## utterance character 10``````
we see that there are three new variables, called `makka.pakka`, `tombliboo` and `upsy.daisy`. Notice that the `importList()` function has converted the original character strings into valid R variable names, so the variable corresponding to `"makka-pakka"` is actually `makka.pakka`.112 Nevertheless, even though the names can change, note that each of these variables contains the exact same information as the original elements of the list did. For example:
``````> makka.pakka
[1] "pip" "pip" "onk" "onk"``` ``` | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.04%3A_Extracting_a_Subset_of_a_Vector.txt |
In this section we turn to the question of how to subset a data frame rather than a vector. To that end, the first thing I should point out is that, if all you want to do is subset one of the variables inside the data frame, then as usual the `\$` operator is your friend. For instance, suppose I’m working with the `itng` data frame, and what I want to do is create the `speech.by.char` list. I can use the exact same tricks that I used last time, since what I really want to do is `split()` the `itng\$utterance` vector, using the `itng\$speaker` vector as the grouping variable. However, most of the time what you actually want to do is select several different variables within the data frame (i.e., keep only some of the columns), or maybe a subset of cases (i.e., keep only some of the rows). In order to understand how this works, we need to talk more specifically about data frames and how to subset them.
Using the `subset()` function
There are several different ways to subset a data frame in R, some easier than others. I’ll start by discussing the `subset()` function, which is probably the conceptually simplest way do it. For our purposes there are three different arguments that you’ll be most interested in:
• `x`. The data frame that you want to subset.
• `subset`. A vector of logical values indicating which cases (rows) of the data frame you want to keep. By default, all cases will be retained.
• `select`. This argument indicates which variables (columns) in the data frame you want to keep. This can either be a list of variable names, or a logical vector indicating which ones to keep, or even just a numeric vector containing the relevant column numbers. By default, all variables will be retained.
Let’s start with an example in which I use all three of these arguments. Suppose that I want to subset the `itng` data frame, keeping only the utterances made by Makka-Pakka. What that means is that I need to use the `select` argument to pick out the `utterance` variable, and I also need to use the `subset` variable, to pick out the cases when Makka-Pakka is speaking (i.e., `speaker == "makka-pakka"`). Therefore, the command I need to use is this:
``````df <- subset( x = itng, # data frame is itng
subset = speaker == "makka-pakka", # keep only Makka-Pakkas speech
select = utterance ) # keep only the utterance variable
print( df )``````
``````## utterance
## 7 pip
## 8 pip
## 9 onk
## 10 onk``````
The variable `df` here is still a data frame, but it only contains one variable (called `utterance`) and four cases. Notice that the row numbers are actually the same ones from the original data frame. It’s worth taking a moment to briefly explain this. The reason that this happens is that these “row numbers’ are actually row names. When you create a new data frame from scratch R will assign each row a fairly boring row name, which is identical to the row number. However, when you subset the data frame, each row keeps its original row name. This can be quite useful, since – as in the current example – it provides you a visual reminder of what each row in the new data frame corresponds to in the original data frame. However, if it annoys you, you can change the row names using the `rownames()` function.113
In any case, let’s return to the `subset()` function, and look at what happens when we don’t use all three of the arguments. Firstly, suppose that I didn’t bother to specify the `select` argument. Let’s see what happens:
``````subset( x = itng,
subset = speaker == "makka-pakka" )```
```
``````## speaker utterance
## 7 makka-pakka pip
## 8 makka-pakka pip
## 9 makka-pakka onk
## 10 makka-pakka onk``````
Not surprisingly, R has kept the same cases from the original data set (i.e., rows 7 through 10), but this time it has kept all of the variables from the data frame. Equally unsurprisingly, if I don’t specify the `subset` argument, what we find is that R keeps all of the cases:
``````subset( x = itng,
select = utterance )```
```
``````## utterance
## 1 pip
## 2 pip
## 3 onk
## 4 onk
## 5 ee
## 6 oo
## 7 pip
## 8 pip
## 9 onk
## 10 onk``````
Again, it’s important to note that this output is still a data frame: it’s just a data frame with only a single variable.
Using square brackets: I. Rows and columns
Throughout the book so far, whenever I’ve been subsetting a vector I’ve tended use the square brackets `[]` to do so. But in the previous section when I started talking about subsetting a data frame I used the `subset()` function. As a consequence, you might be wondering whether it is possible to use the square brackets to subset a data frame. The answer, of course, is yes. Not only can you use square brackets for this purpose, as you become more familiar with R you’ll find that this is actually much more convenient than using `subset()`. Unfortunately, the use of square brackets for this purpose is somewhat complicated, and can be very confusing to novices. So be warned: this section is more complicated than it feels like it “should” be. With that warning in place, I’ll try to walk you through it slowly. For this section, I’ll use a slightly different data set, namely the `garden` data frame that is stored in the `"nightgarden2.Rdata"` file.
``````load("./rbook-master/data/nightgarden2.Rdata" )
garden``````
``````## speaker utterance line
## case.1 upsy-daisy pip 1
## case.2 upsy-daisy pip 2
## case.3 tombliboo ee 5
## case.4 makka-pakka pip 7
## case.5 makka-pakka onk 9``````
As you can see, the `garden` data frame contains 3 variables and 5 cases, and this time around I’ve used the `rownames()` function to attach slightly verbose labels to each of the cases. Moreover, let’s assume that what we want to do is to pick out rows 4 and 5 (the two cases when Makka-Pakka is speaking), and columns 1 and 2 (variables `speaker` and `utterance`).
How shall we do this? As usual, there’s more than one way. The first way is based on the observation that, since a data frame is basically a table, every element in the data frame has a row number and a column number. So, if we want to pick out a single element, we have to specify the row number and a column number within the square brackets. By convention, the row number comes first. So, for the data frame above, which has 5 rows and 3 columns, the numerical indexing scheme looks like this:
``````knitr::kable(data.frame(stringsAsFactors=FALSE, row = c("1","2","3", "4", "5"), col1 = c("[1,1]", "[2,1]", "[3,1]", "[4,1]",
"[5,1]"), col2 = c("[1,2]", "[2,2]", "[3,2]", "[4,2]", "[5,2]"), col3 = c("[1,3]", "[2,3]", "[3,3]", "[4,3]", "[5,3]")))```
```
row col1 col2 col3
1 [1,1] [1,2] [1,3]
2 [2,1] [2,2] [2,3]
3 [3,1] [3,2] [3,3]
4 [4,1] [4,2] [4,3]
5 [5,1] [5,2] [5,3]
If I want the 3rd case of the 2nd variable, what I would type is `garden[3,2]`, and R would print out some output showing that, this element corresponds to the utterance `"ee"`. However, let’s hold off from actually doing that for a moment, because there’s something slightly counterintuitive about the specifics of what R does under those circumstances (see Section 7.5.4). Instead, let’s aim to solve our original problem, which is to pull out two rows (4 and 5) and two columns (1 and 2). This is fairly simple to do, since R allows us to specify multiple rows and multiple columns. So let’s try that:
``garden[ 4:5, 1:2 ]``
``````## speaker utterance
## case.4 makka-pakka pip
## case.5 makka-pakka onk``````
Clearly, that’s exactly what we asked for: the output here is a data frame containing two variables and two cases. Note that I could have gotten the same answer if I’d used the `c()` function to produce my vectors rather than the `:` operator. That is, the following command is equivalent to the last one:
``garden[ c(4,5), c(1,2) ]``
``````## speaker utterance
## case.4 makka-pakka pip
## case.5 makka-pakka onk```
```
It’s just not as pretty. However, if the columns and rows that you want to keep don’t happen to be next to each other in the original data frame, then you might find that you have to resort to using commands like `garden[ c(2,4,5), c(1,3) ]` to extract them.
A second way to do the same thing is to use the names of the rows and columns. That is, instead of using the row numbers and column numbers, you use the character strings that are used as the labels for the rows and columns. To apply this idea to our `garden` data frame, we would use a command like this:
``garden[ c("case.4", "case.5"), c("speaker", "utterance") ]``
``````## speaker utterance
## case.4 makka-pakka pip
## case.5 makka-pakka onk``````
Once again, this produces exactly the same output, so I haven’t bothered to show it. Note that, although this version is more annoying to type than the previous version, it’s a bit easier to read, because it’s often more meaningful to refer to the elements by their names rather than their numbers. Also note that you don’t have to use the same convention for the rows and columns. For instance, I often find that the variable names are meaningful and so I sometimes refer to them by name, whereas the row names are pretty arbitrary so it’s easier to refer to them by number. In fact, that’s more or less exactly what’s happening with the `garden` data frame, so it probably makes more sense to use this as the command:
``garden[ 4:5, c("speaker", "utterance") ]``
``````## speaker utterance
## case.4 makka-pakka pip
## case.5 makka-pakka onk```
```
Again, the output is identical.
Finally, both the rows and columns can be indexed using logicals vectors as well. For example, although I claimed earlier that my goal was to extract cases 4 and 5, it’s pretty obvious that what I really wanted to do was select the cases where Makka-Pakka is speaking. So what I could have done is create a logical vector that indicates which cases correspond to Makka-Pakka speaking:
``````is.MP.speaking <- garden\$speaker == "makka-pakka"
is.MP.speaking``````
``## [1] FALSE FALSE FALSE TRUE TRUE``
As you can see, the 4th and 5th elements of this vector are `TRUE` while the others are `FALSE`. Now that I’ve constructed this “indicator” variable, what I can do is use this vector to select the rows that I want to keep:
``garden[ is.MP.speaking, c("speaker", "utterance") ]``
``````## speaker utterance
## case.4 makka-pakka pip
## case.5 makka-pakka onk``````
And of course the output is, yet again, the same.
Using square brackets: II. Some elaborations
There are two fairly useful elaborations on this “rows and columns” approach that I should point out. Firstly, what if you want to keep all of the rows, or all of the columns? To do this, all we have to do is leave the corresponding entry blank, but it is crucial to remember to keep the comma*! For instance, suppose I want to keep all the rows in the `garden` data, but I only want to retain the first two columns. The easiest way do this is to use a command like this:
``garden[ , 1:2 ]``
``````## speaker utterance
## case.1 upsy-daisy pip
## case.2 upsy-daisy pip
## case.3 tombliboo ee
## case.4 makka-pakka pip
## case.5 makka-pakka onk``````
Alternatively, if I want to keep all the columns but only want the last two rows, I use the same trick, but this time I leave the second index blank. So my command becomes:
``garden[ 4:5, ]``
``````## speaker utterance line
## case.4 makka-pakka pip 7
## case.5 makka-pakka onk 9``````
The second elaboration I should note is that it’s still okay to use negative indexes as a way of telling R to delete certain rows or columns. For instance, if I want to delete the 3rd column, then I use this command:
````garden[ , -3 ]`
```
``````## speaker utterance
## case.1 upsy-daisy pip
## case.2 upsy-daisy pip
## case.3 tombliboo ee
## case.4 makka-pakka pip
## case.5 makka-pakka onk``````
whereas if I want to delete the 3rd row, then I’d use this one:
````garden[ -3, ]`
```
``````## speaker utterance line
## case.1 upsy-daisy pip 1
## case.2 upsy-daisy pip 2
## case.4 makka-pakka pip 7
## case.5 makka-pakka onk 9``````
So that’s nice.
Using square brackets: III. Understanding “dropping”
At this point some of you might be wondering why I’ve been so terribly careful to choose my examples in such a way as to ensure that the output always has are multiple rows and multiple columns. The reason for this is that I’ve been trying to hide the somewhat curious “dropping” behaviour that R produces when the output only has a single column. I’ll start by showing you what happens, and then I’ll try to explain it. Firstly, let’s have a look at what happens when the output contains only a single row:
``garden[ 5, ]``
``````## speaker utterance line
## case.5 makka-pakka onk 9``````
This is exactly what you’d expect to see: a data frame containing three variables, and only one case per variable. Okay, no problems so far. What happens when you ask for a single column? Suppose, for instance, I try this as a command:
``garden[ , 3 ]``
Based on everything that I’ve shown you so far, you would be well within your rights to expect to see R produce a data frame containing a single variable (i.e., `line`) and five cases. After all, that is what the `subset()` command does in this situation, and it’s pretty consistent with everything else that I’ve shown you so far about how square brackets work. In other words, you should expect to see this:
`````` line
case.1 1
case.2 2
case.3 5
case.4 7
case.5 9``````
However, that is emphatically not what happens. What you actually get is this:
``garden[ , 3 ]``
``## [1] 1 2 5 7 9``
That output is not a data frame at all! That’s just an ordinary numeric vector containing 5 elements. What’s going on here is that R has “noticed” that the output that we’ve asked for doesn’t really “need” to be wrapped up in a data frame at all, because it only corresponds to a single variable. So what it does is “drop” the output from a data frame containing a single variable, “down” to a simpler output that corresponds to that variable. This behaviour is actually very convenient for day to day usage once you’ve become familiar with it – and I suppose that’s the real reason why R does this – but there’s no escaping the fact that it is deeply confusing to novices. It’s especially confusing because the behaviour appears only for a very specific case: (a) it only works for columns and not for rows, because the columns correspond to variables and the rows do not, and (b) it only applies to the “rows and columns” version of the square brackets, and not to the `subset()` function,114 or to the “just columns” use of the square brackets (next section). As I say, it’s very confusing when you’re just starting out. For what it’s worth, you can suppress this behaviour if you want, by setting `drop = FALSE` when you construct your bracketed expression. That is, you could do something like this:
``garden[ , 3, drop = FALSE ]``
``````## line
## case.1 1
## case.2 2
## case.3 5
## case.4 7
## case.5 9``````
I suppose that helps a little bit, in that it gives you some control over the dropping behaviour, but I’m not sure it helps to make things any easier to understand. Anyway, that’s the “dropping” special case. Fun, isn’t it?
Using square brackets: IV. Columns only
As if the weird “dropping” behaviour wasn’t annoying enough, R actually provides a completely different way of using square brackets to index a data frame. Specifically, if you only give a single index, R will assume you want the corresponding columns, not the rows. Do not be fooled by the fact that this second method also uses square brackets: it behaves differently to the “rows and columns” method that I’ve discussed in the last few sections. Again, what I’ll do is show you what happens first, and then I’ll try to explain why it happens afterwards. To that end, let’s start with the following command:
``garden[ 1:2 ]``
``````## speaker utterance
## case.1 upsy-daisy pip
## case.2 upsy-daisy pip
## case.3 tombliboo ee
## case.4 makka-pakka pip
## case.5 makka-pakka onk``````
As you can see, the output gives me the first two columns, much as if I’d typed `garden[,1:2]`. It doesn’t give me the first two rows, which is what I’d have gotten if I’d used a command like `garden[1:2,]`. Not only that, if I ask for a single column, R does not drop the output:
``garden[3]``
``````## line
## case.1 1
## case.2 2
## case.3 5
## case.4 7
## case.5 9``````
As I said earlier, the only case where dropping occurs by default is when you use the “row and columns” version of the square brackets, and the output happens to correspond to a single column. However, if you really want to force R to drop the output, you can do so using the “double brackets” notation:
``garden[[3]]``
``## [1] 1 2 5 7 9``
Note that R will only allow you to ask for one column at a time using the double brackets. If you try to ask for multiple columns in this way, you get completely different behaviour,115 which may or may not produce an error, but definitely won’t give you the output you’re expecting. The only reason I’m mentioning it at all is that you might run into double brackets when doing further reading, and a lot of books don’t explicitly point out the difference between `[` and `[[`. However, I promise that I won’t be using `[[` anywhere else in this book.
Okay, for those few readers that have persevered with this section long enough to get here without having set fire to the book, I should explain why R has these two different systems for subsetting a data frame (i.e., “row and column” versus “just columns”), and why they behave so differently to each other. I’m not 100% sure about this since I’m still reading through some of the old references that describe the early development of R, but I think the answer relates to the fact that data frames are actually a very strange hybrid of two different kinds of thing. At a low level, a data frame is a list (Section 4.9). I can demonstrate this to you by overriding the normal `print()` function116 and forcing R to print out the `garden` data frame using the default print method rather than the special one that is defined only for data frames. Here’s what we get:
``print.default( garden )``
``````## \$speaker
## [1] upsy-daisy upsy-daisy tombliboo makka-pakka makka-pakka
## Levels: makka-pakka tombliboo upsy-daisy
##
## \$utterance
## [1] pip pip ee pip onk
## Levels: ee onk oo pip
##
## \$line
## [1] 1 2 5 7 9
##
## attr(,"class")
## [1] "data.frame"``````
Apart from the weird part of the output right at the bottom, this is identical to the print out that you get when you print out a list (see Section 4.9). In other words, a data frame is a list. View from this “list based” perspective, it’s clear what `garden[1]` is: it’s the first variable stored in the list, namely `speaker`. In other words, when you use the “just columns” way of indexing a data frame, using only a single index, R assumes that you’re thinking about the data frame as if it were a list of variables. In fact, when you use the `\$` operator you’re taking advantage of the fact that the data frame is secretly a list.
However, a data frame is more than just a list. It’s a very special kind of list where all the variables are of the same length, and the first element in each variable happens to correspond to the first “case” in the data set. That’s why no-one ever wants to see a data frame printed out in the default “list-like” way that I’ve shown in the extract above. In terms of the deeper meaning behind what a data frame is used for, a data frame really does have this rectangular shape to it:
``print( garden )``
``````## speaker utterance line
## case.1 upsy-daisy pip 1
## case.2 upsy-daisy pip 2
## case.3 tombliboo ee 5
## case.4 makka-pakka pip 7
## case.5 makka-pakka onk 9``````
Because of the fact that a data frame is basically a table of data, R provides a second “row and column” method for interacting with the data frame (see Section 7.11.1 for a related example). This method makes much more sense in terms of the high-level table of data interpretation of what a data frame is, and so for the most part it’s this method that people tend to prefer. In fact, throughout the rest of the book I will be sticking to the “row and column” approach (though I will use `\$` a lot), and never again referring to the “just columns” approach. However, it does get used a lot in practice, so I think it’s important that this book explain what’s going on.
And now let us never speak of this again. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.05%3A_Extracting_a_Subset_of_a_Data_Frame.txt |
In this section I discuss a few useful operations that I feel are loosely related to one another: sorting a vector, sorting a data frame, binding two or more vectors together into a data frame (or matrix), and flipping a data frame (or matrix) on its side. They’re all fairly straightforward tasks, at least in comparison to some of the more obnoxious data handling problems that turn up in real life.
Sorting a numeric or character vector
One thing that you often want to do is sort a variable. If it’s a numeric variable you might want to sort in increasing or decreasing order. If it’s a character vector you might want to sort alphabetically, etc. The `sort()` function provides this capability.
``````numbers <- c(2,4,3)
sort( x = numbers )``````
``## [1] 2 3 4``
You can ask for R to sort in decreasing order rather than increasing:
``sort( x = numbers, decreasing = TRUE )``
``## [1] 4 3 2``
And you can ask it to sort text data in alphabetical order:
``````text <- c("aardvark", "zebra", "swing")
sort( text )``````
``## [1] "aardvark" "swing" "zebra"``
That’s pretty straightforward. That being said, it’s important to note that I’m glossing over something here. When you apply `sort()` to a character vector it doesn’t strictly sort into alphabetical order. R actually has a slightly different notion of how characters are ordered (see Section 7.8.5 and Table 7.3), which is more closely related to how computers store text data than to how letters are ordered in the alphabet. However, that’s a topic we’ll discuss later. For now, the only thing I should note is that the `sort()` function doesn’t alter the original variable. Rather, it creates a new, sorted variable as the output. So if I inspect my original `text` variable:
``text``
``## [1] "aardvark" "zebra" "swing"``
I can see that it has remained unchanged.
Sorting a factor
You can also sort factors, but the story here is slightly more subtle because there’s two different ways you can sort a factor: alphabetically (by label) or by factor level. The `sort()` function uses the latter. To illustrate, let’s look at the two different examples. First, let’s create a factor in the usual way:
``````fac <- factor( text )
fac``````
``````## [1] aardvark zebra swing
## Levels: aardvark swing zebra``````
Now let’s sort it:
``sort(fac)``
``````## [1] aardvark swing zebra
## Levels: aardvark swing zebra``````
This looks like it’s sorted things into alphabetical order, but that’s only because the factor levels themselves happen to be alphabetically ordered. Suppose I deliberately define the factor levels in a non-alphabetical order:
``````fac <- factor( text, levels = c("zebra","swing","aardvark") )
fac``````
``````## [1] aardvark zebra swing
## Levels: zebra swing aardvark``````
Now what happens when we try to sort `fac` this time? The answer:
``sort(fac)``
``````## [1] zebra swing aardvark
## Levels: zebra swing aardvark``````
It sorts the data into the numerical order implied by the factor levels, not the alphabetical order implied by the labels attached to those levels. Normally you never notice the distinction, because by default the factor levels are assigned in alphabetical order, but it’s important to know the difference:
Sorting a data frame
The `sort()` function doesn’t work properly with data frames. If you want to sort a data frame the standard advice that you’ll find online is to use the `order()` function (not described in this book) to determine what order the rows should be sorted, and then use square brackets to do the shuffling. There’s nothing inherently wrong with this advice, I just find it tedious. To that end, the `lsr` package includes a function called `sortFrame()` that you can use to do the sorting. The first argument to the function is named (`x`), and should correspond to the data frame that you want sorted. After that, all you do is type a list of the names of the variables that you want to use to do the sorting. For instance, if I type this:
``sortFrame( garden, speaker, line)``
``````## speaker utterance line
## case.4 makka-pakka pip 7
## case.5 makka-pakka onk 9
## case.3 tombliboo ee 5
## case.1 upsy-daisy pip 1
## case.2 upsy-daisy pip 2``````
what R does is first sort by `speaker` (factor level order). Any ties (i.e., data from the same speaker) are then sorted in order of `line` (increasing numerical order). You can use the minus sign to indicate that numerical variables should be sorted in reverse order:
``sortFrame( garden, speaker, -line)``
``````## speaker utterance line
## case.5 makka-pakka onk 9
## case.4 makka-pakka pip 7
## case.3 tombliboo ee 5
## case.2 upsy-daisy pip 2
## case.1 upsy-daisy pip 1``````
As of the current writing, the `sortFrame()` function is under development. I’ve started introducing functionality to allow you to use the `-` sign to non-numeric variables or to make a distinction between sorting factors alphabetically or by factor level. The idea is that you should be able to type in something like this:
``sortFrame( garden, -speaker)``
and have the output correspond to a sort of the `garden` data frame in reverse alphabetical order (or reverse factor level order) of `speaker`. As things stand right now, this will actually work, and it will produce sensible output:
``sortFrame( garden, -speaker)``
``````## speaker utterance line
## case.1 upsy-daisy pip 1
## case.2 upsy-daisy pip 2
## case.3 tombliboo ee 5
## case.4 makka-pakka pip 7
## case.5 makka-pakka onk 9``````
However, I’m not completely convinced that I’ve set this up in the ideal fashion, so this may change a little bit in the future.
Binding vectors together
A not-uncommon task that you might find yourself needing to undertake is to combine several vectors. For instance, let’s suppose we have the following two numeric vectors:
``````cake.1 <- c(100, 80, 0, 0, 0)
cake.2 <- c(100, 100, 90, 30, 10)``````
The numbers here might represent the amount of each of the two cakes that are left at five different time points. Apparently the first cake is tastier, since that one gets devoured faster. We’ve already seen one method for combining these vectors: we could use the `data.frame()` function to convert them into a data frame with two variables, like so:
``````cake.df <- data.frame( cake.1, cake.2 )
cake.df```
```
``````## cake.1 cake.2
## 1 100 100
## 2 80 100
## 3 0 90
## 4 0 30
## 5 0 10``````
Two other methods that I want to briefly refer to are the `rbind()` and `cbind()` functions, which will convert the vectors into a matrix. I’ll discuss matrices properly in Section 7.11.1 but the details don’t matter too much for our current purposes. The `cbind()` function (“column bind”) produces a very similar looking output to the data frame example:
``````cake.mat1 <- cbind( cake.1, cake.2 )
cake.mat1``````
``````## cake.1 cake.2
## [1,] 100 100
## [2,] 80 100
## [3,] 0 90
## [4,] 0 30
## [5,] 0 10``````
but nevertheless it’s important to keep in mind that `cake.mat1` is a matrix rather than a data frame, and so has a few differences from the `cake.df` variable. The `rbind()` function (“row bind”) produces a somewhat different output: it binds the vectors together row-wise rather than column-wise, so the output now looks like this:
``````cake.mat2 <- rbind( cake.1, cake.2 )
cake.mat2```
```
``````## [,1] [,2] [,3] [,4] [,5]
## cake.1 100 80 0 0 0
## cake.2 100 100 90 30 10``````
You can add names to a matrix by using the `rownames()` and `colnames()` functions, and I should also point out that there’s a fancier function in R called `merge()` that supports more complicated “database like” merging of vectors and data frames, but I won’t go into details here.
Binding multiple copies of the same vector together
It is sometimes very useful to bind together multiple copies of the same vector. You could do this using the `rbind` and `cbind` functions, using comands like this one
``````fibonacci <- c( 1,1,2,3,5,8 )
rbind( fibonacci, fibonacci, fibonacci )``````
``````## [,1] [,2] [,3] [,4] [,5] [,6]
## fibonacci 1 1 2 3 5 8
## fibonacci 1 1 2 3 5 8
## fibonacci 1 1 2 3 5 8``````
but that can be pretty annoying, especially if you needs lots of copies. To make this a little easier, the `lsr` package has two additional functions `rowCopy` and `colCopy` that do the same job, but all you have to do is specify the number of copies that you want, instead of typing the name in over and over again. The two arguments you need to specify are `x`, the vector to be copied, and `times`, indicating how many copies should be created:117
``rowCopy( x = fibonacci, times = 3 )``
``````## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 1 1 2 3 5 8
## [2,] 1 1 2 3 5 8
## [3,] 1 1 2 3 5 8``````
Of course, in practice you don’t need to name the arguments all the time. For instance, here’s an example using the `colCopy()` function with the argument names omitted:
``colCopy( fibonacci, 3 )``
``````## [,1] [,2] [,3]
## [1,] 1 1 1
## [2,] 1 1 1
## [3,] 2 2 2
## [4,] 3 3 3
## [5,] 5 5 5
## [6,] 8 8 8``````
Transposing a matrix or data frame
``````load("./rbook-master/data/cakes.Rdata" )
cakes``````
``````## time.1 time.2 time.3 time.4 time.5
## cake.1 100 80 0 0 0
## cake.2 100 100 90 30 10
## cake.3 100 20 20 20 20
## cake.4 100 100 100 100 100``````
And just to make sure you believe me that this is actually a matrix:
``class( cakes )``
``## [1] "matrix"``
Okay, now let’s transpose the matrix:
``````cakes.flipped <- t( cakes )
cakes.flipped``````
``````## cake.1 cake.2 cake.3 cake.4
## time.1 100 100 100 100
## time.2 80 100 20 100
## time.3 0 90 20 100
## time.4 0 30 20 100
## time.5 0 10 20 100``````
The output here is still a matrix:
````class( cakes.flipped )`
```
``## [1] "matrix"``
At this point you should have two questions: (1) how do we do the same thing for data frames? and (2) why should we care about this? Let’s start with the how question. First, I should note that you can transpose a data frame just fine using the `t()` function, but that has the slightly awkward consequence of converting the output from a data frame to a matrix, which isn’t usually what you want. It’s quite easy to convert the output back again, of course,118 but I hate typing two commands when I can do it with one. To that end, the `lsr` package has a simple “convenience” function called `tFrame()` which does exactly the same thing as `t()` but converts the output to a data frame for you. To illustrate this, let’s transpose the `itng` data frame that we used earlier. Here’s the original data frame:
``itng``
``````## speaker utterance
## 1 upsy-daisy pip
## 2 upsy-daisy pip
## 3 upsy-daisy onk
## 4 upsy-daisy onk
## 5 tombliboo ee
## 6 tombliboo oo
## 7 makka-pakka pip
## 8 makka-pakka pip
## 9 makka-pakka onk
## 10 makka-pakka onk``````
and here’s what happens when you transpose it using `tFrame()`:
``tFrame( itng )``
``````## V1 V2 V3 V4 V5 V6
## speaker upsy-daisy upsy-daisy upsy-daisy upsy-daisy tombliboo tombliboo
## utterance pip pip onk onk ee oo
## V7 V8 V9 V10
## speaker makka-pakka makka-pakka makka-pakka makka-pakka
## utterance pip pip onk onk``````
An important point to recognise is that transposing a data frame is not always a sensible thing to do: in fact, I’d go so far as to argue that it’s usually not sensible. It depends a lot on whether the “cases” from your original data frame would make sense as variables, and to think of each of your original “variables” as cases. I think that’s emphatically not true for our `itng` data frame, so I wouldn’t advise doing it in this situation.
That being said, sometimes it really is true. For instance, had we originally stored our `cakes` variable as a data frame instead of a matrix, then it would absolutely be sensible to flip the data frame!119 There are some situations where it is useful to flip your data frame, so it’s nice to know that you can do it. Indeed, that’s the main reason why I have spent so much time talking about this topic. A lot of statistical tools make the assumption that the rows of your data frame (or matrix) correspond to observations, and the columns correspond to the variables. That’s not unreasonable, of course, since that is a pretty standard convention. However, think about our `cakes` example here. This is a situation where you might want do an analysis of the different cakes (i.e. cakes as variables, time points as cases), but equally you might want to do an analysis where you think of the times as being the things of interest (i.e., times as variables, cakes as cases). If so, then it’s useful to know how to flip a matrix or data frame around. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.06%3A_Sorting_Flipping_and_Merging_Data.txt |
One of the most annoying tasks that you need to undertake on a regular basis is that of reshaping a data frame. Framed in the most general way, reshaping the data means taking the data in whatever format it’s given to you, and converting it to the format you need it. Of course, if we’re going to characterise the problem that broadly, then about half of this chapter can probably be thought of as a kind of reshaping. So we’re going to have to narrow things down a little bit. To that end, I’ll talk about a few different tools that you can use for a few different tasks. In particular, I’ll discuss a couple of easy to use (but limited) functions that I’ve included in the `lsr` package. In future versions of the book I plan to expand this discussion to include some of the more powerful tools that are available in R, but I haven’t had the time to do so yet.
Long form and wide form data
The most common format in which you might obtain data is as a “case by variable” layout, commonly known as the wide form of the data.
``````load("./rbook-master/data/repeated.Rdata")
who()``````
``````## -- Name -- -- Class -- -- Size --
## age numeric 11
## age.breaks numeric 4
## age.group factor 11
## age.group2 factor 11
## age.group3 factor 11
## age.labels character 3
## cake.1 numeric 5
## cake.2 numeric 5
## cake.df data.frame 5 x 2
## cake.mat1 matrix 5 x 2
## cake.mat2 matrix 2 x 5
## cakes matrix 4 x 5
## cakes.flipped matrix 5 x 4
## choice data.frame 4 x 10
## df data.frame 4 x 1
## drugs data.frame 10 x 8
## fac factor 3
## fibonacci numeric 6
## garden data.frame 5 x 3
## is.MP.speaking logical 5
## itng data.frame 10 x 2
## itng.table table 3 x 4
## likert.centred numeric 10
## likert.raw numeric 10
## makka.pakka character 4
## numbers numeric 3
## opinion.dir numeric 10
## opinion.strength numeric 10
## some.data numeric 18
## speaker character 10
## speech.by.char list 3
## text character 3
## tombliboo character 2
## upsy.daisy character 4
## utterance character 10``````
To get a sense of what I’m talking about, consider an experiment in which we are interested in the different effects that alcohol and and caffeine have on people’s working memory capacity (WMC) and reaction times (RT). We recruit 10 participants, and measure their WMC and RT under three different conditions: a “no drug” condition, in which they are not under the influence of either caffeine or alcohol, a “caffeine” condition, in which they are under the inflence of caffeine, and an “alcohol” condition, in which… well, you can probably guess. Ideally, I suppose, there would be a fourth condition in which both drugs are administered, but for the sake of simplicity let’s ignore that. The `drugs` data frame gives you a sense of what kind of data you might observe in an experiment like this:
``drugs``
``````## id gender WMC_alcohol WMC_caffeine WMC_no.drug RT_alcohol RT_caffeine
## 1 1 female 3.7 3.7 3.9 488 236
## 2 2 female 6.4 7.3 7.9 607 376
## 3 3 female 4.6 7.4 7.3 643 226
## 4 4 male 6.4 7.8 8.2 684 206
## 5 5 female 4.9 5.2 7.0 593 262
## 6 6 male 5.4 6.6 7.2 492 230
## 7 7 male 7.9 7.9 8.9 690 259
## 8 8 male 4.1 5.9 4.5 486 230
## 9 9 female 5.2 6.2 7.2 686 273
## 10 10 female 6.2 7.4 7.8 645 240
## RT_no.drug
## 1 371
## 2 349
## 3 412
## 4 252
## 5 439
## 6 464
## 7 327
## 8 305
## 9 327
## 10 498``````
This is a data set in “wide form”, in which each participant corresponds to a single row. We have two variables that are characteristics of the subject (i.e., their `id` number and their `gender`) and six variables that refer to one of the two measured variables (WMC or RT) in one of the three testing conditions (alcohol, caffeine or no drug). Because all of the testing conditions (i.e., the three drug types) are applied to all participants, drug type is an example of a within-subject factor.
Reshaping data using `wideToLong()`
The “wide form” of this data set is useful for some situations: it is often very useful to have each row correspond to a single subject. However, it is not the only way in which you might want to organise this data. For instance, you might want to have a separate row for each “testing occasion”. That is, “participant 1 under the influence of alcohol” would be one row, and “participant 1 under the influence of caffeine” would be another row. This way of organising the data is generally referred to as the long form of the data. It’s not too difficult to switch between wide and long form, and I’ll explain how it works in a moment; for now, let’s just have a look at what the long form of this data set looks like:
``````drugs.2 <- wideToLong( data = drugs, within = "drug" )
head(drugs.2)``````
``````## id gender drug WMC RT
## 1 1 female alcohol 3.7 488
## 2 2 female alcohol 6.4 607
## 3 3 female alcohol 4.6 643
## 4 4 male alcohol 6.4 684
## 5 5 female alcohol 4.9 593
## 6 6 male alcohol 5.4 492``````
The `drugs.2` data frame that we just created has 30 rows: each of the 10 participants appears in three separate rows, one corresponding to each of the three testing conditions. And instead of having a variable like `WMC_caffeine` that indicates that we were measuring “WMC” in the “caffeine” condition, this information is now recorded in two separate variables, one called `drug` and another called `WMC`. Obviously, the long and wide forms of the data contain the same information, but they represent quite different ways of organising that information. Sometimes you find yourself needing to analyse data in wide form, and sometimes you find that you need long form. So it’s really useful to know how to switch between the two.
In the example I gave above, I used a function called `wideToLong()` to do the transformation. The `wideToLong()` function is part of the `lsr` package. The key to understanding this function is that it relies on the variable names to do all the work. Notice that the variable names in the `drugs` data frame follow a very clear scheme. Whenever you have a variable with a name like `WMC_caffeine` you know that the variable being measured is “WMC”, and that the specific condition in which it is being measured is the “caffeine” condition. Similarly, you know that `RT_no.drug` refers to the “RT” variable measured in the “no drug” condition. The measured variable comes first (e.g., `WMC`), followed by a separator character (in this case the separator is an underscore, `_`), and then the name of the condition in which it is being measured (e.g., `caffeine`). There are two different prefixes (i.e, the strings before the separator, `WMC`, `RT`) which means that there are two separate variables being measured. There are three different suffixes (i.e., the strings after the separtator, `caffeine`, `alcohol`, `no.drug`) meaning that there are three different levels of the within-subject factor. Finally, notice that the separator string (i.e., `_`) does not appear anywhere in two of the variables (`id`, `gender`), indicating that these are between-subject variables, namely variables that do not vary within participant (e.g., a person’s `gender` is the same regardless of whether they’re under the influence of alcohol, caffeine etc).
Because of the fact that the variable naming scheme here is so informative, it’s quite possible to reshape the data frame without any additional input from the user. For example, in this particular case, you could just type the following:
``wideToLong( drugs )``
``````## id gender within WMC RT
## 1 1 female alcohol 3.7 488
## 2 2 female alcohol 6.4 607
## 3 3 female alcohol 4.6 643
## 4 4 male alcohol 6.4 684
## 5 5 female alcohol 4.9 593
## 6 6 male alcohol 5.4 492
## 7 7 male alcohol 7.9 690
## 8 8 male alcohol 4.1 486
## 9 9 female alcohol 5.2 686
## 10 10 female alcohol 6.2 645
## 11 1 female caffeine 3.7 236
## 12 2 female caffeine 7.3 376
## 13 3 female caffeine 7.4 226
## 14 4 male caffeine 7.8 206
## 15 5 female caffeine 5.2 262
## 16 6 male caffeine 6.6 230
## 17 7 male caffeine 7.9 259
## 18 8 male caffeine 5.9 230
## 19 9 female caffeine 6.2 273
## 20 10 female caffeine 7.4 240
## 21 1 female no.drug 3.9 371
## 22 2 female no.drug 7.9 349
## 23 3 female no.drug 7.3 412
## 24 4 male no.drug 8.2 252
## 25 5 female no.drug 7.0 439
## 26 6 male no.drug 7.2 464
## 27 7 male no.drug 8.9 327
## 28 8 male no.drug 4.5 305
## 29 9 female no.drug 7.2 327
## 30 10 female no.drug 7.8 498``````
This is pretty good, actually. The only think it has gotten wrong here is that it doesn’t know what name to assign to the within-subject factor, so instaed of calling it something sensible like `drug`, it has use the unimaginative name `within`. If you want to ensure that the `wideToLong()` function applies a sensible name, you have to specify the `within` argument, which is just a character string that specifies the name of the within-subject factor. So when I used this command earlier,
``drugs.2 <- wideToLong( data = drugs, within = "drug" )``
all I was doing was telling R to use `drug` as the name of the within subject factor.
Now, as I was hinting earlier, the `wideToLong()` function is very inflexible. It requires that the variable names all follow this naming scheme that I outlined earlier. If you don’t follow this naming scheme it won’t work.120 The only flexibility that I’ve included here is that you can change the separator character by specifying the `sep` argument. For instance, if you were using variable names of the form `WMC/caffeine`, for instance, you could specify that `sep="/"`, using a command like this
``drugs.2 <- wideToLong( data = drugs, within = "drug", sep = "/" )``
and it would still work.
Reshaping data using `longToWide()`
To convert data from long form to wide form, the `lsr` package also includes a function called `longToWide()`. Recall from earlier that the long form of the data (i.e., the `drugs.2` data frame) contains variables named `id`, `gender`, `drug`, `WMC` and `RT`. In order to convert from long form to wide form, all you need to do is indicate which of these variables are measured separately for each condition (i.e., `WMC` and `RT`), and which variable is the within-subject factor that specifies the condition (i.e., `drug`). You do this via a two-sided formula, in which the measured variables are on the left hand side, and the within-subject factor is on the ritght hand side. In this case, the formula would be `WMC + RT ~ drug`. So the command that we would use might look like this:
``longToWide( data=drugs.2, formula= WMC+RT ~ drug )``
``````## id gender WMC_alcohol RT_alcohol WMC_caffeine RT_caffeine WMC_no.drug
## 1 1 female 3.7 488 3.7 236 3.9
## 2 2 female 6.4 607 7.3 376 7.9
## 3 3 female 4.6 643 7.4 226 7.3
## 4 4 male 6.4 684 7.8 206 8.2
## 5 5 female 4.9 593 5.2 262 7.0
## 6 6 male 5.4 492 6.6 230 7.2
## 7 7 male 7.9 690 7.9 259 8.9
## 8 8 male 4.1 486 5.9 230 4.5
## 9 9 female 5.2 686 6.2 273 7.2
## 10 10 female 6.2 645 7.4 240 7.8
## RT_no.drug
## 1 371
## 2 349
## 3 412
## 4 252
## 5 439
## 6 464
## 7 327
## 8 305
## 9 327
## 10 498``````
or, if we chose to omit argument names, we could simplify it to this:
````longToWide( drugs.2, WMC+RT ~ drug )`
```
``````## id gender WMC_alcohol RT_alcohol WMC_caffeine RT_caffeine WMC_no.drug
## 1 1 female 3.7 488 3.7 236 3.9
## 2 2 female 6.4 607 7.3 376 7.9
## 3 3 female 4.6 643 7.4 226 7.3
## 4 4 male 6.4 684 7.8 206 8.2
## 5 5 female 4.9 593 5.2 262 7.0
## 6 6 male 5.4 492 6.6 230 7.2
## 7 7 male 7.9 690 7.9 259 8.9
## 8 8 male 4.1 486 5.9 230 4.5
## 9 9 female 5.2 686 6.2 273 7.2
## 10 10 female 6.2 645 7.4 240 7.8
## RT_no.drug
## 1 371
## 2 349
## 3 412
## 4 252
## 5 439
## 6 464
## 7 327
## 8 305
## 9 327
## 10 498``````
Note that, just like the `wideToLong()` function, the `longToWide()` function allows you to override the default separator character. For instance, if the command I used had been
``longToWide( drugs.2, WMC+RT ~ drug, sep="/" )``
``````## id gender WMC/alcohol RT/alcohol WMC/caffeine RT/caffeine WMC/no.drug
## 1 1 female 3.7 488 3.7 236 3.9
## 2 2 female 6.4 607 7.3 376 7.9
## 3 3 female 4.6 643 7.4 226 7.3
## 4 4 male 6.4 684 7.8 206 8.2
## 5 5 female 4.9 593 5.2 262 7.0
## 6 6 male 5.4 492 6.6 230 7.2
## 7 7 male 7.9 690 7.9 259 8.9
## 8 8 male 4.1 486 5.9 230 4.5
## 9 9 female 5.2 686 6.2 273 7.2
## 10 10 female 6.2 645 7.4 240 7.8
## RT/no.drug
## 1 371
## 2 349
## 3 412
## 4 252
## 5 439
## 6 464
## 7 327
## 8 305
## 9 327
## 10 498``````
the output would contain variables with names like `RT/alcohol` instead of `RT_alcohol`.
Reshaping with multiple within-subject factors
As I mentioned above, the `wideToLong()` and `longToWide()` functions are quite limited in terms of what they can do. However, they do handle a broader range of situations than the one outlined above. Consider the following, fairly simple psychological experiment. I’m interested in the effects of practice on some simple decision making problem. It doesn’t really matter what the problem is, other than to note that I’m interested in two distinct outcome variables. Firstly, I care about people’s accuracy, measured by the proportion of decisions that people make correctly, denoted PC. Secondly, I care about people’s speed, measured by the mean response time taken to make those decisions, denoted MRT. That’s standard in psychological experiments: the speed-accuracy trade-off is pretty ubiquitous, so we generally need to care about both variables.
To look at the effects of practice over the long term, I test each participant on two days, `day1` and `day2`, where for the sake of argument I’ll assume that `day1` and `day2` are about a week apart. To look at the effects of practice over the short term, the testing during each day is broken into two “blocks”, `block1` and `block2`, which are about 20 minutes apart. This isn’t the world’s most complicated experiment, but it’s still a fair bit more complicated than the last one. This time around we have two within-subject factors (i.e., `day` and `block`) and we have two measured variables for each condition (i.e., `PC` and `MRT`). The `choice` data frame shows what the wide form of this kind of data might look like:
``choice``
``````## id gender MRT/block1/day1 MRT/block1/day2 MRT/block2/day1
## 1 1 male 415 400 455
## 2 2 male 500 490 532
## 3 3 female 478 468 499
## 4 4 female 550 502 602
## MRT/block2/day2 PC/block1/day1 PC/block1/day2 PC/block2/day1
## 1 450 79 88 82
## 2 518 83 92 86
## 3 474 91 98 90
## 4 588 75 89 78
## PC/block2/day2
## 1 93
## 2 97
## 3 100
## 4 95``````
Notice that this time around we have variable names of the form `MRT/block1/day2`. As before, the first part of the name refers to the measured variable (response time), but there are now two suffixes, one indicating that the testing took place in block 1, and the other indicating that it took place on day 2. And just to complicate matters, it uses `/` as the separator character rather than `_`. Even so, reshaping this data set is pretty easy. The command to do it is,
``choice.2 <- wideToLong( choice, within=c("block","day"), sep="/" )``
which is pretty much the exact same command we used last time. The only difference here is that, because there are two within-subject factors, the `within` argument is a vector that contains two names. When we look at the long form data frame that this creates, we get this:
``choice.2``
``````## id gender MRT PC block day
## 1 1 male 415 79 block1 day1
## 2 2 male 500 83 block1 day1
## 3 3 female 478 91 block1 day1
## 4 4 female 550 75 block1 day1
## 5 1 male 400 88 block1 day2
## 6 2 male 490 92 block1 day2
## 7 3 female 468 98 block1 day2
## 8 4 female 502 89 block1 day2
## 9 1 male 455 82 block2 day1
## 10 2 male 532 86 block2 day1
## 11 3 female 499 90 block2 day1
## 12 4 female 602 78 block2 day1
## 13 1 male 450 93 block2 day2
## 14 2 male 518 97 block2 day2
## 15 3 female 474 100 block2 day2
## 16 4 female 588 95 block2 day2``````
In this long form data frame we have two between-subject variables (`id` and `gender`), two variables that define our within-subject manipulations (`block` and `day`), and two more contain the measurements we took (`MRT` and `PC`).
To convert this back to wide form is equally straightforward. We use the `longToWide()` function, but this time around we need to alter the formula in order to tell it that we have two within-subject factors. The command is now
``longToWide( choice.2, MRT+PC ~ block+day, sep="/" ) ``
``````## id gender MRT/block1/day1 PC/block1/day1 MRT/block1/day2 PC/block1/day2
## 1 1 male 415 79 400 88
## 2 2 male 500 83 490 92
## 3 3 female 478 91 468 98
## 4 4 female 550 75 502 89
## MRT/block2/day1 PC/block2/day1 MRT/block2/day2 PC/block2/day2
## 1 455 82 450 93
## 2 532 86 518 97
## 3 499 90 474 100
## 4 602 78 588 95``````
and this produces a wide form data set containing the same variables as the original `choice` data frame.
What other options are there?
The advantage to the approach described in the previous section is that it solves a quite specific problem (but a commonly encountered one) with a minimum of fuss. The disadvantage is that the tools are quite limited in scope. They allow you to switch your data back and forth between two different formats that are very common in everyday data analysis. However, there a number of other tools that you can use if need be. Just within the core packages distributed with R there is the `reshape()` function, as well as the `stack()` and `unstack()` functions, all of which can be useful under certain circumstances. And there are of course thousands of packages on CRAN that you can use to help you with different tasks. One popular package for this purpose is the `reshape` package, written by Hadley Wickham (??? for details see Wickham2007). There are two key functions in this package, called `melt()` and `cast()` that are pretty useful for solving a lot of reshaping problems. In a future version of this book I intend to discuss `melt()` and `cast()` in a fair amount of detail. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.07%3A_Reshaping_a_Data_Frame.txt |
Sometimes your data set is quite text heavy. This can be for a lot of different reasons. Maybe the raw data are actually taken from text sources (e.g., newspaper articles), or maybe your data set contains a lot of free responses to survey questions, in which people can write whatever text they like in response to some query. Or maybe you just need to rejig some of the text used to describe nominal scale variables. Regardless of what the reason is, you’ll probably want to know a little bit about how to handle text in R. Some things you already know how to do: I’ve discussed the use of `nchar()` to calculate the number of characters in a string (Section 3.8.1), and a lot of the general purpose tools that I’ve discussed elsewhere (e.g., the `==` operator) have been applied to text data as well as to numeric data. However, because text data is quite rich, and generally not as well structured as numeric data, R provides a lot of additional tools that are quite specific to text. In this section I discuss only those tools that come as part of the base packages, but there are other possibilities out there: the `stringr` package provides a powerful alternative that is a lot more coherent than the basic tools, and is well worth looking into.
Shortening a string
The first task I want to talk about is how to shorten a character string. For example, suppose that I have a vector that contains the names of several different animals:
``animals <- c( "cat", "dog", "kangaroo", "whale" )``
It might be useful in some contexts to extract the first three letters of each word. This is often useful when annotating figures, or when creating variable labels: it’s often very inconvenient to use the full name, so you want to shorten it to a short code for space reasons. The `strtrim()` function can be used for this purpose. It has two arguments: `x` is a vector containing the text to be shortened and `width` specifies the number of characters to keep. When applied to the `animals` data, here’s what we get:
``strtrim( x = animals, width = 3 )` `
``## [1] "cat" "dog" "kan" "wha"``
Note that the only thing that `strtrim()` does is chop off excess characters at the end of a string. It doesn’t insert any whitespace characters to fill them out if the original string is shorter than the `width` argument. For example, if I trim the `animals` data to 4 characters, here’s what I get:
``strtrim( x = animals, width = 4 )``
``## [1] "cat" "dog" "kang" "whal"``
The `"cat"` and `"dog"` strings still only use 3 characters. Okay, but what if you don’t want to start from the first letter? Suppose, for instance, I only wanted to keep the second and third letter of each word. That doesn’t happen quite as often, but there are some situations where you need to do something like that. If that does happen, then the function you need is `substr()`, in which you specify a `start` point and a `stop` point instead of specifying the width. For instance, to keep only the 2nd and 3rd letters of the various `animals`, I can do the following:
``substr( x = animals, start = 2, stop = 3 )``
``## [1] "at" "og" "an" "ha"``
Pasting strings together
Much more commonly, you will need either to glue several character strings together or to pull them apart. To glue several strings together, the `paste()` function is very useful. There are three arguments to the `paste()` function:
• `...` As usual, the dots “match” up against any number of inputs. In this case, the inputs should be the various different strings you want to paste together.
• `sep`. This argument should be a string, indicating what characters R should use as separators, in order to keep each of the original strings separate from each other in the pasted output. By default the value is a single space, `sep = " "`. This is made a little clearer when we look at the examples.
• `collapse`. This is an argument indicating whether the `paste()` function should interpret vector inputs as things to be collapsed, or whether a vector of inputs should be converted into a vector of outputs. The default value is `collapse = NULL` which is interpreted as meaning that vectors should not be collapsed. If you want to collapse vectors into as single string, then you should specify a value for `collapse`. Specifically, the value of `collapse` should correspond to the separator character that you want to use for the collapsed inputs. Again, see the examples below for more details.
That probably doesn’t make much sense yet, so let’s start with a simple example. First, let’s try to paste two words together, like this:
``paste( "hello", "world" )``
``## [1] "hello world"``
Notice that R has inserted a space between the `"hello"` and `"world"`. Suppose that’s not what I wanted. Instead, I might want to use `.` as the separator character, or to use no separator at all. To do either of those, I would need to specify `sep = "."` or `sep = ""`.121 For instance:
``paste( "hello", "world", sep = "." )``
``## [1] "hello.world"``
Now let’s consider a slightly more complicated example. Suppose I have two vectors that I want to `paste()` together. Let’s say something like this:
``````hw <- c( "hello", "world" )
ng <- c( "nasty", "government" )``````
And suppose I want to paste these together. However, if you think about it, this statement is kind of ambiguous. It could mean that I want to do an “element wise” paste, in which all of the first elements get pasted together (`"hello nasty"`) and all the second elements get pasted together (`"world government"`). Or, alternatively, I might intend to collapse everything into one big string (`"hello nasty world government"`). By default, the `paste()` function assumes that you want to do an element-wise paste:
``paste( hw, ng )``
``## [1] "hello nasty" "world government"``
However, there’s nothing stopping you from overriding this default. All you have to do is specify a value for the `collapse` argument, and R will chuck everything into one dirty big string. To give you a sense of exactly how this works, what I’ll do in this next example is specify different values for `sep` and `collapse`:
````paste( hw, ng, sep = ".", collapse = ":::")`
```
``## [1] "hello.nasty:::world.government"``
Splitting strings
At other times you have the opposite problem to the one in the last section: you have a whole lot of text bundled together into a single string that needs to be pulled apart and stored as several different variables. For instance, the data set that you get sent might include a single variable containing someone’s full name, and you need to separate it into first names and last names. To do this in R you can use the `strsplit()` function, and for the sake of argument, let’s assume that the string you want to split up is the following string:
``monkey <- "It was the best of times. It was the blurst of times."``
To use the `strsplit()` function to break this apart, there are three arguments that you need to pay particular attention to:
• `x`. A vector of character strings containing the data that you want to split.
• `split`. Depending on the value of the `fixed` argument, this is either a fixed string that specifies a delimiter, or a regular expression that matches against one or more possible delimiters. If you don’t know what regular expressions are (probably most readers of this book), don’t use this option. Just specify a separator string, just like you would for the `paste()` function.
• `fixed`. Set `fixed = TRUE` if you want to use a fixed delimiter. As noted above, unless you understand regular expressions this is definitely what you want. However, the default value is `fixed = FALSE`, so you have to set it explicitly.
Let’s look at a simple example:
``````monkey.1 <- strsplit( x = monkey, split = " ", fixed = TRUE )
monkey.1``` ```
``````## [[1]]
## [1] "It" "was" "the" "best" "of" "times." "It"
## [8] "was" "the" "blurst" "of" "times."``````
One thing to note in passing is that the output here is a list (you can tell from the part of the output), whose first and only element is a character vector. This is useful in a lot of ways, since it means that you can input a character vector for `x` and then then have the `strsplit()` function split all of them, but it’s kind of annoying when you only have a single input. To that end, it’s useful to know that you can `unlist()` the output:
``unlist( monkey.1 )``
``````## [1] "It" "was" "the" "best" "of" "times." "It"
## [8] "was" "the" "blurst" "of" "times."``````
To understand why it’s important to remember to use the `fixed = TRUE` argument, suppose we wanted to split this into two separate sentences. That is, we want to use `split = "."` as our delimiter string. As long as we tell R to remember to treat this as a fixed separator character, then we get the right answer:
``strsplit( x = monkey, split = ".", fixed = TRUE )``
``````## [[1]]
## [1] "It was the best of times" " It was the blurst of times"``````
However, if we don’t do this, then R will assume that when you typed `split = "."` you were trying to construct a “regular expression”, and as it happens the character `.` has a special meaning within a regular expression. As a consequence, if you forget to include the `fixed = TRUE` part, you won’t get the answers you’re looking for.
Making simple conversions
A slightly different task that comes up quite often is making transformations to text. A simple example of this would be converting text to lower case or upper case, which you can do using the `toupper()` and `tolower()` functions. Both of these functions have a single argument `x` which contains the text that needs to be converted. An example of this is shown below:
``````text <- c( "lIfe", "Impact" )
tolower( x = text )``````
``## [1] "life" "impact"``
A slightly more powerful way of doing text transformations is to use the `chartr()` function, which allows you to specify a “character by character” substitution. This function contains three arguments, `old`, `new` and `x`. As usual `x` specifies the text that needs to be transformed. The `old` and `new` arguments are strings of the same length, and they specify how `x` is to be converted. Every instance of the first character in `old` is converted to the first character in `new` and so on. For instance, suppose I wanted to convert `"albino"` to `"libido"`. To do this, I need to convert all of the `"a"` characters (all 1 of them) in `"albino"` into `"l"` characters (i.e., `a``l`). Additionally, I need to make the substitutions `l``i` and `n``d`. To do so, I would use the following command:
``````old.text <- "albino"
chartr( old = "aln", new = "lid", x = old.text )```
```
``## [1] "libido"``
Applying logical operations to text
In Section 3.9.5 we discussed a very basic text processing tool, namely the ability to use the equality operator `==` to test to see if two strings are identical to each other. However, you can also use other logical operators too. For instance R also allows you to use the `<` and `>` operators to determine which of two strings comes first, alphabetically speaking. Sort of. Actually, it’s a bit more complicated than that, but let’s start with a simple example:
``"cat" < "dog"``
``## [1] TRUE``
In this case, we see that `"cat"` does does come before `"dog"` alphabetically, so R judges the statement to be true. However, if we ask R to tell us if `"cat"` comes before `"anteater"`,
``"cat" < "anteater"``
``## [1] FALSE``
It tell us that the statement is false. So far, so good. But text data is a bit more complicated than the dictionary suggests. What about `"cat"` and `"CAT"`? Which of these comes first? Let’s try it and find out:
``"CAT" < "cat"``
``## [1] FALSE``
In other words, R assumes that uppercase letters come before lowercase ones. Fair enough. No-one is likely to be surprised by that. What you might find surprising is that R assumes that all uppercase letters come before all lowercase ones. That is, while `"anteater" < "zebra"` is a true statement, and the uppercase equivalent `"ANTEATER" < "ZEBRA"` is also true, it is not true to say that `"anteater" < "ZEBRA"`, as the following extract illustrates:
``"anteater" < "ZEBRA"``
``## [1] TRUE``
This may seem slightly counterintuitive. With that in mind, it may help to have a quick look Table 7.3, which lists various text characters in the order that R uses.
Table 7.3: The ordering of various text characters used by the < and > operators, as well as by the sort() function. Not shown is the “space” character, which actually comes rst on the list.
Characters
! " # \$ % & ’ ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ ] ^ _ ‘ a b c d e f g h i j k l m n o p q r s t u v w x y z } | {
One function that I want to make a point of talking about, even though it’s not quite on topic, is the `cat()` function. The `cat()` function is a of mixture of `paste()` and `print()`. That is, what it does is concatenate strings and then print them out. In your own work you can probably survive without it, since `print()` and `paste()` will actually do what you need, but the `cat()` function is so widely used that I think it’s a good idea to talk about it here. The basic idea behind `cat()` is straightforward. Like `paste()`, it takes several arguments as inputs, which it converts to strings, collapses (using a separator character specified using the `sep` argument), and prints on screen. If you want, you can use the `file` argument to tell R to print the output into a file rather than on screen (I won’t do that here). However, it’s important to note that the `cat()` function collapses vectors first, and then concatenates them. That is, notice that when I use `cat()` to combine `hw` and `ng`, I get a different result than if I’d used `paste()`
``cat( hw, ng )``
``## hello world nasty government``
````paste( hw, ng, collapse = " " )`
```
``## [1] "hello nasty world government"``
Notice the difference in the ordering of words. There’s a few additional details that I need to mention about `cat()`. Firstly, `cat()` really is a function for printing, and not for creating text strings to store for later. You can’t assign the output to a variable, as the following example illustrates:
``x <- cat( hw, ng ) ``
````## hello world nasty government`
```
``x``
``## NULL``
Despite my attempt to store the output as a variable, `cat()` printed the results on screen anyway, and it turns out that the variable I created doesn’t contain anything at all.122 Secondly, the `cat()` function makes use of a number of “special” characters. I’ll talk more about these in the next section, but I’ll illustrate the basic point now, using the example of `"\n"` which is interpreted as a “new line” character. For instance, compare the behaviour of `print()` and `cat()` when asked to print the string `"hello\nworld"`:
``print( "hello\nworld" ) # print literally:``
``## [1] "hello\nworld"` `
``cat( "hello\nworld" ) # interpret as newline``
``````## hello
## world``````
In fact, this behaviour is important enough that it deserves a section of its very own…
Using escape characters in text
The previous section brings us quite naturally to a fairly fundamental issue when dealing with strings, namely the issue of delimiters and escape characters. Reduced to its most basic form, the problem we have is that R commands are written using text characters, and our strings also consist of text characters. So, suppose I want to type in the word “hello”, and have R encode it as a string. If I were to just type `hello`, R will think that I’m referring to a variable or a function called `hello` rather than interpret it as a string. The solution that R adopts is to require you to enclose your string by delimiter characters, which can be either double quotes or single quotes. So, when I type `"hello"` or `'hello'` then R knows that it should treat the text in between the quote marks as a character string. However, this isn’t a complete solution to the problem: after all, `"` and `'` are themselves perfectly legitimate text characters, and so we might want to include those in our string as well. For instance, suppose I wanted to encode the name “O’Rourke” as a string. It’s not legitimate for me to type `'O'rourke'` because R is too stupid to realise that “O’Rourke” is a real word. So it will interpret the `'O'` part as a complete string, and then will get confused when it reaches the `Rourke'` part. As a consequence, what you get is an error message:
``````'O'Rourke'
Error: unexpected symbol in "'O'Rourke"``````
To some extent, R offers us a cheap fix to the problem because of the fact that it allows us to use either `"` or `'` as the delimiter character. Although `'O'rourke'` will make R cry, it is perfectly happy with `"O'Rourke"`:
``"O'Rourke"``
``## [1] "O'Rourke"``
This is a real advantage to having two different delimiter characters. Unfortunately, anyone with even the slightest bit of deviousness to them can see the problem with this. Suppose I’m reading a book that contains the following passage,
P.J. O’Rourke says, “Yay, money!”. It’s a joke, but no-one laughs.
and I want to enter this as a string. Neither the `'` or `"` delimiters will solve the problem here, since this string contains both a single quote character and a double quote character. To encode strings like this one, we have to do something a little bit clever.
Table 7.4: Standard escape characters that are evaluated by some text processing commands, including `cat()`. This convention dates back to the development of the C programming language in the 1970s, and as a consequence a lot of these characters make most sense if you pretend that R is actually a typewriter, as explained in the main text. Type `?Quotes` for the corresponding R help file.
Escape.sequence Interpretation
`\n` Newline
`\t` Horizontal Tab
`\v` Vertical Tab
`\b` Backspace
`\r` Carriage Return
`\f` Form feed
`\a` Alert sound
`\` Backslash
`\'` Single quote
`\"` Double quote
The solution to the problem is to designate an escape character, which in this case is `\`, the humble backslash. The escape character is a bit of a sacrificial lamb: if you include a backslash character in your string, R will not treat it as a literal character at all. It’s actually used as a way of inserting “special” characters into your string. For instance, if you want to force R to insert actual quote marks into the string, then what you actually type is `\'` or `\"` (these are called escape sequences). So, in order to encode the string discussed earlier, here’s a command I could use:
``PJ <- "P.J. O\'Rourke says, \"Yay, money!\". It\'s a joke, but no-one laughs."``
Notice that I’ve included the backslashes for both the single quotes and double quotes. That’s actually overkill: since I’ve used `"` as my delimiter, I only needed to do this for the double quotes. Nevertheless, the command has worked, since I didn’t get an error message. Now let’s see what happens when I print it out:
````print( PJ )`
```
``## [1] "P.J. O'Rourke says, \"Yay, money!\". It's a joke, but no-one laughs."``
Hm. Why has R printed out the string using `\"`? For the exact same reason that I needed to insert the backslash in the first place. That is, when R prints out the `PJ` string, it has enclosed it with delimiter characters, and it wants to unambiguously show us which of the double quotes are delimiters and which ones are actually part of the string. Fortunately, if this bugs you, you can make it go away by using the `print.noquote()` function, which will just print out the literal string that you encoded in the first place:
``print.noquote( PJ )``
Typing `cat(PJ)` will produce a similar output.
Introducing the escape character solves a lot of problems, since it provides a mechanism by which we can insert all sorts of characters that aren’t on the keyboard. For instance, as far as a computer is concerned, “new line” is actually a text character. It’s the character that is printed whenever you hit the “return” key on your keyboard. If you want to insert a new line character into your string, you can actually do this by including the escape sequence `\n`. Or, if you want to insert a backslash character, then you can use `\`. A list of the standard escape sequences recognised by R is shown in Table 7.4. A lot of these actually date back to the days of the typewriter (e.g., carriage return), so they might seem a bit counterintuitive to people who’ve never used one. In order to get a sense for what the various escape sequences do, we’ll have to use the `cat()` function, because it’s the only function “dumb” enough to literally print them out:
``````cat( "xxxx\boo" ) # \b is a backspace, so it deletes the preceding x
cat( "xxxx\too" ) # \t is a tab, so it inserts a tab space
cat( "xxxx\noo" ) # \n is a newline character
cat( "xxxx\roo" ) # \r returns you to the beginning of the line``````
And that’s pretty much it. There are a few other escape sequence that R recognises, which you can use to insert arbitrary ASCII or Unicode characters into your string (type `?Quotes` for more details) but I won’t go into details here.
Matching and substituting text
Another task that we often want to solve is find all strings that match a certain criterion, and possibly even to make alterations to the text on that basis. There are several functions in R that allow you to do this, three of which I’ll talk about briefly here: `grep()`, `gsub()` and `sub()`. Much like the `substr()` function that I talked about earlier, all three of these functions are intended to be used in conjunction with regular expressions (see Section 7.8.9 but you can also use them in a simpler fashion, since they all allow you to set `fixed = TRUE`, which means we can ignore all this regular expression rubbish and just use simple text matching.
So, how do these functions work? Let’s start with the `grep()` function. The purpose of this function is to input a vector of character strings `x`, and to extract all those strings that fit a certain pattern. In our examples, I’ll assume that the `pattern` in question is a literal sequence of characters that the string must contain (that’s what `fixed = TRUE` does). To illustrate this, let’s start with a simple data set, a vector that contains the names of three `beers`. Something like this:
``beers <- c( "little creatures", "sierra nevada", "coopers pale" )``
Next, let’s use `grep()` to find out which of these strings contains the substring `"er"`. That is, the `pattern` that we need to match is the fixed string `"er"`, so the command we need to use is:
``grep( pattern = "er", x = beers, fixed = TRUE )``
``## [1] 2 3``
What the output here is telling us is that the second and third elements of `beers` both contain the substring `"er"`. Alternatively, however, we might prefer it if `grep()` returned the actual strings themselves. We can do this by specifying `value = TRUE` in our function call. That is, we’d use a command like this:
````grep( pattern = "er", x = beers, fixed = TRUE, value = TRUE )`
```
``## [1] "sierra nevada" "coopers pale"``
The other two functions that I wanted to mention in this section are `gsub()` and `sub()`. These are both similar in spirit to `grep()` insofar as what they do is search through the input strings (`x`) and find all of the strings that match a `pattern`. However, what these two functions do is replace the pattern with a `replacement` string. The `gsub()` function will replace all instances of the pattern, whereas the `sub()` function just replaces the first instance of it in each string. To illustrate how this works, suppose I want to replace all instances of the letter `"a"` with the string `"BLAH"`. I can do this to the `beers` data using the `gsub()` function:
``gsub( pattern = "a", replacement = "BLAH", x = beers, fixed = TRUE )``
``````## [1] "little creBLAHtures" "sierrBLAH nevBLAHdBLAH"
## [3] "coopers pBLAHle"``````
Notice that all three of the `"a"`s in `"sierra nevada"` have been replaced. In contrast, let’s see what happens when we use the exact same command, but this time using the `sub()` function instead:
``sub( pattern = "a", replacement = "BLAH", x = beers, fixed = TRUE )``
``## [1] "little creBLAHtures" "sierrBLAH nevada" "coopers pBLAHle"``
Only the first `"a"` is changed.
Regular expressions (not really)
There’s one last thing I want to talk about regarding text manipulation, and that’s the concept of a regular expression. Throughout this section we’ve often needed to specify `fixed = TRUE` in order to force R to treat some of our strings as actual strings, rather than as regular expressions. So, before moving on, I want to very briefly explain what regular expressions are. I’m not going to talk at all about how they work or how you specify them, because they’re genuinely complicated and not at all relevant to this book. However, they are extremely powerful tools and they’re quite widely used by people who have to work with lots of text data (e.g., people who work with natural language data), and so it’s handy to at least have a vague idea about what they are. The basic idea is quite simple. Suppose I want to extract all strings in my `beers` vector that contain a vowel followed immediately by the letter `"s"`. That is, I want to finds the beer names that contain either `"as"`, `"es"`, `"is"`, `"os"` or `"us"`. One possibility would be to manually specify all of these possibilities and then match against these as fixed strings one at a time, but that’s tedious. The alternative is to try to write out a single “regular” expression that matches all of these. The regular expression that does this123 is `"[aeiou]s"`, and you can kind of see what the syntax is doing here. The bracketed expression means “any of the things in the middle”, so the expression as a whole means “any of the things in the middle” (i.e. vowels) followed by the letter `"s"`. When applied to our beer names we get this:
``grep( pattern = "[aeiou]s", x = beers, value = TRUE )``
``## [1] "little creatures"``
So it turns out that only `"little creatures"` contains a vowel followed by the letter `"s"`. But of course, had the data contained a beer like `"fosters"`, that would have matched as well because it contains the string `"os"`. However, I deliberately chose not to include it because Fosters is not – in my opinion – a proper beer.124 As you can tell from this example, regular expressions are a neat tool for specifying patterns in text: in this case, “vowel then s”. So they are definitely things worth knowing about if you ever find yourself needing to work with a large body of text. However, since they are fairly complex and not necessary for any of the applications discussed in this book, I won’t talk about them any further. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.08%3A_Working_with_Text.txt |
In this section I’m going to switch topics (again!) and turn to the question of how you can load data from a range of different sources. Throughout this book I’ve assumed that your data are stored as an `.Rdata` file or as a “properly” formatted CSV file. And if so, then the basic tools that I discussed in Section 4.5 should be quite sufficient. However, in real life that’s not a terribly plausible assumption to make, so I’d better talk about some of the other possibilities that you might run into.
Loading data from text files
The first thing I should point out is that if your data are saved as a text file but aren’t quite in the proper CSV format, then there’s still a pretty good chance that the `read.csv()` function (or equivalently, `read.table()`) will be able to open it. You just need to specify a few more of the optional arguments to the function. If you type `?read.csv` you’ll see that the `read.csv()` function actually has several arguments that you can specify. Obviously you need to specify the `file` that you want it to load, but the others all have sensible default values. Nevertheless, you will sometimes need to change them. The ones that I’ve often found myself needing to change are:
• `header`. A lot of the time when you’re storing data as a CSV file, the first row actually contains the column names and not data. If that’s not true, you need to set `header = FALSE`.
• `sep`. As the name “comma separated value” indicates, the values in a row of a CSV file are usually separated by commas. This isn’t universal, however. In Europe the decimal point is typically written as `,` instead of `.` and as a consequence it would be somewhat awkward to use `,` as the separator. Therefore it is not unusual to use `;` over there. At other times, I’ve seen a TAB character used. To handle these cases, we’d need to set `sep = ";"` or `sep = "\t"`.
• `quote`. It’s conventional in CSV files to include a quoting character for textual data. As you can see by looking at the `booksales.csv}` file, this is usually a double quote character, `"`. But sometimes there is no quoting character at all, or you might see a single quote mark `'` used instead. In those cases you’d need to specify `quote = ""` or `quote = "'"`.
• `skip`. It’s actually very common to receive CSV files in which the first few rows have nothing to do with the actual data. Instead, they provide a human readable summary of where the data came from, or maybe they include some technical info that doesn’t relate to the data. To tell R to ignore the first (say) three lines, you’d need to set `skip = 3`
• `na.strings`. Often you’ll get given data with missing values. For one reason or another, some entries in the table are missing. The data file needs to include a “special” string to indicate that the entry is missing. By default R assumes that this string is `NA`, since that’s what it would do, but there’s no universal agreement on what to use in this situation. If the file uses `???` instead, then you’ll need to set `na.strings = "???"`.
It’s kind of nice to be able to have all these options that you can tinker with. For instance, have a look at the data file shown pictured in Figure 7.1. This file contains almost the same data as the last file (except it doesn’t have a header), and it uses a bunch of wacky features that you don’t normally see in CSV files. In fact, it just so happens that I’m going to have to change all five of those arguments listed above in order to load this file. Here’s how I would do it:
``````data <- read.csv( file = "./rbook-master/data/booksales2.csv", # specify the name of the file
header = FALSE, # variable names in the file?
skip = 8, # ignore the first 8 lines
quote = "*", # what indicates text data?
sep = "\t", # what separates different entries?
na.strings = "NFI" ) # what is the code for missing data?``````
If I now have a look at the data I’ve loaded, I see that this is what I’ve got:
``head( data )``
``````## V1 V2 V3 V4
## 1 January 31 0 high
## 2 February 28 100 high
## 3 March 31 200 low
## 4 April 30 50 out
## 5 May 31 NA out
## 6 June 30 0 high``````
Because I told R to expect `*` to be used as the quoting character instead of `"`; to look for tabs (which we write like this: `\t`) instead of commas, and to skip the first 8 lines of the file, it’s basically loaded the right data. However, since `booksales2.csv` doesn’t contain the column names, R has made them up. Showing the kind of imagination I expect from insentient software, R decided to call them `V1`, `V2`, `V3` and `V4`. Finally, because I told it that the file uses “NFI” to denote missing data, R correctly figures out that the sales data for May are actually missing.
In real life you’ll rarely see data this stupidly formatted.125
Loading data from SPSS (and other statistics packages)
The commands listed above are the main ones we’ll need for data files in this book. But in real life we have many more possibilities. For example, you might want to read data files in from other statistics programs. Since SPSS is probably the most widely used statistics package in psychology, it’s worth briefly showing how to open SPSS data files (file extension `.sav`). It’s surprisingly easy. The extract below should illustrate how to do so:
``````library( foreign ) # load the package
X <- read.spss( "./rbook-master/data/datafile.sav" ) # create a list containing the data
X <- as.data.frame( X ) # convert to data frame``````
If you wanted to import from an SPSS file to a data frame directly, instead of importing a list and then converting the list to a data frame, you can do that too:
``X <- read.spss( file = "datafile.sav", to.data.frame = TRUE )``
And that’s pretty much it, at least as far as SPSS goes. As far as other statistical software goes, the `foreign` package provides a wealth of possibilities. To open SAS files, check out the `read.ssd()`and `read.xport()` functions. To open data from Minitab, the `read.mtp()` function is what you’re looking for. For Stata, the `read.dta()` function is what you want. For Systat, the `read.systat()` function is what you’re after.
Loading Excel files
A different problem is posed by Excel files. Despite years of yelling at people for sending data to me encoded in a proprietary data format, I get sent a lot of Excel files. In general R does a pretty good job of opening them, but it’s bit finicky because Microsoft don’t seem to be terribly fond of people using non-Microsoft products, and go to some lengths to make it tricky. If you get an Excel file, my suggestion would be to open it up in Excel (or better yet, OpenOffice, since that’s free software) and then save the spreadsheet as a CSV file. Once you’ve got the data in that format, you can open it using `read.csv()`. However, if for some reason you’re desperate to open the `.xls` or `.xlsx` file directly, then you can use the `read.xls()` function in the `gdata` package:
``````library( gdata ) # load the package
X <- read.xls( "datafile.xlsx" ) # create a data frame``````
This usually works. And if it doesn’t, you’re probably justified in “suggesting” to the person that sent you the file that they should send you a nice clean CSV file instead.
Loading Matlab (& Octave) files
A lot of scientific labs use Matlab as their default platform for scientific computing; or Octave as a free alternative. Opening Matlab data files (file extension `.mat`) slightly more complicated, and if it wasn’t for the fact that Matlab is so very widespread and is an extremely good platform, I wouldn’t mention it. However, since Matlab is so widely used, I think it’s worth discussing briefly how to get Matlab and R to play nicely together. The way to do this is to install the `R.matlab` package (don’t forget to install the dependencies too). Once you’ve installed and loaded the package, you have access to the `readMat()` function. As any Matlab user will know, the `.mat` files that Matlab produces are workspace files, very much like the `.Rdata` files that R produces. So you can’t import a `.mat` file as a data frame. However, you can import it as a list. So, when we do this:
``````library( R.matlab ) # load the package
data <- readMat( "matlabfile.mat" ) # read the data file to a list``````
The `data` object that gets created will be a list, containing one variable for every variable stored in the Matlab file. It’s fairly straightforward, though there are some subtleties that I’m ignoring. In particular, note that if you don’t have the `Rcompression` package, you can’t open Matlab files above the version 6 format. So, if like me you’ve got a recent version of Matlab, and don’t have the `Rcompression` package, you’ll need to save your files using the `-v6` flag otherwise R can’t open them.
Oh, and Octave users? The `foreign` package contains a `read.octave()` command. Just this once, the world makes life easier for you folks than it does for all those cashed-up swanky Matlab bastards.
Saving other kinds of data
Given that I talked extensively about how to load data from non-R files, it might be worth briefly mentioning that R is also pretty good at writing data into other file formats besides it’s own native ones. I won’t discuss them in this book, but the `write.csv()` function can write CSV files, and the `write.foreign()` function (in the `foreign` package) can write SPSS, Stata and SAS files. There are also a lot of low level commands that you can use to write very specific information to a file, so if you really, really needed to you could create your own `write.obscurefiletype()` function, but that’s also a long way beyond the scope of this book. For now, all that I want you to recognise is that this capability is there if you need it.
done yet?
Of course not. If I’ve learned nothing else about R it’s that you’re never bloody done. This listing doesn’t even come close to exhausting the possibilities. Databases are supported by the `RODBC`, `DBI`, and `RMySQL` packages among others. You can open webpages using the `RCurl` package. Reading and writing JSON objects is supported through the `rjson` package. And so on. In a sense, the right question is not so much “can R do this?” so much as “whereabouts in the wilds of CRAN is the damn package that does it?” | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.09%3A_Reading_Unusual_Data_Files.txt |
Sometimes you want to change the variable class. This can happen for all sorts of reasons. Sometimes when you import data from files, it can come to you in the wrong format: numbers sometimes get imported as text, dates usually get imported as text, and many other possibilities besides. Regardless of how you’ve ended up in this situation, there’s a very good chance that sometimes you’ll want to convert a variable from one class into another one. Or, to use the correct term, you want to coerce the variable from one class into another. Coercion is a little tricky, and so I’ll only discuss the very basics here, using a few simple examples.
Firstly, let’s suppose we have a variable `x` that is supposed to be representing a number, but the data file that you’ve been given has encoded it as text. Let’s imagine that the variable is something like this:
``````x <- "100" # the variable
class(x) # what class is it?``````
``## [1] "character"``
Obviously, if I want to do calculations using `x` in its current state, R is going to get very annoyed at me. It thinks that `x` is text, so it’s not going to allow me to try to do mathematics using it! Obviously, we need to coerce `x` from character to numeric. We can do that in a straightforward way by using the `as.numeric()` function:
``````x <- as.numeric(x) # coerce the variable
class(x) # what class is it?``````
````## [1] "numeric"`
```
``x + 1 # hey, addition works!``
``## [1] 101``
Not surprisingly, we can also convert it back again if we need to. The function that we use to do this is the `as.character()` function:
``````x <- as.character(x) # coerce back to text
class(x) # check the class:``````
``## [1] "character"``
However, there’s some fairly obvious limitations: you can’t coerce the string `"hello world"` into a number because, well, there’s isn’t a number that corresponds to it. Or, at least, you can’t do anything useful:
``as.numeric( "hello world" ) # this isn't going to work.``
````## Warning: NAs introduced by coercion`
```
``## [1] NA``
In this case R doesn’t give you an error message; it just gives you a warning, and then says that the data is missing (see Section 4.6.1 for the interpretation of `NA`).
That gives you a feel for how to change between numeric and character data. What about logical data? To cover this briefly, coercing text to logical data is pretty intuitive: you use the `as.logical()` function, and the character strings `"T"`, `"TRUE"`, `"True"` and `"true"` all convert to the logical value of `TRUE`. Similarly `"F"`, `"FALSE"`, `"False"`, and `"false"` all become `FALSE`. All other strings convert to `NA`. When you go back the other way using `as.character()`, `TRUE` converts to `"TRUE"` and `FALSE` converts to `"FALSE"`. Converting numbers to logicals – again using `as.logical()` – is straightforward. Following the convention in the study of logic, the number `0` converts to `FALSE`. Everything else is `TRUE`. Going back using `as.numeric()`, `FALSE` converts to `0` and `TRUE` converts to `1`. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.10%3A_Coercing_Data_from_One_Class_to_Another.txt |
Up to this point we have encountered several different kinds of variables. At the simplest level, we’ve seen numeric data, logical data and character data. However, we’ve also encountered some more complicated kinds of variables, namely factors, formulas, data frames and lists. We’ll see a few more specialised data structures later on in this book, but there’s a few more generic ones that I want to talk about in passing. None of them are central to the rest of the book (and in fact, the only one we’ll even see anywhere else is the matrix), but they do crop up a fair bit in real life.
Matrices
In various different places in this chapter I’ve made reference to an R data structure called a matrix, and mentioned that I’d talk a bit more about matrices later on. That time has come. Much like a data frame, a matrix is basically a big rectangular table of data, and in fact there are quite a few similarities between the two. However, there are also some key differences, so it’s important to talk about matrices in a little detail. Let’s start by using `rbind()` to create a small matrix:126
``````row.1 <- c( 2,3,1 ) # create data for row 1
row.2 <- c( 5,6,7 ) # create data for row 2
M <- rbind( row.1, row.2 ) # row bind them into a matrix
print( M ) # and print it out...``````
``````## [,1] [,2] [,3]
## row.1 2 3 1
## row.2 5 6 7``````
The variable `M` is a matrix, which we can confirm by using the `class()` function. Notice that, when we bound the two vectors together, R retained the names of the original variables as row names. We could delete these if we wanted by typing `rownames(M)<-NULL`, but I generally prefer having meaningful names attached to my variables, so I’ll keep them. In fact, let’s also add some highly unimaginative column names as well:
``````colnames(M) <- c( "col.1", "col.2", "col.3" )
print(M)``````
``````## col.1 col.2 col.3
## row.1 2 3 1
## row.2 5 6 7``````
You can use square brackets to subset a matrix in much the same way that you can for data frames, again specifying a row index and then a column index. For instance, `M[2,3]` pulls out the entry in the 2nd row and 3rd column of the matrix (i.e., `7`), whereas `M[2,]` pulls out the entire 2nd row, and `M[,3]` pulls out the entire 3rd column. However, it’s worth noting that when you pull out a column, R will print the results horizontally, not vertically. The reason for this relates to how matrices (and arrays generally) are implemented. The original matrix `M` is treated as a two-dimensional objects, containing 2 rows and 3 columns. However, whenever you pull out a single row or a single column, the result is considered to be one-dimensional. As far as R is concerned there’s no real reason to distinguish between a one-dimensional object printed vertically (a column) and a one-dimensional object printed horizontally (a row), and it prints them all out horizontally.127 There is also a way of using only a single index, but due to the internal structure to how R defines a matrix, it works very differently to what we saw previously with data frames.
The single-index approach is illustrated in Table 7.5 but I don’t really want to focus on it since we’ll never really need it for this book, and matrices don’t play anywhere near as large a role in this book as data frames do. The reason for these differences is that for this is that, for both data frames and matrices, the “row and column” version exists to allow the human user to interact with the object in the psychologically meaningful way: since both data frames and matrices are basically just tables of data, it’s the same in each case. However, the single-index version is really a method for you to interact with the object in terms of its internal structure, and the internals for data frames and matrices are quite different.
Table 7.5: The row and column version, which is identical to the corresponding indexing scheme for a data frame of the same size.
Row Col.1 Col.2 Col.3
Row 1 [1,1] [1,2] [1,3]
Row 2 [2,1] [2,2] [2,3]
Table 7.5: The single-index version, which is quite different to what we would get with a data frame.
Row Col.1 Col.2 Col.3
Row 1 1 3 5
Row 2 2 4 6
The critical difference between a data frame and a matrix is that, in a data frame, we have this notion that each of the columns corresponds to a different variable: as a consequence, the columns in a data frame can be of different data types. The first column could be numeric, and the second column could contain character strings, and the third column could be logical data. In that sense, there is a fundamental asymmetry build into a data frame, because of the fact that columns represent variables (which can be qualitatively different to each other) and rows represent cases (which cannot). Matrices are intended to be thought of in a different way. At a fundamental level, a matrix really is just one variable: it just happens that this one variable is formatted into rows and columns. If you want a matrix of numeric data, every single element in the matrix must be a number. If you want a matrix of character strings, every single element in the matrix must be a character string. If you try to mix data of different types together, then R will either spit out an error, or quietly coerce the underlying data into a list. If you want to find out what class R secretly thinks the data within the matrix is, you need to do something like this:
````class( M[1] )`
```
``## [1] "numeric"``
You can’t type `class(M)`, because all that will happen is R will tell you that `M` is a matrix: we’re not interested in the class of the matrix itself, we want to know what class the underlying data is assumed to be. Anyway, to give you a sense of how R enforces this, let’s try to change one of the elements of our numeric matrix into a character string:
``````M[1,2] <- "text"
M``````
``````## col.1 col.2 col.3
## row.1 "2" "text" "1"
## row.2 "5" "6" "7"``````
It looks as if R has coerced all of the data in our matrix into character strings. And in fact, if we now typed in `class(M[1])` we’d see that this is exactly what has happened. If you alter the contents of one element in a matrix, R will change the underlying data type as necessary.
There’s only one more thing I want to talk about regarding matrices. The concept behind a matrix is very much a mathematical one, and in mathematics a matrix is a most definitely a two-dimensional object. However, when doing data analysis, we often have reasons to want to use higher dimensional tables (e.g., sometimes you need to cross-tabulate three variables against each other). You can’t do this with matrices, but you can do it with arrays. An array is just like a matrix, except it can have more than two dimensions if you need it to. In fact, as far as R is concerned a matrix is just a special kind of array, in much the same way that a data frame is a special kind of list. I don’t want to talk about arrays too much, but I will very briefly show you an example of what a 3D array looks like. To that end, let’s cross tabulate the `speaker` and `utterance` variables from the `nightgarden.Rdata` data file, but we’ll add a third variable to the cross-tabs this time, a logical variable which indicates whether or not I was still awake at this point in the show:
``dan.awake <- c( TRUE,TRUE,TRUE,TRUE,TRUE,FALSE,FALSE,FALSE,FALSE,FALSE )``
Now that we’ve got all three variables in the workspace (assuming you loaded the `nightgarden.Rdata` data earlier in the chapter) we can construct our three way cross-tabulation, using the `table()` function.
``````xtab.3d <- table( speaker, utterance, dan.awake )
xtab.3d``````
``````## , , dan.awake = FALSE
##
## utterance
## speaker ee onk oo pip
## makka-pakka 0 2 0 2
## tombliboo 0 0 1 0
## upsy-daisy 0 0 0 0
##
## , , dan.awake = TRUE
##
## utterance
## speaker ee onk oo pip
## makka-pakka 0 0 0 0
## tombliboo 1 0 0 0
## upsy-daisy 0 2 0 2``````
Hopefully this output is fairly straightforward: because R can’t print out text in three dimensions, what it does is show a sequence of 2D slices through the 3D table. That is, the `, , dan.awake = FALSE` part indicates that the 2D table that follows below shows the 2D cross-tabulation of `speaker` against `utterance` only for the `dan.awake = FALSE` instances, and so on.128
Ordered factors
One topic that I neglected to mention when discussing factors previously (Section 4.7 is that there are actually two different types of factor in R, unordered factors and ordered factors. An unordered factor corresponds to a nominal scale variable, and all of the factors we’ve discussed so far in this book have been unordered (as will all the factors used anywhere else except in this section). However, it’s often very useful to explicitly tell R that your variable is ordinal scale, and if so you need to declare it to be an ordered factor. For instance, earlier in this chapter we made use of a variable consisting of Likert scale data, which we represented as the `likert.raw` variable:
``likert.raw``
``## [1] 1 7 3 4 4 4 2 6 5 5``
We can declare this to be an ordered factor in by using the `factor()` function, and setting `ordered = TRUE`. To illustrate how this works, let’s create an ordered factor called `likert.ordinal` and have a look at it:
``````likert.ordinal <- factor( x = likert.raw, # the raw data
levels = seq(7,1,-1), # strongest agreement is 1, weakest is 7
ordered = TRUE ) # and it's ordered
print( likert.ordinal )``````
``````## [1] 1 7 3 4 4 4 2 6 5 5
## Levels: 7 < 6 < 5 < 4 < 3 < 2 < 1``````
Notice that when we print out the ordered factor, R explicitly tells us what order the levels come in. Because I wanted to order my levels in terms of increasing strength of agreement, and because a response of 1 corresponded to the strongest agreement and 7 to the strongest disagreement, it was important that I tell R to encode 7 as the lowest value and 1 as the largest. Always check this when creating an ordered factor: it’s very easy to accidentally encode your data “upside down” if you’re not paying attention. In any case, note that we can (and should) attach meaningful names to these factor levels by using the `levels()` function, like this:
``````levels( likert.ordinal ) <- c( "strong.disagree", "disagree", "weak.disagree",
"neutral", "weak.agree", "agree", "strong.agree" )
print( likert.ordinal )``````
``````## [1] strong.agree strong.disagree weak.agree neutral
## [5] neutral neutral agree disagree
## [9] weak.disagree weak.disagree
## 7 Levels: strong.disagree < disagree < weak.disagree < ... < strong.agree``````
One nice thing about using ordered factors is that there are a lot of analyses for which R automatically treats ordered factors differently from unordered factors, and generally in a way that is more appropriate for ordinal data. However, since I don’t discuss that in this book, I won’t go into details. Like so many things in this chapter, my main goal here is to make you aware that R has this capability built into it; so if you ever need to start thinking about ordinal scale variables in more detail, you have at least some idea where to start looking!
Dates and times
Times and dates are very annoying types of data. To a first approximation we can say that there are 365 days in a year, 24 hours in a day, 60 minutes in an hour and 60 seconds in a minute, but that’s not quite correct. The length of the solar day is not exactly 24 hours, and the length of solar year is not exactly 365 days, so we have a complicated system of corrections that have to be made to keep the time and date system working. On top of that, the measurement of time is usually taken relative to a local time zone, and most (but not all) time zones have both a standard time and a daylight savings time, though the date at which the switch occurs is not at all standardised. So, as a form of data, times and dates suck. Unfortunately, they’re also important. Sometimes it’s possible to avoid having to use any complicated system for dealing with times and dates. Often you just want to know what year something happened in, so you can just use numeric data: in quite a lot of situations something as simple as `this.year <- 2011` works just fine. If you can get away with that for your application, this is probably the best thing to do. However, sometimes you really do need to know the actual date. Or, even worse, the actual time. In this section, I’ll very briefly introduce you to the basics of how R deals with date and time data. As with a lot of things in this chapter, I won’t go into details because I don’t use this kind of data anywhere else in the book. The goal here is to show you the basics of what you need to do if you ever encounter this kind of data in real life. And then we’ll all agree never to speak of it again.
To start with, let’s talk about the date. As it happens, modern operating systems are very good at keeping track of the time and date, and can even handle all those annoying timezone issues and daylight savings pretty well. So R takes the quite sensible view that it can just ask the operating system what the date is. We can pull the date using the `Sys.Date()` function:
``````today <- Sys.Date() # ask the operating system for the date
print(today) # display the date``````
``## [1] "2018-12-30"``
Okay, that seems straightforward. But, it does rather look like `today` is just a character string, doesn’t it? That would be a problem, because dates really do have a numeric character to them, and it would be nice to be able to do basic addition and subtraction to them. Well, fear not. If you type in `class(today)`, R will tell you that the class of the `today` variable is `"Date"`. What this means is that, hidden underneath this text string that prints out an actual date, R actually has a numeric representation.129 What that means is that you actually can add and subtract days. For instance, if we add 1 to `today`, R will print out the date for tomorrow:
``today + 1``
``## [1] "2018-12-31"``
Let’s see what happens when we add 365 days:
``today + 365` `
``## [1] "2019-12-30"``
This is particularly handy if you forget that a year is a leap year since in that case you’d probably get it wrong is doing this in your head. R provides a number of functions for working with dates, but I don’t want to talk about them in any detail. I will, however, make passing mention of the `weekdays()` function which will tell you what day of the week a particular date corresponded to, which is extremely convenient in some situations:
``weekdays( today )``
``## [1] "Sunday"``
I’ll also point out that you can use the `as.Date()` to convert various different kinds of data into dates. If the data happen to be strings formatted exactly according to the international standard notation (i.e., `yyyy-mm-dd`) then the conversion is straightforward, because that’s the format that R expects to see by default. You can convert dates from other formats too, but it’s slightly trickier, and beyond the scope of this book.
What about times? Well, times are even more annoying, so much so that I don’t intend to talk about them at all in this book, other than to point you in the direction of some vaguely useful things. R itself does provide you with some tools for handling time data, and in fact there are two separate classes of data that are used to represent times, known by the odd names `POSIXct` and `POSIXlt`. You can use these to work with times if you want to, but for most applications you would probably be better off downloading the `chron` package, which provides some much more user friendly tools for working with times and dates. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.11%3A_Other_Useful_Data_Structures.txt |
To finish this chapter, I have a few topics to discuss that don’t really fit in with any of the other things in this chapter. They’re all kind of useful things to know about, but they are really just “odd topics” that don’t fit with the other examples. Here goes:
problems with floating point arithmetic
If I’ve learned nothing else about transfinite arithmetic (and I haven’t) it’s that infinity is a tedious and inconvenient concept. Not only is it annoying and counterintuitive at times, but it has nasty practical consequences. As we were all taught in high school, there are some numbers that cannot be represented as a decimal number of finite length, nor can they be represented as any kind of fraction between two whole numbers; √2, π and e, for instance. In everyday life we mostly don’t care about this. I’m perfectly happy to approximate π as 3.14, quite frankly. Sure, this does produce some rounding errors from time to time, and if I’d used a more detailed approximation like 3.1415926535 I’d be less likely to run into those issues, but in all honesty I’ve never needed my calculations to be that precise. In other words, although our pencil and paper calculations cannot represent the number π exactly as a decimal number, we humans are smart enough to realise that we don’t care. Computers, unfortunately, are dumb … and you don’t have to dig too deep in order to run into some very weird issues that arise because they can’t represent numbers perfectly. Here is my favourite example:
``0.1 + 0.2 == 0.3``
``## [1] FALSE``
Obviously, R has made a mistake here, because this is definitely the wrong answer. Your first thought might be that R is broken, and you might be considering switching to some other language. But you can reproduce the same error in dozens of different programming languages, so the issue isn’t specific to R. Your next thought might be that it’s something in the hardware, but you can get the same mistake on any machine. It’s something deeper than that.
The fundamental issue at hand is floating point arithmetic, which is a fancy way of saying that computers will always round a number to fixed number of significant digits. The exact number of significant digits that the computer stores isn’t important to us:130 what matters is that whenever the number that the computer is trying to store is very long, you get rounding errors. That’s actually what’s happening with our example above. There are teeny tiny rounding errors that have appeared in the computer’s storage of the numbers, and these rounding errors have in turn caused the internal storage of `0.1 + 0.2` to be a tiny bit different from the internal storage of `0.3`. How big are these differences? Let’s ask R:
````0.1 + 0.2 - 0.3`
```
``## [1] 5.551115e-17``
Very tiny indeed. No sane person would care about differences that small. But R is not a sane person, and the equality operator `==` is very literal minded. It returns a value of `TRUE` only when the two values that it is given are absolutely identical to each other. And in this case they are not. However, this only answers half of the question. The other half of the question is, why are we getting these rounding errors when we’re only using nice simple numbers like 0.1, 0.2 and 0.3? This seems a little counterintuitive. The answer is that, like most programming languages, R doesn’t store numbers using their decimal expansion (i.e., base 10: using digits 0, 1, 2 …, 9). We humans like to write our numbers in base 10 because we have 10 fingers. But computers don’t have fingers, they have transistors; and transistors are built to store 2 numbers not 10. So you can see where this is going: the internal storage of a number in R is based on its binary expansion (i.e., base 2: using digits 0 and 1). And unfortunately, here’s what the binary expansion of 0.1 looks like:
.1(decimal)=.00011001100110011...(binary)
and the pattern continues forever. In other words, from the perspective of your computer, which likes to encode numbers in binary,131 0.1 is not a simple number at all. To a computer, 0.1 is actually an infinitely long binary number! As a consequence, the computer can make minor errors when doing calculations here.
With any luck you now understand the problem, which ultimately comes down to the twin fact that (1) we usually think in decimal numbers and computers usually compute with binary numbers, and (2) computers are finite machines and can’t store infinitely long numbers. The only questions that remain are when you should care and what you should do about it. Thankfully, you don’t have to care very often: because the rounding errors are small, the only practical situation that I’ve seen this issue arise for is when you want to test whether an arithmetic fact holds exactly numbers are identical (e.g., is someone’s response time equal to exactly 2×0.33 seconds?) This is pretty rare in real world data analysis, but just in case it does occur, it’s better to use a test that allows for a small tolerance. That is, if the difference between the two numbers is below a certain threshold value, we deem them to be equal for all practical purposes. For instance, you could do something like this, which asks whether the difference between the two numbers is less than a tolerance of 10−10
``abs( 0.1 + 0.2 - 0.3 ) < 10^-10``
``## [1] TRUE``
To deal with this problem, there is a function called `all.equal()` that lets you test for equality but allows a small tolerance for rounding errors:
``all.equal( 0.1 + 0.2, 0.3 )``
``## [1] TRUE``
recycling rule
There’s one thing that I haven’t mentioned about how vector arithmetic works in R, and that’s the recycling rule. The easiest way to explain it is to give a simple example. Suppose I have two vectors of different length, `x` and `y`, and I want to add them together. It’s not obvious what that actually means, so let’s have a look at what R does:
``````x <- c( 1,1,1,1,1,1 ) # x is length 6
y <- c( 0,1 ) # y is length 2
x + y # now add them:``````
``## [1] 1 2 1 2 1 2``
As you can see from looking at this output, what R has done is “recycle” the value of the shorter vector (in this case `y`) several times. That is, the first element of `x` is added to the first element of `y`, and the second element of `x` is added to the second element of `y`. However, when R reaches the third element of `x` there isn’t any corresponding element in `y`, so it returns to the beginning: thus, the third element of `x` is added to the first element of `y`. This process continues until R reaches the last element of `x`. And that’s all there is to it really. The same recycling rule also applies for subtraction, multiplication and division. The only other thing I should note is that, if the length of the longer vector isn’t an exact multiple of the length of the shorter one, R still does it, but also gives you a warning message:
``````x <- c( 1,1,1,1,1 ) # x is length 5
y <- c( 0,1 ) # y is length 2
x + y # now add them:```
```
``````## Warning in x + y: longer object length is not a multiple of shorter object
## length``````
``## [1] 1 2 1 2 1``
introduction to environments
In this section I want to ask a slightly different question: what is the workspace exactly? This question seems simple, but there’s a fair bit to it. This section can be skipped if you’re not really interested in the technical details. In the description I gave earlier, I talked about the workspace as an abstract location in which R variables are stored. That’s basically true, but it hides a couple of key details. For example, any time you have R open, it has to store lots of things in the computer’s memory, not just your variables. For example, the `who()` function that I wrote has to be stored in memory somewhere, right? If it weren’t I wouldn’t be able to use it. That’s pretty obvious. But equally obviously it’s not in the workspace either, otherwise you should have seen it! Here’s what’s happening. R needs to keep track of a lot of different things, so what it does is organise them into environments, each of which can contain lots of different variables and functions. Your workspace is one such environment. Every package that you have loaded is another environment. And every time you call a function, R briefly creates a temporary environment in which the function itself can work, which is then deleted after the calculations are complete. So, when I type in `search()` at the command line
``search()``
``````## [1] ".GlobalEnv" "package:lsr" "package:stats"
## [4] "package:graphics" "package:grDevices" "package:utils"
## [7] "package:datasets" "package:methods" "Autoloads"
## [10] "package:base"``````
what I’m actually looking at is a sequence of environments. The first one, `".GlobalEnv"` is the technically-correct name for your workspace. No-one really calls it that: it’s either called the workspace or the global environment. And so when you type in `objects()` or `who()` what you’re really doing is listing the contents of `".GlobalEnv"`. But there’s no reason why we can’t look up the contents of these other environments using the `objects()` function (currently `who()` doesn’t support this). You just have to be a bit more explicit in your command. If I wanted to find out what is in the `package:stats` environment (i.e., the environment into which the contents of the `stats` package have been loaded), here’s what I’d get
``head(objects("package:stats"))``
``````## [1] "acf" "acf2AR" "add.scope" "add1" "addmargins"
## [6] "aggregate"``````
where this time I’ve used head() to hide a lot of output because the stats package contains about 500 functions. In fact, you can actually use the environment panel in Rstudio to browse any of your loaded packages (just click on the text that says “Global Environment” and you’ll see a dropdown menu like the one shown in Figure 7.2). The key thing to understand then, is that you can access any of the R variables and functions that are stored in one of these environments, precisely because those are the environments that you have loaded!132
Attaching a data frame
The last thing I want to mention in this section is the `attach()` function, which you often see referred to in introductory R books. Whenever it is introduced, the author of the book usually mentions that the `attach()` function can be used to “attach” the data frame to the search path, so you don’t have to use the `\$` operator. That is, if I use the command `attach(df)` to attach my data frame, I no longer need to type `df\$variable`, and instead I can just type `variable`. This is true as far as it goes, but it’s very misleading and novice users often get led astray by this description, because it hides a lot of critical details.
Here is the very abridged description: when you use the `attach()` function, what R does is create an entirely new environment in the search path, just like when you load a package. Then, what it does is copy all of the variables in your data frame into this new environment. When you do this, however, you end up with two completely different versions of all your variables: one in the original data frame, and one in the new environment. Whenever you make a statement like `df\$variable` you’re working with the variable inside the data frame; but when you just type `variable` you’re working with the copy in the new environment. And here’s the part that really upsets new users: changes to one version are not reflected in the other version. As a consequence, it’s really easy for R to end up with different value stored in the two different locations, and you end up really confused as a result.
To be fair to the writers of the `attach()` function, the help documentation does actually state all this quite explicitly, and they even give some examples of how this can cause confusion at the bottom of the help page. And I can actually see how it can be very useful to create copies of your data in a separate location (e.g., it lets you make all kinds of modifications and deletions to the data without having to touch the original data frame). However, I don’t think it’s helpful for new users, since it means you have to be very careful to keep track of which copy you’re talking about. As a consequence of all this, for the purpose of this book I’ve decided not to use the `attach()` function. It’s something that you can investigate yourself once you’re feeling a little more confident with R, but I won’t do it here. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.12%3A_Miscellaneous_Topics.txt |
Obviously, there’s no real coherence to this chapter. It’s just a grab bag of topics and tricks that can be handy to know about, so the best wrap up I can give here is just to repeat this list:
• Section 7.1. Tabulating data.
• Section 7.2. Transforming or recoding a variable.
• Section 7.3. Some useful mathematical functions.
• Section 7.4. Extracting a subset of a vector.
• Section 7.5. Extracting a subset of a data frame.
• Section 7.6. Sorting, flipping or merging data sets.
• Section 7.7. Reshaping a data frame.
• Section 7.8. Manipulating text.
• Section 7.9. Opening data from different file types.
• Section 7.10. Coercing data from one type to another.
• Section 7.11. Other important data types.
• Section 7.12. Miscellaneous topics.
There are a number of books out there that extend this discussion. A couple of my favourites are Spector (2008) “Data Manipulation with R” and Teetor (2011) “R Cookbook”.
References
Spector, P. 2008. Data Manipulation with R. New York, NY: Springer.
Teetor, P. 2011. R Cookbook. Sebastopol, CA: O’Reilly.
1. The quote comes from Home is the Hangman, published in 1975.
2. As usual, you can assign this output to a variable. If you type `speaker.freq <- table(speaker)` at the command prompt R will store the table as a variable. If you then type `class(speaker.freq)` you’ll see that the output is actually of class `table`. The key thing to note about a table object is that it’s basically a matrix (see Section 7.11.1.
3. It’s worth noting that there’s also a more powerful function called `recode()` function in the `car` package that I won’t discuss in this book but is worth looking into if you’re looking for a bit more flexibility.
4. If you’ve read further into the book, and are re-reading this section, then a good example of this would be someone choosing to do an ANOVA using `age.group3` as the grouping variable, instead of running a regression using `age` as a predictor. There are sometimes good reasons for do this: for instance, if the relationship between `age` and your outcome variable is highly non-linear, and you aren’t comfortable with trying to run non-linear regression! However, unless you really do have a good rationale for doing this, it’s best not to. It tends to introduce all sorts of other problems (e.g., the data will probably violate the normality assumption), and you can lose a lot of power.
5. The real answer is 0: \$10 for a sandwich is a total ripoff so I should go next door and buy noodles.
6. Again, I doubt that’s the right “real world” answer. I suspect that most sandwich shops won’t allow you to pay off your debts to them in sandwiches. But you get the idea.
7. Actually, that’s a bit of a lie: the `log()` function is more flexible than that, and can be used to calculate logarithms in any base. The `log()` function has a `base` argument that you can specify, which has a default value of e. Thus `log10(1000)` is actually equivalent to `log(x = 1000, base = 10)`.
8. It’s also worth checking out the `match()` function
9. It also works on data frames if you ever feel the need to import all of your variables from the data frame into the workspace. This can be useful at times, though it’s not a good idea if you have large data sets or if you’re working with multiple data sets at once. In particular, if you do this, never forget that you now have two copies of all your variables, one in the workspace and another in the data frame.
10. You can do this yourself using the `make.names()` function. In fact, this is itself a handy thing to know about. For example, if you want to convert the names of the variables in the `speech.by.char` list into valid R variable names, you could use a command like this: `names(speech.by.char) <- make.names(names(speech.by.char))`. However, I won’t go into details here.
11. Conveniently, if you type `rownames(df) <- NULL` R will renumber all the rows from scratch. For the `df` data frame, the labels that currently run from 7 to 10 will be changed to go from 1 to 4.
12. Actually, you can make the `subset()` function behave this way by using the optional `drop` argument, but by default `subset()` does not drop, which is probably more sensible and more intuitive to novice users.
13. Specifically, recursive indexing, a handy tool in some contexts but not something that I want to discuss here.
14. Remember, `print()` is generic: see Section 4.11.
15. Note for advanced users: both of these functions are just wrappers to the `matrix()` function, which is pretty flexible in terms of the ability to convert vectors into matrices. Also, while I’m on this topic, I’ll briefly mention the fact that if you’re a Matlab user and looking for an equivalent of Matlab’s `repmat()` function, I’d suggest checking out the `matlab` package which contains R versions of a lot of handy Matlab functions.
16. The function you need for that is called `as.data.frame()`.
17. In truth, I suspect that most of the cases when you can sensibly flip a data frame occur when all of the original variables are measurements of the same type (e.g., all variables are response times), and if so you could easily have chosen to encode your data as a matrix instead of as a data frame. But since people do sometimes prefer to work with data frames, I’ve written the `tFrame()` function for the sake of convenience. I don’t really think it’s something that is needed very often.
18. This limitation is deliberate, by the way: if you’re getting to the point where you want to do something more complicated, you should probably start learning how to use `reshape()`, `cast()` and `melt()` or some of other the more advanced tools. The `wideToLong()` and `longToWide()` functions are included only to help you out when you’re first starting to use R.
19. To be honest, it does bother me a little that the default value of `sep` is a space. Normally when I want to paste strings together I don’t want any separator character, so I’d prefer it if the default were `sep=""`. To that end, it’s worth noting that there’s also a `paste0()` function, which is identical to `paste()` except that it always assumes that `sep=""`. Type `?paste` for more information about this.
20. Note that you can capture the output from `cat()` if you want to, but you have to be sneaky and use the `capture.output()` function. For example, the command `x <- capture.output(cat(hw,ng))` would work just fine.
21. Sigh. For advanced users: R actually supports two different ways of specifying regular expressions. One is the POSIX standard, the other is to use Perl-style regular expressions. The default is generally POSIX. If you understand regular expressions, that probably made sense to you. If not, don’t worry. It’s not important.
22. I thank Amy Perfors for this example.
23. If you’re lucky.
24. You can also use the `matrix()` command itself, but I think the “binding” approach is a little more intuitive.
25. This has some interesting implications for how matrix algebra is implemented in R (which I’ll admit I initially found odd), but that’s a little beyond the scope of this book. However, since there will be a small proportion of readers that do care, I’ll quickly outline the basic thing you need to get used to: when multiplying a matrix by a vector (or one-dimensional array) using the `\%*\%` operator R will attempt to interpret the vector (or 1D array) as either a row-vector or column-vector, depending on whichever one makes the multiplication work. That is, suppose M is the 2×3 matrix, and v is a 1×3 row vector. It is impossible to multiply Mv, since the dimensions don’t conform, but you can multiply by the corresponding column vector, Mvt. So, if I set `v <- M[2,]` and then try to calculate `M \%*\% v`, which you’d think would fail, it actually works because R treats the one dimensional array as if it were a column vector for the purposes of matrix multiplication. Note that if both objects are one dimensional arrays/vectors, this leads to ambiguity since vvt (inner product) and vtv (outer product) yield different answers. In this situation, the `\%*\%` operator returns the inner product not the outer product. To understand all the details, check out the help documentation.
26. I should note that if you type `class(xtab.3d)` you’ll discover that this is a `"table"` object rather than an `"array"` object. However, this labelling is only skin deep. The underlying data structure here is actually an array. Advanced users may wish to check this using the command `class(unclass(xtab.3d))`, but it’s not important for our purposes. All I really want to do in this section is show you what the output looks like when you encounter a 3D array.
27. Date objects are coded as the number of days that have passed since January 1, 1970.
28. For advanced users: type `?double` for more information.
29. Or at least, that’s the default. If all your numbers are integers (whole numbers), then you can explicitly tell R to store them as integers by adding an `L` suffix at the end of the number. That is, an assignment like `x <- 2L` tells R to assign `x` a value of 2, and to store it as an integer rather than as a binary expansion. Type `?integer` for more details.
30. For advanced users: that’s a little over simplistic in two respects. First, it’s a terribly imprecise way of talking about scoping. Second, it might give you the impression that all the variables in question are actually loaded into memory. That’s not quite true, since that would be very wasteful of memory. Instead R has a “lazy loading” mechanism, in which what R actually does is create a “promise” to load those objects if they’re actually needed. For details, check out the `delayedAssign()` function. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/07%3A_Pragmatic_Matters/7.13%3A_Summary.txt |
Machine dreams hold a special vertigo.
–William Gibson133
Up to this point in the book I’ve tried hard to avoid using the word “programming” too much because – at least in my experience – it’s a word that can cause a lot of fear. For one reason or another, programming (like mathematics and statistics) is often perceived by people on the “outside” as a black art, a magical skill that can be learned only by some kind of super-nerd. I think this is a shame. It’s certainly true that advanced programming is a very specialised skill: several different skills actually, since there’s quite a lot of different kinds of programming out there. However, the basics of programming aren’t all that hard, and you can accomplish a lot of very impressive things just using those basics.
With that in mind, the goal of this chapter is to discuss a few basic programming concepts and how to apply them in R. However, before I do, I want to make one further attempt to point out just how non-magical programming really is, via one very simple observation: you already know how to do it. Stripped to its essentials, programming is nothing more (and nothing less) than the process of writing out a bunch of instructions that a computer can understand. To phrase this slightly differently, when you write a computer program, you need to write it in a programming language that the computer knows how to interpret. R is one such language. Although I’ve been having you type all your commands at the command prompt, and all the commands in this book so far have been shown as if that’s what I were doing, it’s also quite possible (and as you’ll see shortly, shockingly easy) to write a program using these R commands. In other words, if this is the first time reading this book, then you’re only one short chapter away from being able to legitimately claim that you can program in R, albeit at a beginner’s level.
08: Basic Programming
Computer programs come in quite a few different forms: the kind of program that we’re mostly interested in from the perspective of everyday data analysis using R is known as a script. The idea behind a script is that, instead of typing your commands into the R console one at a time, instead you write them all in a text file. Then, once you’ve finished writing them and saved the text file, you can get R to execute all the commands in your file by using the `source()` function. In a moment I’ll show you exactly how this is done, but first I’d better explain why you should care.
scripts?
Before discussing scripting and programming concepts in any more detail, it’s worth stopping to ask why you should bother. After all, if you look at the R commands that I’ve used everywhere else this book, you’ll notice that they’re all formatted as if I were typing them at the command line. Outside this chapter you won’t actually see any scripts. Do not be fooled by this. The reason that I’ve done it that way is purely for pedagogical reasons. My goal in this book is to teach statistics and to teach R. To that end, what I’ve needed to do is chop everything up into tiny little slices: each section tends to focus on one kind of statistical concept, and only a smallish number of R functions. As much as possible, I want you to see what each function does in isolation, one command at a time. By forcing myself to write everything as if it were being typed at the command line, it imposes a kind of discipline on me: it prevents me from piecing together lots of commands into one big script. From a teaching (and learning) perspective I think that’s the right thing to do… but from a data analysis perspective, it is not. When you start analysing real world data sets, you will rapidly find yourself needing to write scripts.
To understand why scripts are so very useful, it may be helpful to consider the drawbacks to typing commands directly at the command prompt. The approach that we’ve been adopting so far, in which you type commands one at a time, and R sits there patiently in between commands, is referred to as the interactive style. Doing your data analysis this way is rather like having a conversation … a very annoying conversation between you and your data set, in which you and the data aren’t directly speaking to each other, and so you have to rely on R to pass messages back and forth. This approach makes a lot of sense when you’re just trying out a few ideas: maybe you’re trying to figure out what analyses are sensible for your data, or maybe just you’re trying to remember how the various R functions work, so you’re just typing in a few commands until you get the one you want. In other words, the interactive style is very useful as a tool for exploring your data. However, it has a number of drawbacks:
• It’s hard to save your work effectively. You can save the workspace, so that later on you can load any variables you created. You can save your plots as images. And you can even save the history or copy the contents of the R console to a file. Taken together, all these things let you create a reasonably decent record of what you did. But it does leave a lot to be desired. It seems like you ought to be able to save a single file that R could use (in conjunction with your raw data files) and reproduce everything (or at least, everything interesting) that you did during your data analysis.
• It’s annoying to have to go back to the beginning when you make a mistake. Suppose you’ve just spent the last two hours typing in commands. Over the course of this time you’ve created lots of new variables and run lots of analyses. Then suddenly you realise that there was a nasty typo in the first command you typed, so all of your later numbers are wrong. Now you have to fix that first command, and then spend another hour or so combing through the R history to try and recreate what you did.
• You can’t leave notes for yourself. Sure, you can scribble down some notes on a piece of paper, or even save a Word document that summarises what you did. But what you really want to be able to do is write down an English translation of your R commands, preferably right “next to” the commands themselves. That way, you can look back at what you’ve done and actually remember what you were doing. In the simple exercises we’ve engaged in so far, it hasn’t been all that hard to remember what you were doing or why you were doing it, but only because everything we’ve done could be done using only a few commands, and you’ve never been asked to reproduce your analysis six months after you originally did it! When your data analysis starts involving hundreds of variables, and requires quite complicated commands to work, then you really, really need to leave yourself some notes to explain your analysis to, well, yourself.
• It’s nearly impossible to reuse your analyses later, or adapt them to similar problems. Suppose that, sometime in January, you are handed a difficult data analysis problem. After working on it for ages, you figure out some really clever tricks that can be used to solve it. Then, in September, you get handed a really similar problem. You can sort of remember what you did, but not very well. You’d like to have a clean record of what you did last time, how you did it, and why you did it the way you did. Something like that would really help you solve this new problem.
• It’s hard to do anything except the basics. There’s a nasty side effect of these problems. Typos are inevitable. Even the best data analyst in the world makes a lot of mistakes. So the chance that you’ll be able to string together dozens of correct R commands in a row are very small. So unless you have some way around this problem, you’ll never really be able to do anything other than simple analyses.
• It’s difficult to share your work other people. Because you don’t have this nice clean record of what R commands were involved in your analysis, it’s not easy to share your work with other people. Sure, you can send them all the data files you’ve saved, and your history and console logs, and even the little notes you wrote to yourself, but odds are pretty good that no-one else will really understand what’s going on (trust me on this: I’ve been handed lots of random bits of output from people who’ve been analysing their data, and it makes very little sense unless you’ve got the original person who did the work sitting right next to you explaining what you’re looking at)
Ideally, what you’d like to be able to do is something like this… Suppose you start out with a data set `myrawdata.csv`. What you want is a single document – let’s call it `mydataanalysis.R` – that stores all of the commands that you’ve used in order to do your data analysis. Kind of similar to the R history but much more focused. It would only include the commands that you want to keep for later. Then, later on, instead of typing in all those commands again, you’d just tell R to run all of the commands that are stored in `mydataanalysis.R`. Also, in order to help you make sense of all those commands, what you’d want is the ability to add some notes or comments within the file, so that anyone reading the document for themselves would be able to understand what each of the commands actually does. But these comments wouldn’t get in the way: when you try to get R to run `mydataanalysis.R` it would be smart enough would recognise that these comments are for the benefit of humans, and so it would ignore them. Later on you could tweak a few of the commands inside the file (maybe in a new file called `mynewdatanalaysis.R`) so that you can adapt an old analysis to be able to handle a new problem. And you could email your friends and colleagues a copy of this file so that they can reproduce your analysis themselves.
In other words, what you want is a script.
first script
Okay then. Since scripts are so terribly awesome, let’s write one. To do this, open up a simple text editing program, like TextEdit (on a Mac) or Notebook (on Windows). Don’t use a fancy word processing program like Microsoft Word or OpenOffice: use the simplest program you can find. Open a new text document, and type some R commands, hitting enter after each command. Let’s try using `x <- "hello world"` and `print(x)` as our commands. Then save the document as `hello.R`, and remember to save it as a plain text file: don’t save it as a word document or a rich text file. Just a boring old plain text file. Also, when it asks you where to save the file, save it to whatever folder you’re using as your working directory in R. At this point, you should be looking at something like Figure 8.1. And if so, you have now successfully written your first R program. Because I don’t want to take screenshots for every single script, I’m going to present scripts using extracts formatted as follows:
``````## --- hello.R
x <- "hello world"
print(x)``````
The line at the top is the filename, and not part of the script itself. Below that, you can see the two R commands that make up the script itself. Next to each command I’ve included the line numbers. You don’t actually type these into your script, but a lot of text editors (including the one built into Rstudio that I’ll show you in a moment) will show line numbers, since it’s a very useful convention that allows you to say things like “line 1 of the script creates a new variable, and line 2 prints it out”.
So how do we run the script? Assuming that the `hello.R` file has been saved to your working directory, then you can run the script using the following command:
``source( "hello.R" )``
If the script file is saved in a different directory, then you need to specify the path to the file, in exactly the same way that you would have to when loading a data file using `load()`. In any case, when you type this command, R opens up the script file: it then reads each command in the file in the same order that they appear in the file, and executes those commands in that order. The simple script that I’ve shown above contains two commands. The first one creates a variable `x` and the second one prints it on screen. So, when we run the script, this is what we see on screen:
``source("./rbook-master/scripts/hello.R")``
``## [1] "hello world"``
If we inspect the workspace using a command like `who()` or `objects()`, we discover that R has created the new variable `x` within the workspace, and not surprisingly `x` is a character string containing the text `"hello world"`. And just like that, you’ve written your first program R. It really is that simple.
Using Rstudio to write scripts
In the example above I assumed that you were writing your scripts using a simple text editor. However, it’s usually more convenient to use a text editor that is specifically designed to help you write scripts. There’s a lot of these out there, and experienced programmers will all have their own personal favourites. For our purposes, however, we can just use the one built into Rstudio. To create new script file in R studio, go to the “File” menu, select the “New” option, and then click on “R script”. This will open a new window within the “source” panel. Then you can type the commands you want (or code as it is generally called when you’re typing the commands into a script file) and save it when you’re done. The nice thing about using Rstudio to do this is that it automatically changes the colour of the text to indicate which parts of the code are comments and which are parts are actual R commands (these colours are called syntax highlighting, but they’re not actually part of the file – it’s just Rstudio trying to be helpful. To see an example of this, let’s open up our `hello.R` script in Rstudio. To do this, go to the “File” menu again, and select “Open…”. Once you’ve opened the file, you should be looking at something like Figure 8.2. As you can see (if you’re looking at this book in colour) the character string “hello world” is highlighted in green.
Using Rstudio for your text editor is convenient for other reasons too. Notice in the top right hand corner of Figure 8.2 there’s a little button that reads “Source”? If you click on that, Rstudio will construct the relevant `source()` command for you, and send it straight to the R console. So you don’t even have to type in the `source()` command, which actually I think is a great thing, because it really bugs me having to type all those extra keystrokes every time I want to run my script. Anyway, Rstudio provide several other convenient little tools to help make scripting easier, but I won’t discuss them here.134
Commenting your script
When writing up your data analysis as a script, one thing that is generally a good idea is to include a lot of comments in the code. That way, if someone else tries to read it (or if you come back to it several days, weeks, months or years later) they can figure out what’s going on. As a beginner, I think it’s especially useful to comment thoroughly, partly because it gets you into the habit of commenting the code, and partly because the simple act of typing in an explanation of what the code does will help you keep it clear in your own mind what you’re trying to achieve. To illustrate this idea, consider the following script:
``````## --- itngscript.R
# A script to analyse nightgarden.Rdata_
# author: Dan Navarro_
# date: 22/11/2011_
# Load the data, and tell the user that this is what we're
# doing.
cat( "loading data from nightgarden.Rdata...\n" )
load( "./rbook-master/data/nightgarden.Rdata" )
# Create a cross tabulation and print it out:
cat( "tabulating data...\n" )
itng.table <- table( speaker, utterance )
print( itng.table )``````
You’ll notice that I’ve gone a bit overboard with my commenting: at the top of the script I’ve explained the purpose of the script, who wrote it, and when it was written. Then, throughout the script file itself I’ve added a lot of comments explaining what each section of the code actually does. In real life people don’t tend to comment this thoroughly, but the basic idea is a very good one: you really do want your script to explain itself. Nevertheless, as you’d expect R completely ignores all of the commented parts. When we run this script, this is what we see on screen:
``````## --- itngscript.R
# A script to analyse nightgarden.Rdata
# author: Dan Navarro
# date: 22/11/2011
# Load the data, and tell the user that this is what we're
# doing.
cat( "loading data from nightgarden.Rdata...\n" )``````
````## loading data from nightgarden.Rdata...`
```
``````load( "./rbook-master/data/nightgarden.Rdata" )
# Create a cross tabulation and print it out:
cat( "tabulating data...\n" )``````
``## tabulating data...``
``````itng.table <- table( speaker, utterance )
print( itng.table )``````
``````## utterance
## speaker ee onk oo pip
## makka-pakka 0 2 0 2
## tombliboo 1 0 1 0
## upsy-daisy 0 2 0 2``````
Even here, notice that the script announces its behaviour. The first two lines of the output tell us a lot about what the script is actually doing behind the scenes (the code do to this corresponds to the two `cat()` commands on lines 8 and 12 of the script). It’s usually a pretty good idea to do this, since it helps ensure that the output makes sense when the script is executed.
Differences between scripts and the command line
For the most part, commands that you insert into a script behave in exactly the same way as they would if you typed the same thing in at the command line. The one major exception to this is that if you want a variable to be printed on screen, you need to explicitly tell R to print it. You can’t just type the name of the variable. For example, our original `hello.R` script produced visible output. The following script does not:
``````## --- silenthello.R
x <- "hello world"
x``````
It does still create the variable `x` when you `source()` the script, but it won’t print anything on screen.
However, apart from the fact that scripts don’t use “auto-printing” as it’s called, there aren’t a lot of differences in the underlying mechanics. There are a few stylistic differences though. For instance, if you want to load a package at the command line, you would generally use the `library()` function. If you want do to it from a script, it’s conventional to use `require()` instead. The two commands are basically identical, the only difference being that if the package doesn’t exist, `require()` produces a warning whereas `library()` gives you an error. Stylistically, what this means is that if the `require()` command fails in your script, R will boldly continue on and try to execute the rest of the script. Often that’s what you’d like to see happen, so it’s better to use `require()`. Clearly, however, you can get by just fine using the `library()` command for everyday usage.
Done!
At this point, you’ve learned the basics of scripting. You are now officially allowed to say that you can program in R, though you probably shouldn’t say it too loudly. There’s a lot more to learn, but nevertheless, if you can write scripts like these then what you are doing is in fact basic programming. The rest of this chapter is devoted to introducing some of the key commands that you need in order to make your programs more powerful; and to help you get used to thinking in terms of scripts, for the rest of this chapter I’ll write up most of my extracts as scripts. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/08%3A_Basic_Programming/8.01%3A_Scripts.txt |
The description I gave earlier for how a script works was a tiny bit of a lie. Specifically, it’s not necessarily the case that R starts at the top of the file and runs straight through to the end of the file. For all the scripts that we’ve seen so far that’s exactly what happens, and unless you insert some commands to explicitly alter how the script runs, that is what will always happen. However, you actually have quite a lot of flexibility in this respect. Depending on how you write the script, you can have R repeat several commands, or skip over different commands, and so on. This topic is referred to as flow control, and the first concept to discuss in this respect is the idea of a loop. The basic idea is very simple: a loop is a block of code (i.e., a sequence of commands) that R will execute over and over again until some termination criterion is met. Looping is a very powerful idea. There are three different ways to construct a loop in R, based on the `while`, `for` and `repeat` functions. I’ll only discuss the first two in this book.
`while` loop
A `while` loop is a simple thing. The basic format of the loop looks like this:
`````` while ( CONDITION ) {
STATEMENT1
STATEMENT2
ETC
}``````
The code corresponding to CONDITION needs to produce a logical value, either `TRUE` or `FALSE`. Whenever R encounters a `while` statement, it checks to see if the CONDITION is `TRUE`. If it is, then R goes on to execute all of the commands inside the curly brackets, proceeding from top to bottom as usual. However, when it gets to the bottom of those statements, it moves back up to the `while` statement. Then, like the mindless automaton it is, it checks to see if the CONDITION is `TRUE`. If it is, then R goes on to execute all … well, you get the idea. This continues endlessly until at some point the CONDITION turns out to be `FALSE`. Once that happens, R jumps to the bottom of the loop (i.e., to the `}` character), and then continues on with whatever commands appear next in the script.
To start with, let’s keep things simple, and use a `while` loop to calculate the smallest multiple of 17 that is greater than or equal to 1000. This is a very silly example since you can actually calculate it using simple arithmetic operations, but the point here isn’t to do something novel. The point is to show how to write a `while` loop. Here’s the script:
``````## --- whileexample.R
x <- 0
while ( x < 1000 ) {
x <- x + 17
}
print( x )``````
When we run this script, R starts at the top and creates a new variable called `x` and assigns it a value of 0. It then moves down to the loop, and “notices” that the condition here is `x < 1000`. Since the current value of `x` is zero, the condition is true, so it enters the body of the loop (inside the curly braces). There’s only one command here135 which instructs R to increase the value of `x` by 17. R then returns to the top of the loop, and rechecks the condition. The value of `x` is now 17, but that’s still less than 1000, so the loop continues. This cycle will continue for a total of 59 iterations, until finally `x` reaches a value of 1003 (i.e., 59×17=1003). At this point, the loop stops, and R finally reaches line 5 of the script, prints out the value of `x` on screen, and then halts. Let’s watch:
``source( "./rbook-master/scripts/whileexample.R" )``
``## [1] 1003``
Truly fascinating stuff.
`for` loop
The `for` loop is also pretty simple, though not quite as simple as the `while` loop. The basic format of this loop goes like this:
`````` for ( VAR in VECTOR ) {
STATEMENT1
STATEMENT2
ETC
}``````
In a `for` loop, R runs a fixed number of iterations. We have a VECTOR which has several elements, each one corresponding to a possible value of the variable VAR. In the first iteration of the loop, VAR is given a value corresponding to the first element of VECTOR; in the second iteration of the loop VAR gets a value corresponding to the second value in VECTOR; and so on. Once we’ve exhausted all of the values in VECTOR, the loop terminates and the flow of the program continues down the script.
Once again, let’s use some very simple examples. Firstly, here is a program that just prints out the word “hello” three times and then stops:
``````## --- forexample.R
for ( i in 1:3 ) {
print( "hello" )
}``````
This is the simplest example of a `for` loop. The vector of possible values for the `i` variable just corresponds to the numbers from 1 to 3. Not only that, the body of the loop doesn’t actually depend on `i` at all. Not surprisingly, here’s what happens when we run it:
``source( "./rbook-master/scripts/forexample.R" )``
``````## [1] "hello"
## [1] "hello"
## [1] "hello"``````
However, there’s nothing that stops you from using something non-numeric as the vector of possible values, as the following example illustrates. This time around, we’ll use a character vector to control our loop, which in this case will be a vector of `words`. And what we’ll do in the loop is get R to convert the word to upper case letters, calculate the length of the word, and print it out. Here’s the script:
``````## --- forexample2.R
#the words_
words <- c("it","was","the","dirty","end","of","winter")
#loop over the words_
for ( w in words ) {
w.length <- nchar( w ) # calculate the number of letters_
W <- toupper( w ) # convert the word to upper case letters_
msg <- paste( W, "has", w.length, "letters" ) # a message to print_
print( msg ) # print it_
}``````
And here’s the output:
````source( "./rbook-master/scripts/forexample2.R" )`
```
``````## [1] "IT has 2 letters"
## [1] "WAS has 3 letters"
## [1] "THE has 3 letters"
## [1] "DIRTY has 5 letters"
## [1] "END has 3 letters"
## [1] "OF has 2 letters"
## [1] "WINTER has 6 letters"``````
Again, pretty straightforward I hope.
more realistic example of a loop
To give you a sense of how you can use a loop in a more complex situation, let’s write a simple script to simulate the progression of a mortgage. Suppose we have a nice young couple who borrow \$300000 from the bank, at an annual interest rate of 5%. The mortgage is a 30 year loan, so they need to pay it off within 360 months total. Our happy couple decide to set their monthly mortgage payment at \$1600 per month. Will they pay off the loan in time or not? Only time will tell.136 Or, alternatively, we could simulate the whole process and get R to tell us. The script to run this is a fair bit more complicated.
``````## --- mortgage.R
# set up
month <- 0 # count the number of months
balance <- 300000 # initial mortgage balance
payments <- 1600 # monthly payments
interest <- 0.05 # 5% interest rate per year
total.paid <- 0 # track what you've paid the bank
# convert annual interest to a monthly multiplier
monthly.multiplier <- (1+interest) ^ (1/12)
# keep looping until the loan is paid off...
while ( balance > 0 ) {
# do the calculations for this month
month <- month + 1 # one more month
balance <- balance * monthly.multiplier # add the interest
balance <- balance - payments # make the payments
total.paid <- total.paid + payments # track the total paid
# print the results on screen
cat( "month", month, ": balance", round(balance), "\n")
} # end of loop
# print the total payments at the end
cat("total payments made", total.paid, "\n" )``````
To explain what’s going on, let’s go through it carefully. In the first block of code (under `#set up`) all we’re doing is specifying all the variables that define the problem. The loan starts with a `balance` of \$300,000 owed to the bank on `month` zero, and at that point in time the `total.paid` money is nothing. The couple is making monthly `payments` of \$1600, at an annual `interest` rate of 5%. Next, we convert the annual percentage interest into a monthly multiplier. That is, the number that you have to multiply the current balance by each month in order to produce an annual interest rate of 5%. An annual interest rate of 5% implies that, if no payments were made over 12 months the balance would end up being 1.05 times what it was originally, so the annual multiplier is 1.05. To calculate the monthly multiplier, we need to calculate the 12th root of 1.05 (i.e., raise 1.05 to the power of 1/12). We store this value in as the `monthly.multiplier` variable, which as it happens corresponds to a value of about 1.004. All of which is a rather long winded way of saying that the annual interest rate of 5% corresponds to a monthly interest rate of about 0.4%.
Anyway… all of that is really just setting the stage. It’s not the interesting part of the script. The interesting part (such as it is) is the loop. The `while` statement on tells R that it needs to keep looping until the `balance` reaches zero (or less, since it might be that the final payment of \$1600 pushes the balance below zero). Then, inside the body of the loop, we have two different blocks of code. In the first bit, we do all the number crunching. Firstly we increase the value `month` by 1. Next, the bank charges the interest, so the `balance` goes up. Then, the couple makes their monthly payment and the `balance` goes down. Finally, we keep track of the total amount of money that the couple has paid so far, by adding the `payments` to the running tally. After having done all this number crunching, we tell R to issue the couple with a very terse monthly statement, which just indicates how many months they’ve been paying the loan and how much money they still owe the bank. Which is rather rude of us really. I’ve grown attached to this couple and I really feel they deserve better than that. But, that’s banks for you.
In any case, the key thing here is the tension between the increase in `balance` on and the decrease. As long as the decrease is bigger, then the balance will eventually drop to zero and the loop will eventually terminate. If not, the loop will continue forever! This is actually very bad programming on my part: I really should have included something to force R to stop if this goes on too long. However, I haven’t shown you how to evaluate “if” statements yet, so we’ll just have to hope that the author of the book has rigged the example so that the code actually runs. Hm. I wonder what the odds of that are? Anyway, assuming that the loop does eventually terminate, there’s one last line of code that prints out the total amount of money that the couple handed over to the bank over the lifetime of the loan.
Now that I’ve explained everything in the script in tedious detail, let’s run it and see what happens:
``source( "./rbook-master/scripts/mortgage.R" )``
``````## month 1 : balance 299622
## month 2 : balance 299243
## month 3 : balance 298862
## month 4 : balance 298480
## month 5 : balance 298096
## month 6 : balance 297710
## month 7 : balance 297323
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## total payments made 569600``````
So our nice young couple have paid off their \$300,000 loan in just 4 months shy of the 30 year term of their loan, at a bargain basement price of \$568,046 (since 569600 - 1554 = 568046). A happy ending! | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/08%3A_Basic_Programming/8.02%3A_Loops.txt |
A second kind of flow control that programming languages provide is the ability to evaluate conditional statements. Unlike loops, which can repeat over and over again, a conditional statement only executes once, but it can switch between different possible commands depending on a CONDITION that is specified by the programmer. The power of these commands is that they allow the program itself to make choices, and in particular, to make different choices depending on the context in which the program is run. The most prominent of example of a conditional statement is the `if` statement, and the accompanying `else` statement. The basic format of an `if` statement in R is as follows:
``````if ( CONDITION ) {
STATEMENT1
STATEMENT2
ETC
}``````
And the execution of the statement is pretty straightforward. If the CONDITION is true, then R will execute the statements contained in the curly braces. If the CONDITION is false, then it dose not. If you want to, you can extend the `if` statement to include an `else` statement as well, leading to the following syntax:
`````` if ( CONDITION ) {
STATEMENT1
STATEMENT2
ETC
} else {
STATEMENT3
STATEMENT4
ETC
} ``````
As you’d expect, the interpretation of this version is similar. If the CONDITION is true, then the contents of the first block of code (i.e., STATEMENT1, STATEMENT2, ETC) are executed; but if it is false, then the contents of the second block of code (i.e., STATEMENT3, STATEMENT4, ETC) are executed instead.
To give you a feel for how you can use `if` and `else` to do something useful, the example that I’ll show you is a script that prints out a different message depending on what day of the week you run it. We can do this making use of some of the tools that we discussed in Section 7.11.3. Here’s the script:
``````## --- ifelseexample.R
# find out what day it is...
today <- Sys.Date() # pull the date from the system clock
day <- weekdays( today ) # what day of the week it is_
# now make a choice depending on the day...
if ( day == "Monday" ) {
print( "I don't like Mondays" )
} else {
print( "I'm a happy little automaton" )
}``````
``## [1] "I'm a happy little automaton"``
Since today happens to be a Sunday, when I run the script here’s what happens:
``source( "./rbook-master/scripts/ifelseexample.R" )``
``## [1] "I'm a happy little automaton"``
There are other ways of making conditional statements in R. In particular, the `ifelse()` function and the `switch()` functions can be very useful in different contexts. However, my main aim in this chapter is to briefly cover the very basics, so I’ll move on.
8.04: Writing Functions
In this section I want to talk about functions again. Functions were introduced in Section 3.5, but you’ve learned a lot about R since then, so we can talk about them in more detail. In particular, I want to show you how to create your own. To stick with the same basic framework that I used to describe loops and conditionals, here’s the syntax that you use to create a function:
`````` FNAME <- function ( ARG1, ARG2, ETC ) {
STATEMENT1
STATEMENT2
ETC
return( VALUE )
}``````
What this does is create a function with the name FNAME, which has arguments ARG1, ARG2 and so forth. Whenever the function is called, R executes the statements in the curly braces, and then outputs the contents of VALUE to the user. Note, however, that R does not execute the commands inside the function in the workspace. Instead, what it does is create a temporary local environment: all the internal statements in the body of the function are executed there, so they remain invisible to the user. Only the final results in the VALUE are returned to the workspace.
To give a simple example of this, let’s create a function called `quadruple()` which multiplies its inputs by four. In keeping with the approach taken in the rest of the chapter, I’ll use a script to do this:
``````## --- functionexample.R
quadruple <- function(x) {
y <- x*4
return(y)
} ``````
When we run this script, as follows
``source( "./rbook-master/scripts/functionexample.R" )``
nothing appears to have happened, but there is a new object created in the workspace called `quadruple`. Not surprisingly, if we ask R to tell us what kind of object it is, it tells us that it is a function:
``class( quadruple )``
``## [1] "function"``
And now that we’ve created the `quadruple()` function, we can call it just like any other function And if I want to store the output as a variable, I can do this:
``````my.var <- quadruple(10)
print(my.var)``````
``## [1] 40``
An important thing to recognise here is that the two internal variables that the `quadruple()` function makes use of, `x` and `y`, stay internal. That is, if we inspect the contents of the workspace,
````library(lsr)`
```
``## Warning: package 'lsr' was built under R version 3.5.2``
````who()`
```
``````## -- Name -- -- Class -- -- Size --
## balance numeric 1
## day character 1
## i integer 1
## interest numeric 1
## itng.table table 3 x 4
## month numeric 1
## monthly.multiplier numeric 1
## msg character 1
## my.var numeric 1
## payments numeric 1
## quadruple function
## speaker character 10
## today Date 1
## total.paid numeric 1
## utterance character 10
## w character 1
## W character 1
## w.length integer 1
## words character 7
## x numeric 1``````
we see everything in our workspace from this chapter including the `quadruple()` function itself, as well as the `my.var` variable that we just created.
Now that we know how to create our own functions in R, it’s probably a good idea to talk a little more about some of the other properties of functions that I’ve been glossing over. To start with, let’s take this opportunity to type the name of the function at the command line without the parentheses:
````quadruple`
```
``````## function (x)
## {
## y <- x * 4
## return(y)
## }``````
As you can see, when you type the name of a function at the command line, R prints out the underlying source code that we used to define the function in the first place. In the case of the `quadruple()` function, this is quite helpful to us – we can read this code and actually see what the function does. For other functions, this is less helpful, as we saw back in Section 3.5 when we tried typing `citation` rather than `citation()`.
Function arguments revisited
Okay, now that we are starting to get a sense for how functions are constructed, let’s have a look at two, slightly more complicated functions that I’ve created. The source code for these functions is contained within the `functionexample2.R` and `functionexample3.R` scripts. Let’s start by looking at the first one:
``````## --- functionexample2.R
pow <- function( x, y = 1) {
out <- x^y # raise x to the power y
return( out )
}``````
and if we type `source("functionexample2.R")` to load the `pow()` function into our workspace, then we can make use of it. As you can see from looking at the code for this function, it has two arguments `x` and `y`, and all it does is raise `x` to the power of `y`. For instance, this command
``pow(x=3, y=2)` `
``## [1] 9``
calculates the value of 32. The interesting thing about this function isn’t what it does, since R already has has perfectly good mechanisms for calculating powers. Rather, notice that when I defined the function, I specified `y=1` when listing the arguments? That’s the default value for `y`. So if we enter a command without specifying a value for `y`, then the function assumes that we want `y=1`:
``pow( x=3 )``
``## [1] 3``
However, since I didn’t specify any default value for `x` when I defined the `pow()` function, we always need to input a value for `x`. If we don’t R will spit out an error message.
So now you know how to specify default values for an argument. The other thing I should point out while I’m on this topic is the use of the `...` argument. The `...` argument is a special construct in R which is only used within functions. It is used as a way of matching against multiple user inputs: in other words, `...` is used as a mechanism to allow the user to enter as many inputs as they like. I won’t talk at all about the low-level details of how this works at all, but I will show you a simple example of a function that makes use of it. To that end, consider the following script:
``````## --- functionexample3.R
doubleMax <- function( ... ) {
max.val <- max( ... ) # find the largest value in ...
out <- 2 * max.val # double it
return( out )
}``````
When we type `source("functionexample3.R")`, R creates the `doubleMax()` function. You can type in as many inputs as you like. The `doubleMax()` function identifies the largest value in the inputs, by passing all the user inputs to the `max()` function, and then doubles it. For example:
````doubleMax( 1,2,5 )`
```
``## [1] 10``
There’s more to functions than this
There’s a lot of other details to functions that I’ve hidden in my description in this chapter. Experienced programmers will wonder exactly how the “scoping rules” work in R,137 or want to know how to use a function to create variables in other environments138, or if function objects can be assigned as elements of a list139 and probably hundreds of other things besides. However, I don’t want to have this discussion get too cluttered with details, so I think it’s best – at least for the purposes of the current book – to stop here. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/08%3A_Basic_Programming/8.03%3A_Conditional_Statements.txt |
There’s one last topic I want to discuss in this chapter. In addition to providing the explicit looping structures via `while` and `for`, R also provides a collection of functions for implicit loops. What I mean by this is that these are functions that carry out operations very similar to those that you’d normally use a loop for. However, instead of typing out the whole loop, the whole thing is done with a single command. The main reason why this can be handy is that – due to the way that R is written – these implicit looping functions are usually about to do the same calculations much faster than the corresponding explicit loops. In most applications that beginners might want to undertake, this probably isn’t very important, since most beginners tend to start out working with fairly small data sets and don’t usually need to undertake extremely time consuming number crunching. However, because you often see these functions referred to in other contexts, it may be useful to very briefly discuss a few of them.
The first and simplest of these functions is `sapply()`. The two most important arguments to this function are `X`, which specifies a vector containing the data, and `FUN`, which specifies the name of a function that should be applied to each element of the data vector. The following example illustrates the basics of how it works:
``````words <- c("along", "the", "loom", "of", "the", "land")
sapply( X = words, FUN = nchar )``````
``````## along the loom of the land
## 5 3 4 2 3 4``````
Notice how similar this is to the second example of a `for` loop in Section 8.2.2. The `sapply()` function has implicitly looped over the elements of `words`, and for each such element applied the `nchar()` function to calculate the number of letters in the corresponding word.
The second of these functions is `tapply()`, which has three key arguments. As before `X` specifies the data, and `FUN` specifies a function. However, there is also an `INDEX` argument which specifies a grouping variable.140 What the `tapply()` function does is loop over all of the different values that appear in the `INDEX` variable. Each such value defines a group: the `tapply()` function constructs the subset of `X` that corresponds to that group, and then applies the function `FUN` to that subset of the data. This probably sounds a little abstract, so let’s consider a specific example, using the `nightgarden.Rdata` file that we used in Chapter 7.
``````gender <- c( "male","male","female","female","male" )
age <- c( 10,12,9,11,13 )
tapply( X = age, INDEX = gender, FUN = mean )``````
``````## female male
## 10.00000 11.66667``````
In this extract, what we’re doing is using `gender` to define two different groups of people, and using their `ages` as the data. We then calculate the `mean()` of the ages, separately for the males and the females. A closely related function is `by()`. It actually does the same thing as `tapply()`, but the output is formatted a bit differently. This time around the three arguments are called `data`, `INDICES` and `FUN`, but they’re pretty much the same thing. An example of how to use the `by()` function is shown in the following extract:
``by( data = age, INDICES = gender, FUN = mean )``
``````## gender: female
## [1] 10
## --------------------------------------------------------
## gender: male
## [1] 11.66667``````
The `tapply()` and `by()` functions are quite handy things to know about, and are pretty widely used. However, although I do make passing reference to the `tapply()` later on, I don’t make much use of them in this book.
Before moving on, I should mention that there are several other functions that work along similar lines, and have suspiciously similar names: `lapply`, `mapply`, `apply`, `vapply`, `rapply` and `eapply`. However, none of these come up anywhere else in this book, so all I wanted to do here is draw your attention to the fact that they exist. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/08%3A_Basic_Programming/8.05%3A_Implicit_Loops.txt |
In this chapter I talked about several key programming concepts, things that you should know about if you want to start converting your simple scripts into full fledged programs:
• Writing and using scripts (Section 8.1).
• Using loops (Section 8.2) and implicit loops (Section 8.5).
• Making conditional statements (Section 8.3)
• Writing your own functions (Section 8.4)
As always, there are lots of things I’m ignoring in this chapter. It takes a lot of work to become a proper programmer, just as it takes a lot of work to be a proper psychologist or a proper statistician, and this book is certainly not going to provide you with all the tools you need to make that step. However, you’d be amazed at how much you can achieve using only the tools that I’ve covered up to this point. Loops, conditionals and functions are very powerful things, especially when combined with the various tools discussed in Chapters 3, 4 and 7. Believe it or not, you’re off to a pretty good start just by having made it to this point. If you want to keep going, there are (as always!) several other books you might want to look at. One that I’ve read and enjoyed is “A first course in statistical programming with R” Braun and Murdoch (2007), but quite a few people have suggested to me that “The art of programming with R” Matloff and Matloff (2011) is worth the effort too.
References
Braun, John, and Duncan J Murdoch. 2007. A First Course in Statistical Programming with R. Cambridge University Press Cambridge.
Matloff, Norman, and Norman S Matloff. 2011. The Art of R Programming: A Tour of Statistical Software Design. No Starch Press.
1. The quote comes from Count Zero (1986)
2. Okay, I lied. Sue me. One of the coolest features of Rstudio is the support for R Markdown, which lets you embed R code inside a Markdown document, and you can automatically publish your R Markdown to the web on Rstudio’s servers. If you’re the kind of nerd interested in this sort of thing, it’s really nice. And, yes, since I’m also that kind of nerd, of course I’m aware that iPython notebooks do the same thing and that R just nicked their idea. So what? It’s still cool. And anyway, this book isn’t called Learning Statistics with Python now, is it? Hm. Maybe I should write a Python version…
3. As an aside: if there’s only a single command that you want to include inside your loop, then you don’t actually need to bother including the curly braces at all. However, until you’re comfortable programming in R I’d advise always using them, even when you don’t have to.
4. Okay, fine. This example is still a bit ridiculous, in three respects. Firstly, the bank absolutely will not let the couple pay less than the amount required to terminate the loan in 30 years. Secondly, a constant interest rate of 30 years is hilarious. Thirdly, you can solve this much more efficiently than through brute force simulation. However, we’re not exactly in the business of being realistic or efficient here.
5. Lexical scope.
6. The `assign()` function.
7. Yes.
8. Or a list of such variables. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/08%3A_Basic_Programming/8.06%3A_Summary.txt |
[God] has afforded us only the twilight … of Probability.
– John Locke
Up to this point in the book, we’ve discussed some of the key ideas in experimental design, and we’ve talked a little about how you can summarise a data set. To a lot of people, this is all there is to statistics: it’s about calculating averages, collecting all the numbers, drawing pictures, and putting them all in a report somewhere. Kind of like stamp collecting, but with numbers. However, statistics covers much more than that. In fact, descriptive statistics is one of the smallest parts of statistics, and one of the least powerful. The bigger and more useful part of statistics is that it provides that let you make inferences about data.
Once you start thinking about statistics in these terms – that statistics is there to help us draw inferences from data – you start seeing examples of it everywhere. For instance, here’s a tiny extract from a newspaper article in the Sydney Morning Herald (30 Oct 2010):
“I have a tough job,” the Premier said in response to a poll which found her government is now the most unpopular Labor administration in polling history, with a primary vote of just 23 per cent.
This kind of remark is entirely unremarkable in the papers or in everyday life, but let’s have a think about what it entails. A polling company has conducted a survey, usually a pretty big one because they can afford it. I’m too lazy to track down the original survey, so let’s just imagine that they called 1000 NSW voters at random, and 230 (23%) of those claimed that they intended to vote for the ALP. For the 2010 Federal election, the Australian Electoral Commission reported 4,610,795 enrolled voters in NSW; so the opinions of the remaining 4,609,795 voters (about 99.98% of voters) remain unknown to us. Even assuming that no-one lied to the polling company the only thing we can say with 100% confidence is that the true ALP primary vote is somewhere between 230/4610795 (about 0.005%) and 4610025/4610795 (about 99.83%). So, on what basis is it legitimate for the polling company, the newspaper, and the readership to conclude that the ALP primary vote is only about 23%?
The answer to the question is pretty obvious: if I call 1000 people at random, and 230 of them say they intend to vote for the ALP, then it seems very unlikely that these are the only 230 people out of the entire voting public who actually intend to do so. In other words, we assume that the data collected by the polling company is pretty representative of the population at large. But how representative? Would we be surprised to discover that the true ALP primary vote is actually 24%? 29%? 37%? At this point everyday intuition starts to break down a bit. No-one would be surprised by 24%, and everybody would be surprised by 37%, but it’s a bit hard to say whether 29% is plausible. We need some more powerful tools than just looking at the numbers and guessing.
Inferential statistics provides the tools that we need to answer these sorts of questions, and since these kinds of questions lie at the heart of the scientific enterprise, they take up the lions share of every introductory course on statistics and research methods. However, the theory of statistical inference is built on top of probability theory. And it is to probability theory that we must now turn. This discussion of probability theory is basically background: there’s not a lot of statistics per se in this chapter, and you don’t need to understand this material in as much depth as the other chapters in this part of the book. Nevertheless, because probability theory does underpin so much of statistics, it’s worth covering some of the basics.
09: Introduction to Probability
Before we start talking about probability theory, it’s helpful to spend a moment thinking about the relationship between probability and statistics. The two disciplines are closely related but they’re not identical. Probability theory is “the doctrine of chances”. It’s a branch of mathematics that tells you how often different kinds of events will happen. For example, all of these questions are things you can answer using probability theory:
• What are the chances of a fair coin coming up heads 10 times in a row?
• If I roll two six sided dice, how likely is it that I’ll roll two sixes?
• How likely is it that five cards drawn from a perfectly shuffled deck will all be hearts?
• What are the chances that I’ll win the lottery?
Notice that all of these questions have something in common. In each case the “truth of the world” is known, and my question relates to the “what kind of events” will happen. In the first question I know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. In the second question, I know that the chance of rolling a 6 on a single die is 1 in 6. In the third question I know that the deck is shuffled properly. And in the fourth question, I know that the lottery follows specific rules. You get the idea. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. The underlying model can be quite simple. For instance, in the coin flipping example, we can write down the model like this:
P(heads)=0.5
which you can read as “the probability of heads is 0.5”. As we’ll see later, in the same way that percentages are numbers that range from 0% to 100%, probabilities are just numbers that range from 0 to 1. When using this probability model to answer the first question, I don’t actually know exactly what’s going to happen. Maybe I’ll get 10 heads, like the question says. But maybe I’ll get three heads. That’s the key thing: in probability theory, the model is known, but the data are not.
So that’s probability. What about statistics? Statistical questions work the other way around. In statistics, we do not know the truth about the world. All we have is the data, and it is from the data that we want to learn the truth about the world. Statistical questions tend to look more like these:
• If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me?
• If five cards off the top of the deck are all hearts, how likely is it that the deck was shuffled? - If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?
This time around, the only thing we have are data. What I know is that I saw my friend flip the coin 10 times and it came up heads every time. And what I want to infer is whether or not I should conclude that what I just saw was actually a fair coin being flipped 10 times in a row, or whether I should suspect that my friend is playing a trick on me. The data I have look like this:
``H H H H H H H H H H H``
and what I’m trying to do is work out which “model of the world” I should put my trust in. If the coin is fair, then the model I should adopt is one that says that the probability of heads is 0.5; that is, P(heads)=0.5. If the coin is not fair, then I should conclude that the probability of heads is not 0.5, which we would write as P(heads)≠0.5. In other words, the statistical inference problem is to figure out which of these probability models is right. Clearly, the statistical question isn’t the same as the probability question, but they’re deeply connected to one another. Because of this, a good introduction to statistical theory will start with a discussion of what probability is and how it works. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/09%3A_Introduction_to_Probability/9.01%3A_How_are_Probability_and_Statistics_Different.txt |
Let’s start with the first of these questions. What is “probability”? It might seem surprising to you, but while statisticians and mathematicians (mostly) agree on what the rules of probability are, there’s much less of a consensus on what the word really means. It seems weird because we’re all very comfortable using words like “chance”, “likely”, “possible” and “probable”, and it doesn’t seem like it should be a very difficult question to answer. If you had to explain “probability” to a five year old, you could do a pretty good job. But if you’ve ever had that experience in real life, you might walk away from the conversation feeling like you didn’t quite get it right, and that (like many everyday concepts) it turns out that you don’t really know what it’s all about.
So I’ll have a go at it. Let’s suppose I want to bet on a soccer game between two teams of robots, Arduino Arsenal and C Milan. After thinking about it, I decide that there is an 80% probability that Arduino Arsenal winning. What do I mean by that? Here are three possibilities…
• They’re robot teams, so I can make them play over and over again, and if I did that, Arduino Arsenal would win 8 out of every 10 games on average.
• For any given game, I would only agree that betting on this game is only “fair” if a \$1 bet on C Milan gives a \$5 payoff (i.e. I get my \$1 back plus a \$4 reward for being correct), as would a \$4 bet on Arduino Arsenal (i.e., my \$4 bet plus a \$1 reward).
• My subjective “belief” or “confidence” in an Arduino Arsenal victory is four times as strong as my belief in a C Milan victory.
Each of these seems sensible. However they’re not identical, and not every statistician would endorse all of them. The reason is that there are different statistical ideologies (yes, really!) and depending on which one you subscribe to, you might say that some of those statements are meaningless or irrelevant. In this section, I give a brief introduction the two main approaches that exist in the literature. These are by no means the only approaches, but they’re the two big ones.
frequentist view
The first of the two major approaches to probability, and the more dominant one in statistics, is referred to as the frequentist view, and it defines probability as a long-run frequency. Suppose we were to try flipping a fair coin, over and over again. By definition, this is a coin that has P(H)=0.5. What might we observe? One possibility is that the first 20 flips might look like this:
``T,H,H,H,H,T,T,H,H,H,H,T,H,H,T,T,T,T,T,H``
In this case 11 of these 20 coin flips (55%) came up heads. Now suppose that I’d been keeping a running tally of the number of heads (which I’ll call NH) that I’ve seen, across the first N flips, and calculate the proportion of heads NH/N every time. Here’s what I’d get (I did literally flip coins to produce this!):
number.of.flips number.of.heads proportion
1 0 0.00
2 1 0.50
3 2 0.67
4 3 0.75
5 4 0.80
6 4 0.67
7 4 0.57
8 5 0.63
9 6 0.67
10 7 0.70
11 8 0.73
12 8 0.67
13 9 0.69
14 10 0.71
15 10 0.67
16 10 0.63
17 10 0.59
18 10 0.56
19 10 0.53
20 11 0.55
Notice that at the start of the sequence, the proportion of heads fluctuates wildly, starting at .00 and rising as high as .80. Later on, one gets the impression that it dampens out a bit, with more and more of the values actually being pretty close to the “right” answer of .50. This is the frequentist definition of probability in a nutshell: flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N→∞), the proportion of heads will converge to 50%. There are some subtle technicalities that the mathematicians care about, but qualitatively speaking, that’s how the frequentists define probability. Unfortunately, I don’t have an infinite number of coins, or the infinite patience required to flip a coin an infinite number of times. However, I do have a computer, and computers excel at mindless repetitive tasks. So I asked my computer to simulate flipping a coin 1000 times, and then drew a picture of what happens to the proportion NH/N as N increases. Actually, I did it four times, just to make sure it wasn’t a fluke. The results are shown in Figure 9.1. As you can see, the proportion of observed heads eventually stops fluctuating, and settles down; when it does, the number at which it finally settles is the true probability of heads.
The frequentist definition of probability has some desirable characteristics. Firstly, it is objective: the probability of an event is necessarily grounded in the world. The only way that probability statements can make sense is if they refer to (a sequence of) events that occur in the physical universe.142 Secondly, it is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer. However, it also has undesirable characteristics. Firstly, infinite sequences don’t exist in the physical world. Suppose you picked up a coin from your pocket and started to flip it. Every time it lands, it impacts on the ground. Each impact wears the coin down a bit; eventually, the coin will be destroyed. So, one might ask whether it really makes sense to pretend that an “infinite” sequence of coin flips is even a meaningful concept, or an objective one. We can’t say that an “infinite sequence” of events is a real thing in the physical universe, because the physical universe doesn’t allow infinite anything. More seriously, the frequentist definition has a narrow scope. There are lots of things out there that human beings are happy to assign probability to in everyday language, but cannot (even in theory) be mapped onto a hypothetical sequence of events. For instance, if a meteorologist comes on TV and says, “the probability of rain in Adelaide on 2 November 2048 is 60%” we humans are happy to accept this. But it’s not clear how to define this in frequentist terms. There’s only one city of Adelaide, and only 2 November 2048. There’s no infinite sequence of events here, just a once-off thing. Frequentist probability genuinely forbids us from making probability statements about a single event. From the frequentist perspective, it will either rain tomorrow or it will not; there is no “probability” that attaches to a single non-repeatable event. Now, it should be said that there are some very clever tricks that frequentists can use to get around this. One possibility is that what the meteorologist means is something like this: “There is a category of days for which I predict a 60% chance of rain; if we look only across those days for which I make this prediction, then on 60% of those days it will actually rain”. It’s very weird and counterintuitive to think of it this way, but you do see frequentists do this sometimes. And it will come up later in this book (see Section 10.5).
Bayesian view
The Bayesian view of probability is often called the subjectivist view, and it is a minority view among statisticians, but one that has been steadily gaining traction for the last several decades. There are many flavours of Bayesianism, making hard to say exactly what “the” Bayesian view is. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. From that perspective, probabilities don’t exist in the world, but rather in the thoughts and assumptions of people and other intelligent beings. However, in order for this approach to work, we need some way of operationalising “degree of belief”. One way that you can do this is to formalise it in terms of “rational gambling”, though there are many other ways. Suppose that I believe that there’s a 60% probability of rain tomorrow. If someone offers me a bet: if it rains tomorrow, then I win \$5, but if it doesn’t rain then I lose \$5. Clearly, from my perspective, this is a pretty good bet. On the other hand, if I think that the probability of rain is only 40%, then it’s a bad bet to take. Thus, we can operationalise the notion of a “subjective probability” in terms of what bets I’m willing to accept.
What are the advantages and disadvantages to the Bayesian approach? The main advantage is that it allows you to assign probabilities to any event you want to. You don’t need to be limited to those events that are repeatable. The main disadvantage (to many people) is that we can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an intelligent agent out there that believes in things. To many people this is uncomfortable: it seems to make probability arbitrary. While the Bayesian approach does require that the agent in question be rational (i.e., obey the rules of probability), it does allow everyone to have their own beliefs; I can believe the coin is fair and you don’t have to, even though we’re both rational. The frequentist view doesn’t allow any two observers to attribute different probabilities to the same event: when that happens, then at least one of them must be wrong. The Bayesian view does not prevent this from occurring. Two observers with different background knowledge can legitimately hold different beliefs about the same event. In short, where the frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to), the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).
What’s the difference? And who is right?
Now that you’ve seen each of these two views independently, it’s useful to make sure you can compare the two. Go back to the hypothetical robot soccer game at the start of the section. What do you think a frequentist and a Bayesian would say about these three statements? Which statement would a frequentist say is the correct definition of probability? Which one would a Bayesian do? Would some of these statements be meaningless to a frequentist or a Bayesian? If you’ve understood the two perspectives, you should have some sense of how to answer those questions.
Okay, assuming you understand the different, you might be wondering which of them is right? Honestly, I don’t know that there is a right answer. As far as I can tell there’s nothing mathematically incorrect about the way frequentists think about sequences of events, and there’s nothing mathematically incorrect about the way that Bayesians define the beliefs of a rational agent. In fact, when you dig down into the details, Bayesians and frequentists actually agree about a lot of things. Many frequentist methods lead to decisions that Bayesians agree a rational agent would make. Many Bayesian methods have very good frequentist properties.
For the most part, I’m a pragmatist so I’ll use any statistical method that I trust. As it turns out, that makes me prefer Bayesian methods, for reasons I’ll explain towards the end of the book, but I’m not fundamentally opposed to frequentist methods. Not everyone is quite so relaxed. For instance, consider Sir Ronald Fisher, one of the towering figures of 20th century statistics and a vehement opponent to all things Bayesian, whose paper on the mathematical foundations of statistics referred to Bayesian probability as “an impenetrable jungle [that] arrests progress towards precision of statistical concepts” Fisher (1922b). Or the psychologist Paul Meehl, who suggests that relying on frequentist methods could turn you into “a potent but sterile intellectual rake who leaves in his merry path a long train of ravished maidens but no viable scientific offspring” Meehl (1967). The history of statistics, as you might gather, is not devoid of entertainment.
In any case, while I personally prefer the Bayesian view, the majority of statistical analyses are based on the frequentist approach. My reasoning is pragmatic: the goal of this book is to cover roughly the same territory as a typical undergraduate stats class in psychology, and if you want to understand the statistical tools used by most psychologists, you’ll need a good grasp of frequentist methods. I promise you that this isn’t wasted effort. Even if you end up wanting to switch to the Bayesian perspective, you really should read through at least one book on the “orthodox” frequentist view. And since R is the most widely used statistical language for Bayesians, you might as well read a book that uses R. Besides, I won’t completely ignore the Bayesian perspective. Every now and then I’ll add some commentary from a Bayesian point of view, and I’ll revisit the topic in more depth in Chapter 17. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/09%3A_Introduction_to_Probability/9.02%3A_What_Does_Probability_Mean.txt |
Ideological arguments between Bayesians and frequentists notwithstanding, it turns out that people mostly agree on the rules that probabilities should obey. There are lots of different ways of arriving at these rules. The most commonly used approach is based on the work of Andrey Kolmogorov, one of the great Soviet mathematicians of the 20th century. I won’t go into a lot of detail, but I’ll try to give you a bit of a sense of how it works. And in order to do so, I’m going to have to talk about my pants.
Introducing probability distributions
One of the disturbing truths about my life is that I only own 5 pairs of pants: three pairs of jeans, the bottom half of a suit, and a pair of tracksuit pants. Even sadder, I’ve given them names: I call them X1, X2, X3, X4 and X5. I really do: that’s why they call me Mister Imaginative. Now, on any given day, I pick out exactly one of pair of pants to wear. Not even I’m so stupid as to try to wear two pairs of pants, and thanks to years of training I never go outside without wearing pants anymore. If I were to describe this situation using the language of probability theory, I would refer to each pair of pants (i.e., each X) as an elementary event. The key characteristic of elementary events is that every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events. Like I said, these days I always wear exactly one pair of pants, so my pants satisfy this constraint. Similarly, the set of all possible events is called a sample space. Granted, some people would call it a “wardrobe”, but that’s because they’re refusing to think about my pants in probabilistic terms. Sad.
Okay, now that we have a sample space (a wardrobe), which is built from lots of possible elementary events (pants), what we want to do is assign a probability of one of these elementary events. For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur. So, for example, if P(X)=0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)=1 it means that event X is certain to occur (i.e., I always wear those pants). For probability values in the middle, it means that I sometimes wear those pants. For instance, if P(X)=0.5 it means that I wear those pants half of the time.
At this point, we’re almost done. The last thing we need to recognise is that “something always happens”. Every time I put on pants, I really do end up wearing pants (crazy, right?). What this somewhat trite statement means, in probabilistic terms, is that the probabilities of the elementary events need to add up to 1. This is known as the law of total probability, not that any of us really care. More importantly, if these requirements are satisfied, then what we have is a probability distribution. For example, this is an example of a probability distribution
Which.pants Blue.jeans Grey.jeans Black.jeans Black.suit Blue.tracksuit
Label X1 X2 X3 X4 X5
Probability P(X1)=.5 P(X2)=.3 P(X3)=.1 P(X4)=0 P(X5)=.1
Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1. Awesome. We can even draw a nice bar graph (see Section 6.7) to visualise this distribution, as shown in Figure ??. And at this point, we’ve all achieved something. You’ve learned what a probability distribution is, and I’ve finally managed to find a way to create a graph that focuses entirely on my pants. Everyone wins!
The only other thing that I need to point out is that probability theory allows you to talk about non elementary events as well as elementary ones. The easiest way to illustrate the concept is with an example. In the pants example, it’s perfectly legitimate to refer to the probability that I wear jeans. In this scenario, the “Dan wears jeans” event said to have happened as long as the elementary event that actually did occur is one of the appropriate ones; in this case “blue jeans”, “black jeans” or “grey jeans”. In mathematical terms, we defined the “jeans” event E to correspond to the set of elementary events (X1,X2,X3). If any of these elementary events occurs, then E is also said to have occurred. Having decided to write down the definition of the E this way, it’s pretty straightforward to state what the probability P(E) is: we just add everything up. In this particular case
P(E)=P(X1)+P(X2)+P(X3)
and, since the probabilities of blue, grey and black jeans respectively are .5, .3 and .1, the probability that I wear jeans is equal to .9.
At this point you might be thinking that this is all terribly obvious and simple and you’d be right. All we’ve really done is wrap some basic mathematics around a few common sense intuitions. However, from these simple beginnings it’s possible to construct some extremely powerful mathematical tools. I’m definitely not going to go into the details in this book, but what I will do is list – in Table 9.1 – some of the other rules that probabilities satisfy. These rules can be derived from the simple assumptions that I’ve outlined above, but since we don’t actually use these rules for anything in this book, I won’t do so here.
Table 9.1: Some basic rules that probabilities must satisfy. You don’t really need to know these rules in order to understand the analyses that we’ll talk about later in the book, but they are important if you want to understand probability theory a bit more deeply.
English Notation NANA Formula
Not A P(¬A) = 1−P(A)
A or B P(A∪B) = P(A)+P(B)−P(A∩B)
A and B P(A∩B) = P(A|B)P(B) | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/09%3A_Introduction_to_Probability/9.03%3A_Basic_Probability_Theory.txt |
As you might imagine, probability distributions vary enormously, and there’s an enormous range of distributions out there. However, they aren’t all equally important. In fact, the vast majority of the content in this book relies on one of five distributions: the binomial distribution, the normal distribution, the t distribution, the χ2 (“chi-square”) distribution and the F distribution. Given this, what I’ll do over the next few sections is provide a brief introduction to all five of these, paying special attention to the binomial and the normal. I’ll start with the binomial distribution, since it’s the simplest of the five.
Introducing the binomial
The theory of probability originated in the attempt to describe how games of chance work, so it seems fitting that our discussion of the binomial distribution should involve a discussion of rolling dice and flipping coins. Let’s imagine a simple “experiment”: in my hot little hand I’m holding 20 identical six-sided dice. On one face of each die there’s a picture of a skull; the other five faces are all blank. If I proceed to roll all 20 dice, what’s the probability that I’ll get exactly 4 skulls? Assuming that the dice are fair, we know that the chance of any one die coming up skulls is 1 in 6; to say this another way, the skull probability for a single die is approximately .167. This is enough information to answer our question, so let’s have a look at how it’s done.
As usual, we’ll want to introduce some names and some notation. We’ll let N denote the number of dice rolls in our experiment; which is often referred to as the size parameter of our binomial distribution. We’ll also use θ to refer to the the probability that a single die comes up skulls, a quantity that is usually called the success probability of the binomial.143 Finally, we’ll use X to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a random variable. In any case, now that we have all this terminology and notation, we can use it to state the problem a little more precisely. The quantity that we want to calculate is the probability that X=4 given that we know that θ=.167 and N=20. The general “form” of the thing I’m interested in calculating could be written as
P(X | θ,N)
and we’re interested in the special case where X=4, θ=.167 and N=20. There’s only one more piece of notation I want to refer to before moving on to discuss the solution to the problem. If I want to say that X is generated randomly from a binomial distribution with parameters θ and N, the notation I would use is as follows:
X∼Binomial(θ,N)
Yeah, yeah. I know what you’re thinking: notation, notation, notation. Really, who cares? Very few readers of this book are here for the notation, so I should probably move on and talk about how to use the binomial distribution. I’ve included the formula for the binomial distribution in Table 9.2, since some readers may want to play with it themselves, but since most people probably don’t care that much and because we don’t need the formula in this book, I won’t talk about it in any detail. Instead, I just want to show you what the binomial distribution looks like. To that end, Figure 9.3 plots the binomial probabilities for all possible values of X for our dice rolling experiment, from X=0 (no skulls) all the way up to X=20 (all skulls). Note that this is basically a bar chart, and is no different to the “pants probability” plot I drew in Figure 9.2. On the horizontal axis we have all the possible events, and on the vertical axis we can read off the probability of each of those events. So, the probability of rolling 4 skulls out of 20 times is about 0.20 (the actual answer is 0.2022036, as we’ll see in a moment). In other words, you’d expect that to happen about 20% of the times you repeated this experiment.
Working with the binomial distribution in R
Although some people find it handy to know the formulas in Table 9.2, most people just want to know how to use the distributions without worrying too much about the maths. To that end, R has a function called dbinom() that calculates binomial probabilities for us. The main arguments to the function are
• x. This is a number, or vector of numbers, specifying the outcomes whose probability you’re trying to calculate.
• size. This is a number telling R the size of the experiment.
• prob. This is the success probability for any one trial in the experiment.
So, in order to calculate the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6 … well, the command I would use is simply this:
dbinom( x = 4, size = 20, prob = 1/6 )
## [1] 0.2022036
To give you a feel for how the binomial distribution changes when we alter the values of θ and N, let’s suppose that instead of rolling dice, I’m actually flipping coins. This time around, my experiment involves flipping a fair coin repeatedly, and the outcome that I’m interested in is the number of heads that I observe. In this scenario, the success probability is now θ=1/2. Suppose I were to flip the coin N=20 times. In this example, I’ve changed the success probability, but kept the size of the experiment the same. What does this do to our binomial distribution? Well, as Figure 9.4 shows, the main effect of this is to shift the whole distribution, as you’d expect. Okay, what if we flipped a coin N=100 times? Well, in that case, we get Figure 9.5. The distribution stays roughly in the middle, but there’s a bit more variability in the possible outcomes.
knitr::kable(data.frame(stringsAsFactors=FALSE, Binomial = c("$P(X | \theta, N) = \displaystyle\dfrac{N!}{X! (N-X)!} \theta^X (1-\theta)^{N-X}$"),
Normal = c("$p(X | \mu, \sigma) = \displaystyle\dfrac{1}{\sqrt{2\pi}\sigma} \exp \left( -\dfrac{(X - \mu)^2} {2\sigma^2} \right)$ ")), caption = "Formulas for the binomial and normal distributions. We don't really use these
formulas for anything in this book, but they're pretty important for more advanced work, so I thought it might be best
to put them here in a table, where they can't get in the way of the text. In the equation for the binomial, $X!$ is the
factorial function (i.e., multiply all whole numbers from 1 to $X$), and for the normal distribution \"exp\" refers to
the exponential function, which we discussed in the Chapter on Data Handling. If these equations don't make a lot of
sense to you, don't worry too much about them.")
Table 9.2: Formulas for the binomial and normal distributions. We don’t really use these formulas for anything in this book, but they’re pretty important for more advanced work, so I thought it might be best to put them here in a table, where they can’t get in the way of the text. In the equation for the binomial, X! is the factorial function (i.e., multiply all whole numbers from 1 to X), and for the normal distribution “exp” refers to the exponential function, which we discussed in the Chapter on Data Handling. If these equations don’t make a lot of sense to you, don’t worry too much about them.
Binomial Normal
$P(X | \theta, N)=\dfrac{N !}{X !(N-X) !} \theta^{X}(1-\theta)^{N-X} \nonumber$
$p(X | \mu, \sigma)=\dfrac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\dfrac{(X-\mu)^{2}}{2 \sigma^{2}}\right) \nonumber$
Table 9.3: The naming system for R probability distribution functions. Every probability distribution implemented in R is actually associated with four separate functions, and there is a pretty standardised way for naming these functions.
What.it.does Prefix Normal.distribution Binomial.distribution
probability (density) of d dnorm() dbinom()
cumulative probability of p dnorm() pbinom()
generate random number from r rnorm() rbinom()
q qnorm() qbinom() q qnorm() qbinom(
At this point, I should probably explain the name of the dbinom() function. Obviously, the “binom” part comes from the fact that we’re working with the binomial distribution, but the “d” prefix is probably a bit of a mystery. In this section I’ll give a partial explanation: specifically, I’ll explain why there is a prefix. As for why it’s a “d” specifically, you’ll have to wait until the next section. What’s going on here is that R actually provides four functions in relation to the binomial distribution. These four functions are dbinom(), pbinom(), rbinom() and qbinom(), and each one calculates a different quantity of interest. Not only that, R does the same thing for every probability distribution that it implements. No matter what distribution you’re talking about, there’s a d function, a p function, a q function and a r function. This is illustrated in Table 9.3, using the binomial distribution and the normal distribution as examples.
Let’s have a look at what all four functions do. Firstly, all four versions of the function require you to specify the size and prob arguments: no matter what you’re trying to get R to calculate, it needs to know what the parameters are. However, they differ in terms of what the other argument is, and what the output is. So let’s look at them one at a time.
• The d form we’ve already seen: you specify a particular outcome x, and the output is the probability of obtaining exactly that outcome. (the “d” is short for density, but ignore that for now).
• The p form calculates the cumulative probability. You specify a particular quantile q, and it tells you the probability of obtaining an outcome smaller than or equal to q.
• The q form calculates the quantiles of the distribution. You specify a probability value p, and gives you the corresponding percentile. That is, the value of the variable for which there’s a probability p of obtaining an outcome lower than that value.
• The r form is a random number generator: specifically, it generates n random outcomes from the distribution.
This is a little abstract, so let’s look at some concrete examples. Again, we’ve already covered dbinom() so let’s focus on the other three versions. We’ll start with pbinom(), and we’ll go back to the skull-dice example. Again, I’m rolling 20 dice, and each die has a 1 in 6 chance of coming up skulls. Suppose, however, that I want to know the probability of rolling 4 or fewer skulls. If I wanted to, I could use the dbinom() function to calculate the exact probability of rolling 0 skulls, 1 skull, 2 skulls, 3 skulls and 4 skulls and then add these up, but there’s a faster way. Instead, I can calculate this using the pbinom() function. Here’s the command:
pbinom( q= 4, size = 20, prob = 1/6)
## [1] 0.7687492
In other words, there is a 76.9% chance that I will roll 4 or fewer skulls. Or, to put it another way, R is telling us that a value of 4 is actually the 76.9th percentile of this binomial distribution.
Next, let’s consider the qbinom() function. Let’s say I want to calculate the 75th percentile of the binomial distribution. If we’re sticking with our skulls example, I would use the following command to do this:
qbinom( p = 0.75, size = 20, prob = 1/6)
## [1] 4
Hm. There’s something odd going on here. Let’s think this through. What the qbinom() function appears to be telling us is that the 75th percentile of the binomial distribution is 4, even though we saw from the pbinom() function that 4 is actually the 76.9th percentile. And it’s definitely the pbinom() function that is correct. I promise. The weirdness here comes from the fact that our binomial distribution doesn’t really have a 75th percentile. Not really. Why not? Well, there’s a 56.7% chance of rolling 3 or fewer skulls (you can type pbinom(3, 20, 1/6) to confirm this if you want), and a 76.9% chance of rolling 4 or fewer skulls. So there’s a sense in which the 75th percentile should lie “in between” 3 and 4 skulls. But that makes no sense at all! You can’t roll 20 dice and get 3.9 of them come up skulls. This issue can be handled in different ways: you could report an in between value (or interpolated value, to use the technical name) like 3.9, you could round down (to 3) or you could round up (to 4). The qbinom() function rounds upwards: if you ask for a percentile that doesn’t actually exist (like the 75th in this example), R finds the smallest value for which the the percentile rank is at least what you asked for. In this case, since the “true” 75th percentile (whatever that would mean) lies somewhere between 3 and 4 skulls, R rounds up and gives you an answer of 4. This subtlety is tedious, I admit, but thankfully it’s only an issue for discrete distributions like the binomial (see Section 2.2.5 for a discussion of continuous versus discrete). The other distributions that I’ll talk about (normal, t, χ2 and F) are all continuous, and so R can always return an exact quantile whenever you ask for it.
Finally, we have the random number generator. To use the rbinom() function, you specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution. So, for instance, suppose I were to repeat my die rolling experiment 100 times. I could get R to simulate the results of these experiments by using the following command:
rbinom( n = 100, size = 20, prob = 1/6 )
## [1] 3 3 3 2 3 3 3 2 3 3 6 2 5 3 1 1 4 7 5 3 3 6 3 4 3 4 5 3 3 3 7 4 5 1 2
## [36] 1 2 4 2 5 5 4 4 3 1 3 0 2 3 2 2 2 2 1 3 4 5 0 3 2 5 1 2 3 1 5 2 4 3 2
## [71] 1 2 1 5 2 3 3 2 3 3 4 2 1 2 6 2 3 2 3 3 6 2 1 1 3 3 1 5 4 3
As you can see, these numbers are pretty much what you’d expect given the distribution shown in Figure 9.3. Most of the time I roll somewhere between 1 to 5 skulls. There are a lot of subtleties associated with random number generation using a computer,144 but for the purposes of this book we don’t need to worry too much about them. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/09%3A_Introduction_to_Probability/9.04%3A_The_Binomial_Distribution.txt |
While the binomial distribution is conceptually the simplest distribution to understand, it’s not the most important one. That particular honour goes to the normal distribution, which is also referred to as “the bell curve” or a “Gaussian distribution”. A normal distribution is described using two parameters, the mean of the distribution μ and the standard deviation of the distribution σ. The notation that we sometimes use to say that a variable X is normally distributed is as follows:
X∼Normal(μ,σ)
Of course, that’s just notation. It doesn’t tell us anything interesting about the normal distribution itself. As was the case with the binomial distribution, I have included the formula for the normal distribution in this book, because I think it’s important enough that everyone who learns statistics should at least look at it, but since this is an introductory text I don’t want to focus on it, so I’ve tucked it away in Table 9.2. Similarly, the R functions for the normal distribution are `dnorm()`, `pnorm()`, `qnorm()` and `rnorm()`. However, they behave in pretty much exactly the same way as the corresponding functions for the binomial distribution, so there’s not a lot that you need to know. The only thing that I should point out is that the argument names for the parameters are `mean` and `sd`. In pretty much every other respect, there’s nothing else to add.
Instead of focusing on the maths, let’s try to get a sense for what it means for a variable to be normally distributed. To that end, have a look at Figure 9.6, which plots a normal distribution with mean μ=0 and standard deviation σ=1. You can see where the name “bell curve” comes from: it looks a bit like a bell. Notice that, unlike the plots that I drew to illustrate the binomial distribution, the picture of the normal distribution in Figure 9.6 shows a smooth curve instead of “histogram-like” bars. This isn’t an arbitrary choice: the normal distribution is continuous, whereas the binomial is discrete. For instance, in the die rolling example from the last section, it was possible to get 3 skulls or 4 skulls, but impossible to get 3.9 skulls. The figures that I drew in the previous section reflected this fact: in Figure 9.3, for instance, there’s a bar located at X=3 and another one at X=4, but there’s nothing in between. Continuous quantities don’t have this constraint. For instance, suppose we’re talking about the weather. The temperature on a pleasant Spring day could be 23 degrees, 24 degrees, 23.9 degrees, or anything in between since temperature is a continuous variable, and so a normal distribution might be quite appropriate for describing Spring temperatures.145
With this in mind, let’s see if we can’t get an intuition for how the normal distribution works. Firstly, let’s have a look at what happens when we play around with the parameters of the distribution. To that end, Figure 9.7 plots normal distributions that have different means, but have the same standard deviation. As you might expect, all of these distributions have the same “width”. The only difference between them is that they’ve been shifted to the left or to the right. In every other respect they’re identical. In contrast, if we increase the standard deviation while keeping the mean constant, the peak of the distribution stays in the same place, but the distribution gets wider, as you can see in Figure 9.8. Notice, though, that when we widen the distribution, the height of the peak shrinks. This has to happen: in the same way that the heights of the bars that we used to draw a discrete binomial distribution have to sum to 1, the total area under the curve for the normal distribution must equal 1. Before moving on, I want to point out one important characteristic of the normal distribution. Irrespective of what the actual mean and standard deviation are, 68.3% of the area falls within 1 standard deviation of the mean. Similarly, 95.4% of the distribution falls within 2 standard deviations of the mean, and 99.7% of the distribution is within 3 standard deviations. This idea is illustrated in Figure ??.
Probability density
There’s something I’ve been trying to hide throughout my discussion of the normal distribution, something that some introductory textbooks omit completely. They might be right to do so: this “thing” that I’m hiding is weird and counterintuitive even by the admittedly distorted standards that apply in statistics. Fortunately, it’s not something that you need to understand at a deep level in order to do basic statistics: rather, it’s something that starts to become important later on when you move beyond the basics. So, if it doesn’t make complete sense, don’t worry: try to make sure that you follow the gist of it.
Throughout my discussion of the normal distribution, there’s been one or two things that don’t quite make sense. Perhaps you noticed that the y-axis in these figures is labelled “Probability Density” rather than density. Maybe you noticed that I used p(X) instead of P(X) when giving the formula for the normal distribution. Maybe you’re wondering why R uses the “d” prefix for functions like `dnorm()`. And maybe, just maybe, you’ve been playing around with the `dnorm()` function, and you accidentally typed in a command like this:
``dnorm( x = 1, mean = 1, sd = 0.1 )``
``## [1] 3.989423``
And if you’ve done the last part, you’re probably very confused. I’ve asked R to calculate the probability that `x = 1`, for a normally distributed variable with `mean = 1` and standard deviation `sd = 0.1`; and it tells me that the probability is 3.99. But, as we discussed earlier, probabilities can’t be larger than 1. So either I’ve made a mistake, or that’s not a probability.
As it turns out, the second answer is correct. What we’ve calculated here isn’t actually a probability: it’s something else. To understand what that something is, you have to spend a little time thinking about what it really means to say that X is a continuous variable. Let’s say we’re talking about the temperature outside. The thermometer tells me it’s 23 degrees, but I know that’s not really true. It’s not exactly 23 degrees. Maybe it’s 23.1 degrees, I think to myself. But I know that that’s not really true either, because it might actually be 23.09 degrees. But, I know that… well, you get the idea. The tricky thing with genuinely continuous quantities is that you never really know exactly what they are.
Now think about what this implies when we talk about probabilities. Suppose that tomorrow’s maximum temperature is sampled from a normal distribution with mean 23 and standard deviation 1. What’s the probability that the temperature will be exactly 23 degrees? The answer is “zero”, or possibly, “a number so close to zero that it might as well be zero”. Why is this? It’s like trying to throw a dart at an infinitely small dart board: no matter how good your aim, you’ll never hit it. In real life you’ll never get a value of exactly 23. It’ll always be something like 23.1 or 22.99998 or something. In other words, it’s completely meaningless to talk about the probability that the temperature is exactly 23 degrees. However, in everyday language, if I told you that it was 23 degrees outside and it turned out to be 22.9998 degrees, you probably wouldn’t call me a liar. Because in everyday language, “23 degrees” usually means something like “somewhere between 22.5 and 23.5 degrees”. And while it doesn’t feel very meaningful to ask about the probability that the temperature is exactly 23 degrees, it does seem sensible to ask about the probability that the temperature lies between 22.5 and 23.5, or between 20 and 30, or any other range of temperatures.
The point of this discussion is to make clear that, when we’re talking about continuous distributions, it’s not meaningful to talk about the probability of a specific value. However, what we can talk about is the probability that the value lies within a particular range of values. To find out the probability associated with a particular range, what you need to do is calculate the “area under the curve”. We’ve seen this concept already: in Figures 9.9 and (fig:sdnorm1b), the shaded areas shown depict genuine probabilities (e.g., in Figure 9.9 it shows the probability of observing a value that falls within 1 standard deviation of the mean).
Okay, so that explains part of the story. I’ve explained a little bit about how continuous probability distributions should be interpreted (i.e., area under the curve is the key thing), but I haven’t actually explained what the `dnorm()` function actually calculates. Equivalently, what does the formula for p(x) that I described earlier actually mean? Obviously, p(x) doesn’t describe a probability, but what is it? The name for this quantity p(x) is a probability density, and in terms of the plots we’ve been drawing, it corresponds to the height of the curve. The densities themselves aren’t meaningful in and of themselves: but they’re “rigged” to ensure that the area under the curve is always interpretable as genuine probabilities. To be honest, that’s about as much as you really need to know for now.146 | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/09%3A_Introduction_to_Probability/9.05%3A_The_Normal_Distribution.txt |
The normal distribution is the distribution that statistics makes most use of (for reasons to be discussed shortly), and the binomial distribution is a very useful one for lots of purposes. But the world of statistics is filled with probability distributions, some of which we’ll run into in passing. In particular, the three that will appear in this book are the t distribution, the χ2 distribution and the F distribution. I won’t give formulas for any of these, or talk about them in too much detail, but I will show you some pictures.
• The t distribution is a continuous distribution that looks very similar to a normal distribution, but has heavier tails: see Figure 9.13. This distribution tends to arise in situations where you think that the data actually follow a normal distribution, but you don’t know the mean or standard deviation. As you might expect, the relevant R functions are `dt()`, `pt()`, `qt()` and `rt()`, and we’ll run into this distribution again in Chapter 13.
• The χ2 distribution is another distribution that turns up in lots of different places. The situation in which we’ll see it is when doing categorical data analysis (Chapter 12), but it’s one of those things that actually pops up all over the place. When you dig into the maths (and who doesn’t love doing that?), it turns out that the main reason why the χ2 distribution turns up all over the place is that, if you have a bunch of variables that are normally distributed, square their values and then add them up (a procedure referred to as taking a “sum of squares”), this sum has a χ2 distribution. You’d be amazed how often this fact turns out to be useful. Anyway, here’s what a χ2 distribution looks like: Figure 9.14. Once again, the R commands for this one are pretty predictable: `dchisq()`, `pchisq()`, `qchisq()`, `rchisq()`.
• The F distribution looks a bit like a χ2 distribution, and it arises whenever you need to compare two χ2 distributions to one another. Admittedly, this doesn’t exactly sound like something that any sane person would want to do, but it turns out to be very important in real world data analysis. Remember when I said that χ2 turns out to be the key distribution when we’re taking a “sum of squares”? Well, what that means is if you want to compare two different “sums of squares”, you’re probably talking about something that has an F distribution. Of course, as yet I still haven’t given you an example of anything that involves a sum of squares, but I will… in Chapter 14. And that’s where we’ll run into the F distribution. Oh, and here’s a picture: Figure 9.15. And of course we can get R to do things with F distributions just by using the commands `df()`, `pf()`, `qf()` and `rf()`.
Because these distributions are all tightly related to the normal distribution and to each other, and because they are will turn out to be the important distributions when doing inferential statistics later in this book, I think it’s useful to do a little demonstration using R, just to “convince ourselves” that these distributions really are related to each other in the way that they’re supposed to be. First, we’ll use the `rnorm()` function to generate 1000 normally-distributed observations:
``````normal.a <- rnorm( n=1000, mean=0, sd=1 )
print(head(normal.a))```
```
``## [1] -0.4728528 -0.4483396 -0.5134192 2.1540478 -0.5104661 0.3013308``
So the `normal.a` variable contains 1000 numbers that are normally distributed, and have mean 0 and standard deviation 1, and the actual print out of these numbers goes on for rather a long time. Note that, because the default parameters of the `rnorm()` function are `mean=0` and `sd=1`, I could have shortened the command to `rnorm( n=1000 )`. In any case, what we can do is use the `hist()` function to draw a histogram of the data, like so:
``hist( normal.a ) ``
If you do this, you should see something similar to Figure ??. Your plot won’t look quite as pretty as the one in the figure, of course, because I’ve played around with all the formatting (see Chapter 6), and I’ve also plotted the true distribution of the data as a solid black line (i.e., a normal distribution with mean 0 and standard deviation 1) so that you can compare the data that we just generated to the true distribution.
In the previous example all I did was generate lots of normally distributed observations using `rnorm()` and then compared those to the true probability distribution in the figure (using `dnorm()` to generate the black line in the figure, but I didn’t show the commmands for that). Now let’s try something trickier. We’ll try to generate some observations that follow a chi-square distribution with 3 degrees of freedom, but instead of using `rchisq()`, we’ll start with variables that are normally distributed, and see if we can exploit the known relationships between normal and chi-square distributions to do the work. As I mentioned earlier, a chi-square distribution with k degrees of freedom is what you get when you take k normally-distributed variables (with mean 0 and standard deviation 1), square them, and add them up. Since we want a chi-square distribution with 3 degrees of freedom, we’ll need to supplement our `normal.a` data with two more sets of normally-distributed observations, imaginatively named `normal.b` and `normal.c`:
``````normal.b <- rnorm( n=1000 ) # another set of normally distributed data
normal.c <- rnorm( n=1000 ) # and another!``````
Now that we’ve done that, the theory says we should square these and add them together, like this
``chi.sq.3 <- (normal.a)^2 + (normal.b)^2 + (normal.c)^2``
and the resulting `chi.sq.3` variable should contain 1000 observations that follow a chi-square distribution with 3 degrees of freedom. You can use the `hist()` function to have a look at these observations yourself, using a command like this,
``hist( chi.sq.3 )``
and you should obtain a result that looks pretty similar to the chi-square plot in Figure ??. Once again, the plot that I’ve drawn is a little fancier: in addition to the histogram of `chi.sq.3`, I’ve also plotted a chi-square distribution with 3 degrees of freedom. It’s pretty clear that – even though I used `rnorm()` to do all the work rather than `rchisq()` – the observations stored in the `chi.sq.3` variable really do follow a chi-square distribution. Admittedly, this probably doesn’t seem all that interesting right now, but later on when we start encountering the chi-square distribution in Chapter 12, it will be useful to understand the fact that these distributions are related to one another.
We can extend this demonstration to the t distribution and the F distribution. Earlier, I implied that the t distribution is related to the normal distribution when the standard deviation is unknown. That’s certainly true, and that’s the what we’ll see later on in Chapter 13, but there’s a somewhat more precise relationship between the normal, chi-square and t distributions. Suppose we “scale” our chi-square data by dividing it by the degrees of freedom, like so
``scaled.chi.sq.3 <- chi.sq.3 / 3``
We then take a set of normally distributed variables and divide them by (the square root of) our scaled chi-square variable which had df=3, and the result is a t distribution with 3 degrees of freedom:
``````normal.d <- rnorm( n=1000 ) # yet another set of normally distributed data
t.3 <- normal.d / sqrt( scaled.chi.sq.3 ) # divide by square root of scaled chi-square to get t``````
If we plot the histogram of `t.3`, we end up with something that looks very similar to the t distribution in Figure ??. Similarly, we can obtain an F distribution by taking the ratio between two scaled chi-square distributions. Suppose, for instance, we wanted to generate data from an F distribution with 3 and 20 degrees of freedom. We could do this using `df()`, but we could also do the same thing by generating two chi-square variables, one with 3 degrees of freedom, and the other with 20 degrees of freedom. As the example with `chi.sq.3` illustrates, we can actually do this using `rnorm()` if we really want to, but this time I’ll take a short cut:
``````chi.sq.20 <- rchisq( 1000, 20) # generate chi square data with df = 20...
scaled.chi.sq.20 <- chi.sq.20 / 20 # scale the chi square variable...
F.3.20 <- scaled.chi.sq.3 / scaled.chi.sq.20 # take the ratio of the two chi squares...
hist( F.3.20 ) # ... and draw a picture``````
The resulting `F.3.20` variable does in fact store variables that follow an F distribution with 3 and 20 degrees of freedom. This is illustrated in Figure ??, which plots the histgram of the observations stored in `F.3.20` against the true F distribution with df1=3 and df2=20. Again, they match.
Okay, time to wrap this section up. We’ve seen three new distributions: χ2, t and F. They’re all continuous distributions, and they’re all closely related to the normal distribution. I’ve talked a little bit about the precise nature of this relationship, and shown you some R commands that illustrate this relationship. The key thing for our purposes, however, is not that you have a deep understanding of all these different distributions, nor that you remember the precise relationships between them. The main thing is that you grasp the basic idea that these distributions are all deeply related to one another, and to the normal distribution. Later on in this book, we’re going to run into data that are normally distributed, or at least assumed to be normally distributed. What I want you to understand right now is that, if you make the assumption that your data are normally distributed, you shouldn’t be surprised to see χ2, t and F distributions popping up all over the place when you start trying to do your data analysis. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/09%3A_Introduction_to_Probability/9.06%3A_Other_Useful_Distributions.txt |
In this chapter we’ve talked about probability. We’ve talked what probability means, and why statisticians can’t agree on what it means. We talked about the rules that probabilities have to obey. And we introduced the idea of a probability distribution, and spent a good chunk of the chapter talking about some of the more important probability distributions that statisticians work with. The section by section breakdown looks like this:
• Probability theory versus statistics (Section 9.1)
• Frequentist versus Bayesian views of probability (Section 9.2)
• Basics of probability theory (Section 9.3)
• Binomial distribution (Section 9.4), normal distribution (Section 9.5), and others (Section 9.6)
As you’d expect, my coverage is by no means exhaustive. Probability theory is a large branch of mathematics in its own right, entirely separate from its application to statistics and data analysis. As such, there are thousands of books written on the subject and universities generally offer multiple classes devoted entirely to probability theory. Even the “simpler” task of documenting standard probability distributions is a big topic. I’ve described five standard probability distributions in this chapter, but sitting on my bookshelf I have a 45-chapter book called “Statistical Distributions” Evans, Hastings, and Peacock (2011) that lists a lot more than that. Fortunately for you, very little of this is necessary. You’re unlikely to need to know dozens of statistical distributions when you go out and do real world data analysis, and you definitely won’t need them for this book, but it never hurts to know that there’s other possibilities out there.
Picking up on that last point, there’s a sense in which this whole chapter is something of a digression. Many undergraduate psychology classes on statistics skim over this content very quickly (I know mine did), and even the more advanced classes will often “forget” to revisit the basic foundations of the field. Most academic psychologists would not know the difference between probability and density, and until recently very few would have been aware of the difference between Bayesian and frequentist probability. However, I think it’s important to understand these things before moving onto the applications. For example, there are a lot of rules about what you’re “allowed” to say when doing statistical inference, and many of these can seem arbitrary and weird. However, they start to make sense if you understand that there is this Bayesian/frequentist distinction. Similarly, in Chapter 13 we’re going to talk about something called the t-test, and if you really want to have a grasp of the mechanics of the t-test it really helps to have a sense of what a t-distribution actually looks like. You get the idea, I hope.
References
Fisher, R. 1922b. “On the Mathematical Foundation of Theoretical Statistics.” Philosophical Transactions of the Royal Society A 222: 309–68.
Meehl, P. H. 1967. “Theory Testing in Psychology and Physics: A Methodological Paradox.” Philosophy of Science 34: 103–15.
Evans, M., N. Hastings, and B. Peacock. 2011. Statistical Distributions (3rd Ed). Wiley.
1. This doesn’t mean that frequentists can’t make hypothetical statements, of course; it’s just that if you want to make a statement about probability, then it must be possible to redescribe that statement in terms of a sequence of potentially observable events, and the relative frequencies of different outcomes that appear within that sequence.
2. Note that the term “success” is pretty arbitrary, and doesn’t actually imply that the outcome is something to be desired. If θ referred to the probability that any one passenger gets injured in a bus crash, I’d still call it the success probability, but that doesn’t mean I want people to get hurt in bus crashes!
3. Since computers are deterministic machines, they can’t actually produce truly random behaviour. Instead, what they do is take advantage of various mathematical functions that share a lot of similarities with true randomness. What this means is that any random numbers generated on a computer are pseudorandom, and the quality of those numbers depends on the specific method used. By default R uses the “Mersenne twister” method. In any case, you can find out more by typing ?Random, but as usual the R help files are fairly dense.
4. In practice, the normal distribution is so handy that people tend to use it even when the variable isn’t actually continuous. As long as there are enough categories (e.g., Likert scale responses to a questionnaire), it’s pretty standard practice to use the normal distribution as an approximation. This works out much better in practice than you’d think.
5. For those readers who know a little calculus, I’ll give a slightly more precise explanation. In the same way that probabilities are non-negative numbers that must sum to 1, probability densities are non-negative numbers that must integrate to 1 (where the integral is taken across all possible values of X). To calculate the probability that X falls between a and b we calculate the definite integral of the density function over the corresponding range, $\int_{a}^{b} p(x) d x$. If you don’t remember or never learned calculus, don’t worry about this. It’s not needed for this book. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/09%3A_Introduction_to_Probability/9.07%3A_Summary.txt |
At the start of the last chapter I highlighted the critical distinction between descriptive statistics and inferential statistics. As discussed in Chapter 5, the role of descriptive statistics is to concisely summarise what we do know. In contrast, the purpose of inferential statistics is to “learn what we do not know from what we do”. Now that we have a foundation in probability theory, we are in a good position to think about the problem of statistical inference. What kinds of things would we like to learn about? And how do we learn them? These are the questions that lie at the heart of inferential statistics, and they are traditionally divided into two “big ideas”: estimation and hypothesis testing. The goal in this chapter is to introduce the first of these big ideas, estimation theory, but I’m going to witter on about sampling theory first because estimation theory doesn’t make sense until you understand sampling. As a consequence, this chapter divides naturally into two parts Sections 10.1 through 10.3 are focused on sampling theory, and Sections 10.4 and 10.5 make use of sampling theory to discuss how statisticians think about estimation.
10: Estimating Unknown Quantities from a Sample
In the prelude to Part I discussed the riddle of induction, and highlighted the fact that all learning requires you to make assumptions. Accepting that this is true, our first task to come up with some fairly general assumptions about data that make sense. This is where sampling theory comes in. If probability theory is the foundations upon which all statistical theory builds, sampling theory is the frame around which you can build the rest of the house. Sampling theory plays a huge role in specifying the assumptions upon which your statistical inferences rely. And in order to talk about “making inferences” the way statisticians think about it, we need to be a bit more explicit about what it is that we’re drawing inferences from (the sample) and what it is that we’re drawing inferences about (the population).
In almost every situation of interest, what we have available to us as researchers is a sample of data. We might have run experiment with some number of participants; a polling company might have phoned some number of people to ask questions about voting intentions; etc. Regardless: the data set available to us is finite, and incomplete. We can’t possibly get every person in the world to do our experiment; a polling company doesn’t have the time or the money to ring up every voter in the country etc. In our earlier discussion of descriptive statistics (Chapter 5, this sample was the only thing we were interested in. Our only goal was to find ways of describing, summarising and graphing that sample. This is about to change.
Defining a population
A sample is a concrete thing. You can open up a data file, and there’s the data from your sample. A population, on the other hand, is a more abstract idea. It refers to the set of all possible people, or all possible observations, that you want to draw conclusions about, and is generally much bigger than the sample. In an ideal world, the researcher would begin the study with a clear idea of what the population of interest is, since the process of designing a study and testing hypotheses about the data that it produces does depend on the population about which you want to make statements. However, that doesn’t always happen in practice: usually the researcher has a fairly vague idea of what the population is and designs the study as best he/she can on that basis.
Sometimes it’s easy to state the population of interest. For instance, in the “polling company” example that opened the chapter, the population consisted of all voters enrolled at the a time of the study – millions of people. The sample was a set of 1000 people who all belong to that population. In most situations the situation is much less simple. In a typical a psychological experiment, determining the population of interest is a bit more complicated. Suppose I run an experiment using 100 undergraduate students as my participants. My goal, as a cognitive scientist, is to try to learn something about how the mind works. So, which of the following would count as “the population”:
• All of the undergraduate psychology students at the University of Adelaide?
• Undergraduate psychology students in general, anywhere in the world?
• Australians currently living?
• Australians of similar ages to my sample?
• Anyone currently alive?
• Any human being, past, present or future?
• Any biological organism with a sufficient degree of intelligence operating in a terrestrial environment?
• Any intelligent being?
Each of these defines a real group of mind-possessing entities, all of which might be of interest to me as a cognitive scientist, and it’s not at all clear which one ought to be the true population of interest. As another example, consider the Wellesley-Croker game that we discussed in the prelude. The sample here is a specific sequence of 12 wins and 0 losses for Wellesley. What is the population?
• All outcomes until Wellesley and Croker arrived at their destination?
• All outcomes if Wellesley and Croker had played the game for the rest of their lives?
• All outcomes if Wellseley and Croker lived forever and played the game until the world ran out of hills?
• All outcomes if we created an infinite set of parallel universes and the Wellesely/Croker pair made guesses about the same 12 hills in each universe?
Again, it’s not obvious what the population is.
Irrespective of how I define the population, the critical point is that the sample is a subset of the population, and our goal is to use our knowledge of the sample to draw inferences about the properties of the population. The relationship between the two depends on the procedure by which the sample was selected. This procedure is referred to as a sampling method, and it is important to understand why it matters.
To keep things simple, let’s imagine that we have a bag containing 10 chips. Each chip has a unique letter printed on it, so we can distinguish between the 10 chips. The chips come in two colours, black and white. This set of chips is the population of interest, and it is depicted graphically on the left of Figure 10.1. As you can see from looking at the picture, there are 4 black chips and 6 white chips, but of course in real life we wouldn’t know that unless we looked in the bag. Now imagine you run the following “experiment”: you shake up the bag, close your eyes, and pull out 4 chips without putting any of them back into the bag. First out comes the a chip (black), then the c chip (white), then j (white) and then finally b (black). If you wanted, you could then put all the chips back in the bag and repeat the experiment, as depicted on the right hand side of Figure 10.1. Each time you get different results, but the procedure is identical in each case. The fact that the same procedure can lead to different results each time, we refer to it as a random process.147 However, because we shook the bag before pulling any chips out, it seems reasonable to think that every chip has the same chance of being selected. A procedure in which every member of the population has the same chance of being selected is called a simple random sample. The fact that we did not put the chips back in the bag after pulling them out means that you can’t observe the same thing twice, and in such cases the observations are said to have been sampled without replacement.
To help make sure you understand the importance of the sampling procedure, consider an alternative way in which the experiment could have been run. Suppose that my 5-year old son had opened the bag, and decided to pull out four black chips without putting any of them back in the bag. This biased sampling scheme is depicted in Figure 10.2. Now consider the evidentiary value of seeing 4 black chips and 0 white chips. Clearly, it depends on the sampling scheme, does it not? If you know that the sampling scheme is biased to select only black chips, then a sample that consists of only black chips doesn’t tell you very much about the population! For this reason, statisticians really like it when a data set can be considered a simple random sample, because it makes the data analysis much easier.
A third procedure is worth mentioning. This time around we close our eyes, shake the bag, and pull out a chip. This time, however, we record the observation and then put the chip back in the bag. Again we close our eyes, shake the bag, and pull out a chip. We then repeat this procedure until we have 4 chips. Data sets generated in this way are still simple random samples, but because we put the chips back in the bag immediately after drawing them it is referred to as a sample with replacement. The difference between this situation and the first one is that it is possible to observe the same population member multiple times, as illustrated in Figure 10.3.
In my experience, most psychology experiments tend to be sampling without replacement, because the same person is not allowed to participate in the experiment twice. However, most statistical theory is based on the assumption that the data arise from a simple random sample with replacement. In real life, this very rarely matters. If the population of interest is large (e.g., has more than 10 entities!) the difference between sampling with- and without- replacement is too small to be concerned with. The difference between simple random samples and biased samples, on the other hand, is not such an easy thing to dismiss.
Most samples are not simple random samples
As you can see from looking at the list of possible populations that I showed above, it is almost impossible to obtain a simple random sample from most populations of interest. When I run experiments, I’d consider it a minor miracle if my participants turned out to be a random sampling of the undergraduate psychology students at Adelaide university, even though this is by far the narrowest population that I might want to generalise to. A thorough discussion of other types of sampling schemes is beyond the scope of this book, but to give you a sense of what’s out there I’ll list a few of the more important ones:
• Stratified sampling. Suppose your population is (or can be) divided into several different subpopulations, or strata. Perhaps you’re running a study at several different sites, for example. Instead of trying to sample randomly from the population as a whole, you instead try to collect a separate random sample from each of the strata. Stratified sampling is sometimes easier to do than simple random sampling, especially when the population is already divided into the distinct strata. It can also be more efficient that simple random sampling, especially when some of the subpopulations are rare. For instance, when studying schizophrenia it would be much better to divide the population into two148 strata (schizophrenic and not-schizophrenic), and then sample an equal number of people from each group. If you selected people randomly, you would get so few schizophrenic people in the sample that your study would be useless. This specific kind of of stratified sampling is referred to as oversampling because it makes a deliberate attempt to over-represent rare groups.
• Snowball sampling is a technique that is especially useful when sampling from a “hidden” or hard to access population, and is especially common in social sciences. For instance, suppose the researchers want to conduct an opinion poll among transgender people. The research team might only have contact details for a few trans folks, so the survey starts by asking them to participate (stage 1). At the end of the survey, the participants are asked to provide contact details for other people who might want to participate. In stage 2, those new contacts are surveyed. The process continues until the researchers have sufficient data. The big advantage to snowball sampling is that it gets you data in situations that might otherwise be impossible to get any. On the statistical side, the main disadvantage is that the sample is highly non-random, and non-random in ways that are difficult to address. On the real life side, the disadvantage is that the procedure can be unethical if not handled well, because hidden populations are often hidden for a reason. I chose transgender people as an example here to highlight this: if you weren’t careful you might end up outing people who don’t want to be outed (very, very bad form), and even if you don’t make that mistake it can still be intrusive to use people’s social networks to study them. It’s certainly very hard to get people’s informed consent before contacting them, yet in many cases the simple act of contacting them and saying “hey we want to study you” can be hurtful. Social networks are complex things, and just because you can use them to get data doesn’t always mean you should.
• Convenience sampling is more or less what it sounds like. The samples are chosen in a way that is convenient to the researcher, and not selected at random from the population of interest. Snowball sampling is one type of convenience sampling, but there are many others. A common example in psychology are studies that rely on undergraduate psychology students. These samples are generally non-random in two respects: firstly, reliance on undergraduate psychology students automatically means that your data are restricted to a single subpopulation. Secondly, the students usually get to pick which studies they participate in, so the sample is a self selected subset of psychology students not a randomly selected subset. In real life, most studies are convenience samples of one form or another. This is sometimes a severe limitation, but not always.
much does it matter if you don’t have a simple random sample?
Okay, so real world data collection tends not to involve nice simple random samples. Does that matter? A little thought should make it clear to you that it can matter if your data are not a simple random sample: just think about the difference between Figures 10.1 and 10.2. However, it’s not quite as bad as it sounds. Some types of biased samples are entirely unproblematic. For instance, when using a stratified sampling technique you actually know what the bias is because you created it deliberately, often to increase the effectiveness of your study, and there are statistical techniques that you can use to adjust for the biases you’ve introduced (not covered in this book!). So in those situations it’s not a problem.
More generally though, it’s important to remember that random sampling is a means to an end, not the end in itself. Let’s assume you’ve relied on a convenience sample, and as such you can assume it’s biased. A bias in your sampling method is only a problem if it causes you to draw the wrong conclusions. When viewed from that perspective, I’d argue that we don’t need the sample to be randomly generated in every respect: we only need it to be random with respect to the psychologically-relevant phenomenon of interest. Suppose I’m doing a study looking at working memory capacity. In study 1, I actually have the ability to sample randomly from all human beings currently alive, with one exception: I can only sample people born on a Monday. In study 2, I am able to sample randomly from the Australian population. I want to generalise my results to the population of all living humans. Which study is better? The answer, obviously, is study 1. Why? Because we have no reason to think that being “born on a Monday” has any interesting relationship to working memory capacity. In contrast, I can think of several reasons why “being Australian” might matter. Australia is a wealthy, industrialised country with a very well-developed education system. People growing up in that system will have had life experiences much more similar to the experiences of the people who designed the tests for working memory capacity. This shared experience might easily translate into similar beliefs about how to “take a test”, a shared assumption about how psychological experimentation works, and so on. These things might actually matter. For instance, “test taking” style might have taught the Australian participants how to direct their attention exclusively on fairly abstract test materials relative to people that haven’t grown up in a similar environment; leading to a misleading picture of what working memory capacity is.
There are two points hidden in this discussion. Firstly, when designing your own studies, it’s important to think about what population you care about, and try hard to sample in a way that is appropriate to that population. In practice, you’re usually forced to put up with a “sample of convenience” (e.g., psychology lecturers sample psychology students because that’s the least expensive way to collect data, and our coffers aren’t exactly overflowing with gold), but if so you should at least spend some time thinking about what the dangers of this practice might be.
Secondly, if you’re going to criticise someone else’s study because they’ve used a sample of convenience rather than laboriously sampling randomly from the entire human population, at least have the courtesy to offer a specific theory as to how this might have distorted the results. Remember, everyone in science is aware of this issue, and does what they can to alleviate it. Merely pointing out that “the study only included people from group BLAH” is entirely unhelpful, and borders on being insulting to the researchers, who are of course aware of the issue. They just don’t happen to be in possession of the infinite supply of time and money required to construct the perfect sample. In short, if you want to offer a responsible critique of the sampling process, then be helpful. Rehashing the blindingly obvious truisms that I’ve been rambling on about in this section isn’t helpful.
Population parameters and sample statistics
Okay. Setting aside the thorny methodological issues associated with obtaining a random sample and my rather unfortunate tendency to rant about lazy methodological criticism, let’s consider a slightly different issue. Up to this point we have been talking about populations the way a scientist might. To a psychologist, a population might be a group of people. To an ecologist, a population might be a group of bears. In most cases the populations that scientists care about are concrete things that actually exist in the real world. Statisticians, however, are a funny lot. On the one hand, they are interested in real world data and real science in the same way that scientists are. On the other hand, they also operate in the realm of pure abstraction in the way that mathematicians do. As a consequence, statistical theory tends to be a bit abstract in how a population is defined. In much the same way that psychological researchers operationalise our abstract theoretical ideas in terms of concrete measurements (Section 2.1, statisticians operationalise the concept of a “population” in terms of mathematical objects that they know how to work with. You’ve already come across these objects in Chapter 9: they’re called probability distributions.
The idea is quite simple. Let’s say we’re talking about IQ scores. To a psychologist, the population of interest is a group of actual humans who have IQ scores. A statistician “simplifies” this by operationally defining the population as the probability distribution depicted in Figure ??. IQ tests are designed so that the average IQ is 100, the standard deviation of IQ scores is 15, and the distribution of IQ scores is normal. These values are referred to as the population parameters because they are characteristics of the entire population. That is, we say that the population mean μ is 100, and the population standard deviation σ is 15.
``## [1] "n= 100 mean= 99.6064025956605 sd= 16.0047604703873"``
``## [1] "n= 10000 mean= 100.096924966188 sd= 14.9554812898374"``
Now suppose I run an experiment. I select 100 people at random and administer an IQ test, giving me a simple random sample from the population. My sample would consist of a collection of numbers like this:
`` 106 101 98 80 74 ... 107 72 100``
Each of these IQ scores is sampled from a normal distribution with mean 100 and standard deviation 15. So if I plot a histogram of the sample, I get something like the one shown in Figure 10.4b. As you can see, the histogram is roughly the right shape, but it’s a very crude approximation to the true population distribution shown in Figure 10.4a. When I calculate the mean of my sample, I get a number that is fairly close to the population mean 100 but not identical. In this case, it turns out that the people in my sample have a mean IQ of 98.5, and the standard deviation of their IQ scores is 15.9. These sample statistics are properties of my data set, and although they are fairly similar to the true population values, they are not the same. In general, sample statistics are the things you can calculate from your data set, and the population parameters are the things you want to learn about. Later on in this chapter I’ll talk about how you can estimate population parameters using your sample statistics (Section 10.4 and how to work out how confident you are in your estimates (Section 10.5 but before we get to that there’s a few more ideas in sampling theory that you need to know about. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/10%3A_Estimating_Unknown_Quantities_from_a_Sample/10.01%3A_Samples_Populations_and_Sampling.txt |
In the previous section I showed you the results of one fictitious IQ experiment with a sample size of N=100. The results were somewhat encouraging: the true population mean is 100, and the sample mean of 98.5 is a pretty reasonable approximation to it. In many scientific studies that level of precision is perfectly acceptable, but in other situations you need to be a lot more precise. If we want our sample statistics to be much closer to the population parameters, what can we do about it?
The obvious answer is to collect more data. Suppose that we ran a much larger experiment, this time measuring the IQs of 10,000 people. We can simulate the results of this experiment using R. In Section 9.5 I introduced the rnorm() function, which generates random numbers sampled from a normal distribution. For an experiment with a sample size of n = 10000, and a population with mean = 100 and sd = 15, R produces our fake IQ data using these commands:
IQ <- rnorm(n = 10000, mean = 100, sd = 15) # generate IQ scores
IQ <- round(IQ) # IQs are whole numbers!
print(head(IQ))
## [1] 82 91 123 129 104 96
I can compute the mean IQ using the command mean(IQ) and the standard deviation using the command sd(IQ), and I can draw a histgram using hist(). The histogram of this much larger sample is shown in Figure 10.4c. Even a moment’s inspections makes clear that the larger sample is a much better approximation to the true population distribution than the smaller one. This is reflected in the sample statistics: the mean IQ for the larger sample turns out to be 99.9, and the standard deviation is 15.1. These values are now very close to the true population.
I feel a bit silly saying this, but the thing I want you to take away from this is that large samples generally give you better information. I feel silly saying it because it’s so bloody obvious that it shouldn’t need to be said. In fact, it’s such an obvious point that when Jacob Bernoulli – one of the founders of probability theory – formalised this idea back in 1713, he was kind of a jerk about it. Here’s how he described the fact that we all share this intuition:
For even the most stupid of men, by some instinct of nature, by himself and without any instruction (which is a remarkable thing), is convinced that the more observations have been made, the less danger there is of wandering from one’s goal Stigler (1986)
Okay, so the passage comes across as a bit condescending (not to mention sexist), but his main point is correct: it really does feel obvious that more data will give you better answers. The question is, why is this so? Not surprisingly, this intuition that we all share turns out to be correct, and statisticians refer to it as the law of large numbers. The law of large numbers is a mathematical law that applies to many different sample statistics, but the simplest way to think about it is as a law about averages. The sample mean is the most obvious example of a statistic that relies on averaging (because that’s what the mean is… an average), so let’s look at that. When applied to the sample mean, what the law of large numbers states is that as the sample gets larger, the sample mean tends to get closer to the true population mean. Or, to say it a little bit more precisely, as the sample size “approaches” infinity (written as N→∞) the sample mean approaches the population mean ($\bar{X}$→μ).149
I don’t intend to subject you to a proof that the law of large numbers is true, but it’s one of the most important tools for statistical theory. The law of large numbers is the thing we can use to justify our belief that collecting more and more data will eventually lead us to the truth. For any particular data set, the sample statistics that we calculate from it will be wrong, but the law of large numbers tells us that if we keep collecting more data those sample statistics will tend to get closer and closer to the true population parameters. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/10%3A_Estimating_Unknown_Quantities_from_a_Sample/10.02%3A_The_Law_of_Large_Numbers.txt |
The law of large numbers is a very powerful tool, but it’s not going to be good enough to answer all our questions. Among other things, all it gives us is a “long run guarantee”. In the long run, if we were somehow able to collect an infinite amount of data, then the law of large numbers guarantees that our sample statistics will be correct. But as John Maynard Keynes famously argued in economics, a long run guarantee is of little use in real life:
[The] long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task, if in tempestuous seasons they can only tell us, that when the storm is long past, the ocean is flat again. Keynes (1923)
As in economics, so too in psychology and statistics. It is not enough to know that we will eventually arrive at the right answer when calculating the sample mean. Knowing that an infinitely large data set will tell me the exact value of the population mean is cold comfort when my actual data set has a sample size of N=100. In real life, then, we must know something about the behaviour of the sample mean when it is calculated from a more modest data set!
Sampling distribution of the mean
With this in mind, let’s abandon the idea that our studies will have sample sizes of 10000, and consider a very modest experiment indeed. This time around we’ll sample N=5 people and measure their IQ scores. As before, I can simulate this experiment in R using the rnorm() function:
> IQ.1 <- round( rnorm(n=5, mean=100, sd=15 ))
> IQ.1
[1] 90 82 94 99 110
The mean IQ in this sample turns out to be exactly 95. Not surprisingly, this is much less accurate than the previous experiment. Now imagine that I decided to replicate the experiment. That is, I repeat the procedure as closely as possible: I randomly sample 5 new people and measure their IQ. Again, R allows me to simulate the results of this procedure:
> IQ.2 <- round( rnorm(n=5, mean=100, sd=15 ))
> IQ.2
[1] 78 88 111 111 117
This time around, the mean IQ in my sample is 101. If I repeat the experiment 10 times I obtain the results shown in Table ??, and as you can see the sample mean varies from one replication to the next.
NANA Person.1 Person.2 Person.3 Person.4 Person.5 Sample.Mean caption
Replication 1 90 82 94 99 110 95.0 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 2 78 88 111 111 117 101.0 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 3 111 122 91 98 86 101.6 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 4 98 96 119 99 107 103.8 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 5 105 113 103 103 98 104.4 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 6 81 89 93 85 114 92.4 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 7 100 93 108 98 133 106.4 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 8 107 100 105 117 85 102.8 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 9 86 119 108 73 116 100.4 Ten replications of the IQ experiment, each with a sample size of N=5.
Replication 10 95 126 112 120 76 105.8 Ten replications of the IQ experiment, each with a sample size of N=
Now suppose that I decided to keep going in this fashion, replicating this “five IQ scores” experiment over and over again. Every time I replicate the experiment I write down the sample mean. Over time, I’d be amassing a new data set, in which every experiment generates a single data point. The first 10 observations from my data set are the sample means listed in Table ??, so my data set starts out like this:
95.0 101.0 101.6 103.8 104.4 ...
What if I continued like this for 10,000 replications, and then drew a histogram? Using the magical powers of R that’s exactly what I did, and you can see the results in Figure 10.5. As this picture illustrates, the average of 5 IQ scores is usually between 90 and 110. But more importantly, what it highlights is that if we replicate an experiment over and over again, what we end up with is a distribution of sample means! This distribution has a special name in statistics: it’s called the sampling distribution of the mean.
Sampling distributions are another important theoretical idea in statistics, and they’re crucial for understanding the behaviour of small samples. For instance, when I ran the very first “five IQ scores” experiment, the sample mean turned out to be 95. What the sampling distribution in Figure 10.5 tells us, though, is that the “five IQ scores” experiment is not very accurate. If I repeat the experiment, the sampling distribution tells me that I can expect to see a sample mean anywhere between 80 and 120.
Sampling distributions exist for any sample statistic!
One thing to keep in mind when thinking about sampling distributions is that any sample statistic you might care to calculate has a sampling distribution. For example, suppose that each time I replicated the “five IQ scores” experiment I wrote down the largest IQ score in the experiment. This would give me a data set that started out like this:
110 117 122 119 113 ...
Doing this over and over again would give me a very different sampling distribution, namely the sampling distribution of the maximum. The sampling distribution of the maximum of 5 IQ scores is shown in Figure 10.6. Not surprisingly, if you pick 5 people at random and then find the person with the highest IQ score, they’re going to have an above average IQ. Most of the time you’ll end up with someone whose IQ is measured in the 100 to 140 range.
central limit theorem
An illustration of the how sampling distribution of the mean depends on sample size. In each panel, I generated 10,000 samples of IQ data, and calculated the mean IQ observed within each of these data sets. The histograms in these plots show the distribution of these means (i.e., the sampling distribution of the mean). Each individual IQ score was drawn from a normal distribution with mean 100 and standard deviation 15, which is shown as the solid black line).
At this point I hope you have a pretty good sense of what sampling distributions are, and in particular what the sampling distribution of the mean is. In this section I want to talk about how the sampling distribution of the mean changes as a function of sample size. Intuitively, you already know part of the answer: if you only have a few observations, the sample mean is likely to be quite inaccurate: if you replicate a small experiment and recalculate the mean you’ll get a very different answer. In other words, the sampling distribution is quite wide. If you replicate a large experiment and recalculate the sample mean you’ll probably get the same answer you got last time, so the sampling distribution will be very narrow. You can see this visually in Figures 10.7, 10.8 and 10.9: the bigger the sample size, the narrower the sampling distribution gets. We can quantify this effect by calculating the standard deviation of the sampling distribution, which is referred to as the standard error. The standard error of a statistic is often denoted SE, and since we’re usually interested in the standard error of the sample mean, we often use the acronym SEM. As you can see just by looking at the picture, as the sample size N increases, the SEM decreases.
Okay, so that’s one part of the story. However, there’s something I’ve been glossing over so far. All my examples up to this point have been based on the “IQ scores” experiments, and because IQ scores are roughly normally distributed, I’ve assumed that the population distribution is normal. What if it isn’t normal? What happens to the sampling distribution of the mean? The remarkable thing is this: no matter what shape your population distribution is, as N increases the sampling distribution of the mean starts to look more like a normal distribution. To give you a sense of this, I ran some simulations using R. To do this, I started with the “ramped” distribution shown in the histogram in Figure 10.10. As you can see by comparing the triangular shaped histogram to the bell curve plotted by the black line, the population distribution doesn’t look very much like a normal distribution at all. Next, I used R to simulate the results of a large number of experiments. In each experiment I took N=2 samples from this distribution, and then calculated the sample mean. Figure ?? plots the histogram of these sample means (i.e., the sampling distribution of the mean for N=2). This time, the histogram produces a ∩-shaped distribution: it’s still not normal, but it’s a lot closer to the black line than the population distribution in Figure ??. When I increase the sample size to N=4, the sampling distribution of the mean is very close to normal (Figure ??, and by the time we reach a sample size of N=8 it’s almost perfectly normal. In other words, as long as your sample size isn’t tiny, the sampling distribution of the mean will be approximately normal no matter what your population distribution looks like!
# needed for printing
width <- 6
height <- 6
# parameters of the beta
a <- 2
b <- 1
# mean and standard deviation of the beta
s <- sqrt( a*b / (a+b)^2 / (a+b+1) )
m <- a / (a+b)
# define function to draw a plot
plotOne <- function(n,N=50000) {
# generate N random sample means of size n
X <- matrix(rbeta(n*N,a,b),n,N)
X <- colMeans(X)
# plot the data
hist( X, breaks=seq(0,1,.025), border="white", freq=FALSE,
col=ifelse(colour,emphColLight,emphGrey),
xlab="Sample Mean", ylab="", xlim=c(0,1.2),
main=paste("Sample Size =",n), axes=FALSE,
font.main=1, ylim=c(0,5)
)
box()
axis(1)
#axis(2)
# plot the theoretical distribution
lines( x <- seq(0,1.2,.01), dnorm(x,m,s/sqrt(n)),
lwd=2, col="black", type="l"
)
}
for( i in c(1,2,4,8)) {
plotOne(i)}
On the basis of these figures, it seems like we have evidence for all of the following claims about the sampling distribution of the mean:
• The mean of the sampling distribution is the same as the mean of the population
• The standard deviation of the sampling distribution (i.e., the standard error) gets smaller as the sample size increases
• The shape of the sampling distribution becomes normal as the sample size increases
As it happens, not only are all of these statements true, there is a very famous theorem in statistics that proves all three of them, known as the central limit theorem. Among other things, the central limit theorem tells us that if the population distribution has mean μ and standard deviation σ, then the sampling distribution of the mean also has mean μ, and the standard error of the mean is
$\mathrm{SEM}=\dfrac{\sigma}{\sqrt{N}} \nonumber$
Because we divide the population standard devation σ by the square root of the sample size N, the SEM gets smaller as the sample size increases. It also tells us that the shape of the sampling distribution becomes normal.150
This result is useful for all sorts of things. It tells us why large experiments are more reliable than small ones, and because it gives us an explicit formula for the standard error it tells us how much more reliable a large experiment is. It tells us why the normal distribution is, well, normal. In real experiments, many of the things that we want to measure are actually averages of lots of different quantities (e.g., arguably, “general” intelligence as measured by IQ is an average of a large number of “specific” skills and abilities), and when that happens, the averaged quantity should follow a normal distribution. Because of this mathematical law, the normal distribution pops up over and over again in real data. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/10%3A_Estimating_Unknown_Quantities_from_a_Sample/10.03%3A_Sampling_Distributions_and_the_Central_Limit_Theo.txt |
In all the IQ examples in the previous sections, we actually knew the population parameters ahead of time. As every undergraduate gets taught in their very first lecture on the measurement of intelligence, IQ scores are defined to have mean 100 and standard deviation 15. However, this is a bit of a lie. How do we know that IQ scores have a true population mean of 100? Well, we know this because the people who designed the tests have administered them to very large samples, and have then “rigged” the scoring rules so that their sample has mean 100. That’s not a bad thing of course: it’s an important part of designing a psychological measurement. However, it’s important to keep in mind that this theoretical mean of 100 only attaches to the population that the test designers used to design the tests. Good test designers will actually go to some lengths to provide “test norms” that can apply to lots of different populations (e.g., different age groups, nationalities etc).
This is very handy, but of course almost every research project of interest involves looking at a different population of people to those used in the test norms. For instance, suppose you wanted to measure the effect of low level lead poisoning on cognitive functioning in Port Pirie, a South Australian industrial town with a lead smelter. Perhaps you decide that you want to compare IQ scores among people in Port Pirie to a comparable sample in Whyalla, a South Australian industrial town with a steel refinery.151 Regardless of which town you’re thinking about, it doesn’t make a lot of sense simply to assume that the true population mean IQ is 100. No-one has, to my knowledge, produced sensible norming data that can automatically be applied to South Australian industrial towns. We’re going to have to estimate the population parameters from a sample of data. So how do we do this?
Estimating the population mean
Suppose we go to Port Pirie and 100 of the locals are kind enough to sit through an IQ test. The average IQ score among these people turns out to be $\bar{X}$ =98.5. So what is the true mean IQ for the entire population of Port Pirie? Obviously, we don’t know the answer to that question. It could be 97.2, but if could also be 103.5. Our sampling isn’t exhaustive so we cannot give a definitive answer. Nevertheless if I was forced at gunpoint to give a “best guess” I’d have to say 98.5. That’s the essence of statistical estimation: giving a best guess.
In this example, estimating the unknown poulation parameter is straightforward. I calculate the sample mean, and I use that as my estimate of the population mean. It’s pretty simple, and in the next section I’ll explain the statistical justification for this intuitive answer. However, for the moment what I want to do is make sure you recognise that the sample statistic and the estimate of the population parameter are conceptually different things. A sample statistic is a description of your data, whereas the estimate is a guess about the population. With that in mind, statisticians often different notation to refer to them. For instance, if true population mean is denoted μ, then we would use $\hat{\mu}$ to refer to our estimate of the population mean. In contrast, the sample mean is denoted $\bar{X}$ or sometimes m. However, in simple random samples, the estimate of the population mean is identical to the sample mean: if I observe a sample mean of $\bar{X}$ =98.5, then my estimate of the population mean is also $\hat{\mu}$=98.5. To help keep the notation clear, here’s a handy table:
knitr::kable(data.frame(stringsAsFactors=FALSE,
Symbol = c("$\bar{X}$", "$\mu$", "$\hat{\mu}$"),
What.is.it = c("Sample mean", "True population mean",
"Estimate of the population mean"),
Do.we.know.what.it.is = c("Yes calculated from the raw data",
"Almost never known for sure",
"Yes identical to the sample mean")))
Symbol What.is.it Do.we.know.what.it.is
$\bar{X}$ Sample mean Yes calculated from the raw data
μ True population mean Almost never known for sure
$\hat{\mu}$ Estimate of the population mean Yes identical to the sample mean
Estimating the population standard deviation
So far, estimation seems pretty simple, and you might be wondering why I forced you to read through all that stuff about sampling theory. In the case of the mean, our estimate of the population parameter (i.e. $\hat{\mu}$ ) turned out to identical to the corresponding sample statistic (i.e. $\bar{X}$). However, that’s not always true. To see this, let’s have a think about how to construct an estimate of the population standard deviation, which we’ll denote $\hat{\sigma}$. What shall we use as our estimate in this case? Your first thought might be that we could do the same thing we did when estimating the mean, and just use the sample statistic as our estimate. That’s almost the right thing to do, but not quite.
Here’s why. Suppose I have a sample that contains a single observation. For this example, it helps to consider a sample where you have no intutions at all about what the true population values might be, so let’s use something completely fictitious. Suppose the observation in question measures the cromulence of my shoes. It turns out that my shoes have a cromulence of 20. So here’s my sample:
20
This is a perfectly legitimate sample, even if it does have a sample size of N=1. It has a sample mean of 20, and because every observation in this sample is equal to the sample mean (obviously!) it has a sample standard deviation of 0. As a description of the sample this seems quite right: the sample contains a single observation and therefore there is no variation observed within the sample. A sample standard deviation of s=0 is the right answer here. But as an estimate of the population standard deviation, it feels completely insane, right? Admittedly, you and I don’t know anything at all about what “cromulence” is, but we know something about data: the only reason that we don’t see any variability in the sample is that the sample is too small to display any variation! So, if you have a sample size of N=1, it feels like the right answer is just to say “no idea at all”.
Notice that you don’t have the same intuition when it comes to the sample mean and the population mean. If forced to make a best guess about the population mean, it doesn’t feel completely insane to guess that the population mean is 20. Sure, you probably wouldn’t feel very confident in that guess, because you have only the one observation to work with, but it’s still the best guess you can make.
Let’s extend this example a little. Suppose I now make a second observation. My data set now has N=2 observations of the cromulence of shoes, and the complete sample now looks like this:
20, 22
This time around, our sample is just large enough for us to be able to observe some variability: two observations is the bare minimum number needed for any variability to be observed! For our new data set, the sample mean is $\bar{X}$ =21, and the sample standard deviation is s=1. What intuitions do we have about the population? Again, as far as the population mean goes, the best guess we can possibly make is the sample mean: if forced to guess, we’d probably guess that the population mean cromulence is 21. What about the standard deviation? This is a little more complicated. The sample standard deviation is only based on two observations, and if you’re at all like me you probably have the intuition that, with only two observations, we haven’t given the population “enough of a chance” to reveal its true variability to us. It’s not just that we suspect that the estimate is wrong: after all, with only two observations we expect it to be wrong to some degree. The worry is that the error is systematic. Specifically, we suspect that the sample standard deviation is likely to be smaller than the population standard deviation.
This intuition feels right, but it would be nice to demonstrate this somehow. There are in fact mathematical proofs that confirm this intuition, but unless you have the right mathematical background they don’t help very much. Instead, what I’ll do is use R to simulate the results of some experiments. With that in mind, let’s return to our IQ studies. Suppose the true population mean IQ is 100 and the standard deviation is 15. I can use the rnorm() function to generate the the results of an experiment in which I measure N=2 IQ scores, and calculate the sample standard deviation. If I do this over and over again, and plot a histogram of these sample standard deviations, what I have is the sampling distribution of the standard deviation. I’ve plotted this distribution in Figure 10.11. Even though the true population standard deviation is 15, the average of the sample standard deviations is only 8.5. Notice that this is a very different result to what we found in Figure 10.8 when we plotted the sampling distribution of the mean. If you look at that sampling distribution, what you see is that the population mean is 100, and the average of the sample means is also 100.
Now let’s extend the simulation. Instead of restricting ourselves to the situation where we have a sample size of N=2, let’s repeat the exercise for sample sizes from 1 to 10. If we plot the average sample mean and average sample standard deviation as a function of sample size, you get the results shown in Figure 10.12. On the left hand side (panel a), I’ve plotted the average sample mean and on the right hand side (panel b), I’ve plotted the average standard deviation. The two plots are quite different: on average, the average sample mean is equal to the population mean. It is an unbiased estimator, which is essentially the reason why your best estimate for the population mean is the sample mean.152 The plot on the right is quite different: on average, the sample standard deviation s is smaller than the population standard deviation σ. It is a biased estimator. In other words, if we want to make a “best guess” $\hat{\sigma}$ about the value of the population standard deviation σ, we should make sure our guess is a little bit larger than the sample standard deviation s.
The fix to this systematic bias turns out to be very simple. Here’s how it works. Before tackling the standard deviation, let’s look at the variance. If you recall from Section 5.2, the sample variance is defined to be the average of the squared deviations from the sample mean. That is:
$s^{2}=\dfrac{1}{N} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}$
The sample variance s2 is a biased estimator of the population variance σ2. But as it turns out, we only need to make a tiny tweak to transform this into an unbiased estimator. All we have to do is divide by N−1 rather than by N. If we do that, we obtain the following formula:
$\hat{\sigma}\ ^{2}=\dfrac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}$
This is an unbiased estimator of the population variance σ. Moreover, this finally answers the question we raised in Section 5.2. Why did R give us slightly different answers when we used the var() function? Because the var() function calculates $\hat{\sigma}\ ^{2}$ not s2, that’s why. A similar story applies for the standard deviation. If we divide by N−1 rather than N, our estimate of the population standard deviation becomes:
$\hat{\sigma}=\sqrt{\dfrac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}}$
and when we use R’s built in standard deviation function sd(), what it’s doing is calculating $\hat{σ}$, not s.153
One final point: in practice, a lot of people tend to refer to $\hat{σ}$ (i.e., the formula where we divide by N−1) as the sample standard deviation. Technically, this is incorrect: the sample standard deviation should be equal to s (i.e., the formula where we divide by N). These aren’t the same thing, either conceptually or numerically. One is a property of the sample, the other is an estimated characteristic of the population. However, in almost every real life application, what we actually care about is the estimate of the population parameter, and so people always report $\hat{σ}$ rather than s. This is the right number to report, of course, it’s that people tend to get a little bit imprecise about terminology when they write it up, because “sample standard deviation” is shorter than “estimated population standard deviation”. It’s no big deal, and in practice I do the same thing everyone else does. Nevertheless, I think it’s important to keep the two concepts separate: it’s never a good idea to confuse “known properties of your sample” with “guesses about the population from which it came”. The moment you start thinking that s and $\hat{σ}$ are the same thing, you start doing exactly that.
To finish this section off, here’s another couple of tables to help keep things clear:
knitr::kable(data.frame(stringsAsFactors=FALSE,
Symbol = c("$s$", "$\sigma$", "$\hat{\sigma}$", "$s^2$",
"$\sigma^2$", "$\hat{\sigma}^2$"),
What.is.it = c("Sample standard deviation",
"Population standard deviation",
"Estimate of the population standard deviation", "Sample variance",
"Population variance",
"Estimate of the population variance"),
Do.we.know.what.it.is = c("Yes - calculated from the raw data",
"Almost never known for sure",
"Yes - but not the same as the sample standard deviation",
"Yes - calculated from the raw data",
"Almost never known for sure",
"Yes - but not the same as the sample variance")
))
Symbol What.is.it Do.we.know.what.it.is
s Sample standard deviation Yes - calculated from the raw data
σ Population standard deviation Almost never known for sure
$\hat{σ}$ Estimate of the population standard deviation Yes - but not the same as the sample standard deviation
s2 Sample variance Yes - calculated from the raw data
σ2 Population variance Almost never known for sure
$\hat{\sigma}\ ^{2}$ Estimate of the population variance Yes - but not the same as the sample variance | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/10%3A_Estimating_Unknown_Quantities_from_a_Sample/10.04%3A_Estimating_Population_Parameters.txt |
Statistics means never having to say you’re certain – Unknown origin154 but I’ve never found the original source.
Up to this point in this chapter, I’ve outlined the basics of sampling theory which statisticians rely on to make guesses about population parameters on the basis of a sample of data. As this discussion illustrates, one of the reasons we need all this sampling theory is that every data set leaves us with a some of uncertainty, so our estimates are never going to be perfectly accurate. The thing that has been missing from this discussion is an attempt to quantify the amount of uncertainty that attaches to our estimate. It’s not enough to be able guess that, say, the mean IQ of undergraduate psychology students is 115 (yes, I just made that number up). We also want to be able to say something that expresses the degree of certainty that we have in our guess. For example, it would be nice to be able to say that there is a 95% chance that the true mean lies between 109 and 121. The name for this is a confidence interval for the mean.
Armed with an understanding of sampling distributions, constructing a confidence interval for the mean is actually pretty easy. Here’s how it works. Suppose the true population mean is μ and the standard deviation is σ. I’ve just finished running my study that has N participants, and the mean IQ among those participants is $\bar{X}$. We know from our discussion of the central limit theorem (Section 10.3.3 that the sampling distribution of the mean is approximately normal. We also know from our discussion of the normal distribution Section 9.5 that there is a 95% chance that a normally-distributed quantity will fall within two standard deviations of the true mean. To be more precise, we can use the qnorm() function to compute the 2.5th and 97.5th percentiles of the normal distribution
qnorm( p = c(.025, .975) )
## [1] -1.959964 1.959964
Okay, so I lied earlier on. The more correct answer is that 95% chance that a normally-distributed quantity will fall within 1.96 standard deviations of the true mean. Next, recall that the standard deviation of the sampling distribution is referred to as the standard error, and the standard error of the mean is written as SEM. When we put all these pieces together, we learn that there is a 95% probability that the sample mean $\bar{X}$ that we have actually observed lies within 1.96 standard errors of the population mean. Mathematically, we write this as:
μ−(1.96×SEM) ≤ $\bar{X}$ ≤ μ+(1.96×SEM)
where the SEM is equal to σ/$\sqrt{N}$, and we can be 95% confident that this is true. However, that’s not answering the question that we’re actually interested in. The equation above tells us what we should expect about the sample mean, given that we know what the population parameters are. What we want is to have this work the other way around: we want to know what we should believe about the population parameters, given that we have observed a particular sample. However, it’s not too difficult to do this. Using a little high school algebra, a sneaky way to rewrite our equation is like this:
$\bar{X}$−(1.96×SEM) ≤ μ ≤ $\bar{X}$+(1.96×SEM)
What this is telling is is that the range of values has a 95% probability of containing the population mean μ. We refer to this range as a 95% confidence interval, denoted CI95. In short, as long as N is sufficiently large – large enough for us to believe that the sampling distribution of the mean is normal – then we can write this as our formula for the 95% confidence interval:
$\mathrm{CI}_{95}=\bar{X} \pm\left(1.96 \times \dfrac{\sigma}{\sqrt{N}}\right)$
Of course, there’s nothing special about the number 1.96: it just happens to be the multiplier you need to use if you want a 95% confidence interval. If I’d wanted a 70% confidence interval, I could have used the qnorm() function to calculate the 15th and 85th quantiles:
qnorm( p = c(.15, .85) )
## [1] -1.036433 1.036433
and so the formula for CI70 would be the same as the formula for CI95 except that we’d use 1.04 as our magic number rather than 1.96.
slight mistake in the formula
As usual, I lied. The formula that I’ve given above for the 95% confidence interval is approximately correct, but I glossed over an important detail in the discussion. Notice my formula requires you to use the standard error of the mean, SEM, which in turn requires you to use the true population standard deviation σ. Yet, in Section @ref(pointestimates I stressed the fact that we don’t actually know the true population parameters. Because we don’t know the true value of σ, we have to use an estimate of the population standard deviation $\hat{σ}$ instead. This is pretty straightforward to do, but this has the consequence that we need to use the quantiles of the t-distribution rather than the normal distribution to calculate our magic number; and the answer depends on the sample size. When N is very large, we get pretty much the same value using qt() that we would if we used qnorm()
N <- 10000 # suppose our sample size is 10,000
qt( p = .975, df = N-1) # calculate the 97.5th quantile of the t-dist
## [1] 1.960201
But when N is small, we get a much bigger number when we use the t distribution:
N <- 10 # suppose our sample size is 10
qt( p = .975, df = N-1) # calculate the 97.5th quantile of the t-dist
## [1] 2.262157
There’s nothing too mysterious about what’s happening here. Bigger values mean that the confidence interval is wider, indicating that we’re more uncertain about what the true value of μ actually is. When we use the t distribution instead of the normal distribution, we get bigger numbers, indicating that we have more uncertainty. And why do we have that extra uncertainty? Well, because our estimate of the population standard deviation ^σ might be wrong! If it’s wrong, it implies that we’re a bit less sure about what our sampling distribution of the mean actually looks like… and this uncertainty ends up getting reflected in a wider confidence interval.
Interpreting a confidence interval
The hardest thing about confidence intervals is understanding what they mean. Whenever people first encounter confidence intervals, the first instinct is almost always to say that “there is a 95% probabaility that the true mean lies inside the confidence interval”. It’s simple, and it seems to capture the common sense idea of what it means to say that I am “95% confident”. Unfortunately, it’s not quite right. The intuitive definition relies very heavily on your own personal beliefs about the value of the population mean. I say that I am 95% confident because those are my beliefs. In everyday life that’s perfectly okay, but if you remember back to Section 9.2, you’ll notice that talking about personal belief and confidence is a Bayesian idea. Personally (speaking as a Bayesian) I have no problem with the idea that the phrase “95% probability” is allowed to refer to a personal belief. However, confidence intervals are not Bayesian tools. Like everything else in this chapter, confidence intervals are frequentist tools, and if you are going to use frequentist methods then it’s not appropriate to attach a Bayesian interpretation to them. If you use frequentist methods, you must adopt frequentist interpretations!
Okay, so if that’s not the right answer, what is? Remember what we said about frequentist probability: the only way we are allowed to make “probability statements” is to talk about a sequence of events, and to count up the frequencies of different kinds of events. From that perspective, the interpretation of a 95% confidence interval must have something to do with replication. Specifically: if we replicated the experiment over and over again and computed a 95% confidence interval for each replication, then 95% of those intervals would contain the true mean. More generally, 95% of all confidence intervals constructed using this procedure should contain the true population mean. This idea is illustrated in Figure 10.13, which shows 50 confidence intervals constructed for a “measure 10 IQ scores” experiment (top panel) and another 50 confidence intervals for a “measure 25 IQ scores” experiment (bottom panel). A bit fortuitously, across the 100 replications that I simulated, it turned out that exactly 95 of them contained the true mean.
The critical difference here is that the Bayesian claim makes a probability statement about the population mean (i.e., it refers to our uncertainty about the population mean), which is not allowed under the frequentist interpretation of probability because you can’t “replicate” a population! In the frequentist claim, the population mean is fixed and no probabilistic claims can be made about it. Confidence intervals, however, are repeatable so we can replicate experiments. Therefore a frequentist is allowed to talk about the probability that the confidence interval (a random variable) contains the true mean; but is not allowed to talk about the probability that the true population mean (not a repeatable event) falls within the confidence interval.
I know that this seems a little pedantic, but it does matter. It matters because the difference in interpretation leads to a difference in the mathematics. There is a Bayesian alternative to confidence intervals, known as credible intervals. In most situations credible intervals are quite similar to confidence intervals, but in other cases they are drastically different. As promised, though, I’ll talk more about the Bayesian perspective in Chapter 17.
Calculating confidence intervals in R
As far as I can tell, the core packages in R don’t include a simple function for calculating confidence intervals for the mean. They do include a lot of complicated, extremely powerful functions that can be used to calculate confidence intervals associated with lots of different things, such as the confint() function that we’ll use in Chapter 15. But I figure that when you’re first learning statistics, it might be useful to start with something simpler. As a consequence, the lsr package includes a function called ciMean() which you can use to calculate your confidence intervals. There are two arguments that you might want to specify:155
• x. This should be a numeric vector containing the data.
• conf. This should be a number, specifying the confidence level. By default, conf = .95, since 95% confidence intervals are the de facto standard in psychology.
So, for example, if I load the afl24.Rdata file, calculate the confidence interval associated with the mean attendance:
> ciMean( x = afl\$attendance )
2.5% 97.5%
31597.32 32593.12
Hopefully that’s fairly clear.
Plotting confidence intervals in R
There’s several different ways you can draw graphs that show confidence intervals as error bars. I’ll show three versions here, but this certainly doesn’t exhaust the possibilities. In doing so, what I’m assuming is that you want to draw is a plot showing the means and confidence intervals for one variable, broken down by different levels of a second variable. For instance, in our afl data that we discussed earlier, we might be interested in plotting the average attendance by year. I’ll do this using two different functions, bargraph.CI() and lineplot.CI() (both of which are in the sciplot package). Assuming that you’ve installed these packages on your system (see Section 4.2 if you’ve forgotten how to do this), you’ll need to load them. You’ll also need to load the lsr package, because we’ll make use of the ciMean() function to actually calculate the confidence intervals
load( "./rbook-master/data/afl24.Rdata" ) # contains the "afl" data frame
library( sciplot ) # bargraph.CI() and lineplot.CI() functions
library( lsr ) # ciMean() function
Here’s how to plot the means and confidence intervals drawn using bargraph.CI().
bargraph.CI( x.factor = year, # grouping variable
response = attendance, # outcome variable
data = afl, # data frame with the variables
ci.fun= ciMean, # name of the function to calculate CIs
xlab = "Year", # x-axis label
ylab = "Average Attendance" # y-axis label
)
We can use the same arguments when calling the lineplot.CI() function:
lineplot.CI( x.factor = year, # grouping variable
response = attendance, # outcome variable
data = afl, # data frame with the variables
ci.fun= ciMean, # name of the function to calculate CIs
xlab = "Year", # x-axis label
ylab = "Average Attendance" # y-axis label
) | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/10%3A_Estimating_Unknown_Quantities_from_a_Sample/10.05%3A_Estimating_a_Confidence_Interval.txt |
In this chapter I’ve covered two main topics. The first half of the chapter talks about sampling theory, and the second half talks about how we can use sampling theory to construct estimates of the population parameters. The section breakdown looks like this:
• Basic ideas about samples, sampling and populations (Section 10.1)
• Statistical theory of sampling: the law of large numbers (Section 10.2), sampling distributions and the central limit theorem (Section 10.3).
• Estimating means and standard deviations (Section 10.4)
• Estimating a confidence interval (Section 10.5)
As always, there’s a lot of topics related to sampling and estimation that aren’t covered in this chapter, but for an introductory psychology class this is fairly comprehensive I think. For most applied researchers you won’t need much more theory than this. One big question that I haven’t touched on in this chapter is what you do when you don’t have a simple random sample. There is a lot of statistical theory you can draw on to handle this situation, but it’s well beyond the scope of this book.
References
Stigler, S. M. 1986. The History of Statistics. Cambridge, MA: Harvard University Press.
Keynes, John Maynard. 1923. A Tract on Monetary Reform. London: Macmillan; Company.
1. The proper mathematical definition of randomness is extraordinarily technical, and way beyond the scope of this book. We’ll be non-technical here and say that a process has an element of randomness to it whenever it is possible to repeat the process and get different answers each time.
2. Nothing in life is that simple: there’s not an obvious division of people into binary categories like “schizophrenic” and “not schizophrenic”. But this isn’t a clinical psychology text, so please forgive me a few simplifications here and there.
3. Technically, the law of large numbers pertains to any sample statistic that can be described as an average of independent quantities. That’s certainly true for the sample mean. However, it’s also possible to write many other sample statistics as averages of one form or another. The variance of a sample, for instance, can be rewritten as a kind of average and so is subject to the law of large numbers. The minimum value of a sample, however, cannot be written as an average of anything and is therefore not governed by the law of large numbers.
4. As usual, I’m being a bit sloppy here. The central limit theorem is a bit more general than this section implies. Like most introductory stats texts, I’ve discussed one situation where the central limit theorem holds: when you’re taking an average across lots of independent events drawn from the same distribution. However, the central limit theorem is much broader than this. There’s a whole class of things called “U-statistics” for instance, all of which satisfy the central limit theorem and therefore become normally distributed for large sample sizes. The mean is one such statistic, but it’s not the only one.
5. Please note that if you were actually interested in this question, you would need to be a lot more careful than I’m being here. You can’t just compare IQ scores in Whyalla to Port Pirie and assume that any differences are due to lead poisoning. Even if it were true that the only differences between the two towns corresponded to the different refineries (and it isn’t, not by a long shot), you need to account for the fact that people already believe that lead pollution causes cognitive deficits: if you recall back to Chapter 2, this means that there are different demand effects for the Port Pirie sample than for the Whyalla sample. In other words, you might end up with an illusory group difference in your data, caused by the fact that people think that there is a real difference. I find it pretty implausible to think that the locals wouldn’t be well aware of what you were trying to do if a bunch of researchers turned up in Port Pirie with lab coats and IQ tests, and even less plausible to think that a lot of people would be pretty resentful of you for doing it. Those people won’t be as co-operative in the tests. Other people in Port Pirie might be more motivated to do well because they don’t want their home town to look bad. The motivational effects that would apply in Whyalla are likely to be weaker, because people don’t have any concept of “iron ore poisoning” in the same way that they have a concept for “lead poisoning”. Psychology is hard.
6. I should note that I’m hiding something here. Unbiasedness is a desirable characteristic for an estimator, but there are other things that matter besides bias. However, it’s beyond the scope of this book to discuss this in any detail. I just want to draw your attention to the fact that there’s some hidden complexity here.
7. , I’m hiding something else here. In a bizarre and counterintuitive twist, since $\hat{σ}\ ^2$ is an unbiased estimator of σ2, you’d assume that taking the square root would be fine, and $\hat{σ}$ would be an unbiased estimator of σ. Right? Weirdly, it’s not. There’s actually a subtle, tiny bias in $\hat{σ}$. This is just bizarre: $\hat{σ}\ ^2$ is and unbiased estimate of the population variance σ2, but when you take the square root, it turns out that ^σ is a biased estimator of the population standard deviation σ. Weird, weird, weird, right? So, why is $\hat{σ}$ biased? The technical answer is “because non-linear transformations (e.g., the square root) don’t commute with expectation”, but that just sounds like gibberish to everyone who hasn’t taken a course in mathematical statistics. Fortunately, it doesn’t matter for practical purposes. The bias is small, and in real life everyone uses $\hat{σ}$ and it works just fine. Sometimes mathematics is just annoying.
8. This quote appears on a great many t-shirts and websites, and even gets a mention in a few academic papers (e.g., \url{http://www.amstat.org/publications/jse/v10n3/friedman.html
9. As of the current writing, these are the only arguments to the function. However, I am planning to add a bit more functionality to ciMean(). However, regardless of what those future changes might look like, the x and conf arguments will remain the same, and the commands used in this book will still work. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/10%3A_Estimating_Unknown_Quantities_from_a_Sample/10.06%3A_Summary.txt |
The process of induction is the process of assuming the simplest law that can be made to harmonize with our experience. This process, however, has no logical foundation but only a psychological one. It is clear that there are no grounds for believing that the simplest course of events will really happen. It is an hypothesis that the sun will rise tomorrow: and this means that we do not know whether it will rise.
– Ludwig Wittgenstein156
In the last chapter, I discussed the ideas behind estimation, which is one of the two “big ideas” in inferential statistics. It’s now time to turn out attention to the other big idea, which is hypothesis testing. In its most abstract form, hypothesis testing really a very simple idea: the researcher has some theory about the world, and wants to determine whether or not the data actually support that theory. However, the details are messy, and most people find the theory of hypothesis testing to be the most frustrating part of statistics. The structure of the chapter is as follows. Firstly, I’ll describe how hypothesis testing works, in a fair amount of detail, using a simple running example to show you how a hypothesis test is “built”. I’ll try to avoid being too dogmatic while doing so, and focus instead on the underlying logic of the testing procedure.157 Afterwards, I’ll spend a bit of time talking about the various dogmas, rules and heresies that surround the theory of hypothesis testing.
11: Hypothesis Testing
Eventually we all succumb to madness. For me, that day will arrive once I’m finally promoted to full professor. Safely ensconced in my ivory tower, happily protected by tenure, I will finally be able to take leave of my senses (so to speak), and indulge in that most thoroughly unproductive line of psychological research: the search for extrasensory perception (ESP).158
Let’s suppose that this glorious day has come. My first study is a simple one, in which I seek to test whether clairvoyance exists. Each participant sits down at a table, and is shown a card by an experimenter. The card is black on one side and white on the other. The experimenter takes the card away, and places it on a table in an adjacent room. The card is placed black side up or white side up completely at random, with the randomisation occurring only after the experimenter has left the room with the participant. A second experimenter comes in and asks the participant which side of the card is now facing upwards. It’s purely a one-shot experiment. Each person sees only one card, and gives only one answer; and at no stage is the participant actually in contact with someone who knows the right answer. My data set, therefore, is very simple. I have asked the question of N people, and some number X of these people have given the correct response. To make things concrete, let’s suppose that I have tested N=100 people, and X=62 of these got the answer right… a surprisingly large number, sure, but is it large enough for me to feel safe in claiming I’ve found evidence for ESP? This is the situation where hypothesis testing comes in useful. However, before we talk about how to test hypotheses, we need to be clear about what we mean by hypotheses.
Research hypotheses versus statistical hypotheses
The first distinction that you need to keep clear in your mind is between research hypotheses and statistical hypotheses. In my ESP study, my overall scientific goal is to demonstrate that clairvoyance exists. In this situation, I have a clear research goal: I am hoping to discover evidence for ESP. In other situations I might actually be a lot more neutral than that, so I might say that my research goal is to determine whether or not clairvoyance exists. Regardless of how I want to portray myself, the basic point that I’m trying to convey here is that a research hypothesis involves making a substantive, testable scientific claim… if you are a psychologist, then your research hypotheses are fundamentally about psychological constructs. Any of the following would count as research hypotheses:
• Listening to music reduces your ability to pay attention to other things. This is a claim about the causal relationship between two psychologically meaningful concepts (listening to music and paying attention to things), so it’s a perfectly reasonable research hypothesis.
• Intelligence is related to personality. Like the last one, this is a relational claim about two psychological constructs (intelligence and personality), but the claim is weaker: correlational not causal.
• Intelligence is* speed of information processing. This hypothesis has a quite different character: it’s not actually a relational claim at all. It’s an ontological claim about the fundamental character of intelligence (and I’m pretty sure it’s wrong). It’s worth expanding on this one actually: It’s usually easier to think about how to construct experiments to test research hypotheses of the form “does X affect Y?” than it is to address claims like “what is X?” And in practice, what usually happens is that you find ways of testing relational claims that follow from your ontological ones. For instance, if I believe that intelligence is* speed of information processing in the brain, my experiments will often involve looking for relationships between measures of intelligence and measures of speed. As a consequence, most everyday research questions do tend to be relational in nature, but they’re almost always motivated by deeper ontological questions about the state of nature.
Notice that in practice, my research hypotheses could overlap a lot. My ultimate goal in the ESP experiment might be to test an ontological claim like “ESP exists”, but I might operationally restrict myself to a narrower hypothesis like “Some people can `see’ objects in a clairvoyant fashion”. That said, there are some things that really don’t count as proper research hypotheses in any meaningful sense:
• Love is a battlefield. This is too vague to be testable. While it’s okay for a research hypothesis to have a degree of vagueness to it, it has to be possible to operationalise your theoretical ideas. Maybe I’m just not creative enough to see it, but I can’t see how this can be converted into any concrete research design. If that’s true, then this isn’t a scientific research hypothesis, it’s a pop song. That doesn’t mean it’s not interesting – a lot of deep questions that humans have fall into this category. Maybe one day science will be able to construct testable theories of love, or to test to see if God exists, and so on; but right now we can’t, and I wouldn’t bet on ever seeing a satisfying scientific approach to either.
• The first rule of tautology club is the first rule of tautology club. This is not a substantive claim of any kind. It’s true by definition. No conceivable state of nature could possibly be inconsistent with this claim. As such, we say that this is an unfalsifiable hypothesis, and as such it is outside the domain of science. Whatever else you do in science, your claims must have the possibility of being wrong.
• More people in my experiment will say “yes” than “no”. This one fails as a research hypothesis because it’s a claim about the data set, not about the psychology (unless of course your actual research question is whether people have some kind of “yes” bias!). As we’ll see shortly, this hypothesis is starting to sound more like a statistical hypothesis than a research hypothesis.
As you can see, research hypotheses can be somewhat messy at times; and ultimately they are scientific claims. Statistical hypotheses are neither of these two things. Statistical hypotheses must be mathematically precise, and they must correspond to specific claims about the characteristics of the data generating mechanism (i.e., the “population”). Even so, the intent is that statistical hypotheses bear a clear relationship to the substantive research hypotheses that you care about! For instance, in my ESP study my research hypothesis is that some people are able to see through walls or whatever. What I want to do is to “map” this onto a statement about how the data were generated. So let’s think about what that statement would be. The quantity that I’m interested in within the experiment is P("correct"), the true-but-unknown probability with which the participants in my experiment answer the question correctly. Let’s use the Greek letter θ (theta) to refer to this probability. Here are four different statistical hypotheses:
• If ESP doesn’t exist and if my experiment is well designed, then my participants are just guessing. So I should expect them to get it right half of the time and so my statistical hypothesis is that the true probability of choosing correctly is θ=0.5.
• Alternatively, suppose ESP does exist and participants can see the card. If that’s true, people will perform better than chance. The statistical hypotheis would be that θ>0.5.
• A third possibility is that ESP does exist, but the colours are all reversed and people don’t realise it (okay, that’s wacky, but you never know…). If that’s how it works then you’d expect people’s performance to be below chance. This would correspond to a statistical hypothesis that θ<0.5.
• Finally, suppose ESP exists, but I have no idea whether people are seeing the right colour or the wrong one. In that case, the only claim I could make about the data would be that the probability of making the correct answer is not equal to 50. This corresponds to the statistical hypothesis that θ≠0.5.
All of these are legitimate examples of a statistical hypothesis because they are statements about a population parameter and are meaningfully related to my experiment.
What this discussion makes clear, I hope, is that when attempting to construct a statistical hypothesis test the researcher actually has two quite distinct hypotheses to consider. First, he or she has a research hypothesis (a claim about psychology), and this corresponds to a statistical hypothesis (a claim about the data generating population). In my ESP example, these might be
Dan.s.research.hypothesis Dan.s.statistical.hypothesis
ESP.exists θ≠0.5
And the key thing to recognise is this: a statistical hypothesis test is a test of the statistical hypothesis, not the research hypothesis. If your study is badly designed, then the link between your research hypothesis and your statistical hypothesis is broken. To give a silly example, suppose that my ESP study was conducted in a situation where the participant can actually see the card reflected in a window; if that happens, I would be able to find very strong evidence that θ≠0.5, but this would tell us nothing about whether “ESP exists”.
Null hypotheses and alternative hypotheses
So far, so good. I have a research hypothesis that corresponds to what I want to believe about the world, and I can map it onto a statistical hypothesis that corresponds to what I want to believe about how the data were generated. It’s at this point that things get somewhat counterintuitive for a lot of people. Because what I’m about to do is invent a new statistical hypothesis (the “null” hypothesis, H0) that corresponds to the exact opposite of what I want to believe, and then focus exclusively on that, almost to the neglect of the thing I’m actually interested in (which is now called the “alternative” hypothesis, H1). In our ESP example, the null hypothesis is that θ=0.5, since that’s what we’d expect if ESP didn’t exist. My hope, of course, is that ESP is totally real, and so the alternative to this null hypothesis is θ≠0.5. In essence, what we’re doing here is dividing up the possible values of θ into two groups: those values that I really hope aren’t true (the null), and those values that I’d be happy with if they turn out to be right (the alternative). Having done so, the important thing to recognise is that the goal of a hypothesis test is not to show that the alternative hypothesis is (probably) true; the goal is to show that the null hypothesis is (probably) false. Most people find this pretty weird.
The best way to think about it, in my experience, is to imagine that a hypothesis test is a criminal trial159the trial of the null hypothesis. The null hypothesis is the defendant, the researcher is the prosecutor, and the statistical test itself is the judge. Just like a criminal trial, there is a presumption of innocence: the null hypothesis is deemed to be true unless you, the researcher, can prove beyond a reasonable doubt that it is false. You are free to design your experiment however you like (within reason, obviously!), and your goal when doing so is to maximise the chance that the data will yield a conviction… for the crime of being false. The catch is that the statistical test sets the rules of the trial, and those rules are designed to protect the null hypothesis – specifically to ensure that if the null hypothesis is actually true, the chances of a false conviction are guaranteed to be low. This is pretty important: after all, the null hypothesis doesn’t get a lawyer. And given that the researcher is trying desperately to prove it to be false, someone has to protect it. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.01%3A_A_Menagerie_of_Hypotheses.txt |
Before going into details about how a statistical test is constructed, it’s useful to understand the philosophy behind it. I hinted at it when pointing out the similarity between a null hypothesis test and a criminal trial, but I should now be explicit. Ideally, we would like to construct our test so that we never make any errors. Unfortunately, since the world is messy, this is never possible. Sometimes you’re just really unlucky: for instance, suppose you flip a coin 10 times in a row and it comes up heads all 10 times. That feels like very strong evidence that the coin is biased (and it is!), but of course there’s a 1 in 1024 chance that this would happen even if the coin was totally fair. In other words, in real life we always have to accept that there’s a chance that we did the wrong thing. As a consequence, the goal behind statistical hypothesis testing is not to eliminate errors, but to minimise them.
At this point, we need to be a bit more precise about what we mean by “errors”. Firstly, let’s state the obvious: it is either the case that the null hypothesis is true, or it is false; and our test will either reject the null hypothesis or retain it.160 So, as the table below illustrates, after we run the test and make our choice, one of four things might have happened:
retain H0 retain H0
H0 is true correct decision error (type I)
H0 is false error (type II) correct decision
As a consequence there are actually two different types of error here. If we reject a null hypothesis that is actually true, then we have made a type I error. On the other hand, if we retain the null hypothesis when it is in fact false, then we have made a type II error.
Remember how I said that statistical testing was kind of like a criminal trial? Well, I meant it. A criminal trial requires that you establish “beyond a reasonable doubt” that the defendant did it. All of the evidentiary rules are (in theory, at least) designed to ensure that there’s (almost) no chance of wrongfully convicting an innocent defendant. The trial is designed to protect the rights of a defendant: as the English jurist William Blackstone famously said, it is “better that ten guilty persons escape than that one innocent suffer.” In other words, a criminal trial doesn’t treat the two types of error in the same way~… punishing the innocent is deemed to be much worse than letting the guilty go free. A statistical test is pretty much the same: the single most important design principle of the test is to control the probability of a type I error, to keep it below some fixed probability. This probability, which is denoted α, is called the significance level of the test (or sometimes, the size of the test). And I’ll say it again, because it is so central to the whole set-up~… a hypothesis test is said to have significance level α if the type I error rate is no larger than α.
So, what about the type II error rate? Well, we’d also like to keep those under control too, and we denote this probability by β. However, it’s much more common to refer to the power of the test, which is the probability with which we reject a null hypothesis when it really is false, which is 1−β. To help keep this straight, here’s the same table again, but with the relevant numbers added:
retain H0 reject H0
H0 is true 1−α (probability of correct retention) α (type I error rate)
H0 is false β (type II error rate) 1−β (power of the test)
A “powerful” hypothesis test is one that has a small value of β, while still keeping α fixed at some (small) desired level. By convention, scientists make use of three different α levels: .05, .01 and .001. Notice the asymmetry here~… the tests are designed to ensure that the α level is kept small, but there’s no corresponding guarantee regarding β. We’d certainly like the type II error rate to be small, and we try to design tests that keep it small, but this is very much secondary to the overwhelming need to control the type I error rate. As Blackstone might have said if he were a statistician, it is “better to retain 10 false null hypotheses than to reject a single true one”. To be honest, I don’t know that I agree with this philosophy – there are situations where I think it makes sense, and situations where I think it doesn’t – but that’s neither here nor there. It’s how the tests are built. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.02%3A_Two_Types_of_Errors.txt |
At this point we need to start talking specifics about how a hypothesis test is constructed. To that end, let’s return to the ESP example. Let’s ignore the actual data that we obtained, for the moment, and think about the structure of the experiment. Regardless of what the actual numbers are, the form of the data is that X out of N people correctly identified the colour of the hidden card. Moreover, let’s suppose for the moment that the null hypothesis really is true: ESP doesn’t exist, and the true probability that anyone picks the correct colour is exactly θ=0.5. What would we expect the data to look like? Well, obviously, we’d expect the proportion of people who make the correct response to be pretty close to 50%. Or, to phrase this in more mathematical terms, we’d say that X/N is approximately 0.5. Of course, we wouldn’t expect this fraction to be exactly 0.5: if, for example we tested N=100 people, and X=53 of them got the question right, we’d probably be forced to concede that the data are quite consistent with the null hypothesis. On the other hand, if X=99 of our participants got the question right, then we’d feel pretty confident that the null hypothesis is wrong. Similarly, if only X=3 people got the answer right, we’d be similarly confident that the null was wrong. Let’s be a little more technical about this: we have a quantity X that we can calculate by looking at our data; after looking at the value of X, we make a decision about whether to believe that the null hypothesis is correct, or to reject the null hypothesis in favour of the alternative. The name for this thing that we calculate to guide our choices is a test statistic.
Having chosen a test statistic, the next step is to state precisely which values of the test statistic would cause is to reject the null hypothesis, and which values would cause us to keep it. In order to do so, we need to determine what the sampling distribution of the test statistic would be if the null hypothesis were actually true (we talked about sampling distributions earlier in Section 10.3.1). Why do we need this? Because this distribution tells us exactly what values of X our null hypothesis would lead us to expect. And therefore, we can use this distribution as a tool for assessing how closely the null hypothesis agrees with our data.
How do we actually determine the sampling distribution of the test statistic? For a lot of hypothesis tests this step is actually quite complicated, and later on in the book you’ll see me being slightly evasive about it for some of the tests (some of them I don’t even understand myself). However, sometimes it’s very easy. And, fortunately for us, our ESP example provides us with one of the easiest cases. Our population parameter θ is just the overall probability that people respond correctly when asked the question, and our test statistic X is the count of the number of people who did so, out of a sample size of N. We’ve seen a distribution like this before, in Section 9.4: that’s exactly what the binomial distribution describes! So, to use the notation and terminology that I introduced in that section, we would say that the null hypothesis predicts that X is binomially distributed, which is written
X∼Binomial(θ,N)
Since the null hypothesis states that θ=0.5 and our experiment has N=100 people, we have the sampling distribution we need. This sampling distribution is plotted in Figure 11.1. No surprises really: the null hypothesis says that X=50 is the most likely outcome, and it says that we’re almost certain to see somewhere between 40 and 60 correct responses. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.03%3A_Test_Statistics_and_Sampling_Distributions.txt |
Okay, we’re very close to being finished. We’ve constructed a test statistic (X), and we chose this test statistic in such a way that we’re pretty confident that if X is close to N/2 then we should retain the null, and if not we should reject it. The question that remains is this: exactly which values of the test statistic should we associate with the null hypothesis, and which exactly values go with the alternative hypothesis? In my ESP study, for example, I’ve observed a value of X=62. What decision should I make? Should I choose to believe the null hypothesis, or the alternative hypothesis?
Critical regions and critical values
To answer this question, we need to introduce the concept of a critical region for the test statistic X. The critical region of the test corresponds to those values of X that would lead us to reject null hypothesis (which is why the critical region is also sometimes called the rejection region). How do we find this critical region? Well, let’s consider what we know:
• X should be very big or very small in order to reject the null hypothesis.
• If the null hypothesis is true, the sampling distribution of X is Binomial(0.5,N).
• If α=.05, the critical region must cover 5% of this sampling distribution.
It’s important to make sure you understand this last point: the critical region corresponds to those values of X for which we would reject the null hypothesis, and the sampling distribution in question describes the probability that we would obtain a particular value of X if the null hypothesis were actually true. Now, let’s suppose that we chose a critical region that covers 20% of the sampling distribution, and suppose that the null hypothesis is actually true. What would be the probability of incorrectly rejecting the null? The answer is of course 20%. And therefore, we would have built a test that had an α level of 0.2. If we want α=.05, the critical region is only allowed to cover 5% of the sampling distribution of our test statistic.
As it turns out, those three things uniquely solve the problem: our critical region consists of the most extreme values, known as the tails of the distribution. This is illustrated in Figure 11.2. As it turns out, if we want α=.05, then our critical regions correspond to X≤40 and X≥60.161 That is, if the number of people saying “true” is between 41 and 59, then we should retain the null hypothesis. If the number is between 0 to 40 or between 60 to 100, then we should reject the null hypothesis. The numbers 40 and 60 are often referred to as the critical values, since they define the edges of the critical region.
At this point, our hypothesis test is essentially complete: (1) we choose an α level (e.g., α=.05, (2) come up with some test statistic (e.g., X) that does a good job (in some meaningful sense) of comparing H0 to H1, (3) figure out the sampling distribution of the test statistic on the assumption that the null hypothesis is true (in this case, binomial) and then (4) calculate the critical region that produces an appropriate α level (0-40 and 60-100). All that we have to do now is calculate the value of the test statistic for the real data (e.g., X=62) and then compare it to the critical values to make our decision. Since 62 is greater than the critical value of 60, we would reject the null hypothesis. Or, to phrase it slightly differently, we say that the test has produced a significant result.
note on statistical “significance”
Like other occult techniques of divination, the statistical method has a private jargon deliberately contrived to obscure its methods from non-practitioners.
– Attributed to G. O. Ashley162
A very brief digression is in order at this point, regarding the word “significant”. The concept of statistical significance is actually a very simple one, but has a very unfortunate name. If the data allow us to reject the null hypothesis, we say that “the result is statistically significant”, which is often shortened to “the result is significant”. This terminology is rather old, and dates back to a time when “significant” just meant something like “indicated”, rather than its modern meaning, which is much closer to “important”. As a result, a lot of modern readers get very confused when they start learning statistics, because they think that a “significant result” must be an important one. It doesn’t mean that at all. All that “statistically significant” means is that the data allowed us to reject a null hypothesis. Whether or not the result is actually important in the real world is a very different question, and depends on all sorts of other things.
difference between one sided and two sided tests
There’s one more thing I want to point out about the hypothesis test that I’ve just constructed. If we take a moment to think about the statistical hypotheses I’ve been using,
H0:θ=.5
H1:θ≠.5
we notice that the alternative hypothesis covers both the possibility that θ<.5 and the possibility that θ>.5. This makes sense if I really think that ESP could produce better-than-chance performance or worse-than-chance performance (and there are some people who think that). In statistical language, this is an example of a two-sided test. It’s called this because the alternative hypothesis covers the area on both “sides” of the null hypothesis, and as a consequence the critical region of the test covers both tails of the sampling distribution (2.5% on either side if α=.05), as illustrated earlier in Figure 11.2.
However, that’s not the only possibility. It might be the case, for example, that I’m only willing to believe in ESP if it produces better than chance performance. If so, then my alternative hypothesis would only covers the possibility that θ>.5, and as a consequence the null hypothesis now becomes θ≤.5:
H0:θ≤.5
H1:θ>.5
When this happens, we have what’s called a one-sided test, and when this happens the critical region only covers one tail of the sampling distribution. This is illustrated in Figure 11.3. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.04%3A_Making_Decisions.txt |
In one sense, our hypothesis test is complete; we’ve constructed a test statistic, figured out its sampling distribution if the null hypothesis is true, and then constructed the critical region for the test. Nevertheless, I’ve actually omitted the most important number of all: the p value. It is to this topic that we now turn. There are two somewhat different ways of interpreting a p value, one proposed by Sir Ronald Fisher and the other by Jerzy Neyman. Both versions are legitimate, though they reflect very different ways of thinking about hypothesis tests. Most introductory textbooks tend to give Fisher’s version only, but I think that’s a bit of a shame. To my mind, Neyman’s version is cleaner, and actually better reflects the logic of the null hypothesis test. You might disagree though, so I’ve included both. I’ll start with Neyman’s version…
softer view of decision making
One problem with the hypothesis testing procedure that I’ve described is that it makes no distinction at all between a result this “barely significant” and those that are “highly significant”. For instance, in my ESP study the data I obtained only just fell inside the critical region - so I did get a significant effect, but was a pretty near thing. In contrast, suppose that I’d run a study in which X=97 out of my N=100 participants got the answer right. This would obviously be significant too, but my a much larger margin; there’s really no ambiguity about this at all. The procedure that I described makes no distinction between the two. If I adopt the standard convention of allowing α=.05 as my acceptable Type I error rate, then both of these are significant results.
This is where the p value comes in handy. To understand how it works, let’s suppose that we ran lots of hypothesis tests on the same data set: but with a different value of α in each case. When we do that for my original ESP data, what we’d get is something like this
Value of α Reject the null?
0.05 Yes
0.04 Yes
0.03 Yes
0.02 No
0.01 No
When we test ESP data (X=62 successes out of N=100 observations) using α levels of .03 and above, we’d always find ourselves rejecting the null hypothesis. For α levels of .02 and below, we always end up retaining the null hypothesis. Therefore, somewhere between .02 and .03 there must be a smallest value of α that would allow us to reject the null hypothesis for this data. This is the p value; as it turns out the ESP data has p=.021. In short:
p is defined to be the smallest Type I error rate (α) that you have to be willing to tolerate if you want to reject the null hypothesis.
If it turns out that p describes an error rate that you find intolerable, then you must retain the null. If you’re comfortable with an error rate equal to p, then it’s okay to reject the null hypothesis in favour of your preferred alternative.
In effect, p is a summary of all the possible hypothesis tests that you could have run, taken across all possible α values. And as a consequence it has the effect of “softening” our decision process. For those tests in which p≤α you would have rejected the null hypothesis, whereas for those tests in which p>α you would have retained the null. In my ESP study I obtained X=62, and as a consequence I’ve ended up with p=.021. So the error rate I have to tolerate is 2.1%. In contrast, suppose my experiment had yielded X=97. What happens to my p value now? This time it’s shrunk to p=1.36×10−25, which is a tiny, tiny163 Type I error rate. For this second case I would be able to reject the null hypothesis with a lot more confidence, because I only have to be “willing” to tolerate a type I error rate of about 1 in 10 trillion trillion in order to justify my decision to reject.
probability of extreme data
The second definition of the p-value comes from Sir Ronald Fisher, and it’s actually this one that you tend to see in most introductory statistics textbooks. Notice how, when I constructed the critical region, it corresponded to the tails (i.e., extreme values) of the sampling distribution? That’s not a coincidence: almost all “good” tests have this characteristic (good in the sense of minimising our type II error rate, β). The reason for that is that a good critical region almost always corresponds to those values of the test statistic that are least likely to be observed if the null hypothesis is true. If this rule is true, then we can define the p-value as the probability that we would have observed a test statistic that is at least as extreme as the one we actually did get. In other words, if the data are extremely implausible according to the null hypothesis, then the null hypothesis is probably wrong.
common mistake
Okay, so you can see that there are two rather different but legitimate ways to interpret the p value, one based on Neyman’s approach to hypothesis testing and the other based on Fisher’s. Unfortunately, there is a third explanation that people sometimes give, especially when they’re first learning statistics, and it is absolutely and completely wrong. This mistaken approach is to refer to the p value as “the probability that the null hypothesis is true”. It’s an intuitively appealing way to think, but it’s wrong in two key respects: (1) null hypothesis testing is a frequentist tool, and the frequentist approach to probability does not allow you to assign probabilities to the null hypothesis… according to this view of probability, the null hypothesis is either true or it is not; it cannot have a “5% chance” of being true. (2) even within the Bayesian approach, which does let you assign probabilities to hypotheses, the p value would not correspond to the probability that the null is true; this interpretation is entirely inconsistent with the mathematics of how the p value is calculated. Put bluntly, despite the intuitive appeal of thinking this way, there is no justification for interpreting a p value this way. Never do it. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.05%3A_The___p___value_of_a_test.txt |
When writing up the results of a hypothesis test, there’s usually several pieces of information that you need to report, but it varies a fair bit from test to test. Throughout the rest of the book I’ll spend a little time talking about how to report the results of different tests (see Section 12.1.9 for a particularly detailed example), so that you can get a feel for how it’s usually done. However, regardless of what test you’re doing, the one thing that you always have to do is say something about the p value, and whether or not the outcome was significant.
The fact that you have to do this is unsurprising; it’s the whole point of doing the test. What might be surprising is the fact that there is some contention over exactly how you’re supposed to do it. Leaving aside those people who completely disagree with the entire framework underpinning null hypothesis testing, there’s a certain amount of tension that exists regarding whether or not to report the exact p value that you obtained, or if you should state only that p<α for a significance level that you chose in advance (e.g., p<.05).
issue
To see why this is an issue, the key thing to recognise is that p values are terribly convenient. In practice, the fact that we can compute a p value means that we don’t actually have to specify any α level at all in order to run the test. Instead, what you can do is calculate your p value and interpret it directly: if you get p=.062, then it means that you’d have to be willing to tolerate a Type I error rate of 6.2% to justify rejecting the null. If you personally find 6.2% intolerable, then you retain the null. Therefore, the argument goes, why don’t we just report the actual p value and let the reader make up their own minds about what an acceptable Type I error rate is? This approach has the big advantage of “softening” the decision making process – in fact, if you accept the Neyman definition of the p value, that’s the whole point of the p value. We no longer have a fixed significance level of α=.05 as a bright line separating “accept” from “reject” decisions; and this removes the rather pathological problem of being forced to treat p=.051 in a fundamentally different way to p=.049.
This flexibility is both the advantage and the disadvantage to the p value. The reason why a lot of people don’t like the idea of reporting an exact p value is that it gives the researcher a bit too much freedom. In particular, it lets you change your mind about what error tolerance you’re willing to put up with after you look at the data. For instance, consider my ESP experiment. Suppose I ran my test, and ended up with a p value of .09. Should I accept or reject? Now, to be honest, I haven’t yet bothered to think about what level of Type I error I’m “really” willing to accept. I don’t have an opinion on that topic. But I do have an opinion about whether or not ESP exists, and I definitely have an opinion about whether my research should be published in a reputable scientific journal. And amazingly, now that I’ve looked at the data I’m starting to think that a 9% error rate isn’t so bad, especially when compared to how annoying it would be to have to admit to the world that my experiment has failed. So, to avoid looking like I just made it up after the fact, I now say that my α is .1: a 10% type I error rate isn’t too bad, and at that level my test is significant! I win.
In other words, the worry here is that I might have the best of intentions, and be the most honest of people, but the temptation to just “shade” things a little bit here and there is really, really strong. As anyone who has ever run an experiment can attest, it’s a long and difficult process, and you often get very attached to your hypotheses. It’s hard to let go and admit the experiment didn’t find what you wanted it to find. And that’s the danger here. If we use the “raw” p-value, people will start interpreting the data in terms of what they want to believe, not what the data are actually saying… and if we allow that, well, why are we bothering to do science at all? Why not let everyone believe whatever they like about anything, regardless of what the facts are? Okay, that’s a bit extreme, but that’s where the worry comes from. According to this view, you really must specify your α value in advance, and then only report whether the test was significant or not. It’s the only way to keep ourselves honest.
proposed solutions
In practice, it’s pretty rare for a researcher to specify a single α level ahead of time. Instead, the convention is that scientists rely on three standard significance levels: .05, .01 and .001. When reporting your results, you indicate which (if any) of these significance levels allow you to reject the null hypothesis. This is summarised in Table 11.1. This allows us to soften the decision rule a little bit, since p<.01 implies that the data meet a stronger evidentiary standard than p<.05 would. Nevertheless, since these levels are fixed in advance by convention, it does prevent people choosing their α level after looking at the data.
Table 11.1: A commonly adopted convention for reporting p values: in many places it is conventional to report one of four different things (e.g., p<.05) as shown below. I’ve included the “significance stars” notation (i.e., a * indicates p<.05) because you sometimes see this notation produced by statistical software. It’s also worth noting that some people will write n.s. (not significant) rather than p>.05.
Usual notation Signif. stars Signif. stars The null is…
p>.05 NA The test wasn’t significant Retained
p<.05 * The test was significant at \$= .05 but not at α=.01 or α=.001.\$ Rejected
p<.01 ** The test was significant at α=.05 and α=.01 but not at \$= .001 Rejected
p<.001 *** The test was significant at all levels Rejected
Nevertheless, quite a lot of people still prefer to report exact p values. To many people, the advantage of allowing the reader to make up their own mind about how to interpret p=.06 outweighs any disadvantages. In practice, however, even among those researchers who prefer exact p values it is quite common to just write p<.001 instead of reporting an exact value for small p. This is in part because a lot of software doesn’t actually print out the p value when it’s that small (e.g., SPSS just writes p=.000 whenever p<.001), and in part because a very small p value can be kind of misleading. The human mind sees a number like .0000000001 and it’s hard to suppress the gut feeling that the evidence in favour of the alternative hypothesis is a near certainty. In practice however, this is usually wrong. Life is a big, messy, complicated thing: and every statistical test ever invented relies on simplifications, approximations and assumptions. As a consequence, it’s probably not reasonable to walk away from any statistical analysis with a feeling of confidence stronger than p<.001 implies. In other words, p<.001 is really code for “as far as this test is concerned, the evidence is overwhelming.”
In light of all this, you might be wondering exactly what you should do. There’s a fair bit of contradictory advice on the topic, with some people arguing that you should report the exact p value, and other people arguing that you should use the tiered approach illustrated in Table 11.1. As a result, the best advice I can give is to suggest that you look at papers/reports written in your field and see what the convention seems to be. If there doesn’t seem to be any consistent pattern, then use whichever method you prefer.
11.07: Running the Hypothesis Test in Practice
At this point some of you might be wondering if this is a “real” hypothesis test, or just a toy example that I made up. It’s real. In the previous discussion I built the test from first principles, thinking that it was the simplest possible problem that you might ever encounter in real life. However, this test already exists: it’s called the binomial test, and it’s implemented by an R function called `binom.test()`. To test the null hypothesis that the response probability is one-half `p = .5`,164 using data in which `x = 62` of `n = 100` people made the correct response, here’s how to do it in R:
``binom.test( x=62, n=100, p=.5 )``
``````##
## Exact binomial test
##
## data: 62 and 100
## number of successes = 62, number of trials = 100, p-value =
## 0.02098
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
## 0.5174607 0.7152325
## sample estimates:
## probability of success
## 0.62``````
Right now, this output looks pretty unfamiliar to you, but you can see that it’s telling you more or less the right things. Specifically, the p-value of 0.02 is less than the usual choice of α=.05, so you can reject the null. We’ll talk a lot more about how to read this sort of output as we go along; and after a while you’ll hopefully find it quite easy to read and understand. For now, however, I just wanted to make the point that R contains a whole lot of functions corresponding to different kinds of hypothesis test. And while I’ll usually spend quite a lot of time explaining the logic behind how the tests are built, every time I discuss a hypothesis test the discussion will end with me showing you a fairly simple R command that you can use to run the test in practice. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.06%3A_Reporting_the_Results_of_a_Hypothesis_Test.txt |
In previous sections I’ve emphasised the fact that the major design principle behind statistical hypothesis testing is that we try to control our Type I error rate. When we fix α=.05 we are attempting to ensure that only 5% of true null hypotheses are incorrectly rejected. However, this doesn’t mean that we don’t care about Type II errors. In fact, from the researcher’s perspective, the error of failing to reject the null when it is actually false is an extremely annoying one. With that in mind, a secondary goal of hypothesis testing is to try to minimise β, the Type II error rate, although we don’t usually talk in terms of minimising Type II errors. Instead, we talk about maximising the power of the test. Since power is defined as 1−β, this is the same thing.
power function
Let’s take a moment to think about what a Type II error actually is. A Type II error occurs when the alternative hypothesis is true, but we are nevertheless unable to reject the null hypothesis. Ideally, we’d be able to calculate a single number β that tells us the Type II error rate, in the same way that we can set α=.05 for the Type I error rate. Unfortunately, this is a lot trickier to do. To see this, notice that in my ESP study the alternative hypothesis actually corresponds to lots of possible values of θ. In fact, the alternative hypothesis corresponds to every value of θ except 0.5. Let’s suppose that the true probability of someone choosing the correct response is 55% (i.e., θ=.55). If so, then the true sampling distribution for X is not the same one that the null hypothesis predicts: the most likely value for X is now 55 out of 100. Not only that, the whole sampling distribution has now shifted, as shown in Figure 11.4. The critical regions, of course, do not change: by definition, the critical regions are based on what the null hypothesis predicts. What we’re seeing in this figure is the fact that when the null hypothesis is wrong, a much larger proportion of the sampling distribution distribution falls in the critical region. And of course that’s what should happen: the probability of rejecting the null hypothesis is larger when the null hypothesis is actually false! However θ=.55 is not the only possibility consistent with the alternative hypothesis. Let’s instead suppose that the true value of θ is actually 0.7. What happens to the sampling distribution when this occurs? The answer, shown in Figure 11.5, is that almost the entirety of the sampling distribution has now moved into the critical region. Therefore, if θ=0.7 the probability of us correctly rejecting the null hypothesis (i.e., the power of the test) is much larger than if θ=0.55. In short, while θ=.55 and θ=.70 are both part of the alternative hypothesis, the Type II error rate is different.
What all this means is that the power of a test (i.e., 1−β) depends on the true value of θ. To illustrate this, I’ve calculated the expected probability of rejecting the null hypothesis for all values of θ, and plotted it in Figure 11.6. This plot describes what is usually called the power function of the test. It’s a nice summary of how good the test is, because it actually tells you the power (1−β) for all possible values of θ. As you can see, when the true value of θ is very close to 0.5, the power of the test drops very sharply, but when it is further away, the power is large.
Effect size
Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned with mice when there are tigers abroad
– George Box 1976
The plot shown in Figure 11.6 captures a fairly basic point about hypothesis testing. If the true state of the world is very different from what the null hypothesis predicts, then your power will be very high; but if the true state of the world is similar to the null (but not identical) then the power of the test is going to be very low. Therefore, it’s useful to be able to have some way of quantifying how “similar” the true state of the world is to the null hypothesis. A statistic that does this is called a measure of effect size (e.g. Cohen 1988; Ellis 2010). Effect size is defined slightly differently in different contexts,165 (and so this section just talks in general terms) but the qualitative idea that it tries to capture is always the same: how big is the difference between the true population parameters, and the parameter values that are assumed by the null hypothesis? In our ESP example, if we let θ0=0.5 denote the value assumed by the null hypothesis, and let θ denote the true value, then a simple measure of effect size could be something like the difference between the true value and null (i.e., θ−θ0), or possibly just the magnitude of this difference, abs(θ−θ0).
big effect size small effect size
significant result difference is real, and of practical importance difference is real, but might not be interesting
non-significant result no effect observed no effect observed
Why calculate effect size? Let’s assume that you’ve run your experiment, collected the data, and gotten a significant effect when you ran your hypothesis test. Isn’t it enough just to say that you’ve gotten a significant effect? Surely that’s the point of hypothesis testing? Well, sort of. Yes, the point of doing a hypothesis test is to try to demonstrate that the null hypothesis is wrong, but that’s hardly the only thing we’re interested in. If the null hypothesis claimed that θ=.5, and we show that it’s wrong, we’ve only really told half of the story. Rejecting the null hypothesis implies that we believe that θ≠.5, but there’s a big difference between θ=.51 and θ=.8. If we find that θ=.8, then not only have we found that the null hypothesis is wrong, it appears to be very wrong. On the other hand, suppose we’ve successfully rejected the null hypothesis, but it looks like the true value of θ is only .51 (this would only be possible with a large study). Sure, the null hypothesis is wrong, but it’s not at all clear that we actually care, because the effect size is so small. In the context of my ESP study we might still care, since any demonstration of real psychic powers would actually be pretty cool166, but in other contexts a 1% difference isn’t very interesting, even if it is a real difference. For instance, suppose we’re looking at differences in high school exam scores between males and females, and it turns out that the female scores are 1% higher on average than the males. If I’ve got data from thousands of students, then this difference will almost certainly be statistically significant, but regardless of how small the p value is it’s just not very interesting. You’d hardly want to go around proclaiming a crisis in boys education on the basis of such a tiny difference would you? It’s for this reason that it is becoming more standard (slowly, but surely) to report some kind of standard measure of effect size along with the the results of the hypothesis test. The hypothesis test itself tells you whether you should believe that the effect you have observed is real (i.e., not just due to chance); the effect size tells you whether or not you should care.
Increasing the power of your study
Not surprisingly, scientists are fairly obsessed with maximising the power of their experiments. We want our experiments to work, and so we want to maximise the chance of rejecting the null hypothesis if it is false (and of course we usually want to believe that it is false!) As we’ve seen, one factor that influences power is the effect size. So the first thing you can do to increase your power is to increase the effect size. In practice, what this means is that you want to design your study in such a way that the effect size gets magnified. For instance, in my ESP study I might believe that psychic powers work best in a quiet, darkened room; with fewer distractions to cloud the mind. Therefore I would try to conduct my experiments in just such an environment: if I can strengthen people’s ESP abilities somehow, then the true value of θ will go up167 and therefore my effect size will be larger. In short, clever experimental design is one way to boost power; because it can alter the effect size.
Unfortunately, it’s often the case that even with the best of experimental designs you may have only a small effect. Perhaps, for example, ESP really does exist, but even under the best of conditions it’s very very weak. Under those circumstances, your best bet for increasing power is to increase the sample size. In general, the more observations that you have available, the more likely it is that you can discriminate between two hypotheses. If I ran my ESP experiment with 10 participants, and 7 of them correctly guessed the colour of the hidden card, you wouldn’t be terribly impressed. But if I ran it with 10,000 participants and 7,000 of them got the answer right, you would be much more likely to think I had discovered something. In other words, power increases with the sample size. This is illustrated in Figure 11.7, which shows the power of the test for a true parameter of θ=0.7, for all sample sizes N from 1 to 100, where I’m assuming that the null hypothesis predicts that θ0=0.5.
``````## [1] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.11837800
## [7] 0.08257300 0.05771362 0.19643626 0.14945203 0.11303734 0.25302172
## [13] 0.20255096 0.16086106 0.29695959 0.24588947 0.38879291 0.33269435
## [19] 0.28223844 0.41641377 0.36272868 0.31341925 0.43996501 0.38859619
## [25] 0.51186665 0.46049782 0.41129777 0.52752694 0.47870819 0.58881596
## [31] 0.54162450 0.49507894 0.59933871 0.55446069 0.65155826 0.60907715
## [37] 0.69828554 0.65867614 0.61815357 0.70325017 0.66542910 0.74296156
## [43] 0.70807163 0.77808343 0.74621569 0.71275488 0.78009449 0.74946571
## [49] 0.81000236 0.78219322 0.83626633 0.81119597 0.78435605 0.83676444
## [55] 0.81250680 0.85920268 0.83741123 0.87881491 0.85934395 0.83818214
## [61] 0.87858194 0.85962510 0.89539581 0.87849413 0.91004390 0.89503851
## [67] 0.92276845 0.90949768 0.89480727 0.92209753 0.90907263 0.93304809
## [73] 0.92153987 0.94254237 0.93240638 0.92108426 0.94185449 0.93185881
## [79] 0.95005094 0.94125189 0.95714694 0.94942195 0.96327866 0.95651332
## [85] 0.94886329 0.96265653 0.95594208 0.96796884 0.96208909 0.97255504
## [91] 0.96741721 0.97650832 0.97202770 0.97991117 0.97601093 0.97153910
## [97] 0.97944717 0.97554675 0.98240749 0.97901142``````
Because power is important, whenever you’re contemplating running an experiment it would be pretty useful to know how much power you’re likely to have. It’s never possible to know for sure, since you can’t possibly know what your effect size is. However, it’s often (well, sometimes) possible to guess how big it should be. If so, you can guess what sample size you need! This idea is called power analysis, and if it’s feasible to do it, then it’s very helpful, since it can tell you something about whether you have enough time or money to be able to run the experiment successfully. It’s increasingly common to see people arguing that power analysis should be a required part of experimental design, so it’s worth knowing about. I don’t discuss power analysis in this book, however. This is partly for a boring reason and partly for a substantive one. The boring reason is that I haven’t had time to write about power analysis yet. The substantive one is that I’m still a little suspicious of power analysis. Speaking as a researcher, I have very rarely found myself in a position to be able to do one – it’s either the case that (a) my experiment is a bit non-standard and I don’t know how to define effect size properly, (b) I literally have so little idea about what the effect size will be that I wouldn’t know how to interpret the answers. Not only that, after extensive conversations with someone who does stats consulting for a living (my wife, as it happens), I can’t help but notice that in practice the only time anyone ever asks her for a power analysis is when she’s helping someone write a grant application. In other words, the only time any scientist ever seems to want a power analysis in real life is when they’re being forced to do it by bureaucratic process. It’s not part of anyone’s day to day work. In short, I’ve always been of the view that while power is an important concept, power analysis is not as useful as people make it sound, except in the rare cases where (a) someone has figured out how to calculate power for your actual experimental design and (b) you have a pretty good idea what the effect size is likely to be. Maybe other people have had better experiences than me, but I’ve personally never been in a situation where both (a) and (b) were true. Maybe I’ll be convinced otherwise in the future, and probably a future version of this book would include a more detailed discussion of power analysis, but for now this is about as much as I’m comfortable saying about the topic. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.08%3A_Effect_Size_Sample_Size_and_Power.txt |
What I’ve described to you in this chapter is the orthodox framework for null hypothesis significance testing (NHST). Understanding how NHST works is an absolute necessity, since it has been the dominant approach to inferential statistics ever since it came to prominence in the early 20th century. It’s what the vast majority of working scientists rely on for their data analysis, so even if you hate it you need to know it. However, the approach is not without problems. There are a number of quirks in the framework, historical oddities in how it came to be, theoretical disputes over whether or not the framework is right, and a lot of practical traps for the unwary. I’m not going to go into a lot of detail on this topic, but I think it’s worth briefly discussing a few of these issues.
Neyman versus Fisher
The first thing you should be aware of is that orthodox NHST is actually a mash-up of two rather different approaches to hypothesis testing, one proposed by Sir Ronald Fisher and the other proposed by Jerzy Neyman (for a historical summary see Lehmann 2011). The history is messy because Fisher and Neyman were real people whose opinions changed over time, and at no point did either of them offer “the definitive statement” of how we should interpret their work many decades later. That said, here’s a quick summary of what I take these two approaches to be.
First, let’s talk about Fisher’s approach. As far as I can tell, Fisher assumed that you only had the one hypothesis (the null), and what you want to do is find out if the null hypothesis is inconsistent with the data. From his perspective, what you should do is check to see if the data are “sufficiently unlikely” according to the null. In fact, if you remember back to our earlier discussion, that’s how Fisher defines the p-value. According to Fisher, if the null hypothesis provided a very poor account of the data, you could safely reject it. But, since you don’t have any other hypotheses to compare it to, there’s no way of “accepting the alternative” because you don’t necessarily have an explicitly stated alternative. That’s more or less all that there was to it.
In contrast, Neyman thought that the point of hypothesis testing was as a guide to action, and his approach was somewhat more formal than Fisher’s. His view was that there are multiple things that you could do (accept the null or accept the alternative) and the point of the test was to tell you which one the data support. From this perspective, it is critical to specify your alternative hypothesis properly. If you don’t know what the alternative hypothesis is, then you don’t know how powerful the test is, or even which action makes sense. His framework genuinely requires a competition between different hypotheses. For Neyman, the p value didn’t directly measure the probability of the data (or data more extreme) under the null, it was more of an abstract description about which “possible tests” were telling you to accept the null, and which “possible tests” were telling you to accept the alternative.
As you can see, what we have today is an odd mishmash of the two. We talk about having both a null hypothesis and an alternative (Neyman), but usually168 define the p value in terms of exreme data (Fisher), but we still have α values (Neyman). Some of the statistical tests have explicitly specified alternatives (Neyman) but others are quite vague about it (Fisher). And, according to some people at least, we’re not allowed to talk about accepting the alternative (Fisher). It’s a mess: but I hope this at least explains why it’s a mess.
Bayesians versus frequentists
Earlier on in this chapter I was quite emphatic about the fact that you cannot interpret the p value as the probability that the null hypothesis is true. NHST is fundamentally a frequentist tool (see Chapter 9) and as such it does not allow you to assign probabilities to hypotheses: the null hypothesis is either true or it is not. The Bayesian approach to statistics interprets probability as a degree of belief, so it’s totally okay to say that there is a 10% chance that the null hypothesis is true: that’s just a reflection of the degree of confidence that you have in this hypothesis. You aren’t allowed to do this within the frequentist approach. Remember, if you’re a frequentist, a probability can only be defined in terms of what happens after a large number of independent replications (i.e., a long run frequency). If this is your interpretation of probability, talking about the “probability” that the null hypothesis is true is complete gibberish: a null hypothesis is either true or it is false. There’s no way you can talk about a long run frequency for this statement. To talk about “the probability of the null hypothesis” is as meaningless as “the colour of freedom”. It doesn’t have one!
Most importantly, this isn’t a purely ideological matter. If you decide that you are a Bayesian and that you’re okay with making probability statements about hypotheses, you have to follow the Bayesian rules for calculating those probabilities. I’ll talk more about this in Chapter 17, but for now what I want to point out to you is the p value is a terrible approximation to the probability that H0 is true. If what you want to know is the probability of the null, then the p value is not what you’re looking for!
Traps
As you can see, the theory behind hypothesis testing is a mess, and even now there are arguments in statistics about how it “should” work. However, disagreements among statisticians are not our real concern here. Our real concern is practical data analysis. And while the “orthodox” approach to null hypothesis significance testing has many drawbacks, even an unrepentant Bayesian like myself would agree that they can be useful if used responsibly. Most of the time they give sensible answers, and you can use them to learn interesting things. Setting aside the various ideologies and historical confusions that we’ve discussed, the fact remains that the biggest danger in all of statistics is thoughtlessness. I don’t mean stupidity, here: I literally mean thoughtlessness. The rush to interpret a result without spending time thinking through what each test actually says about the data, and checking whether that’s consistent with how you’ve interpreted it. That’s where the biggest trap lies.
To give an example of this, consider the following example (see Gelman and Stern 2006). Suppose I’m running my ESP study, and I’ve decided to analyse the data separately for the male participants and the female participants. Of the male participants, 33 out of 50 guessed the colour of the card correctly. This is a significant effect (p=.03). Of the female participants, 29 out of 50 guessed correctly. This is not a significant effect (p=.32). Upon observing this, it is extremely tempting for people to start wondering why there is a difference between males and females in terms of their psychic abilities. However, this is wrong. If you think about it, we haven’t actually run a test that explicitly compares males to females. All we have done is compare males to chance (binomial test was significant) and compared females to chance (binomial test was non significant). If we want to argue that there is a real difference between the males and the females, we should probably run a test of the null hypothesis that there is no difference! We can do that using a different hypothesis test,169 but when we do that it turns out that we have no evidence that males and females are significantly different (p=.54). Now do you think that there’s anything fundamentally different between the two groups? Of course not. What’s happened here is that the data from both groups (male and female) are pretty borderline: by pure chance, one of them happened to end up on the magic side of the p=.05 line, and the other one didn’t. That doesn’t actually imply that males and females are different. This mistake is so common that you should always be wary of it: the difference between significant and not-significant is not evidence of a real difference – if you want to say that there’s a difference between two groups, then you have to test for that difference!
The example above is just that: an example. I’ve singled it out because it’s such a common one, but the bigger picture is that data analysis can be tricky to get right. Think about what it is you want to test, why you want to test it, and whether or not the answers that your test gives could possibly make any sense in the real world. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.09%3A_Some_Issues_to_Consider.txt |
Null hypothesis testing is one of the most ubiquitous elements to statistical theory. The vast majority of scientific papers report the results of some hypothesis test or another. As a consequence it is almost impossible to get by in science without having at least a cursory understanding of what a p-value means, making this one of the most important chapters in the book. As usual, I’ll end the chapter with a quick recap of the key ideas that we’ve talked about:
• Research hypotheses and statistical hypotheses. Null and alternative hypotheses. (Section 11.1).
• Type 1 and Type 2 errors (Section 11.2)
• Test statistics and sampling distributions (Section 11.3)
• Hypothesis testing as a decision making process (Section 11.4)
• p-values as “soft” decisions (Section 11.5)
• Writing up the results of a hypothesis test (Section 11.6)
• Effect size and power (Section 11.8)
• A few issues to consider regarding hypothesis testing (Section 11.9)
Later in the book, in Chapter 17, I’ll revisit the theory of null hypothesis tests from a Bayesian perspective, and introduce a number of new tools that you can use if you aren’t particularly fond of the orthodox approach. But for now, though, we’re done with the abstract statistical theory, and we can start discussing specific data analysis tools.
References
Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Lawrence Erlbaum.
Ellis, P. D. 2010. The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results. Cambridge, UK: Cambridge University Press.
Lehmann, Erich L. 2011. Fisher, Neyman, and the Creation of Classical Statistics. Springer.
Gelman, A., and H. Stern. 2006. “The Difference Between ‘Significant’ and ‘Not Significant’ Is Not Itself Statistically Significant.” The American Statistician 60: 328–31.
1. The quote comes from Wittgenstein’s (1922) text, Tractatus Logico-Philosphicus.
2. A technical note. The description below differs subtly from the standard description given in a lot of introductory texts. The orthodox theory of null hypothesis testing emerged from the work of Sir Ronald Fisher and Jerzy Neyman in the early 20th century; but Fisher and Neyman actually had very different views about how it should work. The standard treatment of hypothesis testing that most texts use is a hybrid of the two approaches. The treatment here is a little more Neyman-style than the orthodox view, especially as regards the meaning of the p value.
3. My apologies to anyone who actually believes in this stuff, but on my reading of the literature on ESP, it’s just not reasonable to think this is real. To be fair, though, some of the studies are rigorously designed; so it’s actually an interesting area for thinking about psychological research design. And of course it’s a free country, so you can spend your own time and effort proving me wrong if you like, but I wouldn’t think that’s a terribly practical use of your intellect.
4. This analogy only works if you’re from an adversarial legal system like UK/US/Australia. As I understand these things, the French inquisitorial system is quite different.
5. An aside regarding the language you use to talk about hypothesis testing. Firstly, one thing you really want to avoid is the word “prove”: a statistical test really doesn’t prove that a hypothesis is true or false. Proof implies certainty, and as the saying goes, statistics means never having to say you’re certain. On that point almost everyone would agree. However, beyond that there’s a fair amount of confusion. Some people argue that you’re only allowed to make statements like “rejected the null”, “failed to reject the null”, or possibly “retained the null”. According to this line of thinking, you can’t say things like “accept the alternative” or “accept the null”. Personally I think this is too strong: in my opinion, this conflates null hypothesis testing with Karl Popper’s falsificationist view of the scientific process. While there are similarities between falsificationism and null hypothesis testing, they aren’t equivalent. However, while I personally think it’s fine to talk about accepting a hypothesis (on the proviso that “acceptance” doesn’t actually mean that it’s necessarily true, especially in the case of the null hypothesis), many people will disagree. And more to the point, you should be aware that this particular weirdness exists, so that you’re not caught unawares by it when writing up your own results.
6. Strictly speaking, the test I just constructed has α=.057, which is a bit too generous. However, if I’d chosen 39 and 61 to be the boundaries for the critical region, then the critical region only covers 3.5% of the distribution. I figured that it makes more sense to use 40 and 60 as my critical values, and be willing to tolerate a 5.7% type I error rate, since that’s as close as I can get to a value of α=.05.
7. The internet seems fairly convinced that Ashley said this, though I can’t for the life of me find anyone willing to give a source for the claim.
8. That’s p=.000000000000000000000000136 for folks that don’t like scientific notation!
9. Note that the `p` here has nothing to do with a p value. The `p` argument in the `binom.test()` function corresponds to the probability of making a correct response, according to the null hypothesis. In other words, it’s the θ value.
10. There’s an R package called `compute.es` that can be used for calculating a very broad range of effect size measures; but for the purposes of the current book we won’t need it: all of the effect size measures that I’ll talk about here have functions in the `lsr` package
11. Although in practice a very small effect size is worrying, because even very minor methodological flaws might be responsible for the effect; and in practice no experiment is perfect, so there are always methodological issues to worry about.
12. Notice that the true population parameter θ doesn’t necessarily correspond to an immutable fact of nature. In this context θ is just the true probability that people would correctly guess the colour of the card in the other room. As such the population parameter can be influenced by all sorts of things. Of course, this is all on the assumption that ESP actually exists!
13. Although this book describes both Neyman’s and Fisher’s definition of the p value, most don’t. Most introductory textbooks will only give you the Fisher version.
14. In this case, the Pearson chi-square test of independence (Chapter 12; `chisq.test()` in R) is what we use; see also the `prop.test()` function. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/11%3A_Hypothesis_Testing/11.10%3A_Summary.txt |
Now that we’ve got the basic theory behind hypothesis testing, it’s time to start looking at specific tests that are commonly used in psychology. So where should we start? Not every textbook agrees on where to start, but I’m going to start with “χ2 tests” (this chapter) and “t-tests” (Chapter 13). Both of these tools are very frequently used in scientific practice, and while they’re not as powerful as “analysis of variance” (Chapter 14) and “regression” (Chapter 15) they’re much easier to understand.
The term “categorical data” is just another name for “nominal scale data”. It’s nothing that we haven’t already discussed, it’s just that in the context of data analysis people tend to use the term “categorical data” rather than “nominal scale data”. I don’t know why. In any case, categorical data analysis refers to a collection of tools that you can use when your data are nominal scale. However, there are a lot of different tools that can be used for categorical data analysis, and this chapter only covers a few of the more common ones.
12: Categorical Data Analysis
The χ2 goodness-of-fit test is one of the oldest hypothesis tests around: it was invented by Karl Pearson around the turn of the century (Pearson 1900), with some corrections made later by Sir Ronald Fisher (Fisher 1922a). To introduce the statistical problem that it addresses, let’s start with some psychology…
cards data
Over the years, there have been a lot of studies showing that humans have a lot of difficulties in simulating randomness. Try as we might to “act” random, we think in terms of patterns and structure, and so when asked to “do something at random”, what people actually do is anything but random. As a consequence, the study of human randomness (or non-randomness, as the case may be) opens up a lot of deep psychological questions about how we think about the world. With this in mind, let’s consider a very simple study. Suppose I asked people to imagine a shuffled deck of cards, and mentally pick one card from this imaginary deck “at random”. After they’ve chosen one card, I ask them to mentally select a second one. For both choices, what we’re going to look at is the suit (hearts, clubs, spades or diamonds) that people chose. After asking, say, N=200 people to do this, I’d like to look at the data and figure out whether or not the cards that people pretended to select were really random. The data are contained in the randomness.Rdata file, which contains a single data frame called cards. Let’s take a look:
library( lsr )
load( "./rbook-master/data/randomness.Rdata" )
str(cards)
## 'data.frame': 200 obs. of 3 variables:
## $id : Factor w/ 200 levels "subj1","subj10",..: 1 112 124 135 146 157 168 179 190 2 ... ##$ choice_1: Factor w/ 4 levels "clubs","diamonds",..: 4 2 3 4 3 1 3 2 4 2 ...
## $choice_2: Factor w/ 4 levels "clubs","diamonds",..: 1 1 1 1 4 3 2 1 1 4 ... As you can see, the cards data frame contains three variables, an id variable that assigns a unique identifier to each participant, and the two variables choice_1 and choice_2 that indicate the card suits that people chose. Here’s the first few entries in the data frame: head( cards ) ## id choice_1 choice_2 ## 1 subj1 spades clubs ## 2 subj2 diamonds clubs ## 3 subj3 hearts clubs ## 4 subj4 spades clubs ## 5 subj5 hearts spades ## 6 subj6 clubs hearts For the moment, let’s just focus on the first choice that people made. We’ll use the table() function to count the number of times that we observed people choosing each suit. I’ll save the table to a variable called observed, for reasons that will become clear very soon: observed <- table( cards$choice_1 )
observed
##
## clubs diamonds hearts spades
## 35 51 64 50
That little frequency table is quite helpful. Looking at it, there’s a bit of a hint that people might be more likely to select hearts than clubs, but it’s not completely obvious just from looking at it whether that’s really true, or if this is just due to chance. So we’ll probably have to do some kind of statistical analysis to find out, which is what I’m going to talk about in the next section.
Excellent. From this point on, we’ll treat this table as the data that we’re looking to analyse. However, since I’m going to have to talk about this data in mathematical terms (sorry!) it might be a good idea to be clear about what the notation is. In R, if I wanted to pull out the number of people that selected diamonds, I could do it by name by typing observed["diamonds"] but, since "diamonds" is second element of the observed vector, it’s equally effective to refer to it as observed[2]. The mathematical notation for this is pretty similar, except that we shorten the human-readable word “observed” to the letter O, and we use subscripts rather than brackets: so the second observation in our table is written as observed[2] in R, and is written as O2 in maths. The relationship between the English descriptions, the R commands, and the mathematical symbols are illustrated below:
label index i math. symbol R command the value
clubs ♣ 1 O1 observed[1] 35
diamonds ♢ 2 O2 observed[2] 51
hearts ♡ 3 O3 observed[3] 64
spades ♠ 4 O4 observed[4] 50
Hopefully that’s pretty clear. It’s also worth nothing that mathematicians prefer to talk about things in general rather than specific things, so you’ll also see the notation Oi, which refers to the number of observations that fall within the i-th category (where i could be 1, 2, 3 or 4). Finally, if we want to refer to the set of all observed frequencies, statisticians group all of observed values into a vector, which I’ll refer to as O.
O=(O1,O2,O3,O4)
Again, there’s nothing new or interesting here: it’s just notation. If I say that O = (35,51,64,50) all I’m doing is describing the table of observed frequencies (i.e., observed), but I’m referring to it using mathematical notation, rather than by referring to an R variable.
null hypothesis and the alternative hypothesis
As the last section indicated, our research hypothesis is that “people don’t choose cards randomly”. What we’re going to want to do now is translate this into some statistical hypotheses, and construct a statistical test of those hypotheses. The test that I’m going to describe to you is Pearson’s χ2 goodness of fit test, and as is so often the case, we have to begin by carefully constructing our null hypothesis. In this case, it’s pretty easy. First, let’s state the null hypothesis in words:
H0
All four suits are chosen with equal probability
Now, because this is statistics, we have to be able to say the same thing in a mathematical way. To do this, let’s use the notation Pj to refer to the true probability that the j-th suit is chosen. If the null hypothesis is true, then each of the four suits has a 25% chance of being selected: in other words, our null hypothesis claims that P1=.25, P2=.25, P3=.25 and finally that P4=.25. However, in the same way that we can group our observed frequencies into a vector O that summarises the entire data set, we can use P to refer to the probabilities that correspond to our null hypothesis. So if I let the vector P=(P1,P2,P3,P4) refer to the collection of probabilities that describe our null hypothesis, then we have
H0:P=(.25,.25,.25,.25)
In this particular instance, our null hypothesis corresponds to a vector of probabilities P in which all of the probabilities are equal to one another. But this doesn’t have to be the case. For instance, if the experimental task was for people to imagine they were drawing from a deck that had twice as many clubs as any other suit, then the null hypothesis would correspond to something like P=(.4,.2,.2,.2). As long as the probabilities are all positive numbers, and they all sum to 1, them it’s a perfectly legitimate choice for the null hypothesis. However, the most common use of the goodness of fit test is to test a null hypothesis that all of the categories are equally likely, so we’ll stick to that for our example.
What about our alternative hypothesis, H1? All we’re really interested in is demonstrating that the probabilities involved aren’t all identical (that is, people’s choices weren’t completely random). As a consequence, the “human friendly” versions of our hypotheses look like this:
H0 H1
All four suits are chosen with equal probability
and the “mathematician friendly” version is
At least one of the suit-choice probabilities isn’t .25
H0 H1
P=(.25,.25,.25,.25) P≠(.25,.25,.25,.25)
Conveniently, the mathematical version of the hypotheses looks quite similar to an R command defining a vector. So maybe what I should do is store the P vector in R as well, since we’re almost certainly going to need it later. And because I’m so imaginative, I’ll call this R vector probabilities,
probabilities <- c(clubs = .25, diamonds = .25, hearts = .25, spades = .25)
probabilities
## clubs diamonds hearts spades
## 0.25 0.25 0.25 0.25
“goodness of fit” test statistic
At this point, we have our observed frequencies O and a collection of probabilities P corresponding the null hypothesis that we want to test. We’ve stored these in R as the corresponding variables observed and probabilities. What we now want to do is construct a test of the null hypothesis. As always, if we want to test H0 against H1, we’re going to need a test statistic. The basic trick that a goodness of fit test uses is to construct a test statistic that measures how “close” the data are to the null hypothesis. If the data don’t resemble what you’d “expect” to see if the null hypothesis were true, then it probably isn’t true. Okay, if the null hypothesis were true, what would we expect to see? Or, to use the correct terminology, what are the expected frequencies. There are N=200 observations, and (if the null is true) the probability of any one of them choosing a heart is P3=.25, so I guess we’re expecting 200×.25=50 hearts, right? Or, more specifically, if we let Ei refer to “the number of category i responses that we’re expecting if the null is true”, then
Ei=N×Pi
This is pretty easy to calculate in R:
N <- 200 # sample size
expected <- N * probabilities # expected frequencies
expected
## clubs diamonds hearts spades
## 50 50 50 50
None of which is very surprising: if there are 200 observation that can fall into four categories, and we think that all four categories are equally likely, then on average we’d expect to see 50 observations in each category, right?
Now, how do we translate this into a test statistic? Clearly, what we want to do is compare the expected number of observations in each category (Ei) with the observed number of observations in that category (Oi). And on the basis of this comparison, we ought to be able to come up with a good test statistic. To start with, let’s calculate the difference between what the null hypothesis expected us to find and what we actually did find. That is, we calculate the “observed minus expected” difference score, Oi−Ei. This is illustrated in the following table.
expected frequency Ei 50 50 50 50
observed frequency Oi 35 51 64 50
difference score Oi−Ei -15 1 14 0
The same calculations can be done in R, using our expected and observed variables:
observed - expected
##
## clubs diamonds hearts spades
## -15 1 14 0
Regardless of whether we do the calculations by hand or whether we do them in R, it’s clear that people chose more hearts and fewer clubs than the null hypothesis predicted. However, a moment’s thought suggests that these raw differences aren’t quite what we’re looking for. Intuitively, it feels like it’s just as bad when the null hypothesis predicts too few observations (which is what happened with hearts) as it is when it predicts too many (which is what happened with clubs). So it’s a bit weird that we have a negative number for clubs and a positive number for heards. One easy way to fix this is to square everything, so that we now calculate the squared differences, (Ei−Oi)2. As before, we could do this by hand, but it’s easier to do it in R…
(observed - expected)^2
##
## clubs diamonds hearts spades
## 225 1 196 0
Now we’re making progress. What we’ve got now is a collection of numbers that are big whenever the null hypothesis makes a bad prediction (clubs and hearts), but are small whenever it makes a good one (diamonds and spades). Next, for some technical reasons that I’ll explain in a moment, let’s also divide all these numbers by the expected frequency Ei, so we’re actually calculating $\dfrac{\left(E_{i}-O_{i}\right)^{2}}{E_{i}}$. Since Ei=50 for all categories in our example, it’s not a very interesting calculation, but let’s do it anyway. The R command becomes:
(observed - expected)^2 / expected
##
## clubs diamonds hearts spades
## 4.50 0.02 3.92 0.00
In effect, what we’ve got here are four different “error” scores, each one telling us how big a “mistake” the null hypothesis made when we tried to use it to predict our observed frequencies. So, in order to convert this into a useful test statistic, one thing we could do is just add these numbers up. The result is called the goodness of fit statistic, conventionally referred to either as X2 or GOF. We can calculate it using this command in R
sum( (observed - expected)^2 / expected )
## [1] 8.44
The formula for this statistic looks remarkably similar to the R command. If we let k refer to the total number of categories (i.e., k=4 for our cards data), then the X2 statistic is given by:
$X^{2}=\sum_{i=1}^{k} \dfrac{\left(O_{i}-E_{i}\right)^{2}}{E_{i}}$
Intuitively, it’s clear that if X2 is small, then the observed data Oi are very close to what the null hypothesis predicted Ei, so we’re going to need a large X2 statistic in order to reject the null. As we’ve seen from our calculations, in our cards data set we’ve got a value of X2=8.44. So now the question becomes, is this a big enough value to reject the null?
sampling distribution of the GOF statistic (advanced)
To determine whether or not a particular value of X2 is large enough to justify rejecting the null hypothesis, we’re going to need to figure out what the sampling distribution for X2 would be if the null hypothesis were true. So that’s what I’m going to do in this section. I’ll show you in a fair amount of detail how this sampling distribution is constructed, and then – in the next section – use it to build up a hypothesis test. If you want to cut to the chase and are willing to take it on faith that the sampling distribution is a chi-squared (χ2) distribution with k−1 degrees of freedom, you can skip the rest of this section. However, if you want to understand why the goodness of fit test works the way it does, read on…
Okay, let’s suppose that the null hypothesis is actually true. If so, then the true probability that an observation falls in the i-th category is Pi – after all, that’s pretty much the definition of our null hypothesis. Let’s think about what this actually means. If you think about it, this is kind of like saying that “nature” makes the decision about whether or not the observation ends up in category i by flipping a weighted coin (i.e., one where the probability of getting a head is Pj). And therefore, we can think of our observed frequency Oi by imagining that nature flipped N of these coins (one for each observation in the data set)… and exactly Oi of them came up heads. Obviously, this is a pretty weird way to think about the experiment. But what it does (I hope) is remind you that we’ve actually seen this scenario before. It’s exactly the same set up that gave rise to the binomial distribution in Section 9.4. In other words, if the null hypothesis is true, then it follows that our observed frequencies were generated by sampling from a binomial distribution:
Oi∼Binomial(Pi,N)
Now, if you remember from our discussion of the central limit theorem (Section 10.3.3), the binomial distribution starts to look pretty much identical to the normal distribution, especially when N is large and when Pi isn’t too close to 0 or 1. In other words as long as N×Pi is large enough – or, to put it another way, when the expected frequency Ei is large enough – the theoretical distribution of Oi is approximately normal. Better yet, if Oi is normally distributed, then so is (Oi−Ei)/$\sqrt{E_{i}}$ … since Ei is a fixed value, subtracting off Ei and dividing by $\sqrt{E_{i}}$ changes the mean and standard deviation of the normal distribution; but that’s all it does. Okay, so now let’s have a look at what our goodness of fit statistic actually is. What we’re doing is taking a bunch of things that are normally-distributed, squaring them, and adding them up. Wait. We’ve seen that before too! As we discussed in Section 9.6, when you take a bunch of things that have a standard normal distribution (i.e., mean 0 and standard deviation 1), square them, then add them up, then the resulting quantity has a chi-square distribution. So now we know that the null hypothesis predicts that the sampling distribution of the goodness of fit statistic is a chi-square distribution. Cool.
There’s one last detail to talk about, namely the degrees of freedom. If you remember back to Section 9.6, I said that if the number of things you’re adding up is k, then the degrees of freedom for the resulting chi-square distribution is k. Yet, what I said at the start of this section is that the actual degrees of freedom for the chi-square goodness of fit test is k−1. What’s up with that? The answer here is that what we’re supposed to be looking at is the number of genuinely independent things that are getting added together. And, as I’ll go on to talk about in the next section, even though there’s k things that we’re adding, only k−1 of them are truly independent; and so the degrees of freedom is actually only k−1. That’s the topic of the next section.170
Degrees of freedom
When I introduced the chi-square distribution in Section 9.6, I was a bit vague about what “degrees of freedom” actually means. Obviously, it matters: looking Figure 12.1 you can see that if we change the degrees of freedom, then the chi-square distribution changes shape quite substantially. But what exactly is it? Again, when I introduced the distribution and explained its relationship to the normal distribution, I did offer an answer… it’s the number of “normally distributed variables” that I’m squaring and adding together. But, for most people, that’s kind of abstract, and not entirely helpful. What we really need to do is try to understand degrees of freedom in terms of our data. So here goes.
The basic idea behind degrees of freedom is quite simple: you calculate it by counting up the number of distinct “quantities” that are used to describe your data; and then subtracting off all of the “constraints” that those data must satisfy.171 This is a bit vague, so let’s use our cards data as a concrete example. We describe out data using four numbers, O1, O2, O3 and O4 corresponding to the observed frequencies of the four different categories (hearts, clubs, diamonds, spades). These four numbers are the random outcomes of our experiment. But, my experiment actually has a fixed constraint built into it: the sample size N.172 That is, if we know how many people chose hearts, how many chose diamonds and how many chose clubs; then we’d be able to figure out exactly how many chose spades. In other words, although our data are described using four numbers, they only actually correspond to 4−1=3 degrees of freedom. A slightly different way of thinking about it is to notice that there are four probabilities that we’re interested in (again, corresponding to the four different categories), but these probabilities must sum to one, which imposes a constraint. Therefore, the degrees of freedom is 4−1=3. Regardless of whether you want to think about it in terms of the observed frequencies or in terms of the probabilities, the answer is the same. In general, when running the chi-square goodness of fit test for an experiment involving k groups, then the degrees of freedom will be k−1.
Testing the null hypothesis
The final step in the process of constructing our hypothesis test is to figure out what the rejection region is. That is, what values of X2 would lead is to reject the null hypothesis. As we saw earlier, large values of X2 imply that the null hypothesis has done a poor job of predicting the data from our experiment, whereas small values of X2 imply that it’s actually done pretty well. Therefore, a pretty sensible strategy would be to say there is some critical value, such that if X2 is bigger than the critical value we reject the null; but if X2 is smaller than this value we retain the null. In other words, to use the language we introduced in Chapter @ref(hypothesistesting the chi-squared goodness of fit test is always a one-sided test. Right, so all we have to do is figure out what this critical value is. And it’s pretty straightforward. If we want our test to have significance level of α=.05 (that is, we are willing to tolerate a Type I error rate of 5%), then we have to choose our critical value so that there is only a 5% chance that X2 could get to be that big if the null hypothesis is true. That is to say, we want the 95th percentile of the sampling distribution. This is illustrated in Figure 12.2.
Ah, but – I hear you ask – how do I calculate the 95th percentile of a chi-squared distribution with k−1 degrees of freedom? If only R had some function, called… oh, I don’t know, qchisq() … that would let you calculate this percentile (see Chapter 9 if you’ve forgotten). Like this…
qchisq( p = .95, df = 3 )
## [1] 7.814728
So if our X2 statistic is bigger than 7.81 or so, then we can reject the null hypothesis. Since we actually calculated that before (i.e., X2=8.44) we can reject the null. If we want an exact p-value, we can calculate it using the pchisq() function:
pchisq( q = 8.44, df = 3, lower.tail = FALSE )
## [1] 0.03774185
This is hopefully pretty straightforward, as long as you recall that the “p” form of the probability distribution functions in R always calculates the probability of getting a value of less than the value you entered (in this case 8.44). We want the opposite: the probability of getting a value of 8.44 or more. That’s why I told R to use the upper tail, not the lower tail. That said, it’s usually easier to calculate the p-value this way:
1-pchisq( q = 8.44, df = 3 )
## [1] 0.03774185
So, in this case we would reject the null hypothesis, since p<.05. And that’s it, basically. You now know “Pearson’s χ2 test for the goodness of fit”. Lucky you.
Doing the test in R
Gosh darn it. Although we did manage to do everything in R as we were going through that little example, it does rather feel as if we’re typing too many things into the magic computing box. And I hate typing. Not surprisingly, R provides a function that will do all of these calculations for you. In fact, there are several different ways of doing it. The one that most people use is the chisq.test() function, which comes with every installation of R. I’ll show you how to use the chisq.test() function later on (in Section @ref(chisq.test), but to start out with I’m going to show you the goodnessOfFitTest() function in the lsr package, because it produces output that I think is easier for beginners to understand. It’s pretty straightforward: our raw data are stored in the variable cards$choice_1, right? If you want to test the null hypothesis that all four suits are equally likely, then (assuming you have the lsr package loaded) all you have to do is type this: goodnessOfFitTest( cards$choice_1 )
##
## Chi-square test against specified probabilities
##
## Data variable: cards$choice_1 ## ## Hypotheses: ## null: true probabilities are as specified ## alternative: true probabilities differ from those specified ## ## Descriptives: ## observed freq. expected freq. specified prob. ## clubs 35 50 0.25 ## diamonds 51 50 0.25 ## hearts 64 50 0.25 ## spades 50 50 0.25 ## ## Test results: ## X-squared statistic: 8.44 ## degrees of freedom: 3 ## p-value: 0.038 R then runs the test, and prints several lines of text. I’ll go through the output line by line, so that you can make sure that you understand what you’re looking at. The first two lines are just telling you things you already know: Chi-square test against specified probabilities Data variable: cards$choice_1
The first line tells us what kind of hypothesis test we ran, and the second line tells us the name of the variable that we ran it on. After that comes a statement of what the null and alternative hypotheses are:
Hypotheses:
null: true probabilities are as specified
alternative: true probabilities differ from those specified
For a beginner, it’s kind of handy to have this as part of the output: it’s a nice reminder of what your null and alternative hypotheses are. Don’t get used to seeing this though. The vast majority of hypothesis tests in R aren’t so kind to novices. Most R functions are written on the assumption that you already understand the statistical tool that you’re using, so they don’t bother to include an explicit statement of the null and alternative hypothesis. The only reason that goodnessOfFitTest() actually does give you this is that I wrote it with novices in mind.
The next part of the output shows you the comparison between the observed frequencies and the expected frequencies:
Descriptives:
observed freq. expected freq. specified prob.
clubs 35 50 0.25
diamonds 51 50 0.25
hearts 64 50 0.25
spades 50 50 0.25
The first column shows what the observed frequencies were, the second column shows the expected frequencies according to the null hypothesis, and the third column shows you what the probabilities actually were according to the null. For novice users, I think this is helpful: you can look at this part of the output and check that it makes sense: if it doesn’t you might have typed something incorrecrtly.
The last part of the output is the “important” stuff: it’s the result of the hypothesis test itself. There are three key numbers that need to be reported: the value of the X2 statistic, the degrees of freedom, and the p-value:
Test results:
X-squared statistic: 8.44
degrees of freedom: 3
p-value: 0.038
Notice that these are the same numbers that we came up with when doing the calculations the long way.
Specifying a different null hypothesis
At this point you might be wondering what to do if you want to run a goodness of fit test, but your null hypothesis is not that all categories are equally likely. For instance, let’s suppose that someone had made the theoretical prediction that people should choose red cards 60% of the time, and black cards 40% of the time (I’ve no idea why you’d predict that), but had no other preferences. If that were the case, the null hypothesis would be to expect 30% of the choices to be hearts, 30% to be diamonds, 20% to be spades and 20% to be clubs. This seems like a silly theory to me, and it’s pretty easy to test it using our data. All we need to do is specify the probabilities associated with the null hypothesis. We create a vector like this:
nullProbs <- c(clubs = .2, diamonds = .3, hearts = .3, spades = .2)
nullProbs
## clubs diamonds hearts spades
## 0.2 0.3 0.3 0.2
Now that we have an explicitly specified null hypothesis, we include it in our command. This time round I’ll use the argument names properly. The data variable corresponds to the argument x, and the probabilities according to the null hypothesis correspond to the argument p. So our command is:
goodnessOfFitTest( x = cards$choice_1, p = nullProbs ) ## ## Chi-square test against specified probabilities ## ## Data variable: cards$choice_1
##
## Hypotheses:
## null: true probabilities are as specified
## alternative: true probabilities differ from those specified
##
## Descriptives:
## observed freq. expected freq. specified prob.
## clubs 35 40 0.2
## diamonds 51 60 0.3
## hearts 64 60 0.3
## spades 50 40 0.2
##
## Test results:
## X-squared statistic: 4.742
## degrees of freedom: 3
## p-value: 0.192
As you can see the null hypothesis and the expected frequencies are different to what they were last time. As a consequence our X2 test statistic is different, and our p-value is different too. Annoyingly, the p-value is .192, so we can’t reject the null hypothesis. Sadly, despite the fact that the null hypothesis corresponds to a very silly theory, these data don’t provide enough evidence against it.
report the results of the test
So now you know how the test works, and you know how to do the test using a wonderful magic computing box. The next thing you need to know is how to write up the results. After all, there’s no point in designing and running an experiment and then analysing the data if you don’t tell anyone about it! So let’s now talk about what you need to do when reporting your analysis. Let’s stick with our card-suits example. If I wanted to write this result up for a paper or something, the conventional way to report this would be to write something like this:
Of the 200 participants in the experiment, 64 selected hearts for their first choice, 51 selected diamonds, 50 selected spades, and 35 selected clubs. A chi-square goodness of fit test was conducted to test whether the choice probabilities were identical for all four suits. The results were significant (χ2(3)=8.44,p<.05), suggesting that people did not select suits purely at random.
This is pretty straightforward, and hopefully it seems pretty unremarkable. That said, there’s a few things that you should note about this description:
• The statistical test is preceded by the descriptive statistics. That is, I told the reader something about what the data look like before going on to do the test. In general, this is good practice: always remember that your reader doesn’t know your data anywhere near as well as you do. So unless you describe it to them properly, the statistical tests won’t make any sense to them, and they’ll get frustrated and cry.
• The description tells you what the null hypothesis being tested is. To be honest, writers don’t always do this, but it’s often a good idea in those situations where some ambiguity exists; or when you can’t rely on your readership being intimately familiar with the statistical tools that you’re using. Quite often the reader might not know (or remember) all the details of the test that your using, so it’s a kind of politeness to “remind” them! As far as the goodness of fit test goes, you can usually rely on a scientific audience knowing how it works (since it’s covered in most intro stats classes). However, it’s still a good idea to be explicit about stating the null hypothesis (briefly!) because the null hypothesis can be different depending on what you’re using the test for. For instance, in the cards example my null hypothesis was that all the four suit probabilities were identical (i.e., P1=P2=P3=P4=0.25), but there’s nothing special about that hypothesis. I could just as easily have tested the null hypothesis that P1=0.7 and P2=P3=P4=0.1 using a goodness of fit test. So it’s helpful to the reader if you explain to them what your null hypothesis was. Also, notice that I described the null hypothesis in words, not in maths. That’s perfectly acceptable. You can describe it in maths if you like, but since most readers find words easier to read than symbols, most writers tend to describe the null using words if they can.
• A “stat block” is included. When reporting the results of the test itself, I didn’t just say that the result was significant, I included a “stat block” (i.e., the dense mathematical-looking part in the parentheses), which reports all the “raw” statistical data. For the chi-square goodness of fit test, the information that gets reported is the test statistic (that the goodness of fit statistic was 8.44), the information about the distribution used in the test (χ2 with 3 degrees of freedom, which is usually shortened to χ2(3)), and then the information about whether the result was significant (in this case p<.05). The particular information that needs to go into the stat block is different for every test, and so each time I introduce a new test I’ll show you what the stat block should look like.173 However the general principle is that you should always provide enough information so that the reader could check the test results themselves if they really wanted to.
• The results are interpreted. In addition to indicating that the result was significant, I provided an interpretation of the result (i.e., that people didn’t choose randomly). This is also a kindness to the reader, because it tells them something about what they should believe about what’s going on in your data. If you don’t include something like this, it’s really hard for your reader to understand what’s going on.174
As with everything else, your overriding concern should be that you explain things to your reader. Always remember that the point of reporting your results is to communicate to another human being. I cannot tell you just how many times I’ve seen the results section of a report or a thesis or even a scientific article that is just gibberish, because the writer has focused solely on making sure they’ve included all the numbers, and forgotten to actually communicate with the human reader.
comment on statistical notation (advanced)
Satan delights equally in statistics and in quoting scripture
– H.G. Wells
If you’ve been reading very closely, and are as much of a mathematical pedant as I am, there is one thing about the way I wrote up the chi-square test in the last section that might be bugging you a little bit. There’s something that feels a bit wrong with writing “χ2(3)=8.44”, you might be thinking. After all, it’s the goodness of fit statistic that is equal to 8.44, so shouldn’t I have written X2=8.44 or maybe GOF=8.44? This seems to be conflating the sampling distribution (i.e., χ2 with df=3) with the test statistic (i.e., X2). Odds are you figured it was a typo, since χ and X look pretty similar. Oddly, it’s not. Writing χ2(3)=8.44 is essentially a highly condensed way of writing “the sampling distribution of the test statistic is χ2(3), and the value of the test statistic is 8.44”.
In one sense, this is kind of stupid. There are lots of different test statistics out there that turn out to have a chi-square sampling distribution: the X2 statistic that we’ve used for our goodness of fit test is only one of many (albeit one of the most commonly encountered ones). In a sensible, perfectly organised world, we’d always have a separate name for the test statistic and the sampling distribution: that way, the stat block itself would tell you exactly what it was that the researcher had calculated. Sometimes this happens. For instance, the test statistic used in the Pearson goodness of fit test is written X2; but there’s a closely related test known as the G-test175 , in which the test statistic is written as G. As it happens, the Pearson goodness of fit test and the G-test both test the same null hypothesis; and the sampling distribution is exactly the same (i.e., chi-square with k−1 degrees of freedom). If I’d done a G-test for the cards data rather than a goodness of fit test, then I’d have ended up with a test statistic of G=8.65, which is slightly different from the X2=8.44 value that I got earlier; and produces a slightly smaller p-value of p=.034. Suppose that the convention was to report the test statistic, then the sampling distribution, and then the p-value. If that were true, then these two situations would produce different stat blocks: my original result would be written X2=8.44, χ2(3),p=.038, whereas the new version using the G-test would be written as G=8.65, χ2(3), p=.034. However, using the condensed reporting standard, the original result is written χ2(3)=8.44, p=.038, and the new one is written χ2(3)=8.65, p=.034, and so it’s actually unclear which test I actually ran.
So why don’t we live in a world in which the contents of the stat block uniquely specifies what tests were ran? The deep reason is that life is messy. We (as users of statistical tools) want it to be nice and neat and organised… we want it to be designed, as if it were a product. But that’s not how life works: statistics is an intellectual discipline just as much as any other one, and as such it’s a massively distributed, partly-collaborative and partly-competitive project that no-one really understands completely. The things that you and I use as data analysis tools weren’t created by an Act of the Gods of Statistics; they were invented by lots of different people, published as papers in academic journals, implemented, corrected and modified by lots of other people, and then explained to students in textbooks by someone else. As a consequence, there’s a lot of test statistics that don’t even have names; and as a consequence they’re just given the same name as the corresponding sampling distribution. As we’ll see later, any test statistic that follows a χ2 distribution is commonly called a “chi-square statistic”; anything that follows a t-distribution is called a “t-statistic” and so on. But, as the X2 versus G example illustrates, two different things with the same sampling distribution are still, well, different.
As a consequence, it’s sometimes a good idea to be clear about what the actual test was that you ran, especially if you’re doing something unusual. If you just say “chi-square test”, it’s not actually clear what test you’re talking about. Although, since the two most common chi-square tests are the goodness of fit test and the independence test (Section 12.2), most readers with stats training can probably guess. Nevertheless, it’s something to be aware of. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.01%3A_The_2_Goodness-of-fit_Test.txt |
GUARDBOT1: Halt!
GUARDBOT2: Be you robot or human?
LEELA: Robot…we be.
FRY: Uh, yup! Just two robots out roboting it up! Eh?
GUARDBOT1: Administer the test.
GUARDBOT2: Which of the following would you most prefer? A: A puppy, B: A pretty flower from your sweetie, or C: A large properly-formatted data file?
GUARDBOT1: Choose!
– Futurama, “Fear of a Bot Planet
The other day I was watching an animated documentary examining the quaint customs of the natives of the planet Chapek 9. Apparently, in order to gain access to their capital city, a visitor must prove that they’re a robot, not a human. In order to determine whether or not visitor is human, they ask whether the visitor prefers puppies, flowers or large, properly formatted data files. “Pretty clever,” I thought to myself “but what if humans and robots have the same preferences? That probably wouldn’t be a very good test then, would it?” As it happens, I got my hands on the testing data that the civil authorities of Chapek 9 used to check this. It turns out that what they did was very simple… they found a bunch of robots and a bunch of humans and asked them what they preferred. I saved their data in a file called chapek9.Rdata, which I can now load and have a quick look at:
load( "./rbook-master/data/chapek9.Rdata" )
str(chapek9)
## 'data.frame': 180 obs. of 2 variables:
## $species: Factor w/ 2 levels "robot","human": 1 2 2 2 1 2 2 1 2 1 ... ##$ choice : Factor w/ 3 levels "puppy","flower",..: 2 3 3 3 3 2 3 3 1 2 ...
Okay, so we have a single data frame called chapek9, which contains two factors, species and choice. As always, it’s nice to have a quick look at the data,
head(chapek9)
## species choice
## 1 robot flower
## 2 human data
## 3 human data
## 4 human data
## 5 robot data
## 6 human flower
and then take a summary(),
summary(chapek9)
## species choice
## robot:87 puppy : 28
## human:93 flower: 43
## data :109
In total there are 180 entries in the data frame, one for each person (counting both robots and humans as “people”) who was asked to make a choice. Specifically, there’s 93 humans and 87 robots; and overwhelmingly the preferred choice is the data file. However, these summaries don’t address the question we’re interested in. To do that, we need a more detailed description of the data. What we want to do is look at the choices broken down by species. That is, we need to cross-tabulate the data (see Section 7.1). There’s quite a few ways to do this, as we’ve seen, but since our data are stored in a data frame, it’s convenient to use the xtabs() function.
chapekFrequencies <- xtabs( ~ choice + species, data = chapek9)
chapekFrequencies
## species
## choice robot human
## puppy 13 15
## flower 30 13
## data 44 65
That’s more or less what we’re after. So, if we add the row and column totals (which is convenient for the purposes of explaining the statistical tests), we would have a table like this,
Robot Human Total
Puppy 13 15 28
Flower 30 13 43
Data file 44 65 109
Total 87 93 180
which actual ly would be a nice way to report the descriptive statistics for this data set. In any case, it’s quite clear that the vast majority of the humans chose the data file, whereas the robots tended to be a lot more even in their preferences. Leaving aside the question of why the humans might be more likely to choose the data file for the moment (which does seem quite odd, admittedly), our first order of business is to determine if the discrepancy between human choices and robot choices in the data set is statistically significant.
Constructing our hypothesis test
How do we analyse this data? Specifically, since my research hypothesis is that “humans and robots answer the question in different ways”, how can I construct a test of the null hypothesis that “humans and robots answer the question the same way”? As before, we begin by establishing some notation to describe the data:
Robot Human Total
Puppy O11 O12 R1
Flower O21 O22 R2
Data file O31 O32 R3
Total C1 C2 N
In this notation we say that Oij is a count (observed frequency) of the number of respondents that are of species j (robots or human) who gave answer i (puppy, flower or data) when asked to make a choice. The total number of observations is written N, as usual. Finally, I’ve used Ri to denote the row totals (e.g., R1 is the total number of people who chose the flower), and Cj to denote the column totals (e.g., C1 is the total number of robots).176
So now let’s think about what the null hypothesis says. If robots and humans are responding in the same way to the question, it means that the probability that “a robot says puppy” is the same as the probability that “a human says puppy”, and so on for the other two possibilities. So, if we use Pij to denote “the probability that a member of species j gives response i” then our null hypothesis is that:
H0: All of the following are true:
P11=P12 (same probability of saying puppy)
P21=P22 (same probability of saying flower) and
P31=P32 (same probability of saying data).
And actually, since the null hypothesis is claiming that the true choice probabilities don’t depend on the species of the person making the choice, we can let Pi refer to this probability: e.g., P1 is the true probability of choosing the puppy.
Next, in much the same way that we did with the goodness of fit test, what we need to do is calculate the expected frequencies. That is, for each of the observed counts Oij, we need to figure out what the null hypothesis would tell us to expect. Let’s denote this expected frequency by Eij. This time, it’s a little bit trickier. If there are a total of Cj people that belong to species j, and the true probability of anyone (regardless of species) choosing option i is Pi, then the expected frequency is just:
$E_{i j}=C_{j} \times P_{i}$
Now, this is all very well and good, but we have a problem. Unlike the situation we had with the goodness of fit test, the null hypothesis doesn’t actually specify a particular value for Pi. It’s something we have to estimate (Chapter 10) from the data! Fortunately, this is pretty easy to do. If 28 out of 180 people selected the flowers, then a natural estimate for the probability of choosing flowers is 28/180, which is approximately .16. If we phrase this in mathematical terms, what we’re saying is that our estimate for the probability of choosing option i is just the row total divided by the total sample size:
$\hat{P}_{i}=\dfrac{R_{i}}{N}$
Therefore, our expected frequency can be written as the product (i.e. multiplication) of the row total and the column total, divided by the total number of observations:177
$E_{i j}=\dfrac{R_{i} \times C_{j}}{N}$
Now that we’ve figured out how to calculate the expected frequencies, it’s straightforward to define a test statistic; following the exact same strategy that we used in the goodness of fit test. In fact, it’s pretty much the same statistic. For a contingency table with r rows and c columns, the equation that defines our X2 statistic is
$X^{2}=\sum_{i=1}^{r} \sum_{j=1}^{c} \dfrac{\left(E_{i j}-O_{i j}\right)^{2}}{E_{i j}}$
The only difference is that I have to include two summation sign (i.e., ∑) to indicate that we’re summing over both rows and columns. As before, large values of X2 indicate that the null hypothesis provides a poor description of the data, whereas small values of X2 suggest that it does a good job of accounting for the data. Therefore, just like last time, we want to reject the null hypothesis if X2 is too large.
Not surprisingly, this statistic is X2 distributed. All we need to do is figure out how many degrees of freedom are involved, which actually isn’t too hard. As I mentioned before, you can (usually) think of the degrees of freedom as being equal to the number of data points that you’re analysing, minus the number of constraints. A contingency table with r rows and c columns contains a total of r×c observed frequencies, so that’s the total number of observations. What about the constraints? Here, it’s slightly trickier. The answer is always the same
df=(r−1)(c−1)
but the explanation for why the degrees of freedom takes this value is different depending on the experimental design. For the sake of argument, let’s suppose that we had honestly intended to survey exactly 87 robots and 93 humans (column totals fixed by the experimenter), but left the row totals free to vary (row totals are random variables). Let’s think about the constraints that apply here. Well, since we deliberately fixed the column totals by Act of Experimenter, we have c constraints right there. But, there’s actually more to it than that. Remember how our null hypothesis had some free parameters (i.e., we had to estimate the Pi values)? Those matter too. I won’t explain why in this book, but every free parameter in the null hypothesis is rather like an additional constraint. So, how many of those are there? Well, since these probabilities have to sum to 1, there’s only r−1 of these. So our total degrees of freedom is:
df=(number of observations)−(number of constraints)
=(rc)−(c+(r−1))
=rc−c−r+1
=(r−1)(c−1)
Alternatively, suppose that the only thing that the experimenter fixed was the total sample size N. That is, we quizzed the first 180 people that we saw, and it just turned out that 87 were robots and 93 were humans. This time around our reasoning would be slightly different, but would still lead is to the same answer. Our null hypothesis still has r−1 free parameters corresponding to the choice probabilities, but it now also has c−1 free parameters corresponding to the species probabilities, because we’d also have to estimate the probability that a randomly sampled person turns out to be a robot.178 Finally, since we did actually fix the total number of observations N, that’s one more constraint. So now we have, rc observations, and (c−1)+(r−1)+1 constraints. What does that give?
df=(number of observations)−(number of constraints)
=rc−((c−1)+(r−1)+1)
=rc−c−r+1
=(r−1)(c−1)
Amazing.
Doing the test in R
Okay, now that we know how the test works, let’s have a look at how it’s done in R. As tempting as it is to lead you through the tedious calculations so that you’re forced to learn it the long way, I figure there’s no point. I already showed you how to do it the long way for the goodness of fit test in the last section, and since the test of independence isn’t conceptually any different, you won’t learn anything new by doing it the long way. So instead, I’ll go straight to showing you the easy way. As always, R lets you do it multiple ways. There’s the chisq.test() function, which I’ll talk about in Section @ref(chisq.test, but first I want to use the associationTest() function in the lsr package, which I think is easier on beginners. It works in the exact same way as the xtabs() function. Recall that, in order to produce the contingency table, we used this command:
xtabs( formula = ~choice+species, data = chapek9 )
## species
## choice robot human
## puppy 13 15
## flower 30 13
## data 44 65
The associationTest() function has exactly the same structure: it needs a formula that specifies which variables you’re cross-tabulating, and the name of a data frame that contains those variables. So the command is just this:
associationTest( formula = ~choice+species, data = chapek9 )
##
## Chi-square test of categorical association
##
## Variables: choice, species
##
## Hypotheses:
## null: variables are independent of one another
## alternative: some contingency exists between variables
##
## Observed contingency table:
## species
## choice robot human
## puppy 13 15
## flower 30 13
## data 44 65
##
## Expected contingency table under the null hypothesis:
## species
## choice robot human
## puppy 13.5 14.5
## flower 20.8 22.2
## data 52.7 56.3
##
## Test results:
## X-squared statistic: 10.722
## degrees of freedom: 2
## p-value: 0.005
##
## Other information:
## estimated effect size (Cramer's v): 0.244
Just like we did with the goodness of fit test, I’ll go through it line by line. The first two lines are, once again, just reminding you what kind of test you ran and what variables were used:
Chi-square test of categorical association
Variables: choice, species
Next, it tells you what the null and alternative hypotheses are (and again, I want to remind you not to get used to seeing these hypotheses written out so explicitly):
Hypotheses:
null: variables are independent of one another
alternative: some contingency exists between variables
Next, it shows you the observed contingency table that is being tested:
Observed contingency table:
species
choice robot human
puppy 13 15
flower 30 13
data 44 65
and it also shows you what the expected frequencies would be if the null hypothesis were true:
Expected contingency table under the null hypothesis:
species
choice robot human
puppy 13.5 14.5
flower 20.8 22.2
data 52.7 56.3
The next part describes the results of the hypothesis test itself:
Test results:
X-squared statistic: 10.722
degrees of freedom: 2
p-value: 0.005
And finally, it reports a measure of effect size:
Other information:
estimated effect size (Cramer's v): 0.244
You can ignore this bit for now. I’ll talk about it in just a moment.
This output gives us enough information to write up the result:
Pearson’s χ2 revealed a significant association between species and choice (χ 2(2)=10.7,p<.01): robots appeared to be more likely to say that they prefer flowers, but the humans were more likely to say they prefer data.
Notice that, once again, I provided a little bit of interpretation to help the human reader understand what’s going on with the data. Later on in my discussion section, I’d provide a bit more context. To illustrate the difference, here’s what I’d probably say later on:
The fact that humans appeared to have a stronger preference for raw data files than robots is somewhat counterintuitive. However, in context it makes some sense: the civil authority on Chapek 9 has an unfortunate tendency to kill and dissect humans when they are identified. As such it seems most likely that the human participants did not respond honestly to the question, so as to avoid potentially undesirable consequences. This should be considered to be a substantial methodological weakness.
This could be classified as a rather extreme example of a reactivity effect, I suppose. Obviously, in this case the problem is severe enough that the study is more or less worthless as a tool for understanding the difference preferences among humans and robots. However, I hope this illustrates the difference between getting a statistically significant result (our null hypothesis is rejected in favour of the alternative), and finding something of scientific value (the data tell us nothing of interest about our research hypothesis due to a big methodological flaw).
Postscript
I later found out the data were made up, and I’d been watching cartoons instead of doing work. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.02%3A_The_2_test_of_independence_%28or_association%29.txt |
Okay, time for a little bit of a digression. I’ve been lying to you a little bit so far. There’s a tiny change that you need to make to your calculations whenever you only have 1 degree of freedom. It’s called the “continuity correction”, or sometimes the Yates correction. Remember what I pointed out earlier: the χ2 test is based on an approximation, specifically on the assumption that binomial distribution starts to look like a normal distribution for large N. One problem with this is that it often doesn’t quite work, especially when you’ve only got 1 degree of freedom (e.g., when you’re doing a test of independence on a 2×2 contingency table). The main reason for this is that the true sampling distribution for the X2 statistic is actually discrete (because you’re dealing with categorical data!) but the χ2 distribution is continuous. This can introduce systematic problems. Specifically, when N is small and when df=1, the goodness of fit statistic tends to be “too big”, meaning that you actually have a bigger α value than you think (or, equivalently, the p values are a bit too small). Yates (1934) suggested a simple fix, in which you redefine the goodness of fit statistic as:
$X^{2}=\sum_{i} \dfrac{\left(\left|E_{i}-O_{i}\right|-0.5\right)^{2}}{E_{i}}$
Basically, he just subtracts off 0.5 everywhere. As far as I can tell from reading Yates’ paper, the correction is basically a hack. It’s not derived from any principled theory: rather, it’s based on an examination of the behaviour of the test, and observing that the corrected version seems to work better. I feel obliged to explain this because you will sometimes see R (or any other software for that matter) introduce this correction, so it’s kind of useful to know what they’re about. You’ll know when it happens, because the R output will explicitly say that it has used a “continuity correction” or “Yates’ correction”.
12.04: Effect Size
As we discussed earlier (Section 11.8), it’s becoming commonplace to ask researchers to report some measure of effect size. So, let’s suppose that you’ve run your chi-square test, which turns out to be significant. So you now know that there is some association between your variables (independence test) or some deviation from the specified probabilities (goodness of fit test). Now you want to report a measure of effect size. That is, given that there is an association/deviation, how strong is it?
There are several different measures that you can choose to report, and several different tools that you can use to calculate them. I won’t discuss all of them,179 but will instead focus on the most commonly reported measures of effect size.
By default, the two measures that people tend to report most frequently are the ϕ statistic and the somewhat superior version, known as Cram'er’s V. Mathematically, they’re very simple. To calculate the ϕ statistic, you just divide your X2 value by the sample size, and take the square root:
$\phi=\sqrt{\dfrac{X^{2}}{N}}$
The idea here is that the ϕ statistic is supposed to range between 0 (no at all association) and 1 (perfect association), but it doesn’t always do this when your contingency table is bigger than 2×2, which is a total pain. For bigger tables it’s actually possible to obtain ϕ>1, which is pretty unsatisfactory. So, to correct for this, people usually prefer to report the V statistic proposed by Cramér (1946). It’s a pretty simple adjustment to ϕ. If you’ve got a contingency table with r rows and c columns, then define k=min(r,c) to be the smaller of the two values. If so, then Cram'er’s V statistic is
$V=\sqrt{\dfrac{X^{2}}{N(k-1)}}$
And you’re done. This seems to be a fairly popular measure, presumably because it’s easy to calculate, and it gives answers that aren’t completely silly: you know that V really does range from 0 (no at all association) to 1 (perfect association).
Calculating V or ϕ is obviously pretty straightforward. So much so that the core packages in R don’t seem to have functions to do it, though other packages do. To save you the time and effort of finding one, I’ve included one in the lsr package, called cramersV(). It takes a contingency table as input, and prints out the measure of effect size:
cramersV( chapekFrequencies )
## [1] 0.244058
However, if you’re using the associationTest() function to do your analysis, then you won’t actually need to use this at all, because it reports the Cram'er’s V statistic as part of the output. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.03%3A_The_Continuity_Correction.txt |
All statistical tests make assumptions, and it’s usually a good idea to check that those assumptions are met. For the chi-square tests discussed so far in this chapter, the assumptions are:
• Expected frequencies are sufficiently large. Remember how in the previous section we saw that the χ2 sampling distribution emerges because the binomial distribution is pretty similar to a normal distribution? Well, like we discussed in Chapter 9 this is only true when the number of observations is sufficiently large. What that means in practice is that all of the expected frequencies need to be reasonably big. How big is reasonably big? Opinions differ, but the default assumption seems to be that you generally would like to see all your expected frequencies larger than about 5, though for larger tables you would probably be okay if at least 80% of the the expected frequencies are above 5 and none of them are below 1. However, from what I’ve been able to discover , these seem to have been proposed as rough guidelines, not hard and fast rules; and they seem to be somewhat conservative [Larntz1978].
• Data are independent of one another. One somewhat hidden assumption of the chi-square test is that you have to genuinely believe that the observations are independent. Here’s what I mean. Suppose I’m interested in proportion of babies born at a particular hospital that are boys. I walk around the maternity wards, and observe 20 girls and only 10 boys. Seems like a pretty convincing difference, right? But later on, it turns out that I’d actually walked into the same ward 10 times, and in fact I’d only seen 2 girls and 1 boy. Not as convincing, is it? My original 30 observations were massively non-independent… and were only in fact equivalent to 3 independent observations. Obviously this is an extreme (and extremely silly) example, but it illustrates the basic issue. Non-independence “stuffs things up”. Sometimes it causes you to falsely reject the null, as the silly hospital example illustrats, but it can go the other way too. To give a slightly less stupid example, let’s consider what would happen if I’d done the cards experiment slightly differently: instead of asking 200 people to try to imagine sampling one card at random, suppose I asked 50 people to select 4 cards. One possibility would be that everyone selects one heart, one club, one diamond and one spade (in keeping with the “representativeness heuristic”; Tversky & Kahneman 1974). This is highly non-random behaviour from people, but in this case, I would get an observed frequency of 50 four all four suits. For this example, the fact that the observations are non-independent (because the four cards that you pick will be related to each other) actually leads to the opposite effect… falsely retaining the null.
If you happen to find yourself in a situation where independence is violated, it may be possible to use the McNemar test (which we’ll discuss) or the Cochran test (which we won’t). Similarly, if your expected cell counts are too small, check out the Fisher exact test. It is to these topics that we now turn. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.05%3A_Assumptions_of_the_Test%28s%29.txt |
When discussing how to do a chi-square goodness of fit test (Section 12.1.7) and the chi-square test of independence (Section 12.2.2), I introduced you to two separate functions in the `lsr` package. We ran our goodness of fit tests using the `goodnessOfFitTest()` function, and our tests of independence (or association) using the `associationTest()` function. And both of those functions produced quite detailed output, showing you the relevant descriptive statistics, printing out explicit reminders of what the hypotheses are, and so on. When you’re first starting out, it can be very handy to be given this sort of guidance. However, once you start becoming a bit more proficient in statistics and in R it can start to get very tiresome. A real statistician hardly needs to be told what the null and alternative hypotheses for a chi-square test are, and if an advanced R user wants the descriptive statistics to be printed out, they know how to produce them!
For this reason, the basic `chisq.test()` function in R is a lot more terse in its output, and because the mathematics that underpin the goodness of fit test and the test of independence is basically the same in each case, it can run either test depending on what kind of input it is given. First, here’s the goodness of fit test. Suppose you have the frequency table `observed` that we used earlier,
``observed``
``````##
## clubs diamonds hearts spades
## 35 51 64 50``````
If you want to run the goodness of fit test against the hypothesis that all four suits are equally likely to appear, then all you need to do is input this frequenct table to the `chisq.test()` function:
``chisq.test( x = observed )``
``````##
## Chi-squared test for given probabilities
##
## data: observed
## X-squared = 8.44, df = 3, p-value = 0.03774``````
Notice that the output is very compressed in comparison to the `goodnessOfFitTest()` function. It doesn’t bother to give you any descriptive statistics, it doesn’t tell you what null hypothesis is being tested, and so on. And as long as you already understand the test, that’s not a problem. Once you start getting familiar with R and with statistics, you’ll probably find that you prefer this simple output rather than the rather lengthy output that `goodnessOfFitTest()` produces. Anyway, if you want to change the null hypothesis, it’s exactly the same as before, just specify the probabilities using the `p` argument. For instance:
``chisq.test( x = observed, p = c(.2, .3, .3, .2) )``
``````##
## Chi-squared test for given probabilities
##
## data: observed
## X-squared = 4.7417, df = 3, p-value = 0.1917``````
Again, these are the same numbers that the `goodnessOfFitTest()` function reports at the end of the output. It just hasn’t included any of the other details.
What about a test of independence? As it turns out, the `chisq.test()` function is pretty clever.180 If you input a cross-tabulation rather than a simple frequency table, it realises that you’re asking for a test of independence and not a goodness of fit test. Recall that we already have this cross-tabulation stored as the `chapekFrequencies` variable:
``chapekFrequencies``
``````## species
## choice robot human
## puppy 13 15
## flower 30 13
## data 44 65``````
To get the test of independence, all we have to do is feed this frequency table into the `chisq.test()` function like so:
``chisq.test( chapekFrequencies )``
``````##
## Pearson's Chi-squared test
##
## data: chapekFrequencies
## X-squared = 10.722, df = 2, p-value = 0.004697``````
Again, the numbers are the same as last time, it’s just that the output is very terse and doesn’t really explain what’s going on in the rather tedious way that `associationTest()` does. As before, my intuition is that when you’re just getting started it’s easier to use something like `associationTest()` because it shows you more detail about what’s going on, but later on you’ll probably find that `chisq.test()` is more convenient. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.06%3A_The_Most_Typical_Way_to_Do_Chi-square_Tests_in_R.txt |
What should you do if your cell counts are too small, but you’d still like to test the null hypothesis that the two variables are independent? One answer would be “collect more data”, but that’s far too glib: there are a lot of situations in which it would be either infeasible or unethical do that. If so, statisticians have a kind of moral obligation to provide scientists with better tests. In this instance, Fisher (1922) kindly provided the right answer to the question. To illustrate the basic idea, let’s suppose that we’re analysing data from a field experiment, looking at the emotional status of people who have been accused of witchcraft; some of whom are currently being burned at the stake.181 Unfortunately for the scientist (but rather fortunately for the general populace), it’s actually quite hard to find people in the process of being set on fire, so the cell counts are awfully small in some cases. The `salem.Rdata` file illustrates the point:
``````load("./rbook-master/data/salem.Rdata")
salem.tabs <- table( trial )
print( salem.tabs )``````
``````## on.fire
## happy FALSE TRUE
## FALSE 3 3
## TRUE 10 0``````
Looking at this data, you’d be hard pressed not to suspect that people not on fire are more likely to be happy than people on fire. However, the chi-square test makes this very hard to test because of the small sample size. If I try to do so, R gives me a warning message:
``chisq.test( salem.tabs )``
``````## Warning in chisq.test(salem.tabs): Chi-squared approximation may be
## incorrect```
```
``````##
## Pearson's Chi-squared test with Yates' continuity correction
##
## data: salem.tabs
## X-squared = 3.3094, df = 1, p-value = 0.06888``````
Speaking as someone who doesn’t want to be set on fire, I’d really like to be able to get a better answer than this. This is where Fisher’s exact test comes in very handy.
The Fisher exact test works somewhat differently to the chi-square test (or in fact any of the other hypothesis tests that I talk about in this book) insofar as it doesn’t have a test statistic; it calculates the p-value “directly”. I’ll explain the basics of how the test works for a 2×2 contingency table, though the test works fine for larger tables. As before, let’s have some notation:
Happy Sad Total
Set on fire O11 O12 R1
Not set on fire O21 O22 R2
Total C1 C2 N
In order to construct the test Fisher treats both the row and column totals (R1, R2, C1 and C2) are known, fixed quantities; and then calculates the probability that we would have obtained the observed frequencies that we did (O11, O12, O21 and O22) given those totals. In the notation that we developed in Chapter 9 this is written:
P(O11,O12,O21,O22 | R1,R2,C1,C2)
and as you might imagine, it’s a slightly tricky exercise to figure out what this probability is, but it turns out that this probability is described by a distribution known as the hypergeometric distribution.182 Now that we know this, what we have to do to calculate our p-value is calculate the probability of observing this particular table or a table that is “more extreme”.183 Back in the 1920s, computing this sum was daunting even in the simplest of situations, but these days it’s pretty easy as long as the tables aren’t too big and the sample size isn’t too large. The conceptually tricky issue is to figure out what it means to say that one contingency table is more “extreme” than another. The easiest solution is to say that the table with the lowest probability is the most extreme. This then gives us the p-value.
The implementation of the test in R is via the `fisher.test()` function. Here’s how it is used:
``fisher.test( salem.tabs )``
``````##
## Fisher's Exact Test for Count Data
##
## data: salem.tabs
## p-value = 0.03571
## alternative hypothesis: true odds ratio is not equal to 1
## 95 percent confidence interval:
## 0.000000 1.202913
## sample estimates:
## odds ratio
## 0``````
This is a bit more output than we got from some of our earlier tests. The main thing we’re interested in here is the p-value, which in this case is small enough (p=.036) to justify rejecting the null hypothesis that people on fire are just as happy as people not on fire. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.07%3A_The_Fisher_Exact_Test.txt |
Suppose you’ve been hired to work for the Australian Generic Political Party (AGPP), and part of your job is to find out how effective the AGPP political advertisements are. So, what you do, is you put together a sample of N=100 people, and ask them to watch the AGPP ads. Before they see anything, you ask them if they intend to vote for the AGPP; and then after showing the ads, you ask them again, to see if anyone has changed their minds. Obviously, if you’re any good at your job, you’d also do a whole lot of other things too, but let’s consider just this one simple experiment. One way to describe your data is via the following contingency table:
Before After Total
Yes 30 10 40
No 70 90 160
Total 100 100 200
At first pass, you might think that this situation lends itself to the Pearson χ2 test of independence (as per Section 12.2). However, a little bit of thought reveals that we’ve got a problem: we have 100 participants, but 200 observations. This is because each person has provided us with an answer in both the before column and the after column. What this means is that the 200 observations aren’t independent of each other: if voter A says “yes” the first time and voter B says “no”, then you’d expect that voter A is more likely to say “yes” the second time than voter B! The consequence of this is that the usual χ2 test won’t give trustworthy answers due to the violation of the independence assumption. Now, if this were a really uncommon situation, I wouldn’t be bothering to waste your time talking about it. But it’s not uncommon at all: this is a standard repeated measures design, and none of the tests we’ve considered so far can handle it. Eek.
The solution to the problem was published by McNemar (1947). The trick is to start by tabulating your data in a slightly different way:
Before: Yes Before: No Total
After: Yes 5 5 10
After: No 25 65 90
Total 30 70 100
This is exactly the same data, but it’s been rewritten so that each of our 100 participants appears in only one cell. Because we’ve written our data this way, the independence assumption is now satisfied, and this is a contingency table that we can use to construct an X2 goodness of fit statistic. However, as we’ll see, we need to do it in a slightly nonstandard way. To see what’s going on, it helps to label the entries in our table a little differently:
Before: Yes Before: No Total
After: Yes a b a+b
After: No c d c+d
Total a+c b+d n
Next, let’s think about what our null hypothesis is: it’s that the “before” test and the “after” test have the same proportion of people saying “Yes, I will vote for AGPP”. Because of the way that we have rewritten the data, it means that we’re now testing the hypothesis that the row totals and column totals come from the same distribution. Thus, the null hypothesis in McNemar’s test is that we have “marginal homogeneity”. That is, the row totals and column totals have the same distribution: Pa+Pb=Pa+Pc, and similarly that Pc+Pd=Pb+Pd. Notice that this means that the null hypothesis actually simplifies to Pb=Pc. In other words, as far as the McNemar test is concerned, it’s only the off-diagonal entries in this table (i.e., b and c) that matter! After noticing this, the McNemar test of marginal homogeneity is no different to a usual χ2 test. After applying the Yates correction, our test statistic becomes:
$X^{2}=\dfrac{(|b-c|-0.5)^{2}}{b+c}$
or, to revert to the notation that we used earlier in this chapter:
$X^{2}=\dfrac{\left(\left|O_{12}-O_{21}\right|-0.5\right)^{2}}{O_{12}+O_{21}}$
and this statistic has an (approximately) χ2 distribution with df=1. However, remember that – just like the other χ2 tests – it’s only an approximation, so you need to have reasonably large expected cell counts for it to work.
Doing the McNemar test in R
Now that you know what the McNemar test is all about, lets actually run one. The agpp.Rdata file contains the raw data that I discussed previously, so let’s have a look at it:
load("./rbook-master/data/agpp.Rdata")
str(agpp)
## 'data.frame': 100 obs. of 3 variables:
## $id : Factor w/ 100 levels "subj.1","subj.10",..: 1 13 24 35 46 57 68 79 90 2 ... ##$ response_before: Factor w/ 2 levels "no","yes": 1 2 2 2 1 1 1 1 1 1 ...
## \$ response_after : Factor w/ 2 levels "no","yes": 2 1 1 1 1 1 1 2 1 1 ...
The agpp data frame contains three variables, an id variable that labels each participant in the data set (we’ll see why that’s useful in a moment), a response_before variable that records the person’s answer when they were asked the question the first time, and a response_after variable that shows the answer that they gave when asked the same question a second time. As usual, here’s the first 6 entries:
head(agpp)
## id response_before response_after
## 1 subj.1 no yes
## 2 subj.2 yes no
## 3 subj.3 yes no
## 4 subj.4 yes no
## 5 subj.5 no no
## 6 subj.6 no no
and here’s a summary:
summary(agpp)
## id response_before response_after
## subj.1 : 1 no :70 no :90
## subj.10 : 1 yes:30 yes:10
## subj.100: 1
## subj.11 : 1
## subj.12 : 1
## subj.13 : 1
## (Other) :94
Notice that each participant appears only once in this data frame. When we tabulate this data frame using xtabs(), we get the appropriate table:
right.table <- xtabs( ~ response_before + response_after, data = agpp)
print( right.table )
## response_after
## response_before no yes
## no 65 5
## yes 25 5
and from there, we can run the McNemar test by using the mcnemar.test() function:
mcnemar.test( right.table )
##
## McNemar's Chi-squared test with continuity correction
##
## data: right.table
## McNemar's chi-squared = 12.033, df = 1, p-value = 0.0005226
And we’re done. We’ve just run a McNemar’s test to determine if people were just as likely to vote AGPP after the ads as they were before hand. The test was significant (χ2(1)=12.04,p<.001), suggesting that they were not. And in fact, it looks like the ads had a negative effect: people were less likely to vote AGPP after seeing the ads. Which makes a lot of sense when you consider the quality of a typical political advertisement.
12.09: Whats the Difference Between McNemar and Independence
Let’s go all the way back to the beginning of the chapter, and look at the `cards` data set again. If you recall, the actual experimental design that I described involved people making two choices. Because we have information about the first choice and the second choice that everyone made, we can construct the following contingency table that cross-tabulates the first choice against the second choice.
``````cardChoices <- xtabs( ~ choice_1 + choice_2, data = cards )
cardChoices``````
``````## choice_2
## choice_1 clubs diamonds hearts spades
## clubs 10 9 10 6
## diamonds 20 4 13 14
## hearts 20 18 3 23
## spades 18 13 15 4``````
Suppose I wanted to know whether the choice you make the second time is dependent on the choice you made the first time. This is where a test of independence is useful, and what we’re trying to do is see if there’s some relationship between the rows and columns of this table. Here’s the result:
``chisq.test( cardChoices )``
Alternatively, suppose I wanted to know if on average, the frequencies of suit choices were different the second time than the first time. In that situation, what I’m really trying to see if the row totals in `cardChoices` (i.e., the frequencies for `choice_1`) are different from the column totals (i.e., the frequencies for `choice_2`). That’s when you use the McNemar test:
````mcnemar.test( cardChoices )`
```
``````##
## McNemar's Chi-squared test
##
## data: cardChoices
## McNemar's chi-squared = 16.033, df = 6, p-value = 0.01358``````
Notice that the results are different! These aren’t the same test. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.08%3A_The_McNemar_Test.txt |
The key ideas discussed in this chapter are:
• The chi-square goodness of fit test (Section 12.1) is used when you have a table of observed frequencies of different categories; and the null hypothesis gives you a set of “known” probabilities to compare them to. You can either use the goodnessOfFitTest() function in the lsr package to run this test, or the chisq.test() function.
• The chi-square test of independence (Section 12.2) is used when you have a contingency table (cross-tabulation) of two categorical variables. The null hypothesis is that there is no relationship/association between the variables. You can either use the associationTest() function in the lsr package, or you can use chisq.test().
• Effect size for a contingency table can be measured in several ways (Section 12.4). In particular we noted the Cramer’s V statistic, which can be calculated using cramersV(). This is also part of the output produced by associationTest().
• Both versions of the Pearson test rely on two assumptions: that the expected frequencies are sufficiently large, and that the observations are independent (Section 12.5). The Fisher exact test (Section 12.7) can be used when the expected frequencies are small, fisher.test(x = contingency.table). The McNemar test (Section 12.8) can be used for some kinds of violations of independence, mcnemar.test(x = contingency.table).
If you’re interested in learning more about categorical data analysis, a good first choice would be Agresti (1996) which, as the title suggests, provides an Introduction to Categorical Data Analysis. If the introductory book isn’t enough for you (or can’t solve the problem you’re working on) you could consider Agresti (2002), Categorical Data Analysis. The latter is a more advanced text, so it’s probably not wise to jump straight from this book to that one.
References
Pearson, K. 1900. “On the Criterion That a Given System of Deviations from the Probable in the Case of a Correlated System of Variables Is Such That It Can Be Reasonably Supposed to Have Arisen from Random Sampling.” Philosophical Magazine 50: 157–75.
Fisher, R. A. 1922a. “On the Interpretation of χ2 from Contingency Tables, and the Calculation of p.” Journal of the Royal Statistical Society 84: 87–94.
Yates, F. 1934. “Contingency Tables Involving Small Numbers and the χ2 Test.” Supplement to the Journal of the Royal Statistical Society 1: 217–35.
Cramér, H. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press.
McNemar, Q. 1947. “Note on the Sampling Error of the Difference Between Correlated Proportions or Percentages.” Psychometrika 12: 153–57.
Agresti, A. 1996. An Introduction to Categorical Data Analysis. Hoboken, NJ: Wiley.
Agresti, A. 2002. Categorical Data Analysis. 2nd ed. Hoboken, NJ: Wiley.
1. I should point out that this issue does complicate the story somewhat: I’m not going to cover it in this book, but there’s a sneaky trick that you can do to rewrite the equation for the goodness of fit statistic as a sum over k−1 independent things. When we do so we get the “proper” sampling distribution, which is chi-square with k−1 degrees of freedom. In fact, in order to get the maths to work out properly, you actually have to rewrite things that way. But it’s beyond the scope of an introductory book to show the maths in that much detail: all I wanted to do is give you a sense of why the goodness of fit statistic is associated with the chi-squared distribution.
2. I feel obliged to point out that this is an over-simplification. It works nicely for quite a few situations; but every now and then we’ll come across degrees of freedom values that aren’t whole numbers. Don’t let this worry you too much – when you come across this, just remind yourself that “degrees of freedom” is actually a bit of a messy concept, and that the nice simple story that I’m telling you here isn’t the whole story. For an introductory class, it’s usually best to stick to the simple story: but I figure it’s best to warn you to expect this simple story to fall apart. If I didn’t give you this warning, you might start getting confused when you see df=3.4 or something; and (incorrectly) thinking that you had misunderstood something that I’ve taught you, rather than (correctly) realising that there’s something that I haven’t told you.
3. In practice, the sample size isn’t always fixed… e.g., we might run the experiment over a fixed period of time, and the number of people participating depends on how many people show up. That doesn’t matter for the current purposes.
4. Well, sort of. The conventions for how statistics should be reported tend to differ somewhat from discipline to discipline; I’ve tended to stick with how things are done in psychology, since that’s what I do. But the general principle of providing enough information to the reader to allow them to check your results is pretty universal, I think.
5. To some people, this advice might sound odd, or at least in conflict with the “usual” advice on how to write a technical report. Very typically, students are told that the “results” section of a report is for describing the data and reporting statistical analysis; and the “discussion” section is for providing interpretation. That’s true as far as it goes, but I think people often interpret it way too literally. The way I usually approach it is to provide a quick and simple interpretation of the data in the results section, so that my reader understands what the data are telling us. Then, in the discussion, I try to tell a bigger story; about how my results fit with the rest of the scientific literature. In short; don’t let the “interpretation goes in the discussion” advice turn your results section into incomprehensible garbage. Being understood by your reader is much more important.
6. Complicating matters, the G-test is a special case of a whole class of tests that are known as likelihood ratio tests. I don’t cover LRTs in this book, but they are quite handy things to know about.
7. A technical note. The way I’ve described the test pretends that the column totals are fixed (i.e., the researcher intended to survey 87 robots and 93 humans) and the row totals are random (i.e., it just turned out that 28 people chose the puppy). To use the terminology from my mathematical statistics textbook (Hogg, McKean, and Craig 2005) I should technically refer to this situation as a chi-square test of homogeneity; and reserve the term chi-square test of independence for the situation where both the row and column totals are random outcomes of the experiment. In the initial drafts of this book that’s exactly what I did. However, it turns out that these two tests are identical; and so I’ve collapsed them together.
8. Technically, Eij here is an estimate, so I should probably write it $\hat{E}_{i j}$. But since no-one else does, I won’t either.
9. A problem many of us worry about in real life.
10. Though I do feel that it’s worth mentioning the assocstats() function in the vcd package. If you install and load the vcd package, then a command like assocstats( chapekFrequencies ) will run the χ2 test as well as the likelihood ratio test (not discussed here); and then report three different measures of effect size: ϕ2, Cram'er’s V, and the contingency coefficient (not discussed here)
11. Not really.
12. This example is based on a joke article published in the Journal of Irreproducible Results.
13. The R functions for this distribution are dhyper(), phyper(), qhyper() and rhyper(), though you don’t need them for this book, and I haven’t given you enough information to use these to perform the Fisher exact test the long way.
14. Not surprisingly, the Fisher exact test is motivated by Fisher’s interpretation of a p-value, not Neyman’s! | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/12%3A_Categorical_Data_Analysis/12.10%3A_Summary.txt |
In the previous chapter we covered the situation when your outcome variable is nominal scale and your predictor variable184 is also nominal scale. Lots of real world situations have that character, and so you’ll find that chi-square tests in particular are quite widely used. However, you’re much more likely to find yourself in a situation where your outcome variable is interval scale or higher, and what you’re interested in is whether the average value of the outcome variable is higher in one group or another. For instance, a psychologist might want to know if anxiety levels are higher among parents than non-parents, or if working memory capacity is reduced by listening to music (relative to not listening to music). In a medical context, we might want to know if a new drug increases or decreases blood pressure. An agricultural scientist might want to know whether adding phosphorus to Australian native plants will kill them.185 In all these situations, our outcome variable is a fairly continuous, interval or ratio scale variable; and our predictor is a binary “grouping” variable. In other words, we want to compare the means of the two groups.
The standard answer to the problem of comparing means is to use a t-test, of which there are several varieties depending on exactly what question you want to solve. As a consequence, the majority of this chapter focuses on different types of t-test: one sample t-tests are discussed in Section 13.2, independent samples t-tests are discussed in Sections 13.3 and 13.4, and paired samples t-tests are discussed in Section 13.5. After that, we’ll talk a bit about Cohen’s d, which is the standard measure of effect size for a t-test (Section 13.8). The later sections of the chapter focus on the assumptions of the t-tests, and possible remedies if they are violated. However, before discussing any of these useful things, we’ll start with a discussion of the z-test.
13: Comparing Two Means
In this section I’ll describe one of the most useless tests in all of statistics: the z-test. Seriously – this test is almost never used in real life. Its only real purpose is that, when teaching statistics, it’s a very convenient stepping stone along the way towards the t-test, which is probably the most (over)used tool in all statistics.
inference problem that the test addresses
To introduce the idea behind the z-test, let’s use a simple example. A friend of mine, Dr Zeppo, grades his introductory statistics class on a curve. Let’s suppose that the average grade in his class is 67.5, and the standard deviation is 9.5. Of his many hundreds of students, it turns out that 20 of them also take psychology classes. Out of curiosity, I find myself wondering: do the psychology students tend to get the same grades as everyone else (i.e., mean 67.5) or do they tend to score higher or lower? He emails me the zeppo.Rdata file, which I use to pull up the grades of those students,
load( "./rbook-master/data/zeppo.Rdata" )
print( grades )
## [1] 50 60 60 64 66 66 67 69 70 74 76 76 77 79 79 79 81 82 82 89
and calculate the mean:
mean( grades )
## [1] 72.3
Hm. It might be that the psychology students are scoring a bit higher than normal: that sample mean of $\bar{X}$ = 72.3 is a fair bit higher than the hypothesised population mean of μ=67.5, but on the other hand, a sample size of N=20 isn’t all that big. Maybe it’s pure chance.
To answer the question, it helps to be able to write down what it is that I think I know. Firstly, I know that the sample mean is $\bar{X}$ =72.3. If I’m willing to assume that the psychology students have the same standard deviation as the rest of the class then I can say that the population standard deviation is σ=9.5. I’ll also assume that since Dr Zeppo is grading to a curve, the psychology student grades are normally distributed.
Next, it helps to be clear about what I want to learn from the data. In this case, my research hypothesis relates to the population mean μ for the psychology student grades, which is unknown. Specifically, I want to know if μ=67.5 or not. Given that this is what I know, can we devise a hypothesis test to solve our problem? The data, along with the hypothesised distribution from which they are thought to arise, are shown in Figure 13.1. Not entirely obvious what the right answer is, is it? For this, we are going to need some statistics.
Constructing the hypothesis test
The first step in constructing a hypothesis test is to be clear about what the null and alternative hypotheses are. This isn’t too hard to do. Our null hypothesis, H0, is that the true population mean μ for psychology student grades is 67.5%; and our alternative hypothesis is that the population mean isn’t 67.5%. If we write this in mathematical notation, these hypotheses become,
H0:μ=67.5
H1:μ≠67.5
though to be honest this notation doesn’t add much to our understanding of the problem, it’s just a compact way of writing down what we’re trying to learn from the data. The null hypotheses H0 and the alternative hypothesis H1 for our test are both illustrated in Figure 13.2. In addition to providing us with these hypotheses, the scenario outlined above provides us with a fair amount of background knowledge that might be useful. Specifically, there are two special pieces of information that we can add:
1 The psychology grades are normally distributed. 1 The true standard deviation of these scores σ is known to be 9.5.
For the moment, we’ll act as if these are absolutely trustworthy facts. In real life, this kind of absolutely trustworthy background knowledge doesn’t exist, and so if we want to rely on these facts we’ll just have make the assumption that these things are true. However, since these assumptions may or may not be warranted, we might need to check them. For now though, we’ll keep things simple.
The next step is to figure out what we would be a good choice for a diagnostic test statistic; something that would help us discriminate between H0 and H1. Given that the hypotheses all refer to the population mean μ, you’d feel pretty confident that the sample mean $\bar{X}$ would be a pretty useful place to start. What we could do, is look at the difference between the sample mean $\bar{X}$ and the value that the null hypothesis predicts for the population mean. In our example, that would mean we calculate $\bar{X}$ - 67.5. More generally, if we let μ0 refer to the value that the null hypothesis claims is our population mean, then we’d want to calculate
$\bar{X}-\mu_{0}$
If this quantity equals or is very close to 0, things are looking good for the null hypothesis. If this quantity is a long way away from 0, then it’s looking less likely that the null hypothesis is worth retaining. But how far away from zero should it be for us to reject H0?
To figure that out, we need to be a bit more sneaky, and we’ll need to rely on those two pieces of background knowledge that I wrote down previously, namely that the raw data are normally distributed, and we know the value of the population standard deviation σ. If the null hypothesis is actually true, and the true mean is μ0, then these facts together mean that we know the complete population distribution of the data: a normal distribution with mean μ0 and standard deviation σ. Adopting the notation from Section 9.5, a statistician might write this as:
X∼Normal(μ02)
Okay, if that’s true, then what can we say about the distribution of $\bar{X}$? Well, as we discussed earlier (see Section 10.3.3), the sampling distribution of the mean $\bar{X}$ is also normal, and has mean μ. But the standard deviation of this sampling distribution SE ($\bar{X}$), which is called the standard error of the mean, is
$\operatorname{SE}(\bar{X})=\dfrac{\sigma}{\sqrt{N}}$
In other words, if the null hypothesis is true then the sampling distribution of the mean can be written as follows:
$\bar{X}$∼Normal(μ0,SE($\bar{X}$))
Now comes the trick. What we can do is convert the sample mean $\bar{X}$ into a standard score (Section 5.6). This is conventionally written as z, but for now I’m going to refer to it as $z_{\bar{X}}$. (The reason for using this expanded notation is to help you remember that we’re calculating standardised version of a sample mean, not a standardised version of a single observation, which is what a z-score usually refers to). When we do so, the z-score for our sample mean is
$\ z_{\bar{X}} = {{\bar{X}-\mu_{0}} \over SE(\bar{X})}$
or, equivalently
$\ z_{\bar{X}} = {{\bar{X}-\mu_{0}} \over \sigma/ \sqrt{N} }$
This z-score is our test statistic. The nice thing about using this as our test statistic is that like all z-scores, it has a standard normal distribution:
$\ z_{\bar{X}}$∼Normal(0,1)
(again, see Section 5.6 if you’ve forgotten why this is true). In other words, regardless of what scale the original data are on, the z-statistic iteself always has the same interpretation: it’s equal to the number of standard errors that separate the observed sample mean$\bar{X}$ from the population mean μ0 predicted by the null hypothesis. Better yet, regardless of what the population parameters for the raw scores actually are, the 5% critical regions for z-test are always the same, as illustrated in Figures 13.4 and 13.3. And what this meant, way back in the days where people did all their statistics by hand, is that someone could publish a table like this:
desired α level two-sided test one-sided test
.1 1.644854 1.281552
.05 1.959964 1.644854
.01 2.575829 2.326348
.001 3.290527 3.090232
which in turn meant that researchers could calculate their z-statistic by hand, and then look up the critical value in a text book. That was an incredibly handy thing to be able to do back then, but it’s kind of unnecessary these days, since it’s trivially easy to do it with software like R.
worked example using R
Now, as I mentioned earlier, the z-test is almost never used in practice. It’s so rarely used in real life that the basic installation of R doesn’t have a built in function for it. However, the test is so incredibly simple that it’s really easy to do one manually. Let’s go back to the data from Dr Zeppo’s class. Having loaded the grades data, the first thing I need to do is calculate the sample mean:
sample.mean <- mean( grades )
print( sample.mean )
## [1] 72.3
Then, I create variables corresponding to known population standard deviation (σ=9.5), and the value of the population mean that the null hypothesis specifies (μ0=67.5):
mu.null <- 67.5
sd.true <- 9.5
Let’s also create a variable for the sample size. We could count up the number of observations ourselves, and type N <- 20 at the command prompt, but counting is tedious and repetitive. Let’s get R to do the tedious repetitive bit by using the length() function, which tells us how many elements there are in a vector:
N <- length( grades )
print( N )
## [1] 20
Next, let’s calculate the (true) standard error of the mean:
sem.true <- sd.true / sqrt(N)
print(sem.true)
## [1] 2.124265
And finally, we calculate our z-score:
z.score <- (sample.mean - mu.null) / sem.true
print( z.score )
## [1] 2.259606
At this point, we would traditionally look up the value 2.26 in our table of critical values. Our original hypothesis was two-sided (we didn’t really have any theory about whether psych students would be better or worse at statistics than other students) so our hypothesis test is two-sided (or two-tailed) also. Looking at the little table that I showed earlier, we can see that 2.26 is bigger than the critical value of 1.96 that would be required to be significant at α=.05, but smaller than the value of 2.58 that would be required to be significant at a level of α=.01. Therefore, we can conclude that we have a significant effect, which we might write up by saying something like this:
With a mean grade of 73.2 in the sample of psychology students, and assuming a true population standard deviation of 9.5, we can conclude that the psychology students have significantly different statistics scores to the class average (z=2.26, N=20, p<.05).
However, what if want an exact p-value? Well, back in the day, the tables of critical values were huge, and so you could look up your actual z-value, and find the smallest value of α for which your data would be significant (which, as discussed earlier, is the very definition of a p-value). However, looking things up in books is tedious, and typing things into computers is awesome. So let’s do it using R instead. Now, notice that the α level of a z-test (or any other test, for that matter) defines the total area “under the curve” for the critical region, right? That is, if we set α=.05 for a two-sided test, then the critical region is set up such that the area under the curve for the critical region is .05. And, for the z-test, the critical value of 1.96 is chosen that way because the area in the lower tail (i.e., below −1.96) is exactly .025 and the area under the upper tail (i.e., above 1.96) is exactly .025. So, since our observed z-statistic is 2.26, why not calculate the area under the curve below −2.26 or above 2.26? In R we can calculate this using the pnorm() function. For the upper tail:
upper.area <- pnorm( q = z.score, lower.tail = FALSE )
print( upper.area )
## [1] 0.01192287
The lower.tail = FALSE is me telling R to calculate the area under the curve from 2.26 and upwards. If I’d told it that lower.tail = TRUE, then R would calculate the area from 2.26 and below, and it would give me an answer 0.9880771. Alternatively, to calculate the area from −2.26 and below, we get
lower.area <- pnorm( q = -z.score, lower.tail = TRUE )
print( lower.area )
## [1] 0.01192287
Thus we get our p-value:
p.value <- lower.area + upper.area
print( p.value )
## [1] 0.02384574
Assumptions of the z-test
As I’ve said before, all statistical tests make assumptions. Some tests make reasonable assumptions, while other tests do not. The test I’ve just described – the one sample z-test – makes three basic assumptions. These are:
• Normality. As usually described, the z-test assumes that the true population distribution is normal.186 is often pretty reasonable, and not only that, it’s an assumption that we can check if we feel worried about it (see Section 13.9).
• Independence. The second assumption of the test is that the observations in your data set are not correlated with each other, or related to each other in some funny way. This isn’t as easy to check statistically: it relies a bit on good experimetal design. An obvious (and stupid) example of something that violates this assumption is a data set where you “copy” the same observation over and over again in your data file: so you end up with a massive “sample size”, consisting of only one genuine observation. More realistically, you have to ask yourself if it’s really plausible to imagine that each observation is a completely random sample from the population that you’re interested in. In practice, this assumption is never met; but we try our best to design studies that minimise the problems of correlated data.
• Known standard deviation. The third assumption of the z-test is that the true standard deviation of the population is known to the researcher. This is just stupid. In no real world data analysis problem do you know the standard deviation σ of some population, but are completely ignorant about the mean μ. In other words, this assumption is always wrong.
In view of the stupidity of assuming that σ is known, let’s see if we can live without it. This takes us out of the dreary domain of the z-test, and into the magical kingdom of the t-test, with unicorns and fairies and leprechauns, and um… | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.01%3A_The_one-sample_z-test.txt |
After some thought, I decided that it might not be safe to assume that the psychology student grades necessarily have the same standard deviation as the other students in Dr Zeppo’s class. After all, if I’m entertaining the hypothesis that they don’t have the same mean, then why should I believe that they absolutely have the same standard deviation? In view of this, I should really stop assuming that I know the true value of σ. This violates the assumptions of my z-test, so in one sense I’m back to square one. However, it’s not like I’m completely bereft of options. After all, I’ve still got my raw data, and those raw data give me an estimate of the population standard deviation:
sd( grades )
## [1] 9.520615
In other words, while I can’t say that I know that σ=9.5, I can say that $\hat{\sigma}$=9.52.
Okay, cool. The obvious thing that you might think to do is run a z-test, but using the estimated standard deviation of 9.52 instead of relying on my assumption that the true standard deviation is 9.5. So, we could just type this new number into R and out would come the answer. And you probably wouldn’t be surprised to hear that this would still give us a significant result. This approach is close, but it’s not quite correct. Because we are now relying on an estimate of the population standard deviation, we need to make some adjustment for the fact that we have some uncertainty about what the true population standard deviation actually is. Maybe our data are just a fluke … maybe the true population standard deviation is 11, for instance. But if that were actually true, and we ran the z-test assuming σ=11, then the result would end up being non-significant. That’s a problem, and it’s one we’re going to have to address.
Introducing the t-test
This ambiguity is annoying, and it was resolved in 1908 by a guy called William Sealy Gosset (Student 1908), who was working as a chemist for the Guinness brewery at the time (see Box 1987). Because Guinness took a dim view of its employees publishing statistical analysis (apparently they felt it was a trade secret), he published the work under the pseudonym “A Student”, and to this day, the full name of the t-test is actually Student’s t-test. The key thing that Gosset figured out is how we should accommodate the fact that we aren’t completely sure what the true standard deviation is.187 The answer is that it subtly changes the sampling distribution. In the t-test, our test statistic (now called a t-statistic) is calculated in exactly the same way I mentioned above. If our null hypothesis is that the true mean is μ, but our sample has mean ¯X and our estimate of the population standard deviation is $\hat{\sigma}$, then our t statistic is:
$\ t = {{\bar{X}-\mu} \over \hat{\sigma}/ \sqrt{N} }$
The only thing that has changed in the equation is that instead of using the known true value σ, we use the estimate $\hat{\sigma}$ And if this estimate has been constructed from N observations, then the sampling distribution turns into a t-distribution with N−1 degrees of freedom (df). The t distribution is very similar to the normal distribution, but has “heavier” tails, as discussed earlier in Section 9.6 and illustrated in Figure 13.5. Notice, though, that as df gets larger, the t-distribution starts to look identical to the standard normal distribution. This is as it should be: if you have a sample size of N=70,000,000 then your “estimate” of the standard deviation would be pretty much perfect, right? So, you should expect that for large N, the t-test would behave exactly the same way as a z-test. And that’s exactly what happens!
Doing the test in R
As you might expect, the mechanics of the t-test are almost identical to the mechanics of the z-test. So there’s not much point in going through the tedious exercise of showing you how to do the calculations using low level commands: it’s pretty much identical to the calculations that we did earlier, except that we use the estimated standard deviation (i.e., something like se.est <- sd(grades)), and then we test our hypothesis using the t distribution rather than the normal distribution (i.e. we use pt() rather than pnorm(). And so instead of going through the calculations in tedious detail for a second time, I’ll jump straight to showing you how t-tests are actually done in practice.
The situation with t-tests is very similar to the one we encountered with chi-squared tests in Chapter 12. R comes with one function called t.test() that is very flexible (it can run lots of different kinds of t-tests) and is somewhat terse (the output is quite compressed). Later on in the chapter I’ll show you how to use the t.test() function (Section 13.7), but to start out with I’m going to rely on some simpler functions in the lsr package. Just like last time, what I’ve done is written a few simpler functions, each of which does only one thing. So, if you want to run a one-sample t-test, use the oneSampleTTest() function! It’s pretty straightforward to use: all you need to do is specify x, the variable containing the data, and mu, the true population mean according to the null hypothesis. All you need to type is this:
library(lsr)
oneSampleTTest( x=grades, mu=67.5 )
##
## One sample t-test
##
## Data variable: grades
##
## Descriptive statistics:
## grades
## mean 72.300
## std dev. 9.521
##
## Hypotheses:
## null: population mean equals 67.5
## alternative: population mean not equal to 67.5
##
## Test results:
## t-statistic: 2.255
## degrees of freedom: 19
## p-value: 0.036
##
## Other information:
## two-sided 95% confidence interval: [67.844, 76.756]
## estimated effect size (Cohen's d): 0.504
Easy enough. Now lets go through the output. Just like we saw in the last chapter, I’ve written the functions so that the output is pretty verbose. It tries to describe in a lot of detail what its actually done:
One sample t-test
Data variable: grades
Descriptive statistics:
grades
mean 72.300
std dev. 9.521
Hypotheses:
null: population mean equals 67.5
alternative: population mean not equal to 67.5
Test results:
t-statistic: 2.255
degrees of freedom: 19
p-value: 0.036
Other information:
two-sided 95% confidence interval: [67.844, 76.756]
estimated effect size (Cohen's d): 0.504
Reading this output from top to bottom, you can see it’s trying to lead you through the data analysis process. The first two lines tell you what kind of test was run and what data were used. It then gives you some basic information about the sample: specifically, the sample mean and standard deviation of the data. It then moves towards the inferential statistics part. It starts by telling you what the null and alternative hypotheses were, and then it reports the results of the test: the t-statistic, the degrees of freedom, and the p-value. Finally, it reports two other things you might care about: the confidence interval for the mean, and a measure of effect size (we’ll talk more about effect sizes later).
So that seems straightforward enough. Now what do we do with this output? Well, since we’re pretending that we actually care about my toy example, we’re overjoyed to discover that the result is statistically significant (i.e. p value below 0.05). We could report the result by saying something like this:
With a mean grade of 72.3, the psychology students scored slightly higher than the average grade of 67.5 (t(19)=2.25, p<.05); the 95% confidence interval is [67.8, 76.8].
where t(19) is shorthand notation for a t-statistic that has 19 degrees of freedom. That said, it’s often the case that people don’t report the confidence interval, or do so using a much more compressed form than I’ve done here. For instance, it’s not uncommon to see the confidence interval included as part of the stat block, like this:
t(19)=2.25, p<.05, CI95=[67.8,76.8]
With that much jargon crammed into half a line, you know it must be really smart.188
Assumptions of the one sample t-test
Okay, so what assumptions does the one-sample t-test make? Well, since the t-test is basically a z-test with the assumption of known standard deviation removed, you shouldn’t be surprised to see that it makes the same assumptions as the z-test, minus the one about the known standard deviation. That is
• Normality. We’re still assuming that the the population distribution is normal^[A technical comment… in the same way that we can weaken the assumptions of the z-test so that we’re only talking about the sampling distribution, we can weaken the t test assumptions so that we don’t have to assume normality of the population. However, for the t-test, it’s trickier to do this. As before, we can replace the assumption of population normality with an assumption that the sampling distribution of ¯X is normal. However, remember that we’re also relying on a sample estimate of the standard deviation; and so we also require the sampling distribution of ^σ to be chi-square. That makes things nastier, and this version is rarely used in practice: fortunately, if the population is normal, then both of these two assumptions are met., and as noted earlier, there are standard tools that you can use to check to see if this assumption is met (Section 13.9), and other tests you can do in it’s place if this assumption is violated (Section 13.10).
• Independence. Once again, we have to assume that the observations in our sample are generated independently of one another. See the earlier discussion about the z-test for specifics (Section 13.1.4).
Overall, these two assumptions aren’t terribly unreasonable, and as a consequence the one-sample t-test is pretty widely used in practice as a way of comparing a sample mean against a hypothesised population mean. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.02%3A_The_One-sample_t-test.txt |
Although the one sample t-test has its uses, it’s not the most typical example of a t-test189. A much more common situation arises when you’ve got two different groups of observations. In psychology, this tends to correspond to two different groups of participants, where each group corresponds to a different condition in your study. For each person in the study, you measure some outcome variable of interest, and the research question that you’re asking is whether or not the two groups have the same population mean. This is the situation that the independent samples t-test is designed for.
data
Suppose we have 33 students taking Dr Harpo’s statistics lectures, and Dr Harpo doesn’t grade to a curve. Actually, Dr Harpo’s grading is a bit of a mystery, so we don’t really know anything about what the average grade is for the class as a whole. There are two tutors for the class, Anastasia and Bernadette. There are N1=15 students in Anastasia’s tutorials, and N2=18 in Bernadette’s tutorials. The research question I’m interested in is whether Anastasia or Bernadette is a better tutor, or if it doesn’t make much of a difference. Dr Harpo emails me the course grades, in the harpo.Rdata file. As usual, I’ll load the file and have a look at what variables it contains:
load( "./rbook-master/data/harpo.Rdata" )
str(harpo)
## 'data.frame': 33 obs. of 2 variables:
## $grade: num 65 72 66 74 73 71 66 76 69 79 ... ##$ tutor: Factor w/ 2 levels "Anastasia","Bernadette": 1 2 2 1 1 2 2 2 2 2 ...
As we can see, there’s a single data frame with two variables, grade and tutor. The grade variable is a numeric vector, containing the grades for all N=33 students taking Dr Harpo’s class; the tutor variable is a factor that indicates who each student’s tutor was. The first six observations in this data set are shown below:
head( harpo )
## grade tutor
## 1 65 Anastasia
## 2 72 Bernadette
## 3 66 Bernadette
## 4 74 Anastasia
## 5 73 Anastasia
## 6 71 Bernadette
We can calculate means and standard deviations, using the mean() and sd() functions. Rather than show the R output, here’s a nice little summary table:
mean std dev N
Anastasia’s students 74.53 9.00 15
Bernadette’s students 69.06 5.77 18
To give you a more detailed sense of what’s going on here, I’ve plotted histograms showing the distribution of grades for both tutors (Figure 13.6 and 13.7). Inspection of these histograms suggests that the students in Anastasia’s class may be getting slightly better grades on average, though they also seem a little more variable.
Here is a simpler plot showing the means and corresponding confidence intervals for both groups of students (Figure 13.8).
Introducing the test
The independent samples t-test comes in two different forms, Student’s and Welch’s. The original Student t-test – which is the one I’ll describe in this section – is the simpler of the two, but relies on much more restrictive assumptions than the Welch t-test. Assuming for the moment that you want to run a two-sided test, the goal is to determine whether two “independent samples” of data are drawn from populations with the same mean (the null hypothesis) or different means (the alternative hypothesis). When we say “independent” samples, what we really mean here is that there’s no special relationship between observations in the two samples. This probably doesn’t make a lot of sense right now, but it will be clearer when we come to talk about the paired samples t-test later on. For now, let’s just point out that if we have an experimental design where participants are randomly allocated to one of two groups, and we want to compare the two groups’ mean performance on some outcome measure, then an independent samples t-test (rather than a paired samples t-test) is what we’re after.
Okay, so let’s let μ1 denote the true population mean for group 1 (e.g., Anastasia’s students), and μ2 will be the true population mean for group 2 (e.g., Bernadette’s students),190 and as usual we’ll let $\bar{X}_{1}$ and $\bar{X}_{2}$ denote the observed sample means for both of these groups. Our null hypothesis states that the two population means are identical (μ12) and the alternative to this is that they are not (μ1≠μ2). Written in mathematical-ese, this is…
H012
H11≠μ2
To construct a hypothesis test that handles this scenario, we start by noting that if the null hypothesis is true, then the difference between the population means is exactly zero, μ1−μ2=0 As a consequence, a diagnostic test statistic will be based on the difference between the two sample means. Because if the null hypothesis is true, then we’d expect
$\bar{X}_{1}$ - $\bar{X}_{2}$
to be pretty close to zero. However, just like we saw with our one-sample tests (i.e., the one-sample z-test and the one-sample t-test) we have to be precise about exactly how close to zero this difference
$\ t ={\bar{X}_1 - \bar{X}_2 \over SE}$
We just need to figure out what this standard error estimate actually is. This is a bit trickier than was the case for either of the two tests we’ve looked at so far, so we need to go through it a lot more carefully to understand how it works.
“pooled estimate” of the standard deviation
In the original “Student t-test”, we make the assumption that the two groups have the same population standard deviation: that is, regardless of whether the population means are the same, we assume that the population standard deviations are identical, σ12. Since we’re assuming that the two standard deviations are the same, we drop the subscripts and refer to both of them as σ. How should we estimate this? How should we construct a single estimate of a standard deviation when we have two samples? The answer is, basically, we average them. Well, sort of. Actually, what we do is take a weighed average of the variance estimates, which we use as our pooled estimate of the variance. The weight assigned to each sample is equal to the number of observations in that sample, minus 1. Mathematically, we can write this as
$\ \omega_{1}$=N1−1
$\ \omega_{2}$=N2−1
Now that we’ve assigned weights to each sample, we calculate the pooled estimate of the variance by taking the weighted average of the two variance estimates, $\ \hat{\sigma_1}^2$ and $\ \hat{\sigma_2}^2$
$\ \hat{\sigma_p}^2 ={ \omega_{1}\hat{\sigma_1}^2+\omega_{2}\hat{\sigma_2}^2 \over \omega_{1}+\omega_{2}}$
Finally, we convert the pooled variance estimate to a pooled standard deviation estimate, by taking the square root. This gives us the following formula for $\ \hat{\sigma_p}$,
$\ \hat{\sigma_p} =\sqrt{\omega_1\hat{\sigma_1}^2+\omega_2\hat{\sigma_2}^2\over \omega_1+\omega_2}$
And if you mentally substitute $\ \omega_1$=N1−1 and $\ \omega_2$=N2−1 into this equation you get a very ugly looking formula; a very ugly formula that actually seems to be the “standard” way of describing the pooled standard deviation estimate. It’s not my favourite way of thinking about pooled standard deviations, however.191
same pooled estimate, described differently
I prefer to think about it like this. Our data set actually corresponds to a set of N observations, which are sorted into two groups. So let’s use the notation Xik to refer to the grade received by the i-th student in the k-th tutorial group: that is, X11 is the grade received by the first student in Anastasia’s class, X21 is her second student, and so on. And we have two separate group means $\ \bar{X_1}$ and $\ \bar{X_2}$, which we could “generically” refer to using the notation $\ \bar{X_k}$, i.e., the mean grade for the k-th tutorial group. So far, so good. Now, since every single student falls into one of the two tutorials, and so we can describe their deviation from the group mean as the difference
$\ X_{ik} - \bar{X_k}$
So why not just use these deviations (i.e., the extent to which each student’s grade differs from the mean grade in their tutorial?) Remember, a variance is just the average of a bunch of squared deviations, so let’s do that. Mathematically, we could write it like this:
$\ ∑_{ik} (X_{ik}-\bar{X}_k)^2 \over N$
where the notation “∑ik” is a lazy way of saying “calculate a sum by looking at all students in all tutorials”, since each “ik” corresponds to one student.192 But, as we saw in Chapter 10, calculating the variance by dividing by N produces a biased estimate of the population variance. And previously, we needed to divide by N−1 to fix this. However, as I mentioned at the time, the reason why this bias exists is because the variance estimate relies on the sample mean; and to the extent that the sample mean isn’t equal to the population mean, it can systematically bias our estimate of the variance. But this time we’re relying on two sample means! Does this mean that we’ve got more bias? Yes, yes it does. And does this mean we now need to divide by N−2 instead of N−1, in order to calculate our pooled variance estimate? Why, yes…
$\hat{\sigma}_{p}\ ^{2}=\dfrac{\sum_{i k}\left(X_{i k}-X_{k}\right)^{2}}{N-2}$
Oh, and if you take the square root of this then you get $\ \hat{\sigma_{P}}$, the pooled standard deviation estimate. In other words, the pooled standard deviation calculation is nothing special: it’s not terribly different to the regular standard deviation calculation.
Completing the test
Regardless of which way you want to think about it, we now have our pooled estimate of the standard deviation. From now on, I’ll drop the silly p subscript, and just refer to this estimate as $\ \hat{\sigma}$. Great. Let’s now go back to thinking about the bloody hypothesis test, shall we? Our whole reason for calculating this pooled estimate was that we knew it would be helpful when calculating our standard error estimate. But, standard error of what? In the one-sample t-test, it was the standard error of the sample mean, SE ($\ \bar{X}$), and since SE ($\ \bar{X}=\sigma/ \sqrt{N}$ that’s what the denominator of our t-statistic looked like. This time around, however, we have two sample means. And what we’re interested in, specifically, is the the difference between the two $\ \bar{X_1}$ - $\ \bar{X_2}$. As a consequence, the standard error that we need to divide by is in fact the standard error of the difference between means. As long as the two variables really do have the same standard deviation, then our estimate for the standard error is
$\operatorname{SE}\left(\bar{X}_{1}-\bar{X}_{2}\right)=\hat{\sigma} \sqrt{\dfrac{1}{N_{1}}+\dfrac{1}{N_{2}}}$
and our t-statistic is therefore
$t=\dfrac{\bar{X}_{1}-\bar{X}_{2}}{\operatorname{SE}\left(\bar{X}_{1}-\bar{X}_{2}\right)}$
(shocking, isn’t it?) as long as the null hypothesis is true, and all of the assumptions of the test are met. The degrees of freedom, however, is slightly different. As usual, we can think of the degrees of freedom to be equal to the number of data points minus the number of constraints. In this case, we have N observations (N1 in sample 1, and N2 in sample 2), and 2 constraints (the sample means). So the total degrees of freedom for this test are N−2.
Doing the test in R
Not surprisingly, you can run an independent samples t-test using the t.test() function (Section 13.7), but once again I’m going to start with a somewhat simpler function in the lsr package. That function is unimaginatively called independentSamplesTTest(). First, recall that our data look like this:
head( harpo )
## grade tutor
## 1 65 Anastasia
## 2 72 Bernadette
## 3 66 Bernadette
## 4 74 Anastasia
## 5 73 Anastasia
## 6 71 Bernadette
The outcome variable for our test is the student grade, and the groups are defined in terms of the tutor for each class. So you probably won’t be too surprised to see that we’re going to describe the test that we want in terms of an R formula that reads like this grade ~ tutor. The specific command that we need is:
independentSamplesTTest(
formula = grade ~ tutor, # formula specifying outcome and group variables
data = harpo, # data frame that contains the variables
var.equal = TRUE # assume that the two groups have the same variance
)
##
## Student's independent samples t-test
##
## Outcome variable: grade
## Grouping variable: tutor
##
## Descriptive statistics:
## Anastasia Bernadette
## mean 74.533 69.056
## std dev. 8.999 5.775
##
## Hypotheses:
## null: population means equal for both groups
## alternative: different population means in each group
##
## Test results:
## t-statistic: 2.115
## degrees of freedom: 31
## p-value: 0.043
##
## Other information:
## two-sided 95% confidence interval: [0.197, 10.759]
## estimated effect size (Cohen's d): 0.74
The first two arguments should be familiar to you. The first one is the formula that tells R what variables to use and the second one tells R the name of the data frame that stores those variables. The third argument is not so obvious. By saying var.equal = TRUE, what we’re really doing is telling R to use the Student independent samples t-test. More on this later. For now, lets ignore that bit and look at the output:
The output has a very familiar form. First, it tells you what test was run, and it tells you the names of the variables that you used. The second part of the output reports the sample means and standard deviations for both groups (i.e., both tutorial groups). The third section of the output states the null hypothesis and the alternative hypothesis in a fairly explicit form. It then reports the test results: just like last time, the test results consist of a t-statistic, the degrees of freedom, and the p-value. The final section reports two things: it gives you a confidence interval, and an effect size. I’ll talk about effect sizes later. The confidence interval, however, I should talk about now.
It’s pretty important to be clear on what this confidence interval actually refers to: it is a confidence interval for the difference between the group means. In our example, Anastasia’s students had an average grade of 74.5, and Bernadette’s students had an average grade of 69.1, so the difference between the two sample means is 5.4. But of course the difference between population means might be bigger or smaller than this. The confidence interval reported by the independentSamplesTTest() function tells you that there’s a 95% chance that the true difference between means lies between 0.2 and 10.8.
In any case, the difference between the two groups is significant (just barely), so we might write up the result using text like this:
The mean grade in Anastasia’s class was 74.5% (std dev = 9.0), whereas the mean in Bernadette’s class was 69.1% (std dev = 5.8). A Student’s independent samples t-test showed that this 5.4% difference was significant (t(31)=2.1, p<.05, CI95=[0.2,10.8], d=.74), suggesting that a genuine difference in learning outcomes has occurred.
Notice that I’ve included the confidence interval and the effect size in the stat block. People don’t always do this. At a bare minimum, you’d expect to see the t-statistic, the degrees of freedom and the p value. So you should include something like this at a minimum: t(31)=2.1, p<.05. If statisticians had their way, everyone would also report the confidence interval and probably the effect size measure too, because they are useful things to know. But real life doesn’t always work the way statisticians want it to: you should make a judgment based on whether you think it will help your readers, and (if you’re writing a scientific paper) the editorial standard for the journal in question. Some journals expect you to report effect sizes, others don’t. Within some scientific communities it is standard practice to report confidence intervals, in other it is not. You’ll need to figure out what your audience expects. But, just for the sake of clarity, if you’re taking my class: my default position is that it’s usually worth includng the effect size, but don’t worry about the confidence interval unless the assignment asks you to or implies that you should.
Positive and negative t values
Before moving on to talk about the assumptions of the t-test, there’s one additional point I want to make about the use of t-tests in practice. The first one relates to the sign of the t-statistic (that is, whether it is a positive number or a negative one). One very common worry that students have when they start running their first t-test is that they often end up with negative values for the t-statistic, and don’t know how to interpret it. In fact, it’s not at all uncommon for two people working independently to end up with R outputs that are almost identical, except that one person has a negative t values and the other one has a positive t value. Assuming that you’re running a two-sided test, then the p-values will be identical. On closer inspection, the students will notice that the confidence intervals also have the opposite signs. This is perfectly okay: whenever this happens, what you’ll find is that the two versions of the R output arise from slightly different ways of running the t-test. What’s happening here is very simple. The t-statistic that R is calculating here is always of the form
$t=\dfrac{(\text { mean } 1)-(\text { mean } 2)}{(\mathrm{SE})}$
If “mean 1” is larger than “mean 2” the t statistic will be positive, whereas if “mean 2” is larger then the t statistic will be negative. Similarly, the confidence interval that R reports is the confidence interval for the difference “(mean 1) minus (mean 2)”, which will be the reverse of what you’d get if you were calculating the confidence interval for the difference “(mean 2) minus (mean 1)”.
Okay, that’s pretty straightforward when you think about it, but now consider our t-test comparing Anastasia’s class to Bernadette’s class. Which one should we call “mean 1” and which one should we call “mean 2”. It’s arbitrary. However, you really do need to designate one of them as “mean 1” and the other one as “mean 2”. Not surprisingly, the way that R handles this is also pretty arbitrary. In earlier versions of the book I used to try to explain it, but after a while I gave up, because it’s not really all that important, and to be honest I can never remember myself. Whenever I get a significant t-test result, and I want to figure out which mean is the larger one, I don’t try to figure it out by looking at the t-statistic. Why would I bother doing that? It’s foolish. It’s easier just look at the actual group means, since the R output actually shows them!
Here’s the important thing. Because it really doesn’t matter what R printed out, I usually try to report the t-statistic in such a way that the numbers match up with the text. Here’s what I mean… suppose that what I want to write in my report is “Anastasia’s class had higher grades than Bernadette’s class”. The phrasing here implies that Anastasia’s group comes first, so it makes sense to report the t-statistic as if Anastasia’s class corresponded to group 1. If so, I would write
Anastasia’s class had higher grades than Bernadette’s class (t(31)=2.1,p=.04).
(I wouldn’t actually emphasise the word “higher” in real life, I’m just doing it to emphasise the point that “higher” corresponds to positive t values). On the other hand, suppose the phrasing I wanted to use has Bernadette’s class listed first. If so, it makes more sense to treat her class as group 1, and if so, the write up looks like this:
Bernadette’s class had lower grades than Anastasia’s class (t(31)=−2.1,p=.04).
Because I’m talking about one group having “lower” scores this time around, it is more sensible to use the negative form of the t-statistic. It just makes it read more cleanly.
One last thing: please note that you can’t do this for other types of test statistics. It works for t-tests, but it wouldn’t be meaningful for chi-square testsm F-tests or indeed for most of the tests I talk about in this book. So don’t overgeneralise this advice! I’m really just talking about t-tests here and nothing else!
Assumptions of the test
As always, our hypothesis test relies on some assumptions. So what are they? For the Student t-test there are three assumptions, some of which we saw previously in the context of the one sample t-test (see Section 13.2.3):
• Normality. Like the one-sample t-test, it is assumed that the data are normally distributed. Specifically, we assume that both groups are normally distributed. In Section 13.9 we’ll discuss how to test for normality, and in Section 13.10 we’ll discuss possible solutions.
• Independence. Once again, it is assumed that the observations are independently sampled. In the context of the Student test this has two aspects to it. Firstly, we assume that the observations within each sample are independent of one another (exactly the same as for the one-sample test). However, we also assume that there are no cross-sample dependencies. If, for instance, it turns out that you included some participants in both experimental conditions of your study (e.g., by accidentally allowing the same person to sign up to different conditions), then there are some cross sample dependencies that you’d need to take into account.
• Homogeneity of variance (also called “homoscedasticity”). The third assumption is that the population standard deviation is the same in both groups. You can test this assumption using the Levene test, which I’ll talk about later on in the book (Section 14.7). However, there’s a very simple remedy for this assumption, which I’ll talk about in the next section. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.03%3A_The_Independent_Samples_t-test_%28Student_Test%29.txt |
The biggest problem with using the Student test in practice is the third assumption listed in the previous section: it assumes that both groups have the same standard deviation. This is rarely true in real life: if two samples don’t have the same means, why should we expect them to have the same standard deviation? There’s really no reason to expect this assumption to be true. We’ll talk a little bit about how you can check this assumption later on because it does crop up in a few different places, not just the t-test. But right now I’ll talk about a different form of the t-test (Welch 1947) that does not rely on this assumption. A graphical illustration of what the Welch t test assumes about the data is shown in Figure 13.10, to provide a contrast with the Student test version in Figure 13.9. I’ll admit it’s a bit odd to talk about the cure before talking about the diagnosis, but as it happens the Welch test is the default t-test in R, so this is probably the best place to discuss it.
The Welch test is very similar to the Student test. For example, the t-statistic that we use in the Welch test is calculated in much the same way as it is for the Student test. That is, we take the difference between the sample means, and then divide it by some estimate of the standard error of that difference:
$t=\dfrac{\bar{X}_{1}-\bar{X}_{2}}{\operatorname{SE}\left(\bar{X}_{1}-\bar{X}_{2}\right)}$
The main difference is that the standard error calculations are different. If the two populations have different standard deviations, then it’s a complete nonsense to try to calculate a pooled standard deviation estimate, because you’re averaging apples and oranges.193 But you can still estimate the standard error of the difference between sample means; it just ends up looking different:
$\operatorname{SE}\left(\bar{X}_{1}-\bar{X}_{2}\right)=\sqrt{\dfrac{\hat{\sigma}_{1}^{2}}{N_{1}}+\dfrac{\hat{\sigma}_{2}^{2}}{N_{2}}}$
The reason why it’s calculated this way is beyond the scope of this book. What matters for our purposes is that the t-statistic that comes out of the Welch test is actually somewhat different to the one that comes from the Student test.
The second difference between Welch and Student is that the degrees of freedom are calculated in a very different way. In the Welch test, the “degrees of freedom” doesn’t have to be a whole number any more, and it doesn’t correspond all that closely to the “number of data points minus the number of constraints” heuristic that I’ve been using up to this point. The degrees of freedom are, in fact…
$\mathrm{d} \mathrm{f}=\dfrac{\left(\hat{\sigma}_{1}^{2} / N_{1}+\hat{\sigma}_{2}^{2} / N_{2}\right)^{2}}{\left(\hat{\sigma}_{1}^{2} / N_{1}\right)^{2} /\left(N_{1}-1\right)+\left(\hat{\sigma}_{2}^{2} / N_{2}\right)^{2} /\left(N_{2}-1\right)}$
… which is all pretty straightforward and obvious, right? Well, perhaps not. It doesn’t really matter for out purposes. What matters is that you’ll see that the “df” value that pops out of a Welch test tends to be a little bit smaller than the one used for the Student test, and it doesn’t have to be a whole number.
Doing the test in R
To run a Welch test in R is pretty easy. All you have to do is not bother telling R to assume equal variances. That is, you take the command we used to run a Student’s t-test and drop the var.equal = TRUE bit. So the command for a Welch test becomes:
independentSamplesTTest(
formula = grade ~ tutor, # formula specifying outcome and group variables
data = harpo # data frame that contains the variables
)
##
## Welch's independent samples t-test
##
## Outcome variable: grade
## Grouping variable: tutor
##
## Descriptive statistics:
## Anastasia Bernadette
## mean 74.533 69.056
## std dev. 8.999 5.775
##
## Hypotheses:
## null: population means equal for both groups
## alternative: different population means in each group
##
## Test results:
## t-statistic: 2.034
## degrees of freedom: 23.025
## p-value: 0.054
##
## Other information:
## two-sided 95% confidence interval: [-0.092, 11.048]
## estimated effect size (Cohen's d): 0.724
Not too difficult, right? Not surprisingly, the output has exactly the same format as it did last time too:
The very first line is different, because it’s telling you that its run a Welch test rather than a Student test, and of course all the numbers are a bit different. But I hope that the interpretation of this output should be fairly obvious. You read the output in the same way that you would for the Student test. You’ve got your descriptive statistics, the hypotheses, the test results and some other information. So that’s all pretty easy.
Except, except… our result isn’t significant anymore. When we ran the Student test, we did get a significant effect; but the Welch test on the same data set is not (t(23.03)=2.03, p=.054). What does this mean? Should we panic? Is the sky burning? Probably not. The fact that one test is significant and the other isn’t doesn’t itself mean very much, especially since I kind of rigged the data so that this would happen. As a general rule, it’s not a good idea to go out of your way to try to interpret or explain the difference between a p-value of .049 and a p-value of .051. If this sort of thing happens in real life, the difference in these p-values is almost certainly due to chance. What does matter is that you take a little bit of care in thinking about what test you use. The Student test and the Welch test have different strengths and weaknesses. If the two populations really do have equal variances, then the Student test is slightly more powerful (lower Type II error rate) than the Welch test. However, if they don’t have the same variances, then the assumptions of the Student test are violated and you may not be able to trust it: you might end up with a higher Type I error rate. So it’s a trade off. However, in real life, I tend to prefer the Welch test; because almost no-one actually believes that the population variances are identical.
Assumptions of the test
The assumptions of the Welch test are very similar to those made by the Student t-test (see Section 13.3.8), except that the Welch test does not assume homogeneity of variance. This leaves only the assumption of normality, and the assumption of independence. The specifics of these assumptions are the same for the Welch test as for the Student test. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.04%3A_The_Independent_Samples_t-test_%28Welch_Test%29.txt |
Regardless of whether we’re talking about the Student test or the Welch test, an independent samples t-test is intended to be used in a situation where you have two samples that are, well, independent of one another. This situation arises naturally when participants are assigned randomly to one of two experimental conditions, but it provides a very poor approximation to other sorts of research designs. In particular, a repeated measures design – in which each participant is measured (with respect to the same outcome variable) in both experimental conditions – is not suited for analysis using independent samples t-tests. For example, we might be interested in whether listening to music reduces people’s working memory capacity. To that end, we could measure each person’s working memory capacity in two conditions: with music, and without music. In an experimental design such as this one,194 each participant appears in both groups. This requires us to approach the problem in a different way; by using the paired samples t-test.
data
The data set that we’ll use this time comes from Dr Chico’s class.195 In her class, students take two major tests, one early in the semester and one later in the semester. To hear her tell it, she runs a very hard class, one that most students find very challenging; but she argues that by setting hard assessments, students are encouraged to work harder. Her theory is that the first test is a bit of a “wake up call” for students: when they realise how hard her class really is, they’ll work harder for the second test and get a better mark. Is she right? To test this, let’s have a look at the chico.Rdata file:
load( "./rbook-master/data/chico.Rdata" )
str(chico)
## 'data.frame': 20 obs. of 3 variables:
## $id : Factor w/ 20 levels "student1","student10",..: 1 12 14 15 16 17 18 19 20 2 ... ##$ grade_test1: num 42.9 51.8 71.7 51.6 63.5 58 59.8 50.8 62.5 61.9 ...
## $grade_test2: num 44.6 54 72.3 53.4 63.8 59.3 60.8 51.6 64.3 63.2 ... The data frame chico contains three variables: an id variable that identifies each student in the class, the grade_test1 variable that records the student grade for the first test, and the grade_test2 variable that has the grades for the second test. Here’s the first six students: head( chico ) ## id grade_test1 grade_test2 ## 1 student1 42.9 44.6 ## 2 student2 51.8 54.0 ## 3 student3 71.7 72.3 ## 4 student4 51.6 53.4 ## 5 student5 63.5 63.8 ## 6 student6 58.0 59.3 At a glance, it does seem like the class is a hard one (most grades are between 50% and 60%), but it does look like there’s an improvement from the first test to the second one. If we take a quick look at the descriptive statistics library( psych ) describe( chico ) ## vars n mean sd median trimmed mad min max range skew ## id* 1 20 10.50 5.92 10.5 10.50 7.41 1.0 20.0 19.0 0.00 ## grade_test1 2 20 56.98 6.62 57.7 56.92 7.71 42.9 71.7 28.8 0.05 ## grade_test2 3 20 58.38 6.41 59.7 58.35 6.45 44.6 72.3 27.7 -0.05 ## kurtosis se ## id* -1.38 1.32 ## grade_test1 -0.35 1.48 ## grade_test2 -0.39 1.43 we see that this impression seems to be supported. Across all 20 students196 the mean grade for the first test is 57%, but this rises to 58% for the second test. Although, given that the standard deviations are 6.6% and 6.4% respectively, it’s starting to feel like maybe the improvement is just illusory; maybe just random variation. This impression is reinforced when you see the means and confidence intervals plotted in Figure 13.11. If we were to rely on this plot alone, we’d come to the same conclusion that we got from looking at the descriptive statistics that the describe() function produced. Looking at how wide those confidence intervals are, we’d be tempted to think that the apparent improvement in student performance is pure chance. Nevertheless, this impression is wrong. To see why, take a look at the scatterplot of the grades for test 1 against the grades for test 2. shown in Figure 13.12. In this plot, each dot corresponds to the two grades for a given student: if their grade for test 1 (x co-ordinate) equals their grade for test 2 (y co-ordinate), then the dot falls on the line. Points falling above the line are the students that performed better on the second test. Critically, almost all of the data points fall above the diagonal line: almost all of the students do seem to have improved their grade, if only by a small amount. This suggests that we should be looking at the improvement made by each student from one test to the next, and treating that as our raw data. To do this, we’ll need to create a new variable for the improvement that each student makes, and add it to the chico data frame. The easiest way to do this is as follows: chico$improvement <- chico$grade_test2 - chico$grade_test1
Notice that I assigned the output to a variable called chico$improvement. That has the effect of creating a new variable called improvement inside the chico data frame. So now when I look at the chico data frame, I get an output that looks like this: head( chico ) ## id grade_test1 grade_test2 improvement ## 1 student1 42.9 44.6 1.7 ## 2 student2 51.8 54.0 2.2 ## 3 student3 71.7 72.3 0.6 ## 4 student4 51.6 53.4 1.8 ## 5 student5 63.5 63.8 0.3 ## 6 student6 58.0 59.3 1.3 Now that we’ve created and stored this improvement variable, we can draw a histogram showing the distribution of these improvement scores (using the hist() function), shown in Figure 13.13. When we look at histogram, it’s very clear that there is a real improvement here. The vast majority of the students scored higher on the test 2 than on test 1, reflected in the fact that almost the entire histogram is above zero. In fact, if we use ciMean() to compute a confidence interval for the population mean of this new variable, ciMean( x = chico$improvement )
## 2.5% 97.5%
## [1,] 0.9508686 1.859131
we see that it is 95% certain that the true (population-wide) average improvement would lie between 0.95% and 1.86%. So you can see, qualitatively, what’s going on: there is a real “within student” improvement (everyone improves by about 1%), but it is very small when set against the quite large “between student” differences (student grades vary by about 20% or so).
What is the paired samples t-test?
In light of the previous exploration, let’s think about how to construct an appropriate t test. One possibility would be to try to run an independent samples t-test using grade_test1 and grade_test2 as the variables of interest. However, this is clearly the wrong thing to do: the independent samples t-test assumes that there is no particular relationship between the two samples. Yet clearly that’s not true in this case, because of the repeated measures structure to the data. To use the language that I introduced in the last section, if we were to try to do an independent samples t-test, we would be conflating the within subject differences (which is what we’re interested in testing) with the between subject variability (which we are not).
The solution to the problem is obvious, I hope, since we already did all the hard work in the previous section. Instead of running an independent samples t-test on grade_test1 and grade_test2, we run a one-sample t-test on the within-subject difference variable, improvement. To formalise this slightly, if Xi1 is the score that the i-th participant obtained on the first variable, and Xi2 is the score that the same person obtained on the second one, then the difference score is:
Di=Xi1−Xi2
Notice that the difference scores is variable 1 minus variable 2 and not the other way around, so if we want improvement to correspond to a positive valued difference, we actually want “test 2” to be our “variable 1”. Equally, we would say that μD1−μ2 is the population mean for this difference variable. So, to convert this to a hypothesis test, our null hypothesis is that this mean difference is zero; the alternative hypothesis is that it is not:
H0D=0
H1D≠0
(this is assuming we’re talking about a two-sided test here). This is more or less identical to the way we described the hypotheses for the one-sample t-test: the only difference is that the specific value that the null hypothesis predicts is 0. And so our t-statistic is defined in more or less the same way too. If we let $\ \bar{D}$ denote the mean of the difference scores, then
$t=\dfrac{\bar{D}}{\operatorname{SE}(\bar{D})}$
which is
$t=\dfrac{\bar{D}}{\hat{\sigma}_{D} / \sqrt{N}}$
where $\ \hat{\sigma_D}$ is the standard deviation of the difference scores. Since this is just an ordinary, one-sample t-test, with nothing special about it, the degrees of freedom are still N−1. And that’s it: the paired samples t-test really isn’t a new test at all: it’s a one-sample t-test, but applied to the difference between two variables. It’s actually very simple; the only reason it merits a discussion as long as the one we’ve just gone through is that you need to be able to recognise when a paired samples test is appropriate, and to understand why it’s better than an independent samples t test.
Doing the test in R, part 1
How do you do a paired samples t-test in R. One possibility is to follow the process I outlined above: create a “difference” variable and then run a one sample t-test on that. Since we’ve already created a variable called chico$improvement, let’s do that: oneSampleTTest( chico$improvement, mu=0 )
##
## One sample t-test
##
## Data variable: chico\$improvement
##
## Descriptive statistics:
## improvement
## mean 1.405
## std dev. 0.970
##
## Hypotheses:
## null: population mean equals 0
## alternative: population mean not equal to 0
##
## Test results:
## t-statistic: 6.475
## degrees of freedom: 19
## p-value: <.001
##
## Other information:
## two-sided 95% confidence interval: [0.951, 1.859]
## estimated effect size (Cohen's d): 1.448
The output here is (obviously) formatted exactly the same was as it was the last time we used the oneSampleTTest() function (Section 13.2), and it confirms our intuition. There’s an average improvement of 1.4% from test 1 to test 2, and this is significantly different from 0 (t(19)=6.48, p<.001).
However, suppose you’re lazy and you don’t want to go to all the effort of creating a new variable. Or perhaps you just want to keep the difference between one-sample and paired-samples tests clear in your head. If so, you can use the pairedSamplesTTest() function, also in the lsr package. Let’s assume that your data organised like they are in the chico data frame, where there are two separate variables, one for each measurement. The way to run the test is to input a one-sided formula, just like you did when running a test of association using the associationTest() function in Chapter 12. For the chico data frame, the formula that you need would be ~ grade_time2 + grade_time1. As usual, you’ll also need to input the name of the data frame too. So the command just looks like this:
pairedSamplesTTest(
formula = ~ grade_test2 + grade_test1, # one-sided formula listing the two variables
data = chico # data frame containing the two variables
)
##
## Paired samples t-test
##
## Variables: grade_test2 , grade_test1
##
## Descriptive statistics:
## grade_test2 grade_test1 difference
## mean 58.385 56.980 1.405
## std dev. 6.406 6.616 0.970
##
## Hypotheses:
## null: population means equal for both measurements
## alternative: different population means for each measurement
##
## Test results:
## t-statistic: 6.475
## degrees of freedom: 19
## p-value: <.001
##
## Other information:
## two-sided 95% confidence interval: [0.951, 1.859]
## estimated effect size (Cohen's d): 1.448
The numbers are identical to those that come from the one sample test, which of course they have to be given that the paired samples t-test is just a one sample test under the hood. However, the output is a bit more detailed:
This time around the descriptive statistics block shows you the means and standard deviations for the original variables, as well as for the difference variable (notice that it always defines the difference as the first listed variable mines the second listed one). The null hypothesis and the alternative hypothesis are now framed in terms of the original variables rather than the difference score, but you should keep in mind that in a paired samples test it’s still the difference score being tested. The statistical information at the bottom about the test result is of course the same as before.
Doing the test in R, part 2
The paired samples t-test is a little different from the other t-tests, because it is used in repeated measures designs. For the chico data, every student is “measured” twice, once for the first test, and again for the second test. Back in Section 7.7 I talked about the fact that repeated measures data can be expressed in two standard ways, known as wide form and long form. The chico data frame is in wide form: every row corresponds to a unique person. I’ve shown you the data in that form first because that’s the form that you’re most used to seeing, and it’s also the format that you’re most likely to receive data in. However, the majority of tools in R for dealing with repeated measures data expect to receive data in long form. The paired samples t-test is a bit of an exception that way.
As you make the transition from a novice user to an advanced one, you’re going to have to get comfortable with long form data, and switching between the two forms. To that end, I want to show you how to apply the pairedSamplesTTest() function to long form data. First, let’s use the wideToLong() function to create a long form version of the chico data frame. If you’ve forgotten how the wideToLong() function works, it might be worth your while quickly re-reading Section 7.7. Assuming that you’ve done so, or that you’re already comfortable with data reshaping, I’ll use it to create a new data frame called chico2:
chico2 <- wideToLong( chico, within="time" )
head( chico2 )
## id improvement time grade
## 1 student1 1.7 test1 42.9
## 2 student2 2.2 test1 51.8
## 3 student3 0.6 test1 71.7
## 4 student4 1.8 test1 51.6
## 5 student5 0.3 test1 63.5
## 6 student6 1.3 test1 58.0
As you can see, this has created a new data frame containing three variables: an id variable indicating which person provided the data, a time variable indicating which test the data refers to (i.e., test 1 or test 2), and a grade variable that records what score the person got on that test. Notice that this data frame is in long form: every row corresponds to a unique measurement. Because every person provides two observations (test 1 and test 2), there are two rows for every person. To see this a little more clearly, I’ll use the sortFrame() function to sort the rows of chico2 by id variable (see Section 7.6.3).
chico2 <- sortFrame( chico2, id )
head( chico2 )
## id improvement time grade
## 1 student1 1.7 test1 42.9
## 21 student1 1.7 test2 44.6
## 10 student10 1.3 test1 61.9
## 30 student10 1.3 test2 63.2
## 11 student11 1.4 test1 50.4
## 31 student11 1.4 test2 51.8
As you can see, there are two rows for “student1”: one showing their grade on the first test, the other showing their grade on the second test.197
Okay, suppose that we were given the chico2 data frame to analyse. How would we run our paired samples t-test now? One possibility would be to use the longToWide() function (Section 7.7) to force the data back into wide form, and do the same thing that we did previously. But that’s sort of defeating the point, and besides, there’s an easier way. Let’s think about what how the chico2 data frame is structured: there are three variables here, and they all matter. The outcome measure is stored as the grade, and we effectively have two “groups” of measurements (test 1 and test 2) that are defined by the time points at which a test is given. Finally, because we want to keep track of which measurements should be paired together, we need to know which student obtained each grade, which is what the id variable gives us. So, when your data are presented to you in long form, we would want specify a two-sided formula and a data frame, in the same way that we do for an independent samples t-test: the formula specifies the outcome variable and the groups, so in this case it would be grade ~ time, and the data frame is chico2. However, we also need to tell it the id variable, which in this case is boringly called id. So our command is:
pairedSamplesTTest(
formula = grade ~ time, # two sided formula: outcome ~ group
data = chico2, # data frame
id = "id" # name of the id variable
)
##
## Paired samples t-test
##
## Outcome variable: grade
## Grouping variable: time
## ID variable: id
##
## Descriptive statistics:
## test1 test2 difference
## mean 56.980 58.385 -1.405
## std dev. 6.616 6.406 0.970
##
## Hypotheses:
## null: population means equal for both measurements
## alternative: different population means for each measurement
##
## Test results:
## t-statistic: -6.475
## degrees of freedom: 19
## p-value: <.001
##
## Other information:
## two-sided 95% confidence interval: [-1.859, -0.951]
## estimated effect size (Cohen's d): 1.448
Note that the name of the id variable is "id" and not id. Note that the id variable must be a factor. As of the current writing, you do need to include the quote marks, because the pairedSamplesTTest() function is expecting a character string that specifies the name of a variable. If I ever find the time I’ll try to relax this constraint.
As you can see, it’s a bit more detailed than the output from oneSampleTTest(). It gives you the descriptive statistics for the original variables, states the null hypothesis in a fashion that is a bit more appropriate for a repeated measures design, and then reports all the nuts and bolts from the hypothesis test itself. Not surprisingly the numbers the same as the ones that we saw last time.
One final comment about the pairedSamplesTTest() function. One of the reasons I designed it to be able handle long form and wide form data is that I want you to be get comfortable thinking about repeated measures data in both formats, and also to become familiar with the different ways in which R functions tend to specify models and tests for repeated measures data. With that last point in mind, I want to highlight a slightly different way of thinking about what the paired samples t-test is doing. There’s a sense in which what you’re really trying to do is look at how the outcome variable (grade) is related to the grouping variable (time), after taking account of the fact that there are individual differences between people (id). So there’s a sense in which id is actually a second predictor: you’re trying to predict the grade on the basis of the time and the id. With that in mind, the pairedSamplesTTest() function lets you specify a formula like this one
grade ~ time + (id)
This formula tells R everything it needs to know: the variable on the left (grade) is the outcome variable, the bracketed term on the right (id) is the id variable, and the other term on the right is the grouping variable (time). If you specify your formula that way, then you only need to specify the formula and the data frame, and so you can get away with using a command as simple as this one:
pairedSamplesTTest(
formula = grade ~ time + (id),
data = chico2
)
or you can drop the argument names and just do this:
> pairedSamplesTTest( grade ~ time + (id), chico2 )
These commands will produce the same output as the last one, I personally find this format a lot more elegant. That being said, the main reason for allowing you to write your formulas that way is that they’re quite similar to the way that mixed models (fancy pants repeated measures analyses) are specified in the lme4 package. This book doesn’t talk about mixed models (yet!), but if you go on to learn more statistics you’ll find them pretty hard to avoid, so I’ve tried to lay a little bit of the groundwork here. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.05%3A_The_Paired-samples_t-test.txt |
When introducing the theory of null hypothesis tests, I mentioned that there are some situations when it’s appropriate to specify a one-sided test (see Section 11.4.3). So far, all of the t-tests have been two-sided tests. For instance, when we specified a one sample t-test for the grades in Dr Zeppo’s class, the null hypothesis was that the true mean was 67.5%. The alternative hypothesis was that the true mean was greater than or less than 67.5%. Suppose we were only interested in finding out if the true mean is greater than 67.5%, and have no interest whatsoever in testing to find out if the true mean is lower than 67.5%. If so, our null hypothesis would be that the true mean is 67.5% or less, and the alternative hypothesis would be that the true mean is greater than 67.5%. The `oneSampleTTest()` function lets you do this, by specifying the `one.sided` argument. If you set `one.sided="greater"`, it means that you’re testing to see if the true mean is larger than `mu`. If you set `one.sided="less"`, then you’re testing to see if the true mean is smaller than `mu`. Here’s how it would work for Dr Zeppo’s class:
``oneSampleTTest( x=grades, mu=67.5, one.sided="greater" )``
``````##
## One sample t-test
##
## Data variable: grades
##
## Descriptive statistics:
## grades
## mean 72.300
## std dev. 9.521
##
## Hypotheses:
## null: population mean less than or equal to 67.5
## alternative: population mean greater than 67.5
##
## Test results:
## t-statistic: 2.255
## degrees of freedom: 19
## p-value: 0.018
##
## Other information:
## one-sided 95% confidence interval: [68.619, Inf]
## estimated effect size (Cohen's d): 0.504``````
Notice that there are a few changes from the output that we saw last time. Most important is the fact that the null and alternative hypotheses have changed, to reflect the different test. The second thing to note is that, although the t-statistic and degrees of freedom have not changed, the p-value has. This is because the one-sided test has a different rejection region from the two-sided test. If you’ve forgotten why this is and what it means, you may find it helpful to read back over Chapter 11, and Section 11.4.3 in particular. The third thing to note is that the confidence interval is different too: it now reports a “one-sided” confidence interval rather than a two-sided one. In a two-sided confidence interval, we’re trying to find numbers a and b such that we’re 95% confident that the true mean lies between a and b. In a one-sided confidence interval, we’re trying to find a single number a such that we’re 95% confident that the true mean is greater than a (or less than a if you set `one.sided="less"`).
So that’s how to do a one-sided one sample t-test. However, all versions of the t-test can be one-sided. For an independent samples t test, you could have a one-sided test if you’re only interestd in testing to see if group A has higher scores than group B, but have no interest in finding out if group B has higher scores than group A. Let’s suppose that, for Dr Harpo’s class, you wanted to see if Anastasia’s students had higher grades than Bernadette’s. The `independentSamplesTTest()` function lets you do this, again by specifying the `one.sided` argument. However, this time around you need to specify the name of the group that you’re expecting to have the higher score. In our case, we’d write `one.sided = "Anastasia"`. So the command would be:
``````independentSamplesTTest(
formula = grade ~ tutor,
data = harpo,
one.sided = "Anastasia"
)``````
``````##
## Welch's independent samples t-test
##
## Outcome variable: grade
## Grouping variable: tutor
##
## Descriptive statistics:
## Anastasia Bernadette
## mean 74.533 69.056
## std dev. 8.999 5.775
##
## Hypotheses:
## null: population means are equal, or smaller for group 'Anastasia'
## alternative: population mean is larger for group 'Anastasia'
##
## Test results:
## t-statistic: 2.034
## degrees of freedom: 23.025
## p-value: 0.027
##
## Other information:
## one-sided 95% confidence interval: [0.863, Inf]
## estimated effect size (Cohen's d): 0.724``````
Again, the output changes in a predictable way. The definition of the null and alternative hypotheses has changed, the p-value has changed, and it now reports a one-sided confidence interval rather than a two-sided one.
What about the paired samples t-test? Suppose we wanted to test the hypothesis that grades go up from test 1 to test 2 in Dr Zeppo’s class, and are not prepared to consider the idea that the grades go down. Again, we can use the `one.sided` argument to specify the one-sided test, and it works the same way it does for the independent samples t-test. You need to specify the name of the group whose scores are expected to be larger under the alternative hypothesis. If your data are in wide form, as they are in the `chico` data frame, you’d use this command:
``````pairedSamplesTTest(
formula = ~ grade_test2 + grade_test1,
data = chico,
one.sided = "grade_test2"
)``````
``````##
## Paired samples t-test
##
## Variables: grade_test2 , grade_test1
##
## Descriptive statistics:
## grade_test2 grade_test1 difference
## mean 58.385 56.980 1.405
## std dev. 6.406 6.616 0.970
##
## Hypotheses:
## null: population means are equal, or smaller for measurement 'grade_test2'
## alternative: population mean is larger for measurement 'grade_test2'
##
## Test results:
## t-statistic: 6.475
## degrees of freedom: 19
## p-value: <.001
##
## Other information:
## one-sided 95% confidence interval: [1.03, Inf]
## estimated effect size (Cohen's d): 1.448``````
Yet again, the output changes in a predictable way. The hypotheses have changed, the p-value has changed, and the confidence interval is now one-sided. If your data are in long form, as they are in the `chico2` data frame, it still works the same way. Either of the following commands would work,
``````> pairedSamplesTTest(
formula = grade ~ time,
data = chico2,
id = "id",
one.sided = "test2"
)
> pairedSamplesTTest(
formula = grade ~ time + (id),
data = chico2,
one.sided = "test2"
)``````
and would produce the same answer as the output shown above. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.06%3A_One_Sided_Tests.txt |
In this chapter, we’ve talked about three different kinds of t-test: the one sample test, the independent samples test (Student’s and Welch’s), and the paired samples test. In order to run these different tests, I’ve shown you three different functions: `oneSampleTTest()`, `independentSamplesTTest()` and `pairedSamplesTTest()`. I wrote these as three different functions for two reasons. Firstly, I thought it made sense to have separate functions for each test, in order to help make it clear to beginners that there are different tests. Secondly, I wanted to show you some functions that produced “verbose” output, to help you see what hypotheses are being tested and so on.
However, once you’ve started to become familiar with t-tests and with using R, you might find it easier to use the `t.test()` function. It’s one function, but it can run all four of the different t-tests that we’ve talked about. Here’s how it works. Firstly, suppose you want to run a one sample t-test. To run the test on the `grades` data from Dr Zeppo’s class (Section 13.2), we’d use a command like this:
``t.test( x = grades, mu = 67.5 )``
``````##
## One Sample t-test
##
## data: grades
## t = 2.2547, df = 19, p-value = 0.03615
## alternative hypothesis: true mean is not equal to 67.5
## 95 percent confidence interval:
## 67.84422 76.75578
## sample estimates:
## mean of x
## 72.3``````
The input is the same as for the `oneSampleTTest()`: we specify the sample data using the argument `x`, and the value against which it is to be tested using the argument `mu`. The output is a lot more compressed.
As you can see, it still has all the information you need. It tells you what type of test it ran and the data it tested it on. It gives you the t-statistic, the degrees of freedom and the p-value. And so on. There’s nothing wrong with this output, but in my experience it can be a little confusing when you’re just starting to learn statistics, because it’s a little disorganised. Once you know what you’re looking at though, it’s pretty easy to read off the relevant information.
What about independent samples t-tests? As it happens, the `t.test()` function can be used in much the same way as the `independentSamplesTTest()` function, by specifying a formula, a data frame, and using `var.equal` to indicate whether you want a Student test or a Welch test. If you want to run the Welch test from Section 13.4, then you’d use this command:
``t.test( formula = grade ~ tutor, data = harpo )``
``````##
## Welch Two Sample t-test
##
## data: grade by tutor
## t = 2.0342, df = 23.025, p-value = 0.05361
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.09249349 11.04804904
## sample estimates:
## mean in group Anastasia mean in group Bernadette
## 74.53333 69.05556``````
If you want to do the Student test, it’s exactly the same except that you need to add an additional argument indicating that `var.equal = TRUE`. This is no different to how it worked in the `independentSamplesTTest()` function.
Finally, we come to the paired samples t-test. Somewhat surprisingly, given that most R functions for dealing with repeated measures data require data to be in long form, the `t.test()` function isn’t really set up to handle data in long form. Instead it expects to be given two separate variables, `x` and `y`, and you need to specify `paired=TRUE`. And on top of that, you’d better make sure that the first element of `x` and the first element of `y` actually correspond to the same person! Because it doesn’t ask for an “id” variable. I don’t know why. So, in order to run the paired samples t test on the data from Dr Chico’s class, we’d use this command:
``````t.test( x = chico\$grade_test2, # variable 1 is the "test2" scores
y = chico\$grade_test1, # variable 2 is the "test1" scores
paired = TRUE # paired test
)``````
``````##
## Paired t-test
##
## data: chico\$grade_test2 and chico\$grade_test1
## t = 6.4754, df = 19, p-value = 3.321e-06
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.9508686 1.8591314
## sample estimates:
## mean of the differences
## 1.405``````
Yet again, these are the same numbers that we saw in Section 13.5. Feel free to check. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.07%3A_Using_the_t.test%28%29_Function.txt |
The most commonly used measure of effect size for a t-test is Cohen’s d (Cohen 1988). It’s a very simple measure in principle, with quite a few wrinkles when you start digging into the details. Cohen himself defined it primarily in the context of an independent samples t-test, specifically the Student test. In that context, a natural way of defining the effect size is to divide the difference between the means by an estimate of the standard deviation. In other words, we’re looking to calculate something along the lines of this:
$d=\dfrac{(\text { mean } 1)-(\text { mean } 2)}{\text { std dev }}$
and he suggested a rough guide for interpreting d in Table ??. You’d think that this would be pretty unambiguous, but it’s not; largely because Cohen wasn’t too specific on what he thought should be used as the measure of the standard deviation (in his defence, he was trying to make a broader point in his book, not nitpick about tiny details). As discussed by McGrath and Meyer (2006), there are several different version in common usage, and each author tends to adopt slightly different notation. For the sake of simplicity (as opposed to accuracy) I’ll use d to refer to any statistic that you calculate from the sample, and use δ to refer to a theoretical population effect. Obviously, that does mean that there are several different things all called d. The cohensD() function in the lsr package uses the method argument to distinguish between them, so that’s what I’ll do in the text.
My suspicion is that the only time that you would want Cohen’s d is when you’re running a t-test, and if you’re using the oneSampleTTest, independentSamplesTTest and pairedSamplesTTest() functions to run your t-tests, then you don’t need to learn any new commands, because they automatically produce an estimate of Cohen’s d as part of the output. However, if you’re using t.test() then you’ll need to use the cohensD() function (also in the lsr package) to do the calculations.
d-value rough interpretation
about 0.2 small effect
about 0.5 moderate effect
about 0.8 large effect
Cohen’s d from one sample
The simplest situation to consider is the one corresponding to a one-sample t-test. In this case, the one sample mean $\ \bar{X}$ and one (hypothesised) population mean μo to compare it to. Not only that, there’s really only one sensible way to estimate the population standard deviation: we just use our usual estimate $\ \hat{\sigma}$. Therefore, we end up with the following as the only way to calculate d,
$d=\dfrac{\bar{X}-\mu_{0}}{\hat{\sigma}}$
When writing the cohensD() function, I’ve made some attempt to make it work in a similar way to t.test(). As a consequence, cohensD() can calculate your effect size regardless of which type of t-test you performed. If what you want is a measure of Cohen’s d to accompany a one-sample t-test, there’s only two arguments that you need to care about. These are:
• x. A numeric vector containing the sample data.
• mu. The mean against which the mean of x is compared (default value is mu = 0).
We don’t need to specify what method to use, because there’s only one version of d that makes sense in this context. So, in order to compute an effect size for the data from Dr Zeppo’s class (Section 13.2), we’d type something like this:
cohensD( x = grades, # data are stored in the grades vector
mu = 67.5 # compare students to a mean of 67.5
)
## [1] 0.5041691
and, just so that you can see that there’s nothing fancy going on, the command below shows you how to calculate it if there weren’t no fancypants cohensD() function available:
( mean(grades) - 67.5 ) / sd(grades)
## [1] 0.5041691
Yep, same number. Overall, then, the psychology students in Dr Zeppo’s class are achieving grades (mean = 72.3%) that are about .5 standard deviations higher than the level that you’d expect (67.5%) if they were performing at the same level as other students. Judged against Cohen’s rough guide, this is a moderate effect size.
Cohen’s d from a Student t test
The majority of discussions of Cohen’s d focus on a situation that is analogous to Student’s independent samples t test, and it’s in this context that the story becomes messier, since there are several different versions of d that you might want to use in this situation, and you can use the method argument to the cohensD() function to pick the one you want. To understand why there are multiple versions of d, it helps to take the time to write down a formula that corresponds to the true population effect size δ. It’s pretty straightforward,
$\delta=\dfrac{\mu_{1}-\mu_{2}}{\sigma}$
where, as usual, μ1 and μ2 are the population means corresponding to group 1 and group 2 respectively, and σ is the standard deviation (the same for both populations). The obvious way to estimate δ is to do exactly the same thing that we did in the t-test itself: use the sample means as the top line, and a pooled standard deviation estimate for the bottom line:
$d=\dfrac{\bar{X}_{1}-\bar{X}_{2}}{\hat{\sigma}_{p}}$
where $\ \hat{\sigma_p}$ is the exact same pooled standard deviation measure that appears in the t-test. This is the most commonly used version of Cohen’s d when applied to the outcome of a Student t-test ,and is sometimes referred to as Hedges’ g statistic (Hedges 1981). It corresponds to method = "pooled" in the cohensD() function, and it’s the default.
However, there are other possibilities, which I’ll briefly describe. Firstly, you may have reason to want to use only one of the two groups as the basis for calculating the standard deviation. This approach (often called Glass’ Δ) only makes most sense when you have good reason to treat one of the two groups as a purer reflection of “natural variation” than the other. This can happen if, for instance, one of the two groups is a control group. If that’s what you want, then use method = "x.sd" or method = "y.sd" when using cohensD(). Secondly, recall that in the usual calculation of the pooled standard deviation we divide by N−2 to correct for the bias in the sample variance; in one version of Cohen’s d this correction is omitted. Instead, we divide by N. This version (method = "raw") makes sense primarily when you’re trying to calculate the effect size in the sample; rather than estimating an effect size in the population. Finally, there is a version based on Hedges and Olkin (1985), who point out there is a small bias in the usual (pooled) estimation for Cohen’s d. Thus they introduce a small correction (method = "corrected"), by multiplying the usual value of d by (N−3)/(N−2.25).
In any case, ignoring all those variations that you could make use of if you wanted, let’s have a look at how to calculate the default version. In particular, suppose we look at the data from Dr Harpo’s class (the harpo data frame). The command that we want to use is very similar to the relevant t.test() command, but also specifies a method
cohensD( formula = grade ~ tutor, # outcome ~ group
data = harpo, # data frame
method = "pooled" # which version to calculate?
)
## [1] 0.7395614
This is the version of Cohen’s d that gets reported by the independentSamplesTTest() function whenever it runs a Student t-test.
Cohen’s d from a Welch test
Suppose the situation you’re in is more like the Welch test: you still have two independent samples, but you no longer believe that the corresponding populations have equal variances. When this happens, we have to redefine what we mean by the population effect size. I’ll refer to this new measure as δ′, so as to keep it distinct from the measure δ which we defined previously. What Cohen (1988) suggests is that we could define our new population effect size by averaging the two population variances. What this means is that we get:
$\delta^{\prime}=\dfrac{\mu_{1}-\mu_{2}}{\sigma^{\prime}}$
where
$\sigma^{\prime}=\sqrt{\dfrac{\sigma_{1}^{2}+\sigma_{2}^{2}}{2}}$
This seems quite reasonable, but notice that none of the measures that we’ve discussed so far are attempting to estimate this new quantity. It might just be my own ignorance of the topic, but I’m only aware of one version of Cohen’s d that actually estimates the unequal-variance effect size δ′ rather than the equal-variance effect size δ. All we do to calculate d for this version (method = "unequal") is substitute the sample means $\ \bar{X_1}$ and $\ \bar{X_2}$ and the corrected sample standard deviations $\ \hat{\sigma_1}$ and $\ \hat{\sigma_2}$ into the equation for δ′. This gives us the following equation for d,
$d=\dfrac{\bar{X}_{1}-\bar{X}_{2}}{\sqrt{\dfrac{\hat{\sigma}_{1}\ ^{2}+\hat{\sigma}_{2}\ ^{2}}{2}}}$
as our estimate of the effect size. There’s nothing particularly difficult about calculating this version in R, since all we have to do is change the method argument:
cohensD( formula = grade ~ tutor,
data = harpo,
method = "unequal"
)
## [1] 0.7244995
This is the version of Cohen’s d that gets reported by the independentSamplesTTest() function whenever it runs a Welch t-test.
Cohen’s d from a paired-samples test
Finally, what should we do for a paired samples t-test? In this case, the answer depends on what it is you’re trying to do. If you want to measure your effect sizes relative to the distribution of difference scores, the measure of d that you calculate is just (method = "paired")
$d=\dfrac{\bar{D}}{\hat{\sigma}_{D}}$
where $\ \hat{\sigma_D}$ is the estimate of the standard deviation of the differences. The calculation here is pretty straightforward
cohensD( x = chico$grade_test2, y = chico$grade_test1,
method = "paired"
)
## [1] 1.447952
This is the version of Cohen’s d that gets reported by the pairedSamplesTTest() function. The only wrinkle is figuring out whether this is the measure you want or not. To the extent that you care about the practical consequences of your research, you often want to measure the effect size relative to the original variables, not the difference scores (e.g., the 1% improvement in Dr Chico’s class is pretty small when measured against the amount of between-student variation in grades), in which case you use the same versions of Cohen’s d that you would use for a Student or Welch test. For instance, when we do that for Dr Chico’s class,
cohensD( x = chico$grade_test2, y = chico$grade_test1,
method = "pooled"
)
## [1] 0.2157646
what we see is that the overall effect size is quite small, when assessed on the scale of the original variables. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.08%3A_Effect_Size.txt |
All of the tests that we have discussed so far in this chapter have assumed that the data are normally distributed. This assumption is often quite reasonable, because the central limit theorem (Section 10.3.3) does tend to ensure that many real world quantities are normally distributed: any time that you suspect that your variable is actually an average of lots of different things, there’s a pretty good chance that it will be normally distributed; or at least close enough to normal that you can get away with using t-tests. However, life doesn’t come with guarantees; and besides, there are lots of ways in which you can end up with variables that are highly non-normal. For example, any time you think that your variable is actually the minimum of lots of different things, there’s a very good chance it will end up quite skewed. In psychology, response time (RT) data is a good example of this. If you suppose that there are lots of things that could trigger a response from a human participant, then the actual response will occur the first time one of these trigger events occurs.198 This means that RT data are systematically non-normal. Okay, so if normality is assumed by all the tests, and is mostly but not always satisfied (at least approximately) by real world data, how can we check the normality of a sample? In this section I discuss two methods: QQ plots, and the Shapiro-Wilk test.
plots
## Normally Distributed Data
## skew= -0.02936155
## kurtosis= -0.06035938
##
## Shapiro-Wilk normality test
##
## data: data
## W = 0.99108, p-value = 0.7515
The Shapiro-Wilk statistic associated with the data in Figures 13.14 and 13.15 is W=.99, indicating that no significant departures from normality were detected (p=.73). As you can see, these data form a pretty straight line; which is no surprise given that we sampled them from a normal distribution! In contrast, have a look at the two data sets shown in Figures 13.16, 13.17, 13.18, 13.19. Figures 13.16 and 13.17 show the histogram and a QQ plot for a data set that is highly skewed: the QQ plot curves upwards. Figures 13.18 and 13.19 show the same plots for a heavy tailed (i.e., high kurtosis) data set: in this case, the QQ plot flattens in the middle and curves sharply at either end.
## Skewed Data
## skew= 1.889475
## kurtosis= 4.4396
##
## Shapiro-Wilk normality test
##
## data: data
## W = 0.81758, p-value = 8.908e-10
The skewness of the data in Figures 13.16 and 13.17 is 1.94, and is reflected in a QQ plot that curves upwards. As a consequence, the Shapiro-Wilk statistic is W=.80, reflecting a significant departure from normality (p<.001).
## Heavy-Tailed Data
## skew= -0.05308273
## kurtosis= 7.508765
##
## Shapiro-Wilk normality test
##
## data: data
## W = 0.83892, p-value = 4.718e-09
Figures 13.18 and 13.19 shows the same plots for a heavy tailed data set, again consisting of 100 observations. In this case, the heavy tails in the data produce a high kurtosis (2.80), and cause the QQ plot to flatten in the middle, and curve away sharply on either side. The resulting Shapiro-Wilk statistic is W=.93, again reflecting significant non-normality (p<.001).
One way to check whether a sample violates the normality assumption is to draw a “quantile-quantile” plot (QQ plot). This allows you to visually check whether you’re seeing any systematic violations. In a QQ plot, each observation is plotted as a single dot. The x co-ordinate is the theoretical quantile that the observation should fall in, if the data were normally distributed (with mean and variance estimated from the sample) and on the y co-ordinate is the actual quantile of the data within the sample. If the data are normal, the dots should form a straight line. For instance, lets see what happens if we generate data by sampling from a normal distribution, and then drawing a QQ plot using the R function qqnorm(). The qqnorm() function has a few arguments, but the only one we really need to care about here is y, a vector specifying the data whose normality we’re interested in checking. Here’s the R commands:
normal.data <- rnorm( n = 100 ) # generate N = 100 normally distributed numbers
hist( x = normal.data ) # draw a histogram of these numbers
qqnorm( y = normal.data ) # draw the QQ plot
Shapiro-Wilk tests
Although QQ plots provide a nice way to informally check the normality of your data, sometimes you’ll want to do something a bit more formal. And when that moment comes, the Shapiro-Wilk test (Shapiro and Wilk 1965) is probably what you’re looking for.199 As you’d expect, the null hypothesis being tested is that a set of N observations is normally distributed. The test statistic that it calculates is conventionally denoted as W, and it’s calculated as follows. First, we sort the observations in order of increasing size, and let X1 be the smallest value in the sample, X2 be the second smallest and so on. Then the value of W is given by
$W=\dfrac{\left(\sum_{i=1}^{N} a_{i} X_{i}\right)^{2}}{\sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}}$
where $\ \bar{X}$ is the mean of the observations, and the ai values are … mumble, mumble … something complicated that is a bit beyond the scope of an introductory text.
Because it’s a little hard to explain the maths behind the W statistic, a better idea is to give a broad brush description of how it behaves. Unlike most of the test statistics that we’ll encounter in this book, it’s actually small values of W that indicated departure from normality. The W statistic has a maximum value of 1, which arises when the data look “perfectly normal”. The smaller the value of W, the less normal the data are. However, the sampling distribution for W – which is not one of the standard ones that I discussed in Chapter 9 and is in fact a complete pain in the arse to work with – does depend on the sample size N. To give you a feel for what these sampling distributions look like, I’ve plotted three of them in Figure 13.20. Notice that, as the sample size starts to get large, the sampling distribution becomes very tightly clumped up near W=1, and as a consequence, for larger samples W doesn’t have to be very much smaller than 1 in order for the test to be significant.
To run the test in R, we use the shapiro.test() function. It has only a single argument x, which is a numeric vector containing the data whose normality needs to be tested. For example, when we apply this function to our normal.data, we get the following:
shapiro.test( x = normal.data )
##
## Shapiro-Wilk normality test
##
## data: normal.data
## W = 0.98654, p-value = 0.4076
So, not surprisingly, we have no evidence that these data depart from normality. When reporting the results for a Shapiro-Wilk test, you should (as usual) make sure to include the test statistic W and the p value, though given that the sampling distribution depends so heavily on N it would probably be a politeness to include N as well. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.09%3A_Checking_the_Normality_of_a_Sample.txt |
Okay, suppose your data turn out to be pretty substantially non-normal, but you still want to run something like a t-test? This situation occurs a lot in real life: for the AFL winning margins data, for instance, the Shapiro-Wilk test made it very clear that the normality assumption is violated. This is the situation where you want to use Wilcoxon tests.
Like the t-test, the Wilcoxon test comes in two forms, one-sample and two-sample, and they’re used in more or less the exact same situations as the corresponding t-tests. Unlike the t-test, the Wilcoxon test doesn’t assume normality, which is nice. In fact, they don’t make any assumptions about what kind of distribution is involved: in statistical jargon, this makes them nonparametric tests. While avoiding the normality assumption is nice, there’s a drawback: the Wilcoxon test is usually less powerful than the t-test (i.e., higher Type II error rate). I won’t discuss the Wilcoxon tests in as much detail as the t-tests, but I’ll give you a brief overview.
sample Wilcoxon test
I’ll start by describing the two sample Wilcoxon test (also known as the Mann-Whitney test), since it’s actually simpler than the one sample version. Suppose we’re looking at the scores of 10 people on some test. Since my imagination has now failed me completely, let’s pretend it’s a “test of awesomeness”, and there are two groups of people, “A” and “B”. I’m curious to know which group is more awesome. The data are included in the file `awesome.Rdata`, and like many of the data sets I’ve been using, it contains only a single data frame, in this case called `awesome`. Here’s the data:
``````load("./rbook-master/data/awesome.Rdata")
print( awesome )``````
``````## scores group
## 1 6.4 A
## 2 10.7 A
## 3 11.9 A
## 4 7.3 A
## 5 10.0 A
## 6 14.5 B
## 7 10.4 B
## 8 12.9 B
## 9 11.7 B
## 10 13.0 B``````
As long as there are no ties (i.e., people with the exact same awesomeness score), then the test that we want to do is surprisingly simple. All we have to do is construct a table that compares every observation in group A against every observation in group B. Whenever the group A datum is larger, we place a check mark in the table:
We then count up the number of checkmarks. This is our test statistic, W.200 The actual sampling distribution for W is somewhat complicated, and I’ll skip the details. For our purposes, it’s sufficient to note that the interpretation of W is qualitatively the same as the interpretation of t or z. That is, if we want a two-sided test, then we reject the null hypothesis when W is very large or very small; but if we have a directional (i.e., one-sided) hypothesis, then we only use one or the other.
The structure of the `wilcox.test()` function should feel very familiar to you by now. When you have your data organised in terms of an outcome variable and a grouping variable, then you use the `formula` and `data` arguments, so your command looks like this:
``wilcox.test( formula = scores ~ group, data = awesome)``
``````##
## Wilcoxon rank sum test
##
## data: scores by group
## W = 3, p-value = 0.05556
## alternative hypothesis: true location shift is not equal to 0``````
Just like we saw with the `t.test()` function, there is an `alternative` argument that you can use to switch between two-sided tests and one-sided tests, plus a few other arguments that we don’t need to worry too much about at an introductory level. Similarly, the `wilcox.test()` function allows you to use the `x` and `y` arguments when you have your data stored separately for each group. For instance, suppose we use the data from the `awesome2.Rdata` file:
``````load( "./rbook-master/data/awesome2.Rdata" )
score.A``````
````## [1] 6.4 10.7 11.9 7.3 10.0`
```
``score.B``
``## [1] 14.5 10.4 12.9 11.7 13.0``
When your data are organised like this, then you would use a command like this:
``wilcox.test( x = score.A, y = score.B )``
``````##
## Wilcoxon rank sum test
##
## data: score.A and score.B
## W = 3, p-value = 0.05556
## alternative hypothesis: true location shift is not equal to 0``````
The output that R produces is pretty much the same as last time.
sample Wilcoxon test
What about the one sample Wilcoxon test (or equivalently, the paired samples Wilcoxon test)? Suppose I’m interested in finding out whether taking a statistics class has any effect on the happiness of students. Here’s my data:
``````load( "./rbook-master/data/happy.Rdata" )
print( happiness )``````
``````## before after change
## 1 30 6 -24
## 2 43 29 -14
## 3 21 11 -10
## 4 24 31 7
## 5 23 17 -6
## 6 40 2 -38
## 7 29 31 2
## 8 56 21 -35
## 9 38 8 -30
## 10 16 21 5``````
What I’ve measured here is the happiness of each student `before` taking the class and `after` taking the class; the `change` score is the difference between the two. Just like we saw with the t-test, there’s no fundamental difference between doing a paired-samples test using `before` and `after`, versus doing a one-sample test using the `change` scores. As before, the simplest way to think about the test is to construct a tabulation. The way to do it this time is to take those change scores that are positive valued, and tabulate them against all the complete sample. What you end up with is a table that looks like this:
Counting up the tick marks this time, we get a test statistic of V=7. As before, if our test is two sided, then we reject the null hypothesis when V is very large or very small. As far of running it in R goes, it’s pretty much what you’d expect. For the one-sample version, the command you would use is
``````wilcox.test( x = happiness\$change,
mu = 0
)``````
``````##
## Wilcoxon signed rank test
##
## data: happiness\$change
## V = 7, p-value = 0.03711
## alternative hypothesis: true location is not equal to 0``````
As this shows, we have a significant effect. Evidently, taking a statistics class does have an effect on your happiness. Switching to a paired samples version of the test won’t give us different answers, of course; but here’s the command to do it:
``````wilcox.test( x = happiness\$after,
y = happiness\$before,
paired = TRUE
)``````
``````##
## Wilcoxon signed rank test
##
## data: happiness\$after and happiness\$before
## V = 7, p-value = 0.03711
## alternative hypothesis: true location shift is not equal to 0``` ``` | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.10%3A__Testing_Non-normal_Data_with_Wilcoxon_Tests.txt |
• A one sample t-test is used to compare a single sample mean against a hypothesised value for the population mean. (Section 13.2)
• An independent samples t-test is used to compare the means of two groups, and tests the null hypothesis that they have the same mean. It comes in two forms: the Student test (Section 13.3 assumes that the groups have the same standard deviation, the Welch test (Section 13.4) does not.
• A paired samples t-test is used when you have two scores from each person, and you want to test the null hypothesis that the two scores have the same mean. It is equivalent to taking the difference between the two scores for each person, and then running a one sample t-test on the difference scores. (Section 13.5)
• Effect size calculations for the difference between means can be calculated via the Cohen’s d statistic. (Section 13.8).
• You can check the normality of a sample using QQ plots and the Shapiro-Wilk test. (Section 13.9)
• If your data are non-normal, you can use Wilcoxon tests instead of t-tests. (Section 13.10)
References
Student, A. 1908. “The Probable Error of a Mean.” Biometrika 6: 1–2.
Box, J. F. 1987. “Guinness, Gosset, Fisher, and Small Samples.” Statistical Science 2: 45–52.
Welch, B. L. 1947. “The Generalization of ‘Student’s’ Problem When Several Different Population Variances Are Involved.” Biometrika 34: 28–35.
Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Lawrence Erlbaum.
McGrath, R. E., and G. J. Meyer. 2006. “When Effect Sizes Disagree: The Case of r and d.” Psychological Methods 11: 386–401.
Hedges, L. V. 1981. “Distribution Theory for Glass’s Estimator of Effect Size and Related Estimators.” Journal of Educational Statistics 6: 107–28.
Hedges, L. V., and I. Olkin. 1985. Statistical Methods for Meta-Analysis. New York: Academic Press.
Shapiro, S. S., and M. B. Wilk. 1965. “An Analysis of Variance Test for Normality (Complete Samples).” Biometrika 52: 591–611.
1. We won’t cover multiple predictors until Chapter 15
2. Informal experimentation in my garden suggests that yes, it does. Australian natives are adapted to low phosphorus levels relative to everywhere else on Earth, apparently, so if you’ve bought a house with a bunch of exotics and you want to plant natives, don’t follow my example: keep them separate. Nutrients to European plants are poison to Australian ones. There’s probably a joke in that, but I can’t figure out what it is.
3. Actually this is too strong. Strictly speaking the z test only requires that the sampling distribution of the mean be normally distributed; if the population is normal then it necessarily follows that the sampling distribution of the mean is also normal. However, as we saw when talking about the central limit theorem, it’s quite possible (even commonplace) for the sampling distribution to be normal even if the population distribution itself is non-normal. However, in light of the sheer ridiculousness of the assumption that the true standard deviation is known, there really isn’t much point in going into details on this front!
4. Well, sort of. As I understand the history, Gosset only provided a partial solution: the general solution to the problem was provided by Sir Ronald Fisher.
5. More seriously, I tend to think the reverse is true: I get very suspicious of technical reports that fill their results sections with nothing except the numbers. It might just be that I’m an arrogant jerk, but I often feel like an author that makes no attempt to explain and interpret their analysis to the reader either doesn’t understand it themselves, or is being a bit lazy. Your readers are smart, but not infinitely patient. Don’t annoy them if you can help it.
6. Although it is the simplest, which is why I started with it.
7. A funny question almost always pops up at this point: what the heck is the population being referred to in this case? Is it the set of students actually taking Dr Harpo’s class (all 33 of them)? The set of people who might take the class (an unknown number) of them? Or something else? Does it matter which of these we pick? It’s traditional in an introductory behavioural stats class to mumble a lot at this point, but since I get asked this question every year by my students, I’ll give a brief answer. Technically yes, it does matter: if you change your definition of what the “real world” population actually is, then the sampling distribution of your observed mean ¯X changes too. The t-test relies on an assumption that the observations are sampled at random from an infinitely large population; and to the extent that real life isn’t like that, then the t-test can be wrong. In practice, however, this isn’t usually a big deal: even though the assumption is almost always wrong, it doesn’t lead to a lot of pathological behaviour from the test, so we tend to just ignore it.
8. Yes, I have a “favourite” way of thinking about pooled standard deviation estimates. So what?
9. A more correct notation will be introduced in Chapter 14.
10. Well, I guess you can average apples and oranges, and what you end up with is a delicious fruit smoothie. But no one really thinks that a fruit smoothie is a very good way to describe the original fruits, do they?
11. This design is very similar to the one in Section 12.8 that motivated the McNemar test. This should be no surprise. Both are standard repeated measures designs involving two measurements. The only difference is that this time our outcome variable is interval scale (working memory capacity) rather than a binary, nominal scale variable (a yes-or-no question).
12. At this point we have Drs Harpo, Chico and Zeppo. No prizes for guessing who Dr Groucho is.
13. This is obviously a class being taught at a very small or very expensive university, or else is a postgraduate class. I’ve never taught an intro stats class with less than 350 students.
14. The `sortFrame()` function sorts factor variables like `id` in alphabetical order, which is why it jumps from “student1” to “student10”
15. This is a massive oversimplification.
16. Either that, or the Kolmogorov-Smirnov test, which is probably more traditional than the Shapiro-Wilk, though most things I’ve read seem to suggest Shapiro-Wilk is the better test of normality; although Kolomogorov-Smirnov is a general purpose test of distributional equivalence, so it can be adapted to handle other kinds of distribution tests; in R it’s implemented via the `ks.test()` function.
17. Actually, there are two different versions of the test statistic; they differ from each other by a constant value. The version that I’ve described is the one that R calculates. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/13%3A_Comparing_Two_Means/13.11%3A_Summary.txt |
This chapter introduces one of the most widely used tools in statistics, known as “the analysis of variance”, which is usually referred to as ANOVA. The basic technique was developed by Sir Ronald Fisher in the early 20th century, and it is to him that we owe the rather unfortunate terminology. The term ANOVA is a little misleading, in two respects. Firstly, although the name of the technique refers to variances, ANOVA is concerned with investigating differences in means. Secondly, there are several different things out there that are all referred to as ANOVAs, some of which have only a very tenuous connection to one another. Later on in the book we’ll encounter a range of different ANOVA methods that apply in quite different situations, but for the purposes of this chapter we’ll only consider the simplest form of ANOVA, in which we have several different groups of observations, and we’re interested in finding out whether those groups differ in terms of some outcome variable of interest. This is the question that is addressed by a one-way ANOVA.
The structure of this chapter is as follows: In Section 14.1 I’ll introduce a fictitious data set that we’ll use as a running example throughout the chapter. After introducing the data, I’ll describe the mechanics of how a one-way ANOVA actually works (Section 14.2) and then focus on how you can run one in R (Section 14.3). These two sections are the core of the chapter. The remainder of the chapter discusses a range of important topics that inevitably arise when running an ANOVA, namely how to calculate effect sizes (Section 14.4), post hoc tests and corrections for multiple comparisons (Section 14.5) and the assumptions that ANOVA relies upon (Section 14.6). We’ll also talk about how to check those assumptions and some of the things you can do if the assumptions are violated (Sections 14.7 to 14.10). At the end of the chapter we’ll talk a little about the relationship between ANOVA and other statistical tools (Section 14.11).
14: Comparing Several Means (One-way ANOVA)
There’s a fair bit covered in this chapter, but there’s still a lot missing. Most obviously, I haven’t yet discussed any analog of the paired samples t-test for more than two groups. There is a way of doing this, known as repeated measures ANOVA, which will appear in a later version of this book. I also haven’t discussed how to run an ANOVA when you are interested in more than one grouping variable, but that will be discussed in a lot of detail in Chapter 16. In terms of what we have discussed, the key topics were:
• The basic logic behind how ANOVA works (Section 14.2) and how to run one in R (Section 14.3).
• How to compute an effect size for an ANOVA (Section 14.4)
• Post hoc analysis and corrections for multiple testing (Section 14.5).
• The assumptions made by ANOVA (Section 14.6).
• How to check the homogeneity of variance assumption (Section 14.7) and what to do if it is violated (Section 14.8).
• How to check the normality assumption (Section 14.9 and what to do if it is violated (Section 14.10).
As with all of the chapters in this book, there are quite a few different sources that I’ve relied upon, but the one stand-out text that I’ve been most heavily influenced by is Sahai and Ageel (2000). It’s not a good book for beginners, but it’s an excellent book for more advanced readers who are interested in understanding the mathematics behind ANOVA.
References
Hays, W. L. 1994. Statistics. 5th ed. Fort Worth, TX: Harcourt Brace.
Shaffer, J. P. 1995. “Multiple Hypothesis Testing.” Annual Review of Psychology 46: 561–84.
Hsu, J. C. 1996. Multiple Comparisons: Theory and Methods. London, UK: Chapman; Hall.
Dunn, O.J. 1961. “Multiple Comparisons Among Means.” Journal of the American Statistical Association 56: 52–64.
Holm, S. 1979. “A Simple Sequentially Rejective Multiple Test Procedure.” Scandinavian Journal of Statistics 6: 65–70.
Levene, H. 1960. “Robust Tests for Equality of Variances.” In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, edited by I. Olkin et al, 278–92. Palo Alto, CA: Stanford University Press.
Brown, M. B., and A. B. Forsythe. 1974. “Robust Tests for Equality of Variances.” Journal of the American Statistical Association 69: 364–67.
Welch, B. 1951. “On the Comparison of Several Mean Values: An Alternative Approach.” Biometrika 38: 330–36.
Kruskal, W. H., and W. A. Wallis. 1952. “Use of Ranks in One-Criterion Variance Analysis.” Journal of the American Statistical Association 47: 583–621.
Sahai, H., and M. I. Ageel. 2000. The Analysis of Variance: Fixed, Random and Mixed Models. Boston: Birkhauser.
1. When all groups have the same number of observations, the experimental design is said to be “balanced”. Balance isn’t such a big deal for one-way ANOVA, which is the topic of this chapter. It becomes more important when you start doing more complicated ANOVAs.
2. In a later versions I’m intending to expand on this. But because I’m writing in a rush, and am already over my deadlines, I’ll just briefly note that if you read ahead to Chapter 16 and look at how the “treatment effect” at level k of a factor is defined in terms of the αk values (see Section 16.2), it turns out that Q refers to a weighted mean of the squared treatment effects, $Q=\left(\sum_{k=1}^{G} N_{k} \alpha_{k}^{2}\right) /(G-1)$
3. we want to be sticklers for accuracy, $\ 1 + {2 \over df_2 - 2}$
4. o be precise, party like “it’s 1899 and we’ve got no friends and nothing better to do with our time than do some calculations that wouldn’t have made any sense in 1899 because ANOVA didn’t exist until about the 1920s”.
5. Actually, it also provides a function called anova(), but that works a bit differently, so let’s just ignore it for now.
6. It’s worth noting that you can get the same result by using the command anova( my.anova ).
7. A potentially important footnote – I wrote the etaSquared() function for the lsr package as a teaching exercise, but like all the other functions in the lsr package it hasn’t been exhaustively tested. As of this writing – lsr package version 0.5 – there is at least one known bug in the code. In some cases at least, it doesn’t work (and can give very silly answers) when you set the weights on the observations to something other than uniform. That doesn’t matter at all for this book, since those kinds of analyses are well beyond the scope, but I haven’t had a chance to revisit the package in a long time; it would probably be wise to be very cautious with the use of this function in any context other than very simple introductory analyses. Thanks to Emil Kirkegaard for finding the bug! (Oh, and while I’m here, there’s an interesting blog post by Daniel Lakens suggesting that eta-squared itself is perhaps not the best measure of effect size in real world data analysis: http://daniellakens.blogspot.com.au/2015/06/why-you-should-use-omega-squared.html
8. I should point out that there are other functions in R for running multiple comparisons, and at least one of them works this way: the TukeyHSD() function takes an aov object as its input, and outputs Tukey’s “honestly significant difference” tests. I talk about Tukey’s HSD in Chapter 16.
9. If you do have some theoretical basis for wanting to investigate some comparisons but not others, it’s a different story. In those circumstances you’re not really running “post hoc” analyses at all: you’re making “planned comparisons”. I do talk about this situation later in the book (Section 16.9), but for now I want to keep things simple.
10. It’s worth noting in passing that not all adjustment methods try to do this. What I’ve described here is an approach for controlling “family wise Type I error rate”. However, there are other post hoc tests seek to control the “false discovery rate”, which is a somewhat different thing.
11. There’s also a function called p.adjust() in which you can input a vector of raw p-values, and it will output a vector of adjusted p-values. This can be handy sometimes. I should also note that more advanced users may wish to consider using some of the tools provided by the multcomp package.
12. Note that neither of these figures has been tidied up at all: if you want to create nicer looking graphs it’s always a good idea to use the tools from Chapter 6 to help you draw cleaner looking images.
13. A technical term. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.01%3A_Summary.txt |
Suppose you’ve become involved in a clinical trial in which you are testing a new antidepressant drug called Joyzepam. In order to construct a fair test of the drug’s effectiveness, the study involves three separate drugs to be administered. One is a placebo, and the other is an existing antidepressant / anti-anxiety drug called Anxifree. A collection of 18 participants with moderate to severe depression are recruited for your initial testing. Because the drugs are sometimes administered in conjunction with psychological therapy, your study includes 9 people undergoing cognitive behavioural therapy (CBT) and 9 who are not. Participants are randomly assigned (doubly blinded, of course) a treatment, such that there are 3 CBT people and 3 no-therapy people assigned to each of the 3 drugs. A psychologist assesses the mood of each person after a 3 month run with each drug: and the overall improvement in each person’s mood is assessed on a scale ranging from −5 to +5.
With that as the study design, let’s now look at what we’ve got in the data file:
``````load( "./rbook-master/data/clinicaltrial.Rdata" ) # load data
str(clin.trial)``````
``````## 'data.frame': 18 obs. of 3 variables:
## \$ drug : Factor w/ 3 levels "placebo","anxifree",..: 1 1 1 2 2 2 3 3 3 1 ...
## \$ therapy : Factor w/ 2 levels "no.therapy","CBT": 1 1 1 1 1 1 1 1 1 2 ...
## \$ mood.gain: num 0.5 0.3 0.1 0.6 0.4 0.2 1.4 1.7 1.3 0.6 ...``````
So we have a single data frame called `clin.trial`, containing three variables; `drug`, `therapy` and `mood.gain`. Next, let’s print the data frame to get a sense of what the data actually look like.
``print( clin.trial )``
``````## drug therapy mood.gain
## 1 placebo no.therapy 0.5
## 2 placebo no.therapy 0.3
## 3 placebo no.therapy 0.1
## 4 anxifree no.therapy 0.6
## 5 anxifree no.therapy 0.4
## 6 anxifree no.therapy 0.2
## 7 joyzepam no.therapy 1.4
## 8 joyzepam no.therapy 1.7
## 9 joyzepam no.therapy 1.3
## 10 placebo CBT 0.6
## 11 placebo CBT 0.9
## 12 placebo CBT 0.3
## 13 anxifree CBT 1.1
## 14 anxifree CBT 0.8
## 15 anxifree CBT 1.2
## 16 joyzepam CBT 1.8
## 17 joyzepam CBT 1.3
## 18 joyzepam CBT 1.4``````
For the purposes of this chapter, what we’re really interested in is the effect of `drug` on `mood.gain`. The first thing to do is calculate some descriptive statistics and draw some graphs. In Chapter 5 we discussed a variety of different functions that can be used for this purpose. For instance, we can use the `xtabs()` function to see how many people we have in each group:
``xtabs( ~drug, clin.trial )``
``````## drug
## placebo anxifree joyzepam
## 6 6 6``````
Similarly, we can use the `aggregate()` function to calculate means and standard deviations for the `mood.gain` variable broken down by which `drug` was administered:
``aggregate( mood.gain ~ drug, clin.trial, mean )``
``````## drug mood.gain
## 1 placebo 0.4500000
## 2 anxifree 0.7166667
## 3 joyzepam 1.4833333```
```
``aggregate( mood.gain ~ drug, clin.trial, sd )``
``````## drug mood.gain
## 1 placebo 0.2810694
## 2 anxifree 0.3920034
## 3 joyzepam 0.2136976``````
Finally, we can use `plotmeans()` from the `gplots` package to produce a pretty picture.
``````library(gplots)
plotmeans( formula = mood.gain ~ drug, # plot mood.gain by drug
data = clin.trial, # the data frame
xlab = "Drug Administered", # x-axis label
ylab = "Mood Gain", # y-axis label
n.label = FALSE # don't display sample size
)``````
The results are shown in Figure 14.1, which plots the average mood gain for all three conditions; error bars show 95% confidence intervals. As the plot makes clear, there is a larger improvement in mood for participants in the Joyzepam group than for either the Anxifree group or the placebo group. The Anxifree group shows a larger mood gain than the control group, but the difference isn’t as large.
The question that we want to answer is: are these difference “real”, or are they just due to chance?
``## Warning: package 'gplots' was built under R version 3.5.2``
``````##
## Attaching package: 'gplots'```
```
``````## The following object is masked from 'package:stats':
##
## lowess`````` | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.02%3A_An_Illustrative_Data_Set.txt |
In order to answer the question posed by our clinical trial data, we’re going to run a one-way ANOVA. As usual, I’m going to start by showing you how to do it the hard way, building the statistical tool from the ground up and showing you how you could do it in R if you didn’t have access to any of the cool built-in ANOVA functions. And, as always, I hope you’ll read it carefully, try to do it the long way once or twice to make sure you really understand how ANOVA works, and then – once you’ve grasped the concept – never ever do it this way again.
The experimental design that I described in the previous section strongly suggests that we’re interested in comparing the average mood change for the three different drugs. In that sense, we’re talking about an analysis similar to the t-test (Chapter 13, but involving more than two groups. If we let μP denote the population mean for the mood change induced by the placebo, and let μA and μJ denote the corresponding means for our two drugs, Anxifree and Joyzepam, then the (somewhat pessimistic) null hypothesis that we want to test is that all three population means are identical: that is, neither of the two drugs is any more effective than a placebo. Mathematically, we write this null hypothesis like this:
H0: it is true that μPAJ
As a consequence, our alternative hypothesis is that at least one of the three different treatments is different from the others. It’s a little trickier to write this mathematically, because (as we’ll discuss) there are quite a few different ways in which the null hypothesis can be false. So for now we’ll just write the alternative hypothesis like this:
H1: it is *not* true that μPAJ
This null hypothesis is a lot trickier to test than any of the ones we’ve seen previously. How shall we do it? A sensible guess would be to “do an ANOVA”, since that’s the title of the chapter, but it’s not particularly clear why an “analysis of variances” will help us learn anything useful about the means. In fact, this is one of the biggest conceptual difficulties that people have when first encountering ANOVA. To see how this works, I find it most helpful to start by talking about variances. In fact, what I’m going to do is start by playing some mathematical games with the formula that describes the variance. That is, we’ll start out by playing around with variances, and it will turn out that this gives us a useful tool for investigating means.
formulas for the variance of Y
Firstly, let’s start by introducing some notation. We’ll use G to refer to the total number of groups. For our data set, there are three drugs, so there are G=3 groups. Next, we’ll use N to refer to the total sample size: there are a total of N=18 people in our data set. Similarly, let’s use Nk to denote the number of people in the k-th group. In our fake clinical trial, the sample size is Nk=6 for all three groups.201 Finally, we’ll use Y to denote the outcome variable: in our case, Y refers to mood change. Specifically, we’ll use Yik to refer to the mood change experienced by the i-th member of the k-th group. Similarly, we’ll use $\ \bar{Y}$ to be the average mood change, taken across all 18 people in the experiment, and $\ \bar{Y_k}$ to refer to the average mood change experienced by the 6 people in group k.
Excellent. Now that we’ve got our notation sorted out, we can start writing down formulas. To start with, let’s recall the formula for the variance that we used in Section 5.2, way back in those kinder days when we were just doing descriptive statistics. The sample variance of Y is defined as follows:
$\operatorname{Var}(Y)=\dfrac{1}{N} \sum_{k=1}^{G} \sum_{i=1}^{N_{k}}\left(Y_{i k}-\bar{Y}\right)^{2}$
This formula looks pretty much identical to the formula for the variance in Section 5.2. The only difference is that this time around I’ve got two summations here: I’m summing over groups (i.e., values for k) and over the people within the groups (i.e., values for i). This is purely a cosmetic detail: if I’d instead used the notation Yp to refer to the value of the outcome variable for person p in the sample, then I’d only have a single summation. The only reason that we have a double summation here is that I’ve classified people into groups, and then assigned numbers to people within groups.
A concrete example might be useful here. Let’s consider this table, in which we have a total of N=5 people sorted into G=2 groups. Arbitrarily, let’s say that the “cool” people are group 1, and the “uncool” people are group 2, and it turns out that we have three cool people (N1=3) and two uncool people (N2=2).
name person (p) group group num (k) index in group (i) grumpiness (Yik or Yp)
Ann 1 cool 1 1 20
Ben 2 cool 1 2 55
Cat 3 cool 1 3 21
Dan 4 uncool 2 1 91
Egg 5 uncool 2 2 22
Notice that I’ve constructed two different labelling schemes here. We have a “person” variable p, so it would be perfectly sensible to refer to Yp as the grumpiness of the p-th person in the sample. For instance, the table shows that Dan is the four so we’d say p=4. So, when talking about the grumpiness Y of this “Dan” person, whoever he might be, we could refer to his grumpiness by saying that Yp=91, for person p=4 that is. However, that’s not the only way we could refer to Dan. As an alternative we could note that Dan belongs to the “uncool” group (k=2), and is in fact the first person listed in the uncool group (i=1). So it’s equally valid to refer to Dan’s grumpiness by saying that Yik=91, where k=2 and i=1. In other words, each person p corresponds to a unique ik combination, and so the formula that I gave above is actually identical to our original formula for the variance, which would be
$\operatorname{Var}(Y)=\dfrac{1}{N} \sum_{p=1}^{N}\left(Y_{p}-\bar{Y}\right)^{2}$
In both formulas, all we’re doing is summing over all of the observations in the sample. Most of the time we would just use the simpler Yp notation: the equation using Yp is clearly the simpler of the two. However, when doing an ANOVA it’s important to keep track of which participants belong in which groups, and we need to use the Yik notation to do this.
From variances to sums of squares
Okay, now that we’ve got a good grasp on how the variance is calculated, let’s define something called the total sum of squares, which is denoted SStot. This is very simple: instead of averaging the squared deviations, which is what we do when calculating the variance, we just add them up. So the formula for the total sum of squares is almost identical to the formula for the variance:
$\mathrm{SS}_{t o t}=\sum_{k=1}^{G} \sum_{i=1}^{N_{k}}\left(Y_{i k}-\bar{Y}\right)^{2}$
When we talk about analysing variances in the context of ANOVA, what we’re really doing is working with the total sums of squares rather than the actual variance. One very nice thing about the total sum of squares is that we can break it up into two different kinds of variation. Firstly, we can talk about the within-group sum of squares, in which we look to see how different each individual person is from their own group mean:
$\mathrm{SS}_{w}=\sum_{k=1}^{G} \sum_{i=1}^{N_{k}}\left(Y_{i k}-\bar{Y}_{k}\right)^{2}$
where $\ \bar{Y_k}$ is a group mean. In our example, $\ \bar{Y_k}$ would be the average mood change experienced by those people given the k-th drug. So, instead of comparing individuals to the average of all people in the experiment, we’re only comparing them to those people in the the same group. As a consequence, you’d expect the value of SSw to be smaller than the total sum of squares, because it’s completely ignoring any group differences – that is, the fact that the drugs (if they work) will have different effects on people’s moods.
Next, we can define a third notion of variation which captures only the differences between groups. We do this by looking at the differences between the group means $\ \bar{Y_k}$ and grand mean $\ \bar{Y}$. In order to quantify the extent of this variation, what we do is calculate the between-group sum of squares:
\begin{aligned} \mathrm{SS}_{b} &=\sum_{k=1}^{G} \sum_{i=1}^{N_{k}}\left(\bar{Y}_{k}-\bar{Y}\right)^{2} \ &=\sum_{k=1}^{G} N_{k}\left(\bar{Y}_{k}-\bar{Y}\right)^{2} \end{aligned}
It’s not too difficult to show that the total variation among people in the experiment SStot is actually the sum of the differences between the groups SSb and the variation inside the groups SSw. That is:
SSw+SSb=SStot
Yay.
Okay, so what have we found out? We’ve discovered that the total variability associated with the outcome variable (SStot) can be mathematically carved up into the sum of “the variation due to the differences in the sample means for the different groups” (SSb) plus “all the rest of the variation” (SSw). How does that help me find out whether the groups have different population means? Um. Wait. Hold on a second… now that I think about it, this is exactly what we were looking for. If the null hypothesis is true, then you’d expect all the sample means to be pretty similar to each other, right? And that would imply that you’d expect SSb to be really small, or at least you’d expect it to be a lot smaller than the “the variation associated with everything else”, SSw. Hm. I detect a hypothesis test coming on…
From sums of squares to the F-test
As we saw in the last section, the qualitative idea behind ANOVA is to compare the two sums of squares values SSb and SSw to each other: if the between-group variation is SSb is large relative to the within-group variation SSw then we have reason to suspect that the population means for the different groups aren’t identical to each other. In order to convert this into a workable hypothesis test, there’s a little bit of “fiddling around” needed. What I’ll do is first show you what we do to calculate our test statistic – which is called an F ratio – and then try to give you a feel for why we do it this way.
In order to convert our SS values into an F-ratio, the first thing we need to calculate is the degrees of freedom associated with the SSb and SSw values. As usual, the degrees of freedom corresponds to the number of unique “data points” that contribute to a particular calculation, minus the number of “constraints” that they need to satisfy. For the within-groups variability, what we’re calculating is the variation of the individual observations (N data points) around the group means (G constraints). In contrast, for the between groups variability, we’re interested in the variation of the group means (G data points) around the grand mean (1 constraint). Therefore, the degrees of freedom here are:
dfb=G−1
dfw=N−G
Okay, that seems simple enough. What we do next is convert our summed squares value into a “mean squares” value, which we do by dividing by the degrees of freedom:
$\ MS_b = {SS_b \over df_b}$
$\ MS_w = {SS_w \over df_w}$
Finally, we calculate the F-ratio by dividing the between-groups MS by the within-groups MS:
$\ F={MS_b \over MS_w}$
At a very general level, the intuition behind the F statistic is straightforward: bigger values of F means that the between-groups variation is large, relative to the within-groups variation. As a consequence, the larger the value of F, the more evidence we have against the null hypothesis. But how large does F have to be in order to actually reject H0? In order to understand this, you need a slightly deeper understanding of what ANOVA is and what the mean squares values actually are.
The next section discusses that in a bit of detail, but for readers that aren’t interested in the details of what the test is actually measuring, I’ll cut to the chase. In order to complete our hypothesis test, we need to know the sampling distribution for F if the null hypothesis is true. Not surprisingly, the sampling distribution for the F statistic under the null hypothesis is an F distribution. If you recall back to our discussion of the F distribution in Chapter @ref(probability, the F distribution has two parameters, corresponding to the two degrees of freedom involved: the first one df1 is the between groups degrees of freedom dfb, and the second one df2 is the within groups degrees of freedom dfw.
A summary of all the key quantities involved in a one-way ANOVA, including the formulas showing how they are calculated, is shown in Table 14.1.
Table 14.1: All of the key quantities involved in an ANOVA, organised into a “standard” ANOVA table. The formulas for all quantities (except the p-value, which has a very ugly formula and would be nightmarishly hard to calculate without a computer) are shown.
df sum of squares mean squares F statistic p value
between groups dfb=G−1 $\mathrm{SS}_{b}=\sum_{k=1}^{G} N_{k}\left(\bar{Y}_{k}-\bar{Y}\right)^{2}$ $\mathrm{MS}_{b}=\dfrac{\mathrm{SS}_{b}}{\mathrm{df}_{b}}$ $F=\dfrac{\mathrm{MS}_{b}}{\mathrm{MS}_{w}}$ [complicated]
within groups dfw=N−G $SS_w=\sum_{k=1}^{G} \sum_{i=1}^{N_{k}}\left(Y_{i k}-\bar{Y}_{k}\right)^{2}$ $\mathrm{MS}_{w}=\dfrac{\mathrm{SS}_{w}}{\mathrm{df}_{w}}$ - -
model for the data and the meaning of F (advanced)
At a fundamental level, ANOVA is a competition between two different statistical models, H0 and H1. When I described the null and alternative hypotheses at the start of the section, I was a little imprecise about what these models actually are. I’ll remedy that now, though you probably won’t like me for doing so. If you recall, our null hypothesis was that all of the group means are identical to one another. If so, then a natural way to think about the outcome variable Yik is to describe individual scores in terms of a single population mean μ, plus the deviation from that population mean. This deviation is usually denoted ϵik and is traditionally called the error or residual associated with that observation. Be careful though: just like we saw with the word “significant”, the word “error” has a technical meaning in statistics that isn’t quite the same as its everyday English definition. In everyday language, “error” implies a mistake of some kind; in statistics, it doesn’t (or at least, not necessarily). With that in mind, the word “residual” is a better term than the word “error”. In statistics, both words mean “leftover variability”: that is, “stuff” that the model can’t explain. In any case, here’s what the null hypothesis looks like when we write it as a statistical model:
Yik=μ+ϵik
where we make the assumption (discussed later) that the residual values ϵik are normally distributed, with mean 0 and a standard deviation σ that is the same for all groups. To use the notation that we introduced in Chapter 9 we would write this assumption like this:
ϵik∼Normal(0,σ2)
What about the alternative hypothesis, H1? The only difference between the null hypothesis and the alternative hypothesis is that we allow each group to have a different population mean. So, if we let μk denote the population mean for the k-th group in our experiment, then the statistical model corresponding to H1 is:
Yikkik
where, once again, we assume that the error terms are normally distributed with mean 0 and standard deviation σ. That is, the alternative hypothesis also assumes that
ϵ∼Normal(0,σ2)
Okay, now that we’ve described the statistical models underpinning H0 and H1 in more detail, it’s now pretty straightforward to say what the mean square values are measuring, and what this means for the interpretation of F. I won’t bore you with the proof of this, but it turns out that the within-groups mean square, MSw, can be viewed as an estimator (in the technical sense: Chapter 10 of the error variance σ2. The between-groups mean square MSb is also an estimator; but what it estimates is the error variance plus a quantity that depends on the true differences among the group means. If we call this quantity Q, then we can see that the F-statistic is basically202
$F=\dfrac{\hat{Q}+\hat{\sigma}^{2}}{\hat{\sigma}^{2}}$
where the true value Q=0 if the null hypothesis is true, and Q>0 if the alternative hypothesis is true (e.g. ch. 10 Hays 1994). Therefore, at a bare minimum the F value must be larger than 1 to have any chance of rejecting the null hypothesis. Note that this doesn’t mean that it’s impossible to get an F-value less than 1. What it means is that, if the null hypothesis is true the sampling distribution of the F ratio has a mean of 1,203 and so we need to see F-values larger than 1 in order to safely reject the null.
To be a bit more precise about the sampling distribution, notice that if the null hypothesis is true, both MSb and MSw are estimators of the variance of the residuals ϵik. If those residuals are normally distributed, then you might suspect that the estimate of the variance of ϵik is chi-square distributed… because (as discussed in Section 9.6 that’s what a chi-square distribution is: it’s what you get when you square a bunch of normally-distributed things and add them up. And since the F distribution is (again, by definition) what you get when you take the ratio between two things that are X2 distributed… we have our sampling distribution. Obviously, I’m glossing over a whole lot of stuff when I say this, but in broad terms, this really is where our sampling distribution comes from.
worked example
The previous discussion was fairly abstract, and a little on the technical side, so I think that at this point it might be useful to see a worked example. For that, let’s go back to the clinical trial data that I introduced at the start of the chapter. The descriptive statistics that we calculated at the beginning tell us our group means: an average mood gain of 0.45 for the placebo, 0.72 for Anxifree, and 1.48 for Joyzepam. With that in mind, let’s party like it’s 1899204 and start doing some pencil and paper calculations. I’ll only do this for the first 5 observations, because it’s not bloody 1899 and I’m very lazy. Let’s start by calculating SSw, the within-group sums of squares. First, let’s draw up a nice table to help us with our calculations…
group (k) outcome (Yik)
placebo 0.5
placebo 0.3
placebo 0.1
anxifree 0.6
anxifree 0.4
At this stage, the only thing I’ve included in the table is the raw data itself: that is, the grouping variable (i.e., drug) and outcome variable (i.e. mood.gain) for each person. Note that the outcome variable here corresponds to the Yik value in our equation previously. The next step in the calculation is to write down, for each person in the study, the corresponding group mean; that is, $\ \bar{Y_k}$. This is slightly repetitive, but not particularly difficult since we already calculated those group means when doing our descriptive statistics:
group (k) outcome (Yik) group mean ($\ \bar{Y_k}$
placebo 0.5 0.45
placebo 0.3 0.45
placebo 0.1 0.45
anxifree 0.6 0.72
anxifree 0.4 0.72
Now that we’ve written those down, we need to calculate – again for every person – the deviation from the corresponding group mean. That is, we want to subtract Yik−$\ \bar{Y_k}$. After we’ve done that, we need to square everything. When we do that, here’s what we get:
group (k) outcome (Yik) group mean ($\ \bar{Y_k}$) dev. from group mean (Yik−$\ \bar{Y_k}$) squared deviation ((Yik−$\ \bar{Y_k}$)2)
placebo 0.5 0.45 0.05 0.0025
placebo 0.3 0.45 -0.15 0.0225
placebo 0.1 0.45 -0.35 0.1225
anxifree 0.6 0.72 -0.12 0.0136
anxifree 0.4 0.72 -0.32 0.1003
The last step is equally straightforward. In order to calculate the within-group sum of squares, we just add up the squared deviations across all observations:
\begin{aligned} \mathrm{SS}_{w} &=0.0025+0.0225+0.1225+0.0136+0.1003 \ &=0.2614 \end{aligned}
Of course, if we actually wanted to get the right answer, we’d need to do this for all 18 observations in the data set, not just the first five. We could continue with the pencil and paper calculations if we wanted to, but it’s pretty tedious. Alternatively, it’s not too hard to get R to do it. Here’s how:
outcome <- clin.trial$mood.gain group <- clin.trial$drug
gp.means <- tapply(outcome,group,mean)
gp.means <- gp.means[group]
dev.from.gp.means <- outcome - gp.means
squared.devs <- dev.from.gp.means ^2
It might not be obvious from inspection what these commands are doing: as a general rule, the human brain seems to just shut down when faced with a big block of programming. However, I strongly suggest that – if you’re like me and tend to find that the mere sight of this code makes you want to look away and see if there’s any beer left in the fridge or a game of footy on the telly – you take a moment and look closely at these commands one at a time. Every single one of these commands is something you’ve seen before somewhere else in the book. There’s nothing novel about them (though I’ll have to admit that the tapply() function takes a while to get a handle on), so if you’re not quite sure how these commands work, this might be a good time to try playing around with them yourself, to try to get a sense of what’s happening. On the other hand, if this does seem to make sense, then you won’t be all that surprised at what happens when I wrap these variables in a data frame, and print it out…
Y <- data.frame( group, outcome, gp.means,
dev.from.gp.means, squared.devs )
print(Y, digits = 2)
## group outcome gp.means dev.from.gp.means squared.devs
## 1 placebo 0.5 0.45 0.050 0.0025
## 2 placebo 0.3 0.45 -0.150 0.0225
## 3 placebo 0.1 0.45 -0.350 0.1225
## 4 anxifree 0.6 0.72 -0.117 0.0136
## 5 anxifree 0.4 0.72 -0.317 0.1003
## 6 anxifree 0.2 0.72 -0.517 0.2669
## 7 joyzepam 1.4 1.48 -0.083 0.0069
## 8 joyzepam 1.7 1.48 0.217 0.0469
## 9 joyzepam 1.3 1.48 -0.183 0.0336
## 10 placebo 0.6 0.45 0.150 0.0225
## 11 placebo 0.9 0.45 0.450 0.2025
## 12 placebo 0.3 0.45 -0.150 0.0225
## 13 anxifree 1.1 0.72 0.383 0.1469
## 14 anxifree 0.8 0.72 0.083 0.0069
## 15 anxifree 1.2 0.72 0.483 0.2336
## 16 joyzepam 1.8 1.48 0.317 0.1003
## 17 joyzepam 1.3 1.48 -0.183 0.0336
## 18 joyzepam 1.4 1.48 -0.083 0.0069
If you compare this output to the contents of the table I’ve been constructing by hand, you can see that R has done exactly the same calculations that I was doing, and much faster too. So, if we want to finish the calculations of the within-group sum of squares in R, we just ask for the sum() of the squared.devs variable:
SSw <- sum( squared.devs )
print( SSw )
## [1] 1.391667
Obviously, this isn’t the same as what I calculated, because R used all 18 observations. But if I’d typed sum( squared.devs[1:5] ) instead, it would have given the same answer that I got earlier.
Okay. Now that we’ve calculated the within groups variation, SSw, it’s time to turn our attention to the between-group sum of squares, SSb. The calculations for this case are very similar. The main difference is that, instead of calculating the differences between an observation Yik and a group mean $\ \bar{Y_k}$ for all of the observations, we calculate the differences between the group means $\ \bar{Y_k}$ and the grand mean $\ \bar{Y}$ (in this case 0.88) for all of the groups…
group (k) group mean ($\ \bar{Y_k}$) grand mean ($\ \bar{Y}$) deviation ($\ \bar{Y_k} - \bar{Y}$) squared deviations (($\ \bar{Y_k} - \bar{Y}$)2)
placebo 0.45 0.88 -0.43 0.18
anxifree 0.72 0.88 -0.16 0.03
joyzepam 1.48 0.88 0.60 0.36
However, for the between group calculations we need to multiply each of these squared deviations by Nk, the number of observations in the group. We do this because every observation in the group (all Nk of them) is associated with a between group difference. So if there are six people in the placebo group, and the placebo group mean differs from the grand mean by 0.19, then the total between group variation associated with these six people is 6×0.16=1.14. So we have to extend our little table of calculations…
group (k) squared deviations (($\ \bar{Y_k} - \bar{Y}$)2) sample size (Nk) weighted squared dev (Nk($\ \bar{Y_k} - \bar{Y}$)2)
placebo 0.18 6 1.11
anxifree 0.03 6 0.16
joyzepam 0.36 6 2.18
And so now our between group sum of squares is obtained by summing these “weighted squared deviations” over all three groups in the study:
\begin{aligned} \mathrm{SS}_{b} &=1.11+0.16+2.18 \ &=3.45 \end{aligned}
As you can see, the between group calculations are a lot shorter, so you probably wouldn’t usually want to bother using R as your calculator. However, if you did decide to do so, here’s one way you could do it:
gp.means <- tapply(outcome,group,mean)
grand.mean <- mean(outcome)
dev.from.grand.mean <- gp.means - grand.mean
squared.devs <- dev.from.grand.mean ^2
gp.sizes <- tapply(outcome,group,length)
wt.squared.devs <- gp.sizes * squared.devs
Again, I won’t actually try to explain this code line by line, but – just like last time – there’s nothing in there that we haven’t seen in several places elsewhere in the book, so I’ll leave it as an exercise for you to make sure you understand it. Once again, we can dump all our variables into a data frame so that we can print it out as a nice table:
Y <- data.frame( gp.means, grand.mean, dev.from.grand.mean,
squared.devs, gp.sizes, wt.squared.devs )
print(Y, digits = 2)
## gp.means grand.mean dev.from.grand.mean squared.devs gp.sizes
## placebo 0.45 0.88 -0.43 0.188 6
## anxifree 0.72 0.88 -0.17 0.028 6
## joyzepam 1.48 0.88 0.60 0.360 6
## wt.squared.devs
## placebo 1.13
## anxifree 0.17
## joyzepam 2.16
Clearly, these are basically the same numbers that we got before. There are a few tiny differences, but that’s only because the hand-calculated versions have some small errors caused by the fact that I rounded all my numbers to 2 decimal places at each step in the calculations, whereas R only does it at the end (obviously, R s version is more accurate). Anyway, here’s the R command showing the final step:
SSb <- sum( wt.squared.devs )
print( SSb )
## [1] 3.453333
which is (ignoring the slight differences due to rounding error) the same answer that I got when doing things by hand.
Now that we’ve calculated our sums of squares values, SSb and SSw, the rest of the ANOVA is pretty painless. The next step is to calculate the degrees of freedom. Since we have G=3 groups and N=18 observations in total, our degrees of freedom can be calculated by simple subtraction:
dfb=G−1=2
dfw=N−G=15
Next, since we’ve now calculated the values for the sums of squares and the degrees of freedom, for both the within-groups variability and the between-groups variability, we can obtain the mean square values by dividing one by the other:
$\mathrm{MS}_{b}=\dfrac{\mathrm{SS}_{b}}{\mathrm{df}_{b}}=\dfrac{3.45}{2}=1.73$
$\mathrm{MS}_{w}=\dfrac{\mathrm{SS}_{w}}{\mathrm{df}_{w}}=\dfrac{1.39}{15}=0.09$
We’re almost done. The mean square values can be used to calculate the F-value, which is the test statistic that we’re interested in. We do this by dividing the between-groups MS value by the and within-groups MS value.
$\ F={MS_b \over MS_w} = {1.73 \over 0.09}=18.6$
Woohooo! This is terribly exciting, yes? Now that we have our test statistic, the last step is to find out whether the test itself gives us a significant result. As discussed in Chapter @ref(hypothesistesting, what we really ought to do is choose an α level (i.e., acceptable Type I error rate) ahead of time, construct our rejection region, etc etc. But in practice it’s just easier to directly calculate the p-value. Back in the “old days”, what we’d do is open up a statistics textbook or something and flick to the back section which would actually have a huge lookup table… that’s how we’d “compute” our p-value, because it’s too much effort to do it any other way. However, since we have access to R, I’ll use the pf() function to do it instead. Now, remember that I explained earlier that the F-test is always one sided? And that we only reject the null hypothesis for very large F-values? That means we’re only interested in the upper tail of the F-distribution. The command that you’d use here would be this…
pf( 18.6, df1 = 2, df2 = 15, lower.tail = FALSE)
## [1] 8.672727e-05
Therefore, our p-value comes to 0.0000867, or 8.67×10−5 in scientific notation. So, unless we’re being extremely conservative about our Type I error rate, we’re pretty much guaranteed to reject the null hypothesis.
At this point, we’re basically done. Having completed our calculations, it’s traditional to organise all these numbers into an ANOVA table like the one in Table@reftab:anovatable. For our clinical trial data, the ANOVA table would look like this:
df sum of squares mean squares F-statistic p-value
between groups 2 3.45 1.73 18.6 8.67×10−5
within groups 15 1.39 0.09 - -
These days, you’ll probably never have much reason to want to construct one of these tables yourself, but you will find that almost all statistical software (R included) tends to organise the output of an ANOVA into a table like this, so it’s a good idea to get used to reading them. However, although the software will output a full ANOVA table, there’s almost never a good reason to include the whole table in your write up. A pretty standard way of reporting this result would be to write something like this:
One-way ANOVA showed a significant effect of drug on mood gain (F(2,15)=18.6,p<.001).
Sigh. So much work for one short sentence. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.03%3A_How_ANOVA_Works.txt |
I’m pretty sure I know what you’re thinking after reading the last section, especially if you followed my advice and tried typing all the commands in yourself…. doing the ANOVA calculations yourself sucks. There’s quite a lot of calculations that we needed to do along the way, and it would be tedious to have to do this over and over again every time you wanted to do an ANOVA. One possible solution to the problem would be to take all these calculations and turn them into some R functions yourself. You’d still have to do a lot of typing, but at least you’d only have to do it the one time: once you’ve created the functions, you can reuse them over and over again. However, writing your own functions is a lot of work, so this is kind of a last resort. Besides, it’s much better if someone else does all the work for you…
Using the `aov()`function to specify your ANOVA
To make life easier for you, R provides a function called `aov()`, which – obviously – is an acronym of “Analysis Of Variance.”205 If you type `?aov` and have a look at the help documentation, you’ll see that there are several arguments to the `aov()` function, but the only two that we’re interested in are `formula` and `data`. As we’ve seen in a few places previously, the `formula` argument is what you use to specify the outcome variable and the grouping variable, and the `data` argument is what you use to specify the data frame that stores these variables. In other words, to do the same ANOVA that I laboriously calculated in the previous section, I’d use a command like this:
``aov( formula = mood.gain ~ drug, data = clin.trial ) ``
Actually, that’s not quite the whole story, as you’ll see as soon as you look at the output from this command, which I’ve hidden for the moment in order to avoid confusing you. Before we go into specifics, I should point out that either of these commands will do the same thing:
``````aov( clin.trial\$mood.gain ~ clin.trial\$drug )
aov( mood.gain ~ drug, clin.trial ) ``````
In the first command, I didn’t specify a `data` set, and instead relied on the `\$` operator to tell R how to find the variables. In the second command, I dropped the argument names, which is okay in this case because `formula` is the first argument to the `aov()` function, and `data` is the second one. Regardless of how I specify the ANOVA, I can assign the output of the `aov()` function to a variable, like this for example:
``my.anova <- aov( mood.gain ~ drug, clin.trial ) ``
This is almost always a good thing to do, because there’s lots of useful things that we can do with the `my.anova` variable. So let’s assume that it’s this last command that I used to specify the ANOVA that I’m trying to run, and as a consequence I have this `my.anova` variable sitting in my workspace, waiting for me to do something with it…
Understanding what the `aov()` function produces
Now that we’ve seen how to use the `aov()` function to create `my.anova` we’d better have a look at what this variable actually is. The first thing to do is to check to see what class of variable we’ve created, since it’s kind of interesting in this case. When we do that…
``class( my.anova )``
``## [1] "aov" "lm"``
… we discover that `my.anova` actually has two classes! The first class tells us that it’s an `aov` (analysis of variance) object, but the second tells us that it’s also an `lm` (linear model) object. Later on, we’ll see that this reflects a pretty deep statistical relationship between ANOVA and regression (Chapter 15 and it means that any function that exists in R for dealing with regressions can also be applied to `aov` objects, which is neat; but I’m getting ahead of myself. For now, I want to note that what we’ve created is an `aov` object, and to also make the point that `aov` objects are actually rather complicated beasts. I won’t be trying to explain everything about them, since it’s way beyond the scope of an introductory statistics subject, but to give you a tiny hint of some of the stuff that R stores inside an `aov` object, let’s ask it to print out the `names()` of all the stored quantities…
``names( my.anova )``
``````## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "contrasts" "xlevels" "call" "terms"
## [13] "model"``````
As we go through the rest of the book, I hope that a few of these will become a little more obvious to you, but right now that’s going to look pretty damned opaque. That’s okay. You don’t need to know any of the details about it right now, and most of it you don’t need at all… what you do need to understand is that the `aov()` function does a lot of calculations for you, not just the basic ones that I outlined in the previous sections. What this means is that it’s generally a good idea to create a variable like `my.anova` that stores the output of the `aov()` function… because later on, you can use `my.anova` as an input to lots of other functions: those other functions can pull out bits and pieces from the `aov` object, and calculate various other things that you might need.
Right then. The simplest thing you can do with an `aov` object is to `print()` it out. When we do that, it shows us a few of the key quantities of interest:
``print( my.anova )``
``````## Call:
## aov(formula = mood.gain ~ drug, data = clin.trial)
##
## Terms:
## drug Residuals
## Sum of Squares 3.453333 1.391667
## Deg. of Freedom 2 15
##
## Residual standard error: 0.3045944
## Estimated effects may be unbalanced``````
Specificially, it prints out a reminder of the command that you used when you called `aov()` in the first place, shows you the sums of squares values, the degrees of freedom, and a couple of other quantities that we’re not really interested in right now. Notice, however, that R doesn’t use the names “between-group” and “within-group”. Instead, it tries to assign more meaningful names: in our particular example, the between groups variance corresponds to the effect that the `drug` has on the outcome variable; and the within groups variance is corresponds to the “leftover” variability, so it calls that the residuals. If we compare these numbers to the numbers that I calculated by hand in Section 14.2.5, you can see that they’re identical… the between groups sums of squares is SSb=3.45, the within groups sums of squares is SSw=1.39, and the degrees of freedom are 2 and 15 repectively.
Running the hypothesis tests for the ANOVA
Okay, so we’ve verified that `my.anova` seems to be storing a bunch of the numbers that we’re looking for, but the `print()` function didn’t quite give us the output that we really wanted. Where’s the F-value? The p-value? These are the most important numbers in our hypothesis test, but the `print()` function doesn’t provide them. To get those numbers, we need to use a different function. Instead of asking R to `print()` out the `aov` object, we should have asked for a `summary()` of it.206 When we do that…
``summary( my.anova )``
``````## Df Sum Sq Mean Sq F value Pr(>F)
## drug 2 3.453 1.7267 18.61 8.65e-05 ***
## Residuals 15 1.392 0.0928
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1``````
… we get all of the key numbers that we calculated earlier. We get the sums of squares, the degrees of freedom, the mean squares, the F-statistic, and the p-value itself. These are all identical to the numbers that we calculated ourselves when doing it the long and tedious way, and it’s even organised into the same kind of ANOVA table that I showed in Table 14.1, and then filled out by hand in Section 14.2.5. The only things that are even slightly different is that some of the row and column names are a bit different.
14.05: Effect Size
There’s a few different ways you could measure the effect size in an ANOVA, but the most commonly used measures are η2 (eta squared) and partial η2. For a one way analysis of variance they’re identical to each other, so for the moment I’ll just explain η2. The definition of η2 is actually really simple:
$\eta^{2}=\dfrac{\mathrm{SS}_{b}}{\mathrm{SS}_{t o t}}$
That’s all it is. So when I look at the ANOVA table above, I see that SSb=3.45 and SStot=3.45+1.39=4.84. Thus we get an η2 value of
$\ \eta^2 ={ 3.45 \over 4.84}=0.71$
The interpretation of η2 is equally straightforward: it refers to the proportion of the variability in the outcome variable (mood.gain) that can be explained in terms of the predictor (drug). A value of η2=0 means that there is no relationship at all between the two, whereas a value of η2=1 means that the relationship is perfect. Better yet, the η2 value is very closely related to a squared correlation (i.e., r2). So, if you’re trying to figure out whether a particular value of η2 is big or small, it’s sometimes useful to remember that
$\ \eta = {\sqrt{SS_b \over SS_{tot}}}$
can be interpreted as if it referred to the magnitude of a Pearson correlation. So in our drugs example, the η2 value of .71 corresponds to an η value of $\ \sqrt{.71}$ =.84. If we think about this as being equivalent to a correlation of about .84, we’d conclude that the relationship between drug and mood.gain is strong.
The core packages in R don’t include any functions for calculating η2. However, it’s pretty straightforward to calculate it directly from the numbers in the ANOVA table. In fact, since I’ve already got the SSw and SSb variables lying around from my earlier calculations, I can do this:
SStot <- SSb + SSw # total sums of squares
eta.squared <- SSb / SStot # eta-squared value
print( eta.squared )
## [1] 0.7127623
However, since it can be tedious to do this the long way (especially when we start running more complicated ANOVAs, such as those in Chapter 16 I’ve included an etaSquared() function in the lsr package which will do it for you. For now, the only argument you need to care about is x, which should be the aov object corresponding to your ANOVA. When we do this, what we get as output is this:
etaSquared( x = my.anova )
## eta.sq eta.sq.part
## drug 0.7127623 0.7127623
The output here shows two different numbers. The first one corresponds to the η2 statistic, precisely as described above. The second one refers to “partial η2”, which is a somewhat different measure of effect size that I’ll describe later. For the simple ANOVA that we’ve just run, they’re the same number. But this won’t always be true once we start running more complicated ANOVAs.207 | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.04%3A_Running_an_ANOVA_in_R.txt |
Any time you run an ANOVA with more than two groups, and you end up with a significant effect, the first thing you’ll probably want to ask is which groups are actually different from one another. In our drugs example, our null hypothesis was that all three drugs (placebo, Anxifree and Joyzepam) have the exact same effect on mood. But if you think about it, the null hypothesis is actually claiming three different things all at once here. Specifically, it claims that:
• Your competitor’s drug (Anxifree) is no better than a placebo (i.e., μAP)
• Your drug (Joyzepam) is no better than a placebo (i.e., μJP)
• Anxifree and Joyzepam are equally effective (i.e., μJA)
If any one of those three claims is false, then the null hypothesis is also false. So, now that we’ve rejected our null hypothesis, we’re thinking that at least one of those things isn’t true. But which ones? All three of these propositions are of interest: you certainly want to know if your new drug Joyzepam is better than a placebo, and it would be nice to know how well it stacks up against an existing commercial alternative (i.e., Anxifree). It would even be useful to check the performance of Anxifree against the placebo: even if Anxifree has already been extensively tested against placebos by other researchers, it can still be very useful to check that your study is producing similar results to earlier work.
When we characterise the null hypothesis in terms of these three distinct propositions, it becomes clear that there are eight possible “states of the world” that we need to distinguish between:
possibility: is μPA? is μPJ? is μAJ? which hypothesis?
1 null
2 alternative
3 alternative
4 alternative
5 alternative
6 alternative
7 alternative
8 alternative
By rejecting the null hypothesis, we’ve decided that we don’t believe that #1 is the true state of the world. The next question to ask is, which of the other seven possibilities do we think is right? When faced with this situation, its usually helps to look at the data. For instance, if we look at the plots in Figure 14.1, it’s tempting to conclude that Joyzepam is better than the placebo and better than Anxifree, but there’s no real difference between Anxifree and the placebo. However, if we want to get a clearer answer about this, it might help to run some tests.
Running “pairwise” t-tests
How might we go about solving our problem? Given that we’ve got three separate pairs of means (placebo versus Anxifree, placebo versus Joyzepam, and Anxifree versus Joyzepam) to compare, what we could do is run three separate t-tests and see what happens. There’s a couple of ways that we could do this. One method would be to construct new variables corresponding the groups you want to compare (e.g., `anxifree`, `placebo` and `joyzepam`), and then run a t-test on these new variables:
``````t.test( anxifree, placebo, var.equal = TRUE ) # Student t-test
anxifree <- with(clin.trial, mood.gain[drug == "anxifree"]) # mood change due to anxifree
placebo <- with(clin.trial, mood.gain[drug == "placebo"]) # mood change due to placebo ``````
or, you could use the `subset` argument in the `t.test()` function to select only those observations corresponding to one of the two groups we’re interested in:
``````t.test( formula = mood.gain ~ drug,
data = clin.trial,
subset = drug %in% c("placebo","anxifree"),
var.equal = TRUE
)``````
See Chapter 7 if you’ve forgotten how the `%in%` operator works. Regardless of which version we do, R will print out the results of the t-test, though I haven’t included that output here. If we go on to do this for all possible pairs of variables, we can look to see which (if any) pairs of groups are significantly different to each other. This “lots of t-tests idea” isn’t a bad strategy, though as we’ll see later on there are some problems with it. However, for the moment our bigger problem is that it’s a pain to have to type in such a long command over and over again: for instance, if your experiment has 10 groups, then you have to run 45 t-tests. That’s way too much typing.
To help keep the typing to a minimum, R provides a function called `pairwise.t.test()` that automatically runs all of the t-tests for you. There are three arguments that you need to specify, the outcome variable `x`, the group variable `g`, and the `p.adjust.method` argument, which “adjusts” the p-value in one way or another. I’ll explain p-value adjustment in a moment, but for now we can just set `p.adjust.method = "none"` since we’re not doing any adjustments. For our example, here’s what we do:
``````pairwise.t.test( x = clin.trial\$mood.gain, # outcome variable
g = clin.trial\$drug, # grouping variable
p.adjust.method = "none" # which correction to use?
)```
```
``````##
## Pairwise comparisons using t tests with pooled SD
##
## data: clin.trial\$mood.gain and clin.trial\$drug
##
## placebo anxifree
## anxifree 0.15021 -
## joyzepam 3e-05 0.00056
##
## P value adjustment method: none``````
One thing that bugs me slightly about the `pairwise.t.test()` function is that you can’t just give it an `aov` object, and have it produce this output. After all, I went to all that trouble earlier of getting R to create the `my.anova` variable and – as we saw in Section 14.3.2 – R has actually stored enough information inside it that I should just be able to get it to run all the pairwise tests using `my.anova` as an input. To that end, I’ve included a `posthocPairwiseT()` function in the `lsr` package that lets you do this. The idea behind this function is that you can just input the `aov` object itself,208 and then get the pairwise tests as an output. As of the current writing, `posthocPairwiseT()` is actually just a simple way of calling `pairwise.t.test()` function, but you should be aware that I intend to make some changes to it later on. Here’s an example:
``posthocPairwiseT( x = my.anova, p.adjust.method = "none" )``
``````##
## Pairwise comparisons using t tests with pooled SD
##
## data: mood.gain and drug
##
## placebo anxifree
## anxifree 0.15021 -
## joyzepam 3e-05 0.00056
##
## P value adjustment method: none``````
In later versions, I plan to add more functionality (e.g., adjusted confidence intervals), but for now I think it’s at least kind of useful. To see why, let’s suppose you’ve run your ANOVA and stored the results in `my.anova`, and you’re happy using the Holm correction (the default method in `pairwise.t.test()`, which I’ll explain this in a moment). In that case, all you have to do is type this:
``posthocPairwiseT( my.anova )``
and R will output the test results. Much more convenient, I think.
Corrections for multiple testing
In the previous section I hinted that there’s a problem with just running lots and lots of t-tests. The concern is that when running these analyses, what we’re doing is going on a “fishing expedition”: we’re running lots and lots of tests without much theoretical guidance, in the hope that some of them come up significant. This kind of theory-free search for group differences is referred to as post hoc analysis (“post hoc” being Latin for “after this”).209
It’s okay to run post hoc analyses, but a lot of care is required. For instance, the analysis that I ran in the previous section is actually pretty dangerous: each individual t-test is designed to have a 5% Type I error rate (i.e., α=.05), and I ran three of these tests. Imagine what would have happened if my ANOVA involved 10 different groups, and I had decided to run 45 “post hoc” t-tests to try to find out which ones were significantly different from each other, you’d expect 2 or 3 of them to come up significant by chance alone. As we saw in Chapter 11, the central organising principle behind null hypothesis testing is that we seek to control our Type I error rate, but now that I’m running lots of t-tests at once, in order to determine the source of my ANOVA results, my actual Type I error rate across this whole family of tests has gotten completely out of control.
The usual solution to this problem is to introduce an adjustment to the p-value, which aims to control the total error rate across the family of tests (see Shaffer 1995). An adjustment of this form, which is usually (but not always) applied because one is doing post hoc analysis, is often referred to as a correction for multiple comparisons, though it is sometimes referred to as “simultaneous inference”. In any case, there are quite a few different ways of doing this adjustment. I’ll discuss a few of them in this section and in Section 16.8, but you should be aware that there are many other methods out there (see, e.g., Hsu 1996).
Bonferroni corrections
The simplest of these adjustments is called the Bonferroni correction (Dunn 1961), and it’s very very simple indeed. Suppose that my post hoc analysis consists of m separate tests, and I want to ensure that the total probability of making any Type I errors at all is at most α.210 If so, then the Bonferroni correction just says “multiply all your raw p-values by m”. If we let p denote the original p-value, and let p′j be the corrected value, then the Bonferroni correction tells that:
p′=m×p
And therefore, if you’re using the Bonferroni correction, you would reject the null hypothesis if p′<α. The logic behind this correction is very straightforward. We’re doing m different tests; so if we arrange it so that each test has a Type I error rate of at most α/m, then the total Type I error rate across these tests cannot be larger than α. That’s pretty simple, so much so that in the original paper, the author writes:
The method given here is so simple and so general that I am sure it must have been used before this. I do not find it, however, so can only conclude that perhaps its very simplicity has kept statisticians from realizing that it is a very good method in some situations (pp 52-53 Dunn 1961)
To use the Bonferroni correction in R, you can use the `pairwise.t.test()` function,211 making sure that you set `p.adjust.method = "bonferroni"`. Alternatively, since the whole reason why we’re doing these pairwise tests in the first place is because we have an ANOVA that we’re trying to understand, it’s probably more convenient to use the `posthocPairwiseT()` function in the `lsr` package, since we can use `my.anova` as the input:
``posthocPairwiseT( my.anova, p.adjust.method = "bonferroni")``
``````##
## Pairwise comparisons using t tests with pooled SD
##
## data: mood.gain and drug
##
## placebo anxifree
## anxifree 0.4506 -
## joyzepam 9.1e-05 0.0017
##
## P value adjustment method: bonferroni``````
If we compare these three p-values to those that we saw in the previous section when we made no adjustment at all, it is clear that the only thing that R has done is multiply them by 3.
Holm corrections
Although the Bonferroni correction is the simplest adjustment out there, it’s not usually the best one to use. One method that is often used instead is the Holm correction (Holm 1979). The idea behind the Holm correction is to pretend that you’re doing the tests sequentially; starting with the smallest (raw) p-value and moving onto the largest one. For the j-th largest of the p-values, the adjustment is either
p′j=j×pj
(i.e., the biggest p-value remains unchanged, the second biggest p-value is doubled, the third biggest p-value is tripled, and so on), or
p′j=p′j+1
whichever one is larger. This might sound a little confusing, so let’s go through it a little more slowly. Here’s what the Holm correction does. First, you sort all of your p-values in order, from smallest to largest. For the smallest p-value all you do is multiply it by m, and you’re done. However, for all the other ones it’s a two-stage process. For instance, when you move to the second smallest p value, you first multiply it by m−1. If this produces a number that is bigger than the adjusted p-value that you got last time, then you keep it. But if it’s smaller than the last one, then you copy the last p-value. To illustrate how this works, consider the table below, which shows the calculations of a Holm correction for a collection of five p-values:
raw p rank j p×j Holm p
.001 5 .005 .005
.005 4 .020 .020
.019 3 .057 .057
.022 2 .044 .057
.103 1 .103 .103
Hopefully that makes things clear.
Although it’s a little harder to calculate, the Holm correction has some very nice properties: it’s more powerful than Bonferroni (i.e., it has a lower Type II error rate), but – counterintuitive as it might seem – it has the same Type I error rate. As a consequence, in practice there’s never any reason to use the simpler Bonferroni correction, since it is always outperformed by the slightly more elaborate Holm correction. Because of this, the Holm correction is the default one used by `pairwise.t.test()` and `posthocPairwiseT()`. To run the Holm correction in R, you could specify `p.adjust.method = "Holm"` if you wanted to, but since it’s the default you can just to do this:
``posthocPairwiseT( my.anova )``
``````##
## Pairwise comparisons using t tests with pooled SD
##
## data: mood.gain and drug
##
## placebo anxifree
## anxifree 0.1502 -
## joyzepam 9.1e-05 0.0011
##
## P value adjustment method: holm``````
As you can see, the biggest p-value (corresponding to the comparison between Anxifree and the placebo) is unaltered: at a value of .15, it is exactly the same as the value we got originally when we applied no correction at all. In contrast, the smallest p-value (Joyzepam versus placebo) has been multiplied by three.
Writing up the post hoc test
Finally, having run the post hoc analysis to determine which groups are significantly different to one another, you might write up the result like this:
Post hoc tests (using the Holm correction to adjust p) indicated that Joyzepam produced a significantly larger mood change than both Anxifree (p=.001) and the placebo (p=9.1×10−5). We found no evidence that Anxifree performed better than the placebo (p=.15).
Or, if you don’t like the idea of reporting exact p-values, then you’d change those numbers to p<.01, p<.001 and p>.05 respectively. Either way, the key thing is that you indicate that you used Holm’s correction to adjust the p-values. And of course, I’m assuming that elsewhere in the write up you’ve included the relevant descriptive statistics (i.e., the group means and standard deviations), since these p-values on their own aren’t terribly informative. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.06%3A_Multiple_Comparisons_and_Post_Hoc_Tests.txt |
Like any statistical test, analysis of variance relies on some assumptions about the data. There are three key assumptions that you need to be aware of: normality, homogeneity of variance and independence. If you remember back to Section 14.2.4 – which I hope you at least skimmed even if you didn’t read the whole thing – I described the statistical models underpinning ANOVA, which I wrote down like this:
H0:Yik=μ+ϵik
H1:Yikkik
In these equations μ refers to a single, grand population mean which is the same for all groups, and μk is the population mean for the k-th group. Up to this point we’ve been mostly interested in whether our data are best described in terms of a single grand mean (the null hypothesis) or in terms of different group-specific means (the alternative hypothesis). This makes sense, of course: that’s actually the important research question! However, all of our testing procedures have – implicitly – relied on a specific assumption about the residuals, ϵik, namely that
ϵik∼Normal(0,σ2)
None of the maths works properly without this bit. Or, to be precise, you can still do all the calculations, and you’ll end up with an F-statistic, but you have no guarantee that this F-statistic actually measures what you think it’s measuring, and so any conclusions that you might draw on the basis of the F test might be wrong.
So, how do we check whether this assumption about the residuals is accurate? Well, as I indicated above, there are three distinct claims buried in this one statement, and we’ll consider them separately.
• Normality. The residuals are assumed to be normally distributed. As we saw in Section 13.9, we can assess this by looking at QQ plots or running a Shapiro-Wilk test. I’ll talk about this in an ANOVA context in Section 14.9.
• Homogeneity of variance. Notice that we’ve only got the one value for the population standard deviation (i.e., σ), rather than allowing each group to have it’s own value (i.e., σk). This is referred to as the homogeneity of variance (sometimes called homoscedasticity) assumption. ANOVA assumes that the population standard deviation is the same for all groups. We’ll talk about this extensively in Section 14.7.
• Independence. The independence assumption is a little trickier. What it basically means is that, knowing one residual tells you nothing about any other residual. All of the ϵik values are assumed to have been generated without any “regard for” or “relationship to” any of the other ones. There’s not an obvious or simple way to test for this, but there are some situations that are clear violations of this: for instance, if you have a repeated-measures design, where each participant in your study appears in more than one condition, then independence doesn’t hold; there’s a special relationship between some observations… namely those that correspond to the same person! When that happens, you need to use something like repeated measures ANOVA. I don’t currently talk about repeated measures ANOVA in this book, but it will be included in later versions.
robust is ANOVA?
One question that people often want to know the answer to is the extent to which you can trust the results of an ANOVA if the assumptions are violated. Or, to use the technical language, how robust is ANOVA to violations of the assumptions. Due to deadline constraints I don’t have the time to discuss this topic. This is a topic I’ll cover in some detail in a later version of the book. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.07%3A_Assumptions_of_One-way_ANOVA.txt |
There’s more than one way to skin a cat, as the saying goes, and more than one way to test the homogeneity of variance assumption, too (though for some reason no-one made a saying out of that). The most commonly used test for this that I’ve seen in the literature is the Levene test (Levene 1960), and the closely related Brown-Forsythe test (Brown and Forsythe 1974), both of which I’ll describe here. Alternatively, you could use the Bartlett test, which is implemented in R via the bartlett.test() function, but I’ll leave it as an exercise for the reader to go check that one out if you’re interested.
Levene’s test is shockingly simple. Suppose we have our outcome variable Yik. All we do is define a new variable, which I’ll call Zik, corresponding to the absolute deviation from the group mean:
Zik=|Yik − $\ \bar{Y_k}$|
Okay, what good does this do us? Well, let’s take a moment to think about what Zik actually is, and what we’re trying to test. The value of Zik is a measure of how the i-th observation in the k-th group deviates from its group mean. And our null hypothesis is that all groups have the same variance; that is, the same overall deviations from the group means! So, the null hypothesis in a Levene’s test is that the population means of Z are identical for all groups. Hm. So what we need now is a statistical test of the null hypothesis that all group means are identical. Where have we seen that before? Oh right, that’s what ANOVA is… and so all that the Levene’s test does is run an ANOVA on the new variable Zik.
What about the Brown-Forsythe test? Does that do anything particularly different? Nope. The only change from the Levene’s test is that it constructs the transformed variable Z in a slightly different way, using deviations from the group medians rather than deviations from the group means. That is, for the Brown-Forsythe test,
Zik=|Yik−mediank(Y)|
where mediank(Y) is the median for group k. Regardless of whether you’re doing the standard Levene test or the Brown-Forsythe test, the test statistic – which is sometimes denoted F, but sometimes written as W – is calculated in exactly the same way that the F-statistic for the regular ANOVA is calculated, just using a Zik rather than Yik. With that in mind, let’s just move on and look at how to run the test in R.
Running the Levene’s test in R
Okay, so how do we run the Levene test? Obviously, since the Levene test is just an ANOVA, it would be easy enough to manually create the transformed variable Zik and then use the aov() function to run an ANOVA on that. However, that’s the tedious way to do it. A better way to do run your Levene’s test is to use the leveneTest() function, which is in the car package. As usual, we first load the package
library( car )
## Loading required package: carData
and now that we have, we can run our Levene test. The main argument that you need to specify is y, but you can do this in lots of different ways. Probably the simplest way to do it is actually input the original aov object. Since I’ve got the my.anova variable stored from my original ANOVA, I can just do this:
leveneTest( my.anova )
## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 2 1.4672 0.2618
## 15
If we look at the output, we see that the test is non-significant (F2,15=1.47,p=.26), so it looks like the homogeneity of variance assumption is fine. Remember, although R reports the test statistic as an F-value, it could equally be called W, in which case you’d just write W2,15=1.47. Also, note the part of the output that says center = median. That’s telling you that, by default, the leveneTest() function actually does the Brown-Forsythe test. If you want to use the mean instead, then you need to explicitly set the center argument, like this:
leveneTest( y = my.anova, center = mean )
## Levene's Test for Homogeneity of Variance (center = mean)
## Df F value Pr(>F)
## group 2 1.4497 0.2657
## 15
That being said, in most cases it’s probably best to stick to the default value, since the Brown-Forsythe test is a bit more robust than the original Levene test.
Additional comments
Two more quick comments before I move onto a different topic. Firstly, as mentioned above, there are other ways of calling the leveneTest() function. Although the vast majority of situations that call for a Levene test involve checking the assumptions of an ANOVA (in which case you probably have a variable like my.anova lying around), sometimes you might find yourself wanting to specify the variables directly. Two different ways that you can do this are shown below:
leveneTest(y = mood.gain ~ drug, data = clin.trial) # y is a formula in this case
leveneTest(y = clin.trial$mood.gain, group = clin.trial$drug) # y is the outcome
Secondly, I did mention that it’s possible to run a Levene test just using the aov() function. I don’t want to waste a lot of space on this, but just in case some readers are interested in seeing how this is done, here’s the code that creates the new variables and runs an ANOVA. If you are interested, feel free to run this to verify that it produces the same answers as the Levene test (i.e., with center = mean):
Y <- clin.trial $mood.gain # the original outcome variable, Y G <- clin.trial$ drug # the grouping variable, G
gp.mean <- tapply(Y, G, mean) # calculate group means
Ybar <- gp.mean[G] # group mean associated with each obs
Z <- abs(Y - Ybar) # the transformed variable, Z
summary( aov(Z ~ G) ) # run the ANOVA
## Df Sum Sq Mean Sq F value Pr(>F)
## G 2 0.0616 0.03080 1.45 0.266
## Residuals 15 0.3187 0.02125
That said, I don’t imagine that many people will care about this. Nevertheless, it’s nice to know that you could do it this way if you wanted to. And for those of you who do try it, I think it helps to demystify the test a little bit when you can see – with your own eyes – the way in which Levene’s test relates to ANOVA. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.08%3A_Checking_the_Homogeneity_of_Variance_Assumption.txt |
In our example, the homogeneity of variance assumption turned out to be a pretty safe one: the Levene test came back non-significant, so we probably don’t need to worry. However, in real life we aren’t always that lucky. How do we save our ANOVA when the homogeneity of variance assumption is violated? If you recall from our discussion of t-tests, we’ve seen this problem before. The Student t-test assumes equal variances, so the solution was to use the Welch t-test, which does not. In fact, Welch (1951) also showed how we can solve this problem for ANOVA too (the Welch one-way test). It’s implemented in R using the `oneway.test()` function. The arguments that we’ll need for our example are:
• `formula`. This is the model formula, which (as usual) needs to specify the outcome variable on the left hand side and the grouping variable on the right hand side: i.e., something like `outcome ~ group`.
• `data`. Specifies the data frame containing the variables.
• `var.equal`. If this is `FALSE` (the default) a Welch one-way test is run. If it is `TRUE` then it just runs a regular ANOVA.
The function also has a `subset` argument that lets you analyse only some of the observations and a `na.action` argument that tells it how to handle missing data, but these aren’t necessary for our purposes. So, to run the Welch one-way ANOVA for our example, we would do this:
``oneway.test(mood.gain ~ drug, data = clin.trial)``
``````##
## One-way analysis of means (not assuming equal variances)
##
## data: mood.gain and drug
## F = 26.322, num df = 2.0000, denom df = 9.4932, p-value = 0.000134``````
To understand what’s happening here, let’s compare these numbers to what we got earlier in Section 14.3 when we ran our original ANOVA. To save you the trouble of flicking back, here are those numbers again, this time calculated by setting `var.equal = TRUE` for the `oneway.test()` function:
``oneway.test(mood.gain ~ drug, data = clin.trial, var.equal = TRUE)``
``````##
## One-way analysis of means
##
## data: mood.gain and drug
## F = 18.611, num df = 2, denom df = 15, p-value = 8.646e-05``````
Okay, so originally our ANOVA gave us the result F(2,15)=18.6, whereas the Welch one-way test gave us F(2,9.49)=26.32. In other words, the Welch test has reduced the within-groups degrees of freedom from 15 to 9.49, and the F-value has increased from 18.6 to 26.32.
14.10: Checking the Normality Assumption
Testing the normality assumption is relatively straightforward. We covered most of what you need to know in Section 13.9. The only thing we really need to know how to do is pull out the residuals (i.e., the ϵik values) so that we can draw our QQ plot and run our Shapiro-Wilk test. First, let’s extract the residuals. R provides a function called `residuals()` that will do this for us. If we pass our `my.anova` to this function, it will return the residuals. So let’s do that:
``my.anova.residuals <- residuals( object = my.anova ) # extract the residuals``
We can print them out too, though it’s not exactly an edifying experience. In fact, given that I’m on the verge of putting myself to sleep just typing this, it might be a good idea to skip that step. Instead, let’s draw some pictures and run ourselves a hypothesis test:
``hist( x = my.anova.residuals ) # plot a histogram (similar to Figure @ref{fig:normalityanova}a)``
``qqnorm( y = my.anova.residuals ) # draw a QQ plot (similar to Figure @ref{fig:normalityanova}b)``
``shapiro.test( x = my.anova.residuals ) # run Shapiro-Wilk test``
``````##
## Shapiro-Wilk normality test
##
## data: my.anova.residuals
## W = 0.96019, p-value = 0.6053``````
The histogram and QQ plot are both look pretty normal to me.212 This is supported by the results of our Shapiro-Wilk test (W=.96, p=.61) which finds no indication that normality is violated. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.09%3A_Removing_the_Homogeneity_of_Variance_Assumption.txt |
Now that we’ve seen how to check for normality, we are led naturally to ask what we can do to address violations of normality. In the context of a one-way ANOVA, the easiest solution is probably to switch to a non-parametric test (i.e., one that doesn’t rely on any particular assumption about the kind of distribution involved). We’ve seen non-parametric tests before, in Chapter 13: when you only have two groups, the Wilcoxon test provides the non-parametric alternative that you need. When you’ve got three or more groups, you can use the Kruskal-Wallis rank sum test (Kruskal and Wallis 1952). So that’s the test we’ll talk about next.
logic behind the Kruskal-Wallis test
The Kruskal-Wallis test is surprisingly similar to ANOVA, in some ways. In ANOVA, we started with Yik, the value of the outcome variable for the ith person in the kth group. For the Kruskal-Wallis test, what we’ll do is rank order all of these Yik values, and conduct our analysis on the ranked data. So let’s let Rik refer to the ranking given to the ith member of the kth group. Now, let’s calculate $\ \bar{R_k}$, the average rank given to observations in the kth group:
$\bar{R}_{k}=\dfrac{1}{N_{K}} \sum_{i} R_{i k}$
and let’s also calculate $\ \bar{R}$, the grand mean rank:
$\bar{R}=\dfrac{1}{N} \sum_{i} \sum_{k} R_{i k}$
Now that we’ve done this, we can calculate the squared deviations from the grand mean rank $\ \bar{R}$. When we do this for the individual scores – i.e., if we calculate ($\ R_{ik} - \bar{R})^2$ – what we have is a “nonparametric” measure of how far the ik-th observation deviates from the grand mean rank. When we calculate the squared deviation of the group means from the grand means – i.e., if we calculate ($\ \bar{R_k} - \bar{R})^2$ – then what we have is a nonparametric measure of how much the group deviates from the grand mean rank. With this in mind, let’s follow the same logic that we did with ANOVA, and define our ranked sums of squares measures in much the same way that we did earlier. First, we have our “total ranked sums of squares”:
$\mathrm{RSS}_{t o t}=\sum_{k} \sum_{i}\left(R_{i k}-R\right)^{2}$
and we can define the “between groups ranked sums of squares” like this:
\begin{aligned} \mathrm{RSS}_{b} &=\sum_{k} \sum_{i}\left(\bar{R}_{k}-\bar{R}\right)^{2} \ &=\sum_{k} N_{k}\left(\bar{R}_{k}-\bar{R}\right)^{2} \end{aligned}
So, if the null hypothesis is true and there are no true group differences at all, you’d expect the between group rank sums RSSb to be very small, much smaller than the total rank sums RSStot. Qualitatively this is very much the same as what we found when we went about constructing the ANOVA F-statistic; but for technical reasons the Kruskal-Wallis test statistic, usually denoted K, is constructed in a slightly different way:
$\ K = (N-1)\ x\ {RSS_b \over RSS_{tot}}$
and, if the null hypothesis is true, then the sampling distribution of K is approximately chi-square with G−1 degrees of freedom (where G is the number of groups). The larger the value of K, the less consistent the data are with null hypothesis, so this is a one-sided test: we reject H0 when K is sufficiently large.
Additional details
The description in the previous section illustrates the logic behind the Kruskal-Wallis test. At a conceptual level, this is the right way to think about how the test works. However, from a purely mathematical perspective it’s needlessly complicated. I won’t show you the derivation, but you can use a bit of algebraic jiggery-pokery213 to show that the equation for K can be rewritten as
$K=\dfrac{12}{N(N-1)} \sum_{k} N_{k} \bar{R}_{k}^{2}-3(N+1)$
It’s this last equation that you sometimes see given for K. This is way easier to calculate than the version I described in the previous section, it’s just that it’s totally meaningless to actual humans. It’s probably best to think of K the way I described it earlier… as an analogue of ANOVA based on ranks. But keep in mind that the test statistic that gets calculated ends up with a rather different look to it than the one we used for our original ANOVA.
But wait, there’s more! Dear lord, why is there always more? The story I’ve told so far is only actually true when there are no ties in the raw data. That is, if there are no two observations that have exactly the same value. If there are ties, then we have to introduce a correction factor to these calculations. At this point I’m assuming that even the most diligent reader has stopped caring (or at least formed the opinion that the tie-correction factor is something that doesn’t require their immediate attention). So I’ll very quickly tell you how it’s calculated, and omit the tedious details about why it’s done this way. Suppose we construct a frequency table for the raw data, and let fj be the number of observations that have the j-th unique value. This might sound a bit abstract, so here’s the R code showing a concrete example:
f <- table( clin.trial$mood.gain ) # frequency table for mood gain print(f) # we have some ties ## ## 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.1 1.2 1.3 1.4 1.7 1.8 ## 1 1 2 1 1 2 1 1 1 1 2 2 1 1 Looking at this table, notice that the third entry in the frequency table has a value of 2. Since this corresponds to a mood.gain of 0.3, this table is telling us that two people’s mood increased by 0.3. More to the point, note that we can say that f[3] has a value of 2. Or, in the mathematical notation I introduced above, this is telling us that f3=2. Yay. So, now that we know this, the tie correction factor (TCF) is: $\mathrm{TCF}=1-\dfrac{\sum_{j} f_{j}^{3}-f_{j}}{N^{3}-N}$ The tie-corrected value of the Kruskal-Wallis statistic obtained by dividing the value of K by this quantity: it is this tie-corrected version that R calculates. And at long last, we’re actually finished with the theory of the Kruskal-Wallis test. I’m sure you’re all terribly relieved that I’ve cured you of the existential anxiety that naturally arises when you realise that you don’t know how to calculate the tie-correction factor for the Kruskal-Wallis test. Right? run the Kruskal-Wallis test in R Despite the horror that we’ve gone through in trying to understand what the Kruskal-Wallis test actually does, it turns out that running the test is pretty painless, since R has a function called kruskal.test(). The function is pretty flexible, and allows you to input your data in a few different ways. Most of the time you’ll have data like the clin.trial data set, in which you have your outcome variable mood.gain, and a grouping variable drug. If so, you can call the kruskal.test() function by specifying a formula, and a data frame: kruskal.test(mood.gain ~ drug, data = clin.trial) ## ## Kruskal-Wallis rank sum test ## ## data: mood.gain by drug ## Kruskal-Wallis chi-squared = 12.076, df = 2, p-value = 0.002386 A second way of using the kruskal.test() function, which you probably won’t have much reason to use, is to directly specify the outcome variable and the grouping variable as separate input arguments, x and g: kruskal.test(x = clin.trial$mood.gain, g = clin.trial$drug) ## ## Kruskal-Wallis rank sum test ## ## data: clin.trial$mood.gain and clin.trial\$drug
## Kruskal-Wallis chi-squared = 12.076, df = 2, p-value = 0.002386
This isn’t very interesting, since it’s just plain easier to specify a formula. However, sometimes it can be useful to specify x as a list. What I mean is this. Suppose you actually had data as three separate variables, placebo, anxifree and joyzepam. If that’s the format that your data are in, then it’s convenient to know that you can bundle all three together as a list:
mood.gain <- list( placebo, joyzepam, anxifree )
kruskal.test( x = mood.gain )
And again, this would give you exactly the same results as the command we tried originally.
14.12: On the Relationship Between ANOVA and the Student t
There’s one last thing I want to point out before finishing. It’s something that a lot of people find kind of surprising, but it’s worth knowing about: an ANOVA with two groups is identical to the Student t-test. No, really. It’s not just that they are similar, but they are actually equivalent in every meaningful way. I won’t try to prove that this is always true, but I will show you a single concrete demonstration. Suppose that, instead of running an ANOVA on our `mood.gain ~ drug` model, let’s instead do it using `therapy` as the predictor. If we run this ANOVA, here’s what we get:
``summary( aov( mood.gain ~ therapy, data = clin.trial ))``
``````## Df Sum Sq Mean Sq F value Pr(>F)
## therapy 1 0.467 0.4672 1.708 0.21
## Residuals 16 4.378 0.2736``````
Overall, it looks like there’s no significant effect here at all but, as we’ll see in Chapter @ref(anova2 this is actually a misleading answer! In any case, it’s irrelevant to our current goals: our interest here is in the F-statistic, which is F(1,16)=1.71, and the p-value, which is .21. Since we only have two groups, I didn’t actually need to resort to an ANOVA, I could have just decided to run a Student t-test. So let’s see what happens when I do that:
``t.test( mood.gain ~ therapy, data = clin.trial, var.equal = TRUE )``
``````##
## Two Sample t-test
##
## data: mood.gain by therapy
## t = -1.3068, df = 16, p-value = 0.2098
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.8449518 0.2005073
## sample estimates:
## mean in group no.therapy mean in group CBT
## 0.7222222 1.0444444``````
Curiously, the p-values are identical: once again we obtain a value of p=.21. But what about the test statistic? Having run a t-test instead of an ANOVA, we get a somewhat different answer, namely t(16)=−1.3068. However, there is a fairly straightforward relationship here. If we square the t-statistic
``1.3068 ^ 2``
``## [1] 1.707726``
we get the F-statistic from before. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/14%3A_Comparing_Several_Means_(One-way_ANOVA)/14.11%3A_Removing_the_Normality_Assumption.txt |
The goal in this chapter is to introduce linear regression, the standard tool that statisticians rely on when analysing the relationship between interval scale predictors and interval scale outcomes. Stripped to its bare essentials, linear regression models are basically a slightly fancier version of the Pearson correlation (Section 5.7) though as we’ll see, regression models are much more powerful tools.
15: Linear Regression
Since the basic ideas in regression are closely tied to correlation, we’ll return to the parenthood.Rdata file that we were using to illustrate how correlations work. Recall that, in this data set, we were trying to find out why Dan is so very grumpy all the time, and our working hypothesis was that I’m not getting enough sleep. We drew some scatterplots to help us examine the relationship between the amount of sleep I get, and my grumpiness the following day. The actual scatterplot that we draw is the one shown in Figure 15.1, and as we saw previously this corresponds to a correlation of r=−.90, but what we find ourselves secretly imagining is something that looks closer to Figure 15.2. That is, we mentally draw a straight line through the middle of the data. In statistics, this line that we’re drawing is called a regression line. Notice that – since we’re not idiots – the regression line goes through the middle of the data. We don’t find ourselves imagining anything like the rather silly plot shown in Figure 15.3.
This is not highly surprising: the line that I’ve drawn in Figure 15.3 doesn’t “fit” the data very well, so it doesn’t make a lot of sense to propose it as a way of summarising the data, right? This is a very simple observation to make, but it turns out to be very powerful when we start trying to wrap just a little bit of maths around it. To do so, let’s start with a refresher of some high school maths. The formula for a straight line is usually written like this:
y=mx+c
Or, at least, that’s what it was when I went to high school all those years ago. The two variables are x and y, and we have two coefficients, m and c. The coefficient m represents the slope of the line, and the coefficient c represents the y-intercept of the line. Digging further back into our decaying memories of high school (sorry, for some of us high school was a long time ago), we remember that the intercept is interpreted as “the value of y that you get when x=0”. Similarly, a slope of m means that if you increase the x-value by 1 unit, then the y-value goes up by m units; a negative slope means that the y-value would go down rather than up. Ah yes, it’s all coming back to me now.
Now that we’ve remembered that, it should come as no surprise to discover that we use the exact same formula to describe a regression line. If Y is the outcome variable (the DV) and X is the predictor variable (the IV), then the formula that describes our regression is written like this:
$\ \hat{Y_i} = b_1X_i + b_0$
Hm. Looks like the same formula, but there’s some extra frilly bits in this version. Let’s make sure we understand them. Firstly, notice that I’ve written Xi and Yi rather than just plain old X and Y. This is because we want to remember that we’re dealing with actual data. In this equation, Xi is the value of predictor variable for the ith observation (i.e., the number of hours of sleep that I got on day i of my little study), and Yi is the corresponding value of the outcome variable (i.e., my grumpiness on that day). And although I haven’t said so explicitly in the equation, what we’re assuming is that this formula works for all observations in the data set (i.e., for all i). Secondly, notice that I wrote $\ \hat{Y_i}$ and not Yi. This is because we want to make the distinction between the actual data Yi, and the estimate $\ \hat{Y_i}$ (i.e., the prediction that our regression line is making). Thirdly, I changed the letters used to describe the coefficients from m and c to b1 and b0. That’s just the way that statisticians like to refer to the coefficients in a regression model. I’ve no idea why they chose b, but that’s what they did. In any case b0 always refers to the intercept term, and b1 refers to the slope.
Excellent, excellent. Next, I can’t help but notice that – regardless of whether we’re talking about the good regression line or the bad one – the data don’t fall perfectly on the line. Or, to say it another way, the data Yi are not identical to the predictions of the regression model $\ \hat{Y_i}$. Since statisticians love to attach letters, names and numbers to everything, let’s refer to the difference between the model prediction and that actual data point as a residual, and we’ll refer to it as ϵi.214 Written using mathematics, the residuals are defined as:
$\ \epsilon_i = Y_i - \hat{Y_i}$
which in turn means that we can write down the complete linear regression model as:
$\ Y_i = b_1X_i + b_0 + \epsilon_i$ | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.01%3A_What_Is_a_Linear_Regression_Model.txt |
Okay, now let’s redraw our pictures, but this time I’ll add some lines to show the size of the residual for all observations. When the regression line is good, our residuals (the lengths of the solid black lines) all look pretty small, as shown in Figure 15.4, but when the regression line is a bad one, the residuals are a lot larger, as you can see from looking at Figure 15.5. Hm. Maybe what we “want” in a regression model is small residuals. Yes, that does seem to make sense. In fact, I think I’ll go so far as to say that the “best fitting” regression line is the one that has the smallest residuals. Or, better yet, since statisticians seem to like to take squares of everything why not say that …
The estimated regression coefficients, $\ \hat{b_0}$ and $\hat{b_1}$ are those that minimise the sum of the squared residuals, which we could either write as $\sum_{i}\left(Y_{i}-\hat{Y}_{i}\right)^{2}$ or as $\sum_{i} \epsilon_{i}^{2}$.
Yes, yes that sounds even better. And since I’ve indented it like that, it probably means that this is the right answer. And since this is the right answer, it’s probably worth making a note of the fact that our regression coefficients are estimates (we’re trying to guess the parameters that describe a population!), which is why I’ve added the little hats, so that we get $\ \hat{b_0}$ and $\ \hat{b_1}$ rather than b0 and b1. Finally, I should also note that – since there’s actually more than one way to estimate a regression model – the more technical name for this estimation process is ordinary least squares (OLS) regression.
At this point, we now have a concrete definition for what counts as our “best” choice of regression coefficients, $\ \hat{b_0}$ and $\ \hat{b_1}$. The natural question to ask next is, if our optimal regression coefficients are those that minimise the sum squared residuals, how do we find these wonderful numbers? The actual answer to this question is complicated, and it doesn’t help you understand the logic of regression.215 As a result, this time I’m going to let you off the hook. Instead of showing you how to do it the long and tedious way first, and then “revealing” the wonderful shortcut that R provides you with, let’s cut straight to the chase… and use the lm() function (short for “linear model”) to do all the heavy lifting.
Using the lm() function
The lm() function is a fairly complicated one: if you type ?lm, the help files will reveal that there are a lot of arguments that you can specify, and most of them won’t make a lot of sense to you. At this stage however, there’s really only two of them that you care about, and as it turns out you’ve seen them before:
• formula. A formula that specifies the regression model. For the simple linear regression models that we’ve talked about so far, in which you have a single predictor variable as well as an intercept term, this formula is of the form outcome ~ predictor. However, more complicated formulas are allowed, and we’ll discuss them later.
• data. The data frame containing the variables.
As we saw with aov() in Chapter 14, the output of the lm() function is a fairly complicated object, with quite a lot of technical information buried under the hood. Because this technical information is used by other functions, it’s generally a good idea to create a variable that stores the results of your regression. With this in mind, to run my linear regression, the command I want to use is this:
regression.1 <- lm( formula = dan.grump ~ dan.sleep,
data = parenthood )
Note that I used dan.grump ~ dan.sleep as the formula: in the model that I’m trying to estimate, dan.grump is the outcome variable, and dan.sleep is the predictor variable. It’s always a good idea to remember which one is which! Anyway, what this does is create an “lm object” (i.e., a variable whose class is "lm") called regression.1. Let’s have a look at what happens when we print() it out:
print( regression.1 )
##
## Call:
## lm(formula = dan.grump ~ dan.sleep, data = parenthood)
##
## Coefficients:
## (Intercept) dan.sleep
## 125.956 -8.937
This looks promising. There’s two separate pieces of information here. Firstly, R is politely reminding us what the command was that we used to specify the model in the first place, which can be helpful. More importantly from our perspective, however, is the second part, in which R gives us the intercept $\ \hat{b_0}$ =125.96 and the slope $\ \hat{b_1}$ =−8.94. In other words, the best-fitting regression line that I plotted in Figure 15.2 has this formula:
$\ \hat{Y_i} = -8.94 \ X_i + 125.96$
Interpreting the estimated model
The most important thing to be able to understand is how to interpret these coefficients. Let’s start with $\ \hat{b_1}$, the slope. If we remember the definition of the slope, a regression coefficient of $\ \hat{b_1}$ =−8.94 means that if I increase Xi by 1, then I’m decreasing Yi by 8.94. That is, each additional hour of sleep that I gain will improve my mood, reducing my grumpiness by 8.94 grumpiness points. What about the intercept? Well, since $\ \hat{b_0}$ corresponds to “the expected value of Yi when Xi equals 0”, it’s pretty straightforward. It implies that if I get zero hours of sleep (Xi=0) then my grumpiness will go off the scale, to an insane value of (Yi=125.96). Best to be avoided, I think. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.02%3A_Estimating_a_Linear_Regression_Model.txt |
The simple linear regression model that we’ve discussed up to this point assumes that there’s a single predictor variable that you’re interested in, in this case dan.sleep. In fact, up to this point, every statistical tool that we’ve talked about has assumed that your analysis uses one predictor variable and one outcome variable. However, in many (perhaps most) research projects you actually have multiple predictors that you want to examine. If so, it would be nice to be able to extend the linear regression framework to be able to include multiple predictors. Perhaps some kind of multiple regression model would be in order?
Multiple regression is conceptually very simple. All we do is add more terms to our regression equation. Let’s suppose that we’ve got two variables that we’re interested in; perhaps we want to use both dan.sleep and baby.sleep to predict the dan.grump variable. As before, we let Yi refer to my grumpiness on the i-th day. But now we have two X variables: the first corresponding to the amount of sleep I got and the second corresponding to the amount of sleep my son got. So we’ll let Xi1 refer to the hours I slept on the i-th day, and Xi2 refers to the hours that the baby slept on that day. If so, then we can write our regression model like this:
Yi = b2 Xi2 + b1 Xi1 + b0 + ϵi
As before, ϵi is the residual associated with the i-th observation, $\ \epsilon_i = Y_i - \hat{Y_i}$. In this model, we now have three coefficients that need to be estimated: b0 is the intercept, b1 is the coefficient associated with my sleep, and b2 is the coefficient associated with my son’s sleep. However, although the number of coefficients that need to be estimated has changed, the basic idea of how the estimation works is unchanged: our estimated coefficients $\ \hat{b_0}$, $\ \hat{b_1}$ and $\ \hat{b_2}$ are those that minimise the sum squared residuals.
Doing it in R
Multiple regression in R is no different to simple regression: all we have to do is specify a more complicated formula when using the lm() function. For example, if we want to use both dan.sleep and baby.sleep as predictors in our attempt to explain why I’m so grumpy, then the formula we need is this:
dan.grump ~ dan.sleep + baby.sleep
Notice that, just like last time, I haven’t explicitly included any reference to the intercept term in this formula; only the two predictor variables and the outcome. By default, the lm() function assumes that the model should include an intercept (though you can get rid of it if you want). In any case, I can create a new regression model – which I’ll call regression.2 – using the following command:
regression.2 <- lm( formula = dan.grump ~ dan.sleep + baby.sleep,
data = parenthood )
And just like last time, if we print() out this regression model we can see what the estimated regression coefficients are:
print( regression.2 )
##
## Call:
## lm(formula = dan.grump ~ dan.sleep + baby.sleep, data = parenthood)
##
## Coefficients:
## (Intercept) dan.sleep baby.sleep
## 125.96557 -8.95025 0.01052
The coefficient associated with dan.sleep is quite large, suggesting that every hour of sleep I lose makes me a lot grumpier. However, the coefficient for baby.sleep is very small, suggesting that it doesn’t really matter how much sleep my son gets; not really. What matters as far as my grumpiness goes is how much sleep I get. To get a sense of what this multiple regression model looks like, Figure 15.6 shows a 3D plot that plots all three variables, along with the regression model itself.
Formula for the general case
The equation that I gave above shows you what a multiple regression model looks like when you include two predictors. Not surprisingly, then, if you want more than two predictors all you have to do is add more X terms and more b coefficients. In other words, if you have K predictor variables in the model then the regression equation looks like this:
$Y_{i}=\left(\sum_{k=1}^{K} b_{k} X_{i k}\right)+b_{0}+\epsilon_{i}$ | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.03%3A_Multiple_Linear_Regression.txt |
So we now know how to estimate the coefficients of a linear regression model. The problem is, we don’t yet know if this regression model is any good. For example, the regression.1 model claims that every hour of sleep will improve my mood by quite a lot, but it might just be rubbish. Remember, the regression model only produces a prediction $\ \hat{Y_i}$ about what my mood is like: my actual mood is Yi. If these two are very close, then the regression model has done a good job. If they are very different, then it has done a bad job.
R2 value
Once again, let’s wrap a little bit of mathematics around this. Firstly, we’ve got the sum of the squared residuals:
$\ SS_{res} = \sum_i(Y_i - \hat{Y_i})^2$
which we would hope to be pretty small. Specifically, what we’d like is for it to be very small in comparison to the total variability in the outcome variable,
$\ SS_{tot} = \sum_i(Y_i - \bar{Y})^2$
While we’re here, let’s calculate these values in R. Firstly, in order to make my R commands look a bit more similar to the mathematical equations, I’ll create variables X and Y:
X <- parenthood$dan.sleep # the predictor Y <- parenthood$dan.grump # the outcome
Now that we’ve done this, let’s calculate the $\ \hat{Y}$ values and store them in a variable called Y.pred. For the simple model that uses only a single predictor, regression.1, we would do the following:
Y.pred <- -8.94 * X + 125.97
Okay, now that we’ve got a variable which stores the regression model predictions for how grumpy I will be on any given day, let’s calculate our sum of squared residuals. We would do that using the following command:
SS.resid <- sum( (Y - Y.pred)^2 )
print( SS.resid )
## [1] 1838.722
Wonderful. A big number that doesn’t mean very much. Still, let’s forge boldly onwards anyway, and calculate the total sum of squares as well. That’s also pretty simple:
SS.tot <- sum( (Y - mean(Y))^2 )
print( SS.tot )
## [1] 9998.59
Hm. Well, it’s a much bigger number than the last one, so this does suggest that our regression model was making good predictions. But it’s not very interpretable.
Perhaps we can fix this. What we’d like to do is to convert these two fairly meaningless numbers into one number. A nice, interpretable number, which for no particular reason we’ll call R2. What we would like is for the value of R2 to be equal to 1 if the regression model makes no errors in predicting the data. In other words, if it turns out that the residual errors are zero – that is, if SSres=0 – then we expect R2=1. Similarly, if the model is completely useless, we would like R2 to be equal to 0. What do I mean by “useless”? Tempting as it is demand that the regression model move out of the house, cut its hair and get a real job, I’m probably going to have to pick a more practical definition: in this case, all I mean is that the residual sum of squares is no smaller than the total sum of squares, SSres=SStot. Wait, why don’t we do exactly that? The formula that provides us with out R2 value is pretty simple to write down,
$\ R^2 ={ 1 -{ SS_{res} \over SS_{tot} }}$
and equally simple to calculate in R:
R.squared <- 1 - (SS.resid / SS.tot)
print( R.squared )
## [1] 0.8161018
The R2 value, sometimes called the coefficient of determination216 has a simple interpretation: it is the proportion of the variance in the outcome variable that can be accounted for by the predictor. So in this case, the fact that we have obtained R2=.816 means that the predictor (my.sleep) explains 81.6% of the variance in the outcome (my.grump).
Naturally, you don’t actually need to type in all these commands yourself if you want to obtain the R2 value for your regression model. As we’ll see later on in Section 15.5.3, all you need to do is use the summary() function. However, let’s put that to one side for the moment. There’s another property of R2 that I want to point out.
relationship between regression and correlation
At this point we can revisit my earlier claim that regression, in this very simple form that I’ve discussed so far, is basically the same thing as a correlation. Previously, we used the symbol r to denote a Pearson correlation. Might there be some relationship between the value of the correlation coefficient r and the R2 value from linear regression? Of course there is: the squared correlation r2 is identical to the R2 value for a linear regression with only a single predictor. To illustrate this, here’s the squared correlation:
r <- cor(X, Y) # calculate the correlation
print( r^2 ) # print the squared correlation
## [1] 0.8161027
Yep, same number. In other words, running a Pearson correlation is more or less equivalent to running a linear regression model that uses only one predictor variable.
adjusted R2 value
One final thing to point out before moving on. It’s quite common for people to report a slightly different measure of model performance, known as “adjusted R2”. The motivation behind calculating the adjusted R2 value is the observation that adding more predictors into the model will always call the R2 value to increase (or at least not decrease). The adjusted R2 value introduces a slight change to the calculation, as follows. For a regression model with K predictors, fit to a data set containing N observations, the adjusted R2 is:
$\operatorname{adj} . R^{2}=1-\left(\dfrac{\mathrm{SS}_{r e s}}{\mathrm{SS}_{t o t}} \times \dfrac{N-1}{N-K-1}\right)$
This adjustment is an attempt to take the degrees of freedom into account. The big advantage of the adjusted R2 value is that when you add more predictors to the model, the adjusted R2 value will only increase if the new variables improve the model performance more than you’d expect by chance. The big disadvantage is that the adjusted R2 value can’t be interpreted in the elegant way that R2 can. R2 has a simple interpretation as the proportion of variance in the outcome variable that is explained by the regression model; to my knowledge, no equivalent interpretation exists for adjusted R2.
An obvious question then, is whether you should report R2 or adjusted R2. This is probably a matter of personal preference. If you care more about interpretability, then R2 is better. If you care more about correcting for bias, then adjusted R2 is probably better. Speaking just for myself, I prefer R2: my feeling is that it’s more important to be able to interpret your measure of model performance. Besides, as we’ll see in Section 15.5, if you’re worried that the improvement in R2 that you get by adding a predictor is just due to chance and not because it’s a better model, well, we’ve got hypothesis tests for that. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.04%3A_Quantifying_the_Fit_of_the_Regression_Model.txt |
So far we’ve talked about what a regression model is, how the coefficients of a regression model are estimated, and how we quantify the performance of the model (the last of these, incidentally, is basically our measure of effect size). The next thing we need to talk about is hypothesis tests. There are two different (but related) kinds of hypothesis tests that we need to talk about: those in which we test whether the regression model as a whole is performing significantly better than a null model; and those in which we test whether a particular regression coefficient is significantly different from zero.
At this point, you’re probably groaning internally, thinking that I’m going to introduce a whole new collection of tests. You’re probably sick of hypothesis tests by now, and don’t want to learn any new ones. Me too. I’m so sick of hypothesis tests that I’m going to shamelessly reuse the F-test from Chapter 14 and the t-test from Chapter 13. In fact, all I’m going to do in this section is show you how those tests are imported wholesale into the regression framework.
Testing the model as a whole
Okay, suppose you’ve estimated your regression model. The first hypothesis test you might want to try is one in which the null hypothesis that there is no relationship between the predictors and the outcome, and the alternative hypothesis is that the data are distributed in exactly the way that the regression model predicts. Formally, our “null model” corresponds to the fairly trivial “regression” model in which we include 0 predictors, and only include the intercept term b0
H0:Yi=b0i
If our regression model has K predictors, the “alternative model” is described using the usual formula for a multiple regression model:
$H_{1}: Y_{i}=\left(\sum_{k=1}^{K} b_{k} X_{i k}\right)+b_{0}+\epsilon_{i}$
How can we test these two hypotheses against each other? The trick is to understand that just like we did with ANOVA, it’s possible to divide up the total variance SStot into the sum of the residual variance SSres and the regression model variance SSmod. I’ll skip over the technicalities, since we covered most of them in the ANOVA chapter, and just note that:
SSmod=SStot−SSres
And, just like we did with the ANOVA, we can convert the sums of squares in to mean squares by dividing by the degrees of freedom.
$\mathrm{MS}_{m o d}=\dfrac{\mathrm{SS}_{m o d}}{d f_{m o d}}$
$\mathrm{MS}_{r e s}=\dfrac{\mathrm{SS}_{r e s}}{d f_{r e s}}$
So, how many degrees of freedom do we have? As you might expect, the df associated with the model is closely tied to the number of predictors that we’ve included. In fact, it turns out that dfmod=K. For the residuals, the total degrees of freedom is dfres=N−K−1.
$\ F={MS_{mod} \over MS_{res}}$
and the degrees of freedom associated with this are K and N−K−1. This F statistic has exactly the same interpretation as the one we introduced in Chapter 14. Large F values indicate that the null hypothesis is performing poorly in comparison to the alternative hypothesis. And since we already did some tedious “do it the long way” calculations back then, I won’t waste your time repeating them. In a moment I’ll show you how to do the test in R the easy way, but first, let’s have a look at the tests for the individual regression coefficients.
Tests for individual coefficients
The F-test that we’ve just introduced is useful for checking that the model as a whole is performing better than chance. This is important: if your regression model doesn’t produce a significant result for the F-test then you probably don’t have a very good regression model (or, quite possibly, you don’t have very good data). However, while failing this test is a pretty strong indicator that the model has problems, passing the test (i.e., rejecting the null) doesn’t imply that the model is good! Why is that, you might be wondering? The answer to that can be found by looking at the coefficients for the regression.2 model:
print( regression.2 )
##
## Call:
## lm(formula = dan.grump ~ dan.sleep + baby.sleep, data = parenthood)
##
## Coefficients:
## (Intercept) dan.sleep baby.sleep
## 125.96557 -8.95025 0.01052
I can’t help but notice that the estimated regression coefficient for the baby.sleep variable is tiny (0.01), relative to the value that we get for dan.sleep (-8.95). Given that these two variables are absolutely on the same scale (they’re both measured in “hours slept”), I find this suspicious. In fact, I’m beginning to suspect that it’s really only the amount of sleep that I get that matters in order to predict my grumpiness.
Once again, we can reuse a hypothesis test that we discussed earlier, this time the t-test. The test that we’re interested has a null hypothesis that the true regression coefficient is zero (b=0), which is to be tested against the alternative hypothesis that it isn’t (b≠0). That is:
H0: b=0
H1: b≠0
How can we test this? Well, if the central limit theorem is kind to us, we might be able to guess that the sampling distribution of $\ \hat{b}$, the estimated regression coefficient, is a normal distribution with mean centred on b. What that would mean is that if the null hypothesis were true, then the sampling distribution of $\ \hat{b}$ has mean zero and unknown standard deviation. Assuming that we can come up with a good estimate for the standard error of the regression coefficient, SE ($\ \hat{b}$), then we’re in luck. That’s exactly the situation for which we introduced the one-sample t way back in Chapter 13. So let’s define a t-statistic like this,
$\ t = { \hat{b} \over SE(\hat{b})}$
I’ll skip over the reasons why, but our degrees of freedom in this case are df=N−K−1. Irritatingly, the estimate of the standard error of the regression coefficient, SE($\ \hat{b}$), is not as easy to calculate as the standard error of the mean that we used for the simpler t-tests in Chapter 13. In fact, the formula is somewhat ugly, and not terribly helpful to look at. For our purposes it’s sufficient to point out that the standard error of the estimated regression coefficient depends on both the predictor and outcome variables, and is somewhat sensitive to violations of the homogeneity of variance assumption (discussed shortly).
In any case, this t-statistic can be interpreted in the same way as the t-statistics that we discussed in Chapter 13. Assuming that you have a two-sided alternative (i.e., you don’t really care if b>0 or b<0), then it’s the extreme values of t (i.e., a lot less than zero or a lot greater than zero) that suggest that you should reject the null hypothesis.
Running the hypothesis tests in R
To compute all of the quantities that we have talked about so far, all you need to do is ask for a summary() of your regression model. Since I’ve been using regression.2 as my example, let’s do that:
summary( regression.2 )
##
## Call:
## lm(formula = dan.grump ~ dan.sleep + baby.sleep, data = parenthood)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.0345 -2.2198 -0.4016 2.6775 11.7496
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 125.96557 3.04095 41.423 <2e-16 ***
## dan.sleep -8.95025 0.55346 -16.172 <2e-16 ***
## baby.sleep 0.01052 0.27106 0.039 0.969
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.354 on 97 degrees of freedom
## Multiple R-squared: 0.8161, Adjusted R-squared: 0.8123
## F-statistic: 215.2 on 2 and 97 DF, p-value: < 2.2e-16
The output that this command produces is pretty dense, but we’ve already discussed everything of interest in it, so what I’ll do is go through it line by line. The first line reminds us of what the actual regression model is:
Call:
lm(formula = dan.grump ~ dan.sleep + baby.sleep, data = parenthood)
You can see why this is handy, since it was a little while back when we actually created the regression.2 model, and so it’s nice to be reminded of what it was we were doing. The next part provides a quick summary of the residuals (i.e., the ϵi values),
Residuals:
Min 1Q Median 3Q Max
-11.0345 -2.2198 -0.4016 2.6775 11.7496
which can be convenient as a quick and dirty check that the model is okay. Remember, we did assume that these residuals were normally distributed, with mean 0. In particular it’s worth quickly checking to see if the median is close to zero, and to see if the first quartile is about the same size as the third quartile. If they look badly off, there’s a good chance that the assumptions of regression are violated. These ones look pretty nice to me, so let’s move on to the interesting stuff. The next part of the R output looks at the coefficients of the regression model:
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 125.96557 3.04095 41.423 <2e-16 ***
dan.sleep -8.95025 0.55346 -16.172 <2e-16 ***
baby.sleep 0.01052 0.27106 0.039 0.969
---
Signif. codes: 0 �***� 0.001 �**� 0.01 �*� 0.05 �.� 0.1 � � 1
Each row in this table refers to one of the coefficients in the regression model. The first row is the intercept term, and the later ones look at each of the predictors. The columns give you all of the relevant information. The first column is the actual estimate of b (e.g., 125.96 for the intercept, and -8.9 for the dan.sleep predictor). The second column is the standard error estimate $\ \hat{\sigma_b}$. The third column gives you the t-statistic, and it’s worth noticing that in this table t= $\ \hat{b}$ /SE($\ \hat{b}$) every time. Finally, the fourth column gives you the actual p value for each of these tests.217 The only thing that the table itself doesn’t list is the degrees of freedom used in the t-test, which is always N−K−1 and is listed immediately below, in this line:
Residual standard error: 4.354 on 97 degrees of freedom
The value of df=97 is equal to N−K−1, so that’s what we use for our t-tests. In the final part of the output we have the F-test and the R2 values which assess the performance of the model as a whole
Residual standard error: 4.354 on 97 degrees of freedom
Multiple R-squared: 0.8161, Adjusted R-squared: 0.8123
F-statistic: 215.2 on 2 and 97 DF, p-value: < 2.2e-16
So in this case, the model performs significantly better than you’d expect by chance (F(2,97)=215.2, p<.001), which isn’t all that surprising: the R2=.812 value indicate that the regression model accounts for 81.2% of the variability in the outcome measure. However, when we look back up at the t-tests for each of the individual coefficients, we have pretty strong evidence that the baby.sleep variable has no significant effect; all the work is being done by the dan.sleep variable. Taken together, these results suggest that regression.2 is actually the wrong model for the data: you’d probably be better off dropping the baby.sleep predictor entirely. In other words, the regression.1 model that we started with is the better model. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.05%3A_Hypothesis_Tests_for_Regression_Models.txt |
Hypothesis tests for a single correlation
I don’t want to spend too much time on this, but it’s worth very briefly returning to the point I made earlier, that Pearson correlations are basically the same thing as linear regressions with only a single predictor added to the model. What this means is that the hypothesis tests that I just described in a regression context can also be applied to correlation coefficients. To see this, let’s take a `summary()` of the `regression.1` model:
``summary( regression.1 )``
``````##
## Call:
## lm(formula = dan.grump ~ dan.sleep, data = parenthood)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.025 -2.213 -0.399 2.681 11.750
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 125.9563 3.0161 41.76 <2e-16 ***
## dan.sleep -8.9368 0.4285 -20.85 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.332 on 98 degrees of freedom
## Multiple R-squared: 0.8161, Adjusted R-squared: 0.8142
## F-statistic: 434.9 on 1 and 98 DF, p-value: < 2.2e-16``````
The important thing to note here is the t test associated with the predictor, in which we get a result of t(98)=−20.85, p<.001. Now let’s compare this to the output of a different function, which goes by the name of `cor.test()`. As you might expect, this function runs a hypothesis test to see if the observed correlation between two variables is significantly different from 0. Let’s have a look:
``cor.test( x = parenthood\$dan.sleep, y = parenthood\$dan.grump )``
``````##
## Pearson's product-moment correlation
##
## data: parenthood\$dan.sleep and parenthood\$dan.grump
## t = -20.854, df = 98, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.9340614 -0.8594714
## sample estimates:
## cor
## -0.903384``````
Again, the key thing to note is the line that reports the hypothesis test itself, which seems to be saying that t(98)=−20.85, p<.001. Hm. Looks like it’s exactly the same test, doesn’t it? And that’s exactly what it is. The test for the significance of a correlation is identical to the t test that we run on a coefficient in a regression model.
Hypothesis tests for all pairwise correlations
Okay, one more digression before I return to regression properly. In the previous section I talked about the `cor.test()` function, which lets you run a hypothesis test on a single correlation. The `cor.test()` function is (obviously) an extension of the `cor()` function, which we talked about in Section 5.7. However, the `cor()` function isn’t restricted to computing a single correlation: you can use it to compute all pairwise correlations among the variables in your data set. This leads people to the natural question: can the `cor.test()` function do the same thing? Can we use `cor.test()` to run hypothesis tests for all possible parwise correlations among the variables in a data frame?
The answer is no, and there’s a very good reason for this. Testing a single correlation is fine: if you’ve got some reason to be asking “is A related to B?”, then you should absolutely run a test to see if there’s a significant correlation. But if you’ve got variables A, B, C, D and E and you’re thinking about testing the correlations among all possible pairs of these, a statistician would want to ask: what’s your hypothesis? If you’re in the position of wanting to test all possible pairs of variables, then you’re pretty clearly on a fishing expedition, hunting around in search of significant effects when you don’t actually have a clear research hypothesis in mind. This is dangerous, and the authors of `cor.test()` obviously felt that they didn’t want to support that kind of behaviour.
On the other hand… a somewhat less hardline view might be to argue we’ve encountered this situation before, back in Section 14.5 when we talked about post hoc tests in ANOVA. When running post hoc tests, we didn’t have any specific comparisons in mind, so what we did was apply a correction (e.g., Bonferroni, Holm, etc) in order to avoid the possibility of an inflated Type I error rate. From this perspective, it’s okay to run hypothesis tests on all your pairwise correlations, but you must treat them as post hoc analyses, and if so you need to apply a correction for multiple comparisons. That’s what the `correlate()` function in the `lsr` package does. When we use the `correlate()` function in Section 5.7 all it did was print out the correlation matrix. But you can get it to output the results of all the pairwise tests as well by specifying `test=TRUE`. Here’s what happens with the `parenthood` data:
``library(lsr)``
``## Warning: package 'lsr' was built under R version 3.5.2` `
``correlate(parenthood, test=TRUE)``
``````##
## CORRELATIONS
## ============
## - correlation type: pearson
## - correlations shown only when both variables are numeric
##
## dan.sleep baby.sleep dan.grump day
## dan.sleep . 0.628*** -0.903*** -0.098
## baby.sleep 0.628*** . -0.566*** -0.010
## dan.grump -0.903*** -0.566*** . 0.076
## day -0.098 -0.010 0.076 .
##
## ---
## Signif. codes: . = p < .1, * = p<.05, ** = p<.01, *** = p<.001
##
##
## p-VALUES
## ========
## - total number of tests run: 6
## - correction for multiple testing: holm
##
## dan.sleep baby.sleep dan.grump day
## dan.sleep . 0.000 0.000 0.990
## baby.sleep 0.000 . 0.000 0.990
## dan.grump 0.000 0.000 . 0.990
## day 0.990 0.990 0.990 .
##
##
## SAMPLE SIZES
## ============
##
## dan.sleep baby.sleep dan.grump day
## dan.sleep 100 100 100 100
## baby.sleep 100 100 100 100
## dan.grump 100 100 100 100
## day 100 100 100 100``````
The output here contains three matrices. First it prints out the correlation matrix. Second it prints out a matrix of p-values, using the Holm method218 to correct for multiple comparisons. Finally, it prints out a matrix indicating the sample size (number of pairwise complete cases) that contributed to each correlation.
So there you have it. If you really desperately want to do pairwise hypothesis tests on your correlations, the `correlate()` function will let you do it. But please, please be careful. I can’t count the number of times I’ve had a student panicking in my office because they’ve run these pairwise correlation tests, and they get one or two significant results that don’t make any sense. For some reason, the moment people see those little significance stars appear, they feel compelled to throw away all common sense and assume that the results must correspond to something real that requires an explanation. In most such cases, my experience has been that the right answer is “it’s a Type I error”. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.06%3A_Testing_the_Significance_of_a_Correlation.txt |
Before moving on to discuss the assumptions underlying linear regression and what you can do to check if they’re being met, there’s two more topics I want to briefly discuss, both of which relate to the regression coefficients. The first thing to talk about is calculating confidence intervals for the coefficients; after that, I’ll discuss the somewhat murky question of how to determine which of predictor is most important.
Confidence intervals for the coefficients
Like any population parameter, the regression coefficients b cannot be estimated with complete precision from a sample of data; that’s part of why we need hypothesis tests. Given this, it’s quite useful to be able to report confidence intervals that capture our uncertainty about the true value of b. This is especially useful when the research question focuses heavily on an attempt to find out how strongly variable X is related to variable Y, since in those situations the interest is primarily in the regression weight b. Fortunately, confidence intervals for the regression weights can be constructed in the usual fashion,
$\ CI(b) = \hat{b}\ \pm\ (t_{crit} \ x\ SE(\hat{b}) )$
where SE($\ \hat{b}$) is the standard error of the regression coefficient, and tcrit is the relevant critical value of the appropriate t distribution. For instance, if it’s a 95% confidence interval that we want, then the critical value is the 97.5th quantile of a t distribution with N−K−1 degrees of freedom. In other words, this is basically the same approach to calculating confidence intervals that we’ve used throughout. To do this in R we can use the confint() function. There arguments to this function are
• object. The regression model (lm object) for which confidence intervals are required.
• parm. A vector indicating which coefficients we should calculate intervals for. This can be either a vector of numbers or (more usefully) a character vector containing variable names. By default, all coefficients are included, so usually you don’t bother specifying this argument.
• level. A number indicating the confidence level that should be used. As is usually the case, the default value is 0.95, so you wouldn’t usually need to specify this argument.
So, suppose I want 99% confidence intervals for the coefficients in the regression.2 model. I could do this using the following command:
confint( object = regression.2,
level = .99)
## 0.5 % 99.5 %
## (Intercept) 117.9755724 133.9555593
## dan.sleep -10.4044419 -7.4960575
## baby.sleep -0.7016868 0.7227357
Simple enough.
Calculating standardised regression coefficients
One more thing that you might want to do is to calculate “standardised” regression coefficients, often denoted β. The rationale behind standardised coefficients goes like this. In a lot of situations, your variables are on fundamentally different scales. Suppose, for example, my regression model aims to predict people’s IQ scores, using their educational attainment (number of years of education) and their income as predictors. Obviously, educational attainment and income are not on the same scales: the number of years of schooling can only vary by 10s of years, whereas income would vary by 10,000s of dollars (or more). The units of measurement have a big influence on the regression coefficients: the b coefficients only make sense when interpreted in light of the units, both of the predictor variables and the outcome variable. This makes it very difficult to compare the coefficients of different predictors. Yet there are situations where you really do want to make comparisons between different coefficients. Specifically, you might want some kind of standard measure of which predictors have the strongest relationship to the outcome. This is what standardised coefficients aim to do.
The basic idea is quite simple: the standardised coefficients are the coefficients that you would have obtained if you’d converted all the variables to z-scores before running the regression.219 The idea here is that, by converting all the predictors to z-scores, they all go into the regression on the same scale, thereby removing the problem of having variables on different scales. Regardless of what the original variables were, a β value of 1 means that an increase in the predictor of 1 standard deviation will produce a corresponding 1 standard deviation increase in the outcome variable. Therefore, if variable A has a larger absolute value of β than variable B, it is deemed to have a stronger relationship with the outcome. Or at least that’s the idea: it’s worth being a little cautious here, since this does rely very heavily on the assumption that “a 1 standard deviation change” is fundamentally the same kind of thing for all variables. It’s not always obvious that this is true.
Leaving aside the interpretation issues, let’s look at how it’s calculated. What you could do is standardise all the variables yourself and then run a regression, but there’s a much simpler way to do it. As it turns out, the β coefficient for a predictor X and outcome Y has a very simple formula, namely
$\ \beta_X = {b_X \times {\sigma_X \over \sigma_Y}}$
where σX is the standard deviation of the predictor, and σY is the standard deviation of the outcome variable Y. This makes matters a lot simpler. To make things even simpler, the lsr package includes a function standardCoefs() that computes the β coefficients.
standardCoefs( regression.2 )
## b beta
## dan.sleep -8.95024973 -0.90474809
## baby.sleep 0.01052447 0.00217223
This clearly shows that the dan.sleep variable has a much stronger effect than the baby.sleep variable. However, this is a perfect example of a situation where it would probably make sense to use the original coefficients b rather than the standardised coefficients β. After all, my sleep and the baby’s sleep are already on the same scale: number of hours slept. Why complicate matters by converting these to z-scores?
15.08: Assumptions of Regression
The linear regression model that I’ve been discussing relies on several assumptions. In Section 15.9 we’ll talk a lot more about how to check that these assumptions are being met, but first, let’s have a look at each of them.
• Normality. Like half the models in statistics, standard linear regression relies on an assumption of normality. Specifically, it assumes that the residuals are normally distributed. It’s actually okay if the predictors X and the outcome Y are non-normal, so long as the residuals ϵ are normal. See Section 15.9.3.
• Linearity. A pretty fundamental assumption of the linear regression model is that relationship between X and Y actually be linear! Regardless of whether it’s a simple regression or a multiple regression, we assume that the relatiships involved are linear. See Section 15.9.4.
• Homogeneity of variance. Strictly speaking, the regression model assumes that each residual ϵi is generated from a normal distribution with mean 0, and (more importantly for the current purposes) with a standard deviation σ that is the same for every single residual. In practice, it’s impossible to test the assumption that every residual is identically distributed. Instead, what we care about is that the standard deviation of the residual is the same for all values of $\ \hat{Y}$, and (if we’re being especially paranoid) all values of every predictor X in the model. See Section 15.9.5.
• Uncorrelated predictors. The idea here is that, is a multiple regression model, you don’t want your predictors to be too strongly correlated with each other. This isn’t “technically” an assumption of the regression model, but in practice it’s required. Predictors that are too strongly correlated with each other (referred to as “collinearity”) can cause problems when evaluating the model. See Section 15.9.6
• Residuals are independent of each other. This is really just a “catch all” assumption, to the effect that “there’s nothing else funny going on in the residuals”. If there is something weird (e.g., the residuals all depend heavily on some other unmeasured variable) going on, it might screw things up.
• No “bad” outliers. Again, not actually a technical assumption of the model (or rather, it’s sort of implied by all the others), but there is an implicit assumption that your regression model isn’t being too strongly influenced by one or two anomalous data points; since this raises questions about the adequacy of the model, and the trustworthiness of the data in some cases. See Section 15.9.2. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.07%3A_Regarding_Regression_Coefficients.txt |
The main focus of this section is regression diagnostics, a term that refers to the art of checking that the assumptions of your regression model have been met, figuring out how to fix the model if the assumptions are violated, and generally to check that nothing “funny” is going on. I refer to this as the “art” of model checking with good reason: it’s not easy, and while there are a lot of fairly standardised tools that you can use to diagnose and maybe even cure the problems that ail your model (if there are any, that is!), you really do need to exercise a certain amount of judgment when doing this. It’s easy to get lost in all the details of checking this thing or that thing, and it’s quite exhausting to try to remember what all the different things are. This has the very nasty side effect that a lot of people get frustrated when trying to learn all the tools, so instead they decide not to do any model checking. This is a bit of a worry!
In this section, I describe several different things you can do to check that your regression model is doing what it’s supposed to. It doesn’t cover the full space of things you could do, but it’s still much more detailed than what I see a lot of people doing in practice; and I don’t usually cover all of this in my intro stats class myself. However, I do think it’s important that you get a sense of what tools are at your disposal, so I’ll try to introduce a bunch of them here. Finally, I should note that this section draws quite heavily from the Fox and Weisberg (2011) text, the book associated with the car package. The car package is notable for providing some excellent tools for regression diagnostics, and the book itself talks about them in an admirably clear fashion. I don’t want to sound too gushy about it, but I do think that Fox and Weisberg (2011) is well worth reading.
Three kinds of residuals
The majority of regression diagnostics revolve around looking at the residuals, and by now you’ve probably formed a sufficiently pessimistic theory of statistics to be able to guess that – precisely because of the fact that we care a lot about the residuals – there are several different kinds of residual that we might consider. In particular, the following three kinds of residual are referred to in this section: “ordinary residuals”, “standardised residuals”, and “Studentised residuals”. There is a fourth kind that you’ll see referred to in some of the Figures, and that’s the “Pearson residual”: however, for the models that we’re talking about in this chapter, the Pearson residual is identical to the ordinary residual.
The first and simplest kind of residuals that we care about are ordinary residuals. These are the actual, raw residuals that I’ve been talking about throughout this chapter. The ordinary residual is just the difference between the fitted value $\ \hat{Y_i}$ and the observed value Yi. I’ve been using the notation ϵi to refer to the i-th ordinary residual, and by gum I’m going to stick to it. With this in mind, we have the very simple equation
$\ \epsilon_i = Y_i - \hat{Y_i}$
This is of course what we saw earlier, and unless I specifically refer to some other kind of residual, this is the one I’m talking about. So there’s nothing new here: I just wanted to repeat myself. In any case, you can get R to output a vector of ordinary residuals, you can use a command like this:
residuals( object = regression.2 )
## 1 2 3 4 5 6
## -2.1403095 4.7081942 1.9553640 -2.0602806 0.7194888 -0.4066133
## 7 8 9 10 11 12
## 0.2269987 -1.7003077 0.2025039 3.8524589 3.9986291 -4.9120150
## 13 14 15 16 17 18
## 1.2060134 0.4946578 -2.6579276 -0.3966805 3.3538613 1.7261225
## 19 20 21 22 23 24
## -0.4922551 -5.6405941 -0.4660764 2.7238389 9.3653697 0.2841513
## 25 26 27 28 29 30
## -0.5037668 -1.4941146 8.1328623 1.9787316 -1.5126726 3.5171148
## 31 32 33 34 35 36
## -8.9256951 -2.8282946 6.1030349 -7.5460717 4.5572128 -10.6510836
## 37 38 39 40 41 42
## -5.6931846 6.3096506 -2.1082466 -0.5044253 0.1875576 4.8094841
## 43 44 45 46 47 48
## -5.4135163 -6.2292842 -4.5725232 -5.3354601 3.9950111 2.1718745
## 49 50 51 52 53 54
## -3.4766440 0.4834367 6.2839790 2.0109396 -1.5846631 -2.2166613
## 55 56 57 58 59 60
## 2.2033140 1.9328736 -1.8301204 -1.5401430 2.5298509 -3.3705782
## 61 62 63 64 65 66
## -2.9380806 0.6590736 -0.5917559 -8.6131971 5.9781035 5.9332979
## 67 68 69 70 71 72
## -1.2341956 3.0047669 -1.0802468 6.5174672 -3.0155469 2.1176720
## 73 74 75 76 77 78
## 0.6058757 -2.7237421 -2.2291472 -1.4053822 4.7461491 11.7495569
## 79 80 81 82 83 84
## 4.7634141 2.6620908 -11.0345292 -0.7588667 1.4558227 -0.4745727
## 85 86 87 88 89 90
## 8.9091201 -1.1409777 0.7555223 -0.4107130 0.8797237 -1.4095586
## 91 92 93 94 95 96
## 3.1571385 -3.4205757 -5.7228699 -2.2033958 -3.8647891 0.4982711
## 97 98 99 100
## -5.5249495 4.1134221 -8.2038533 5.6800859
One drawback to using ordinary residuals is that they’re always on a different scale, depending on what the outcome variable is and how good the regression model is. That is, Unless you’ve decided to run a regression model without an intercept term, the ordinary residuals will have mean 0; but the variance is different for every regression. In a lot of contexts, especially where you’re only interested in the pattern of the residuals and not their actual values, it’s convenient to estimate the standardised residuals, which are normalised in such a way as to have standard deviation 1. The way we calculate these is to divide the ordinary residual by an estimate of the (population) standard deviation of these residuals. For technical reasons, mumble mumble, the formula for this is:
$\epsilon_{i}^{\prime}=\dfrac{\epsilon_{i}}{\hat{\sigma} \sqrt{1-h_{i}}}$
where $\ \hat{\sigma}$ in this context is the estimated population standard deviation of the ordinary residuals, and hi is the “hat value” of the ith observation. I haven’t explained hat values to you yet (but have no fear,220 it’s coming shortly), so this won’t make a lot of sense. For now, it’s enough to interpret the standardised residuals as if we’d converted the ordinary residuals to z-scores. In fact, that is more or less the truth, it’s just that we’re being a bit fancier. To get the standardised residuals, the command you want is this:
rstandard( model = regression.2 )
## 1 2 3 4 5 6
## -0.49675845 1.10430571 0.46361264 -0.47725357 0.16756281 -0.09488969
## 7 8 9 10 11 12
## 0.05286626 -0.39260381 0.04739691 0.89033990 0.95851248 -1.13898701
## 13 14 15 16 17 18
## 0.28047841 0.11519184 -0.61657092 -0.09191865 0.77692937 0.40403495
## 19 20 21 22 23 24
## -0.11552373 -1.31540412 -0.10819238 0.62951824 2.17129803 0.06586227
## 25 26 27 28 29 30
## -0.11980449 -0.34704024 1.91121833 0.45686516 -0.34986350 0.81233165
## 31 32 33 34 35 36
## -2.08659993 -0.66317843 1.42930082 -1.77763064 1.07452436 -2.47385780
## 37 38 39 40 41 42
## -1.32715114 1.49419658 -0.49115639 -0.11674947 0.04401233 1.11881912
## 43 44 45 46 47 48
## -1.27081641 -1.46422595 -1.06943700 -1.24659673 0.94152881 0.51069809
## 49 50 51 52 53 54
## -0.81373349 0.11412178 1.47938594 0.46437962 -0.37157009 -0.51609949
## 55 56 57 58 59 60
## 0.51800753 0.44813204 -0.42662358 -0.35575611 0.58403297 -0.78022677
## 61 62 63 64 65 66
## -0.67833325 0.15484699 -0.13760574 -2.05662232 1.40238029 1.37505125
## 67 68 69 70 71 72
## -0.28964989 0.69497632 -0.24945316 1.50709623 -0.69864682 0.49071427
## 73 74 75 76 77 78
## 0.14267297 -0.63246560 -0.51972828 -0.32509811 1.10842574 2.72171671
## 79 80 81 82 83 84
## 1.09975101 0.62057080 -2.55172097 -0.17584803 0.34340064 -0.11158952
## 85 86 87 88 89 90
## 2.10863391 -0.26386516 0.17624445 -0.09504416 0.20450884 -0.32730740
## 91 92 93 94 95 96
## 0.73475640 -0.79400855 -1.32768248 -0.51940736 -0.91512580 0.11661226
## 97 98 99 100
## -1.28069115 0.96332849 -1.90290258 1.31368144
Note that this function uses a different name for the input argument, but it’s still just a linear regression object that the function wants to take as its input here.
The third kind of residuals are Studentised residuals (also called “jackknifed residuals”) and they’re even fancier than standardised residuals. Again, the idea is to take the ordinary residual and divide it by some quantity in order to estimate some standardised notion of the residual, but the formula for doing the calculations this time is subtly different:
$\epsilon_{i}^{*}=\dfrac{\epsilon_{i}}{\hat{\sigma}_{(-i)} \sqrt{1-h_{i}}}$
Notice that our estimate of the standard deviation here is written $\ \hat{\sigma_{-i}}$. What this corresponds to is the estimate of the residual standard deviation that you would have obtained, if you just deleted the ith observation from the data set. This sounds like the sort of thing that would be a nightmare to calculate, since it seems to be saying that you have to run N new regression models (even a modern computer might grumble a bit at that, especially if you’ve got a large data set). Fortunately, some terribly clever person has shown that this standard deviation estimate is actually given by the following equation:
$\hat{\sigma}_{(-i)}=\hat{\sigma} \sqrt{\dfrac{N-K-1-\epsilon_{i}^{\prime 2}}{N-K-2}}$
Isn’t that a pip? Anyway, the command that you would use if you wanted to pull out the Studentised residuals for our regression model is
rstudent( model = regression.2 )
## 1 2 3 4 5 6
## -0.49482102 1.10557030 0.46172854 -0.47534555 0.16672097 -0.09440368
## 7 8 9 10 11 12
## 0.05259381 -0.39088553 0.04715251 0.88938019 0.95810710 -1.14075472
## 13 14 15 16 17 18
## 0.27914212 0.11460437 -0.61459001 -0.09144760 0.77533036 0.40228555
## 19 20 21 22 23 24
## -0.11493461 -1.32043609 -0.10763974 0.62754813 2.21456485 0.06552336
## 25 26 27 28 29 30
## -0.11919416 -0.34546127 1.93818473 0.45499388 -0.34827522 0.81089646
## 31 32 33 34 35 36
## -2.12403286 -0.66125192 1.43712830 -1.79797263 1.07539064 -2.54258876
## 37 38 39 40 41 42
## -1.33244515 1.50388257 -0.48922682 -0.11615428 0.04378531 1.12028904
## 43 44 45 46 47 48
## -1.27490649 -1.47302872 -1.07023828 -1.25020935 0.94097261 0.50874322
## 49 50 51 52 53 54
## -0.81230544 0.11353962 1.48863006 0.46249410 -0.36991317 -0.51413868
## 55 56 57 58 59 60
## 0.51604474 0.44627831 -0.42481754 -0.35414868 0.58203894 -0.77864171
## 61 62 63 64 65 66
## -0.67643392 0.15406579 -0.13690795 -2.09211556 1.40949469 1.38147541
## 67 68 69 70 71 72
## -0.28827768 0.69311245 -0.24824363 1.51717578 -0.69679156 0.48878534
## 73 74 75 76 77 78
## 0.14195054 -0.63049841 -0.51776374 -0.32359434 1.10974786 2.81736616
## 79 80 81 82 83 84
## 1.10095270 0.61859288 -2.62827967 -0.17496714 0.34183379 -0.11101996
## 85 86 87 88 89 90
## 2.14753375 -0.26259576 0.17536170 -0.09455738 0.20349582 -0.32579584
## 91 92 93 94 95 96
## 0.73300184 -0.79248469 -1.33298848 -0.51744314 -0.91435205 0.11601774
## 97 98 99 100
## -1.28498273 0.96296745 -1.92942389 1.31867548
Before moving on, I should point out that you don’t often need to manually extract these residuals yourself, even though they are at the heart of almost all regression diagnostics. That is, the residuals(), rstandard() and rstudent() functions are all useful to know about, but most of the time the various functions that run the diagnostics will take care of these calculations for you. Even so, it’s always nice to know how to actually get hold of these things yourself in case you ever need to do something non-standard.
Three kinds of anomalous data
One danger that you can run into with linear regression models is that your analysis might be disproportionately sensitive to a smallish number of “unusual” or “anomalous” observations. I discussed this idea previously in Section 6.5.2 in the context of discussing the outliers that get automatically identified by the boxplot() function, but this time we need to be much more precise. In the context of linear regression, there are three conceptually distinct ways in which an observation might be called “anomalous”. All three are interesting, but they have rather different implications for your analysis.
The first kind of unusual observation is an outlier. The definition of an outlier (in this context) is an observation that is very different from what the regression model predicts. An example is shown in Figure 15.7. In practice, we operationalise this concept by saying that an outlier is an observation that has a very large Studentised residual, $\ {\epsilon_i}^*$. Outliers are interesting: a big outlier might correspond to junk data – e.g., the variables might have been entered incorrectly, or some other defect may be detectable. Note that you shouldn’t throw an observation away just because it’s an outlier. But the fact that it’s an outlier is often a cue to look more closely at that case, and try to find out why it’s so different.
The second way in which an observation can be unusual is if it has high leverage: this happens when the observation is very different from all the other observations. This doesn’t necessarily have to correspond to a large residual: if the observation happens to be unusual on all variables in precisely the same way, it can actually lie very close to the regression line. An example of this is shown in Figure 15.8. The leverage of an observation is operationalised in terms of its hat value, usually written hi. The formula for the hat value is rather complicated221 but its interpretation is not: hi is a measure of the extent to which the i-th observation is “in control” of where the regression line ends up going. You can extract the hat values using the following command:
hatvalues( model = regression.2 )
## 1 2 3 4 5 6
## 0.02067452 0.04105320 0.06155445 0.01685226 0.02734865 0.03129943
## 7 8 9 10 11 12
## 0.02735579 0.01051224 0.03698976 0.01229155 0.08189763 0.01882551
## 13 14 15 16 17 18
## 0.02462902 0.02718388 0.01964210 0.01748592 0.01691392 0.03712530
## 19 20 21 22 23 24
## 0.04213891 0.02994643 0.02099435 0.01233280 0.01853370 0.01804801
## 25 26 27 28 29 30
## 0.06722392 0.02214927 0.04472007 0.01039447 0.01381812 0.01105817
## 31 32 33 34 35 36
## 0.03468260 0.04048248 0.03814670 0.04934440 0.05107803 0.02208177
## 37 38 39 40 41 42
## 0.02919013 0.05928178 0.02799695 0.01519967 0.04195751 0.02514137
## 43 44 45 46 47 48
## 0.04267879 0.04517340 0.03558080 0.03360160 0.05019778 0.04587468
## 49 50 51 52 53 54
## 0.03701290 0.05331282 0.04814477 0.01072699 0.04047386 0.02681315
## 55 56 57 58 59 60
## 0.04556787 0.01856997 0.02919045 0.01126069 0.01012683 0.01546412
## 61 62 63 64 65 66
## 0.01029534 0.04428870 0.02438944 0.07469673 0.04135090 0.01775697
## 67 68 69 70 71 72
## 0.04217616 0.01384321 0.01069005 0.01340216 0.01716361 0.01751844
## 73 74 75 76 77 78
## 0.04863314 0.02158623 0.02951418 0.01411915 0.03276064 0.01684599
## 79 80 81 82 83 84
## 0.01028001 0.02920514 0.01348051 0.01752758 0.05184527 0.04583604
## 85 86 87 88 89 90
## 0.05825858 0.01359644 0.03054414 0.01487724 0.02381348 0.02159418
## 91 92 93 94 95 96
## 0.02598661 0.02093288 0.01982480 0.05063492 0.05907629 0.03682026
## 97 98 99 100
## 0.01817919 0.03811718 0.01945603 0.01373394
In general, if an observation lies far away from the other ones in terms of the predictor variables, it will have a large hat value (as a rough guide, high leverage is when the hat value is more than 2-3 times the average; and note that the sum of the hat values is constrained to be equal to K+1). High leverage points are also worth looking at in more detail, but they’re much less likely to be a cause for concern unless they are also outliers. % guide from Venables and Ripley.
This brings us to our third measure of unusualness, the influence of an observation. A high influence observation is an outlier that has high leverage. That is, it is an observation that is very different to all the other ones in some respect, and also lies a long way from the regression line. This is illustrated in Figure 15.9. Notice the contrast to the previous two figures: outliers don’t move the regression line much, and neither do high leverage points. But something that is an outlier and has high leverage… that has a big effect on the regression line.
That’s why we call these points high influence; and it’s why they’re the biggest worry. We operationalise influence in terms of a measure known as Cook’s distance,
$D_{i}=\dfrac{\epsilon_{i}^{* 2}}{K+1} \times \dfrac{h_{i}}{1-h_{i}}$
Notice that this is a multiplication of something that measures the outlier-ness of the observation (the bit on the left), and something that measures the leverage of the observation (the bit on the right). In other words, in order to have a large Cook’s distance, an observation must be a fairly substantial outlier and have high leverage. In a stunning turn of events, you can obtain these values using the following command:
cooks.distance( model = regression.2 )
## 1 2 3 4 5
## 1.736512e-03 1.740243e-02 4.699370e-03 1.301417e-03 2.631557e-04
## 6 7 8 9 10
## 9.697585e-05 2.620181e-05 5.458491e-04 2.876269e-05 3.288277e-03
## 11 12 13 14 15
## 2.731835e-02 8.296919e-03 6.621479e-04 1.235956e-04 2.538915e-03
## 16 17 18 19 20
## 5.012283e-05 3.461742e-03 2.098055e-03 1.957050e-04 1.780519e-02
## 21 22 23 24 25
## 8.367377e-05 1.649478e-03 2.967594e-02 2.657610e-05 3.448032e-04
## 26 27 28 29 30
## 9.093379e-04 5.699951e-02 7.307943e-04 5.716998e-04 2.459564e-03
## 31 32 33 34 35
## 5.214331e-02 6.185200e-03 2.700686e-02 5.467345e-02 2.071643e-02
## 36 37 38 39 40
## 4.606378e-02 1.765312e-02 4.689817e-02 2.316122e-03 7.012530e-05
## 41 42 43 44 45
## 2.827824e-05 1.076083e-02 2.399931e-02 3.381062e-02 1.406498e-02
## 46 47 48 49 50
## 1.801086e-02 1.561699e-02 4.179986e-03 8.483514e-03 2.444787e-04
## 51 52 53 54 55
## 3.689946e-02 7.794472e-04 1.941235e-03 2.446230e-03 4.270361e-03
## 56 57 58 59 60
## 1.266609e-03 1.824212e-03 4.804705e-04 1.163181e-03 3.187235e-03
## 61 62 63 64 65
## 1.595512e-03 3.703826e-04 1.577892e-04 1.138165e-01 2.827715e-02
## 66 67 68 69 70
## 1.139374e-02 1.231422e-03 2.260006e-03 2.241322e-04 1.028479e-02
## 71 72 73 74 75
## 2.841329e-03 1.431223e-03 3.468538e-04 2.941757e-03 2.738249e-03
## 76 77 78 79 80
## 5.045357e-04 1.387108e-02 4.230966e-02 4.187440e-03 3.861831e-03
## 81 82 83 84 85
## 2.965826e-02 1.838888e-04 2.149369e-03 1.993929e-04 9.168733e-02
## 86 87 88 89 90
## 3.198994e-04 3.262192e-04 4.547383e-05 3.400893e-04 7.881487e-04
## 91 92 93 94 95
## 4.801204e-03 4.493095e-03 1.188427e-02 4.796360e-03 1.752666e-02
## 96 97 98 99 100
## 1.732793e-04 1.012302e-02 1.225818e-02 2.394964e-02 8.010508e-03
As a rough guide, Cook’s distance greater than 1 is often considered large (that’s what I typically use as a quick and dirty rule), though a quick scan of the internet and a few papers suggests that 4/N has also been suggested as a possible rule of thumb.
As hinted above, you don’t usually need to make use of these functions, since you can have R automatically draw the critical plots.222 For the regression.2 model, these are the plots showing Cook’s distance (Figure 15.10) and the more detailed breakdown showing the scatter plot of the Studentised residual against leverage (Figure 15.11). To draw these, we can use the plot() function. When the main argument x to this function is a linear model object, it will draw one of six different plots, each of which is quite useful for doing regression diagnostics. You specify which one you want using the which argument (a number between 1 and 6). If you don’t do this then R will draw all six. The two plots of interest to us in this context are generated using the following commands:
An obvious question to ask next is, if you do have large values of Cook’s distance, what should you do? As always, there’s no hard and fast rules. Probably the first thing to do is to try running the regression with that point excluded and see what happens to the model performance and to the regression coefficients. If they really are substantially different, it’s time to start digging into your data set and your notes that you no doubt were scribbling as your ran your study; try to figure out why the point is so different. If you start to become convinced that this one data point is badly distorting your results, you might consider excluding it, but that’s less than ideal unless you have a solid explanation for why this particular case is qualitatively different from the others and therefore deserves to be handled separately.223 To give an example, let’s delete the observation from day 64, the observation with the largest Cook’s distance for the regression.2 model. We can do this using the subset argument:
lm( formula = dan.grump ~ dan.sleep + baby.sleep, # same formula
data = parenthood, # same data frame...
subset = -64 # ...but observation 64 is deleted
)
##
## Call:
## lm(formula = dan.grump ~ dan.sleep + baby.sleep, data = parenthood,
## subset = -64)
##
## Coefficients:
## (Intercept) dan.sleep baby.sleep
## 126.3553 -8.8283 -0.1319
As you can see, those regression coefficients have barely changed in comparison to the values we got earlier. In other words, we really don’t have any problem as far as anomalous data are concerned.
Checking the normality of the residuals
Like many of the statistical tools we’ve discussed in this book, regression models rely on a normality assumption. In this case, we assume that the residuals are normally distributed. The tools for testing this aren’t fundamentally different to those that we discussed earlier in Section 13.9. Firstly, I firmly believe that it never hurts to draw an old fashioned histogram. The command I use might be something like this:
hist( x = residuals( regression.2 ), # data are the residuals
xlab = "Value of residual", # x-axis label
main = "", # no title
breaks = 20 # lots of breaks
)
The resulting plot is shown in Figure 15.12, and as you can see the plot looks pretty damn close to normal, almost unnaturally so.
I could also run a Shapiro-Wilk test to check, using the shapiro.test() function; the W value of .99, at this sample size, is non-significant (p=.84), again suggesting that the normality assumption isn’t in any danger here. As a third measure, we might also want to draw a QQ-plot using the qqnorm() function. The QQ plot is an excellent one to draw, and so you might not be surprised to discover that it’s one of the regression plots that we can produce using the plot() function:
plot( x = regression.2, which = 2 ) # Figure @ref{fig:regressionplot2}
The output is shown in Figure 15.13, showing the standardised residuals plotted as a function of their theoretical quantiles according to the regression model. The fact that the output appends the model specification to the picture is nice.
Checking the linearity of the relationship
The third thing we might want to test is the linearity of the relationships between the predictors and the outcomes. There’s a few different things that you might want to do in order to check this. Firstly, it never hurts to just plot the relationship between the fitted values $\ \hat{Y_i}$ and the observed values Yi for the outcome variable, as illustrated in Figure 15.14. To draw this we could use the fitted.values() function to extract the $\ \hat{Y_i}$ values in much the same way that we used the residuals() function to extract the ϵi values. So the commands to draw this figure might look like this:
yhat.2 <- fitted.values( object = regression.2 )
plot( x = yhat.2,
y = parenthood\$dan.grump,
xlab = "Fitted Values",
ylab = "Observed Values"
)
One of the reasons I like to draw these plots is that they give you a kind of “big picture view”. If this plot looks approximately linear, then we’re probably not doing too badly (though that’s not to say that there aren’t problems). However, if you can see big departures from linearity here, then it strongly suggests that you need to make some changes.
In any case, in order to get a more detailed picture it’s often more informative to look at the relationship between the fitted values and the residuals themselves. Again, we could draw this plot using low level commands, but there’s an easier way. Just plot() the regression model, and select which = 1:
plot(x = regression.2, which = 1)
The output is shown in Figure 15.15. As you can see, not only does it draw the scatterplot showing the fitted value against the residuals, it also plots a line through the data that shows the relationship between the two. Ideally, this should be a straight, perfectly horizontal line. There’s some hint of curvature here, but it’s not clear whether or not we be concerned.
A somewhat more advanced version of the same plot is produced by the residualPlots() function in the car package. This function not only draws plots comparing the fitted values to the residuals, it does so for each individual predictor. The command is and the resulting plots are shown in Figure 15.16.
residualPlots( model = regression.2 )
## Loading required package: carData
## Test stat Pr(>|Test stat|)
## dan.sleep 2.1604 0.03323 *
## baby.sleep -0.5445 0.58733
## Tukey test 2.1615 0.03066 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note that this function also reports the results of a bunch of curvature tests. For a predictor variable X in some regression model, this test is equivalent to adding a new predictor to the model corresponding to X2, and running the t-test on the b coefficient associated with this new predictor. If it comes up significant, it implies that there is some nonlinear relationship between the variable and the residuals.
The third line here is the Tukey test, which is basically the same test, except that instead of squaring one of the predictors and adding it to the model, you square the fitted-value. In any case, the fact that the curvature tests have come up significant is hinting that the curvature that we can see in Figures 15.15 and 15.16 is genuine;224 although it still bears remembering that the pattern in Figure 15.14 is pretty damn straight: in other words the deviations from linearity are pretty small, and probably not worth worrying about.
In a lot of cases, the solution to this problem (and many others) is to transform one or more of the variables. We discussed the basics of variable transformation in Sections 7.2 and (mathfunc), but I do want to make special note of one additional possibility that I didn’t mention earlier: the Box-Cox transform. The Box-Cox function is a fairly simple one, but it’s very widely used
$f(x, \lambda)=\dfrac{x^{\lambda}-1}{\lambda}$
for all values of λ except λ=0. When λ=0 we just take the natural logarithm (i.e., ln(x)). You can calculate it using the boxCox() function in the car package. Better yet, if what you’re trying to do is convert a data to normal, or as normal as possible, there’s the powerTransform() function in the car package that can estimate the best value of λ. Variable transformation is another topic that deserves a fairly detailed treatment, but (again) due to deadline constraints, it will have to wait until a future version of this book.
Checking the homogeneity of variance
The regression models that we’ve talked about all make a homogeneity of variance assumption: the variance of the residuals is assumed to be constant. The “default” plot that R provides to help with doing this (which = 3 when using plot()) shows a plot of the square root of the size of the residual $\ \sqrt{|\epsilon_i|}$, as a function of the fitted value $\ \hat{Y_i}$. We can produce the plot using the following command,
plot(x = regression.2, which = 3)
and the resulting plot is shown in Figure 15.17. Note that this plot actually uses the standardised residuals (i.e., converted to z scores) rather than the raw ones, but it’s immaterial from our point of view. What we’re looking to see here is a straight, horizontal line running through the middle of the plot.
A slightly more formal approach is to run hypothesis tests. The car package provides a function called ncvTest() (non-constant variance test) that can be used for this purpose (Cook and Weisberg 1983). I won’t explain the details of how it works, other than to say that the idea is that what you do is run a regression to see if there is a relationship between the squared residuals ϵi and the fitted values $\ \hat{Y_i}$, or possibly to run a regression using all of the original predictors instead of just $\ \hat{Y_i}$.225 Using the default settings, the ncvTest() looks for a relationship between $\ \hat{Y_i}$ and the variance of the residuals, making it a straightforward analogue of Figure 15.17. So if we run it for our model,
ncvTest( regression.2 )
## Non-constant Variance Score Test
## Variance formula: ~ fitted.values
## Chisquare = 0.09317511, Df = 1, p = 0.76018
We see that our original impression was right: there’s no violations of homogeneity of variance in this data.
It’s a bit beyond the scope of this chapter to talk too much about how to deal with violations of homogeneity of variance, but I’ll give you a quick sense of what you need to consider. The main thing to worry about, if homogeneity of variance is violated, is that the standard error estimates associated with the regression coefficients are no longer entirely reliable, and so your t tests for the coefficients aren’t quite right either. A simple fix to the problem is to make use of a “heteroscedasticity corrected covariance matrix” when estimating the standard errors. These are often called sandwich estimators, for reasons that only make sense if you understand the maths at a low level226 have implemented as the default in the hccm() function is a tweak on this, proposed by Long and Ervin (2000). This version uses $\Sigma=\operatorname{diag}\left(\epsilon_{i}^{2} /\left(1-h_{i}^{2}\right)\right)$, where hi is the ith hat value. Gosh, regression is fun, isn’t it?] You don’t need to understand what this means (not for an introductory class), but it might help to note that there’s a hccm() function in the car() package that does it. Better yet, you don’t even need to use it. You can use the coeftest() function in the lmtest package, but you need the car package loaded:
library(lmtest)
library(car)
coeftest( regression.2, vcov= hccm )
##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 125.965566 3.247285 38.7910 <2e-16 ***
## dan.sleep -8.950250 0.615820 -14.5339 <2e-16 ***
## baby.sleep 0.010524 0.291565 0.0361 0.9713
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Not surprisingly, these t tests are pretty much identical to the ones that we saw when we used the summary(regression.2) command earlier; because the homogeneity of variance assumption wasn’t violated. But if it had been, we might have seen some more substantial differences.
Checking for collinearity
The last kind of regression diagnostic that I’m going to discuss in this chapter is the use of variance inflation factors (VIFs), which are useful for determining whether or not the predictors in your regression model are too highly correlated with each other. There is a variance inflation factor associated with each predictor Xk in the model, and the formula for the k-th VIF is:
$\mathrm{VIF}_{k}=\dfrac{1}{1-R_{(-k)}^{2}}$
where $\ {R^2}_{(-k)}$ refers to R-squared value you would get if you ran a regression using Xk as the outcome variable, and all the other X variables as the predictors. The idea here is that $\ {R^2}_{(-k)}$ is a very good measure of the extent to which Xk is correlated with all the other variables in the model. Better yet, the square root of the VIF is pretty interpretable: it tells you how much wider the confidence interval for the corresponding coefficient bk is, relative to what you would have expected if the predictors are all nice and uncorrelated with one another. If you’ve only got two predictors, the VIF values are always going to be the same, as we can see if we use the vif() function (car package)…
vif( mod = regression.2 )
## dan.sleep baby.sleep
## 1.651038 1.651038
And since the square root of 1.65 is 1.28, we see that the correlation between our two predictors isn’t causing much of a problem.
To give a sense of how we could end up with a model that has bigger collinearity problems, suppose I were to run a much less interesting regression model, in which I tried to predict the day on which the data were collected, as a function of all the other variables in the data set. To see why this would be a bit of a problem, let’s have a look at the correlation matrix for all four variables:
cor( parenthood )
## dan.sleep baby.sleep dan.grump day
## dan.sleep 1.00000000 0.62794934 -0.90338404 -0.09840768
## baby.sleep 0.62794934 1.00000000 -0.56596373 -0.01043394
## dan.grump -0.90338404 -0.56596373 1.00000000 0.07647926
## day -0.09840768 -0.01043394 0.07647926 1.00000000
We have some fairly large correlations between some of our predictor variables! When we run the regression model and look at the VIF values, we see that the collinearity is causing a lot of uncertainty about the coefficients. First, run the regression…
regression.3 <- lm( day ~ baby.sleep + dan.sleep + dan.grump, parenthood )
and second, look at the VIFs…
vif( regression.3 )
## baby.sleep dan.sleep dan.grump
## 1.651064 6.102337 5.437903
Yep, that’s some mighty fine collinearity you’ve got there. | textbooks/stats/Applied_Statistics/Learning_Statistics_with_R_-_A_tutorial_for_Psychology_Students_and_other_Beginners_(Navarro)/15%3A_Linear_Regression/15.09%3A_Model_Checking.txt |
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