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<p>I would love to get some advice regarding the following please!</p>
<p>I am involved in a study that is looking at cognition across 3 separate groups (1= the clinical group of interest; 2= clinical control group; 3=healthy control group). All participants completed a range of cognitive tests. Preliminary analysis on the entire sample (n=51) showed that there was a general intellect difference between groups (grp 3 was higher than grps 1+2), which meant that any subsequent analysis showing differences on the cognitive tests could be due to intellect alone rather than a true difference. So the sample was specifically matched on IQ, resulting in an adjusted sample size of n=33 (11 per group). </p>
<p>I was originally instructed to perform repeated measures ANOVAs on the refined sample for each of my cognitive tests. As such, the layout of my SPSS datafile shows 11 rows, so there are 3'groups' per variable (e.g., test1_group1; test1_group2; test1_group3). </p>
<p>Now, there were no significant differences for any of these cognitive tests. However, there was a significant difference in terms of mood, and so I would be interested in running a repeated measures ANCOVA to investigate how much mood is impacting on the cognitive results. However, given the layout of my variables, I am becoming confused at how to run the repeated measures ANCOVA using SPSS, as my variable options re: the mood questionnaire produces 3 options (e.g., Mood_group1; mood_group2; mood_group3) - do I choose the mood variable for the group with the significant difference only, or use all 3 variables as covariates (which complicates the output somewhat!)</p>
<p>To add yet another layer of complexity, I consulted with some friends about my analyses. They were confused why I was analysing using repeated measures and suggested I instead alter the layout of my SPSS datasheet so that I would have 33 rows (to match the n=33), and run a series of independent one-way ANOVAs, which would seem to solve my 3 variables per 'person' dilemma re: performing an ANCOVA, but leaves me extremely confused as this seems to make sense, but is not in line with what I was told originally.</p>
<p>I'm sorry this has turned into a longer post than I had intended but hopefully it provides enough background in order for some well-needed advice to be thrown my way!! Any input/ideas would be greatly appreciated!</p> | 73,563 | [
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<p>Say I have the following model:</p>
<p>$$\text{Poisson}(\lambda) \sim \begin{cases}
\lambda_1 & \text{if } t \lt \tau \\
\lambda_2 & \text{if } t \geq \tau
\end{cases}
$$</p>
<p>And I infer the posteriors for $\lambda_1$ and $\lambda_2$ shown below from my data. Is there a Bayesian way of telling (or quantifying) if $\lambda_1$ and $\lambda_2$ are the <strong>same or different</strong>? </p>
<p>Perhaps measuring <strong>the probability that $\lambda_1$ is different from $\lambda_2$</strong>? Or perhaps using KL divergences? </p>
<p>For example, how can I measure $p(\lambda_2 \neq \lambda_1)$, or at least, $p(\lambda_2 \gt \lambda_1)$?</p>
<p>In general, once you have the posteriors shown below (assume <strong>non-zero</strong> PDF values everywhere for both), what is a good way of answering this question?</p>
<p><img src="http://i.stack.imgur.com/jb5HU.png" alt="enter image description here"></p>
<h2>Update</h2>
<p>It seems that this question can be answered in two ways:</p>
<ol>
<li><p>If we have samples of the posteriors, we could look at the fraction of the samples where $\lambda_1 \neq \lambda_2$ (or equivalently $\lambda_2 > \lambda_1$). @Cam.Davidson.Pilon included an answer that would address this problem using such samples.</p></li>
<li><p>Integrating some sort of difference of the posteriors. And that's an important part of my question. <strong>What would that integration look like?</strong> Presumably the sampling approach would approximate this integral, but I would like to know the formulation of this integral.</p></li>
</ol>
<p><strong>Note:</strong> The plots above come from <a href="http://nbviewer.ipython.org/github/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/blob/master/Chapter1_Introduction/Chapter1_Introduction.ipynb" rel="nofollow">this material</a>.</p> | 73,564 | [
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<p>from what you have read or heard about, which is a good book on fuzzy logic/sets/systems? I'm interested in basic of fuzzy systems, fuzzification/defuzzification etc.</p>
<p>Thanks,
Lucian</p> | 47,395 | [
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<p>I do have a big list of numeric values (including duplicates) and I do want to group them into ranges in order to see if how do they distribute.</p>
<p>Let's say there are 1000 values ranging from 0 to 2.000.000 and I do want to group them. </p>
<p>How can I achieve this, preferably in Excel or SQL.</p> | 73,565 | [
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<p>Although I know how to calculate many of the statistical values, I have some difficulties understanding which one should I use in a specific situation. Here is my example:</p>
<p>In my experiment I tried to find out how fast does the Water evaporate from 8 different nonvowen Materials. To achieve this I prepared 4 Testobject's for each Material (that makes a total of 32) and moisturized them. Afterwards i let them dry in a controled enviroment (Laboratory was airconditioned and the Temperature and rel. air humidity was recorded). Every hour I measured the weight of each Testobject. I calculated the relation betwen weight of water and weight of a dry Testobject to be able to compare the results to each other and presented them in a diagram (Water/dry ratio to time diagramm). I made a linear Trendline for each of 32 Testobjects with gaussian least square method. The slope of the Graph represents the Evaporative rate per hour of a given Object. I used Excel to calculate.</p>
<p>I can approximate the observational error for measured Weight and the corresponding measurement time. Given that </p>
<p>-calculated trendline is defined as:</p>
<p>$ m = a \times t + b$</p>
<p>-the measured weight and time are </p>
<p>$m_{i}$ and $t_{i}$</p>
<p>and the vertikal distance between measured data and the trendline is defined as:</p>
<p>$ e_{i} = a \times t_{i} + b - m_{i}$</p>
<p>one can calculate the error of the slope <code>a</code> with following equation:</p>
<p><img src="http://i.stack.imgur.com/BpwGw.png" alt="Error of Slope of a trendline"></p>
<p>So here comes my problem: I have 4 trendlines for each material with 4 slopes each. I calculated an average for each material. All of the slopes have their error. Also each average has its statistical dispersion. I would like to presents those averages with some kind of +- range, like standard deviation or something. What kind of measure of statistical dispersion should i use? </p> | 73,566 | [
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<p>[I'm a stats noob, so please correct if I am going wrong somewhere!]</p>
<p>We run a survey of our customers based on certain events or touchpoints. The responses vary significantly by these touchpoints, so I would like to weight the responses by post stratification.</p>
<p>However, I have no responses from customers who didn't have an "event", and these customers make up more than 60% of the portfolio population! Is there a workaround to this problem - how do I accommodate for non-sampled segments in my analysis? Is imputation a possible solution?</p>
<p>For example:</p>
<p>Let us assume - we are a telephony services provider, and we survey our customers to find out how satisfied they are with certain elements of our service. We have oversampled customers with "dropped calls", because that is more of a "problem area". In addition, we randomly survey some customers who have 3G connections but do not have a dropped call.</p>
<p>Let's add some numbers to this - 1000 customers, 600 with 3G, 400 without. Of these, 100 customers with 3G have a dropped call, and 100 customers without 3G have a dropped call.</p>
<p>In our survey, let us assume we have responses from 150 customers - 50 with 3G and a dropped call, 50 with 3G and no dropped call, and 50 without 3G and dropped calls. If we wanted to understand how satisfied the portfolio of customers is, we could stratify the responses with the following weights:</p>
<pre><code>Customers with 3G and Dropped call (100/1000)/(50/150) = 0.3
Customers with 3G and no Dropped call - ((600-100)/1000)/(50/150) = 1.5
Customers without 3G and Dropped call - (100/1000)/(50/150) = 0.3
Customers without 3G and no Dropped call - ????
</code></pre>
<p>If we do not have responses from this segment, how would we extrapolate the survey to understand the portfolio's satisfaction?</p>
<p>The only thing I can think of is imputing responses from a few customers based on the responses from the other 3 segments - maybe there is a baseline response for a customer, given certain predictor variables, and the presence/absence of 3G and dropped calls will change that response for a given customer? This is an extremely vague idea and I am worried about adding responses to my survey which are "made up"!</p>
<p>Please help! Thanks!</p> | 73,567 | [
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<p>I'm currently trying to use Newey-West standard errors accounting for Heteroskedasticity and Autocorrelation with the sandwich package in R, but i lack understanding as i can not fully grasp the instructions provided in the manual.</p>
<pre><code>library(sandwich)
## fit investment equation
data(Investment)
fm <- lm(RealInv ~ RealGNP + RealInt, data = Investment)
summary(fm)
## Newey & West (1994) (Heteroskedasticity and Autocorrelationconsistent; HAC)
## Covariance Matrix Estimation
NeweyWest(fm)
## The Newey & West (1987) estimator requires specification
## of the lag and suppression of prewhitening
NeweyWest(fm, lag = 4, prewhite = FALSE)
</code></pre>
<p>Unfortunately that's all the information the authors provide. I understand that it is only the estimated HAC Variance-Covariance Matrix but how do you proceed from there to obtain valid standard errors and t-statistics? And is my assumption correct that if you don't specify the lag, the bwNeweyWest function automatically selects the number of relevant lags? </p> | 73,568 | [
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<p>I'm trying to classify my set of data into two classes (introvert / extrovert). I was thinking of using a decision tree at first, but I don't have any potential known results in order to create my decision tree model. Thus I decided to use a <em>k</em>-means clustering algorithm with <em>k</em> = 2.</p>
<ol>
<li><p>Since the clustering algorithm accepts only numeric values, can I use the decision tree algorithm to transform some type of values I have into numeric ones at first (based on some rules I define within the tree) before I start clustering?</p></li>
<li><p>Let's suppose at the end of the algorithm I get my 2 clusters: cluster 1 and cluster 2. How can I classify these two clusters based on my 2 classes? Am I supposed to use supervised or semi-supervised clustering? (I don't know how semi- and supervised clustering work).</p></li>
<li><p>Is there any other simple and efficient classification technique that can satisfy my needs?</p></li>
</ol>
<p>P.S. I'm new to this domain and all your advice and remarks are appreciated.</p> | 35,966 | [
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<p>I would like to analyze my results which have 3 groups with 5 possible outcomes (they are ordinal data by the way) and the expected cell counts are less than 5. Seems like it is not possible to run a analysis larger than a 2X2 tables in SPSS. </p>
<p>Does anyone has a solution to it? Or should I consider using other tests like ordinal regression?</p>
<p>Thanks</p> | 73,569 | [
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<p>Suppose I run some models with following format.The dependent variable as the sum of realized values of a binary random variable over multiple trials, that is "number of yes votes on bills":</p>
<pre><code> Coef. SE z
(Constant) 0.467 0.239 1.950
SPD 0.094 0.174 0.541
#parties -0.218 0.100 -2.192
$\gamma$ 0.012 0.009 1.374
logL -41.21
</code></pre>
<p>and I would like to calculate the first difference effect of increasing the number of parties by one on the expected number of times a state
approves of the federal government’s proposal out of 103 votes.
How can I do that based on my regression output?</p> | 73,570 | [
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<p>Suppose I have random variables $X_i$ which are Poisson distributed with mean $\mu$. I m interested in the sampling distribution of the variable $\frac{X_1+...+X_n}{n}$. We now that as $n$ goes to infinity, the distribution of mean is Gaussian. Assuming that I know the underlying distribution which is Poisson, is there a way to compute a bound for $n$ say $N$ such that if $n>N$ then we have convergence with some small error $\epsilon$ ?</p> | 35,967 | [
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<p>I have already solved this question, but I need some feedback.</p>
<blockquote>
<p>A new method for measuring blood pressure in laboratory mice using radiotelemetry has been proposed. How does this compare with the existing method? You apply both methods to a sample of 16 mice and use <strong>________</strong>.</p>
</blockquote>
<p>I believe a matched-pairs <em>t</em>-test should go in the blank.
Am I right or wrong?
Thanks.</p> | 35,968 | [
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<p>In my current dataset I have quite a few categorical variables.
Most have decent distributions between the categories. 30:40:30 splits etc. where these are percentage of dataset members per category</p>
<p>dataset is ~2000 members</p>
<p>But a fair few have either (I feel) useless or difficult distributions.</p>
<p>Some have distributions like 97:2:1 - to me there are so few members in the low populated categories I would just leave this variable out. There may only be 10-20 members in the 1% category.</p>
<p>Others have distributions where I am not sure what to do.
80:10:5:5 etc
if it makes real world sense, with this option is combining the 10:5:5 portion into one group viable? Ie the new variable has 2 categories comprising 80:20 split in members.</p>
<p>With decision trees I do not think this is too much of a problem as it will effectively do the combining itself.
With other modelling methods where dummy variables are created is the combining (or even removal) of variables an encouraged method as it also helps reduce dimensionality?</p> | 73,571 | [
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<p>Ideally, I would like to know the fraction of the mean of $x$ over the sum of the mean of $x$ and the mean of $y$:</p>
<p>$$\text{Fraction of interest} = \frac{\bar{x}}{\bar{x}+\bar{y}}$$</p>
<p>where $$\bar{x} = \frac{\sum_{i=1}^I x_i }{I}$$.</p>
<p>However, I have the fraction of the geometric mean of $x$ over the sum of the geometric mean of $x$ and the geometric mean of $y$:</p>
<p>$$\text{Fraction I have} = \frac{\prod_{i=1}^I{x_i}}{\prod_{i=1}^I{x_i}+\prod_{i=1}^I{y_i}}$$.</p>
<p>Is there a good interpretation for the fraction I have? I lose intuition when it comes to geometric means. Alternatively, is there a way to connect the two fractions?</p>
<p>Note (thanks to comments notice that): $x_i, y_i, i=1,\ldots,I$ always $>0$.</p>
<p>To further add to this, $x_i$ and $y_i$ are observations from an experiment. It is assumed that $log(x_i)$ and $log(y_i)$ follow a normal distribution. I'm interested in the difference between $x_i$ and $y_i$, across replicates (i.e., I have $R$ experiments where in each I measure $x_i$'s and $y_i$'s) hence it is convenient for me to model $log(x_i/y_i)$ as a normally distributed random variable (with mean and variance as the sample mean and sample variance of $log(x_i/y_i)$ over $I$). I then use a regression model (without going into too much details) where the response in every experiment $r$ of $R$ is the mean $log(x_i/y_i)$ over $I$ (roughly, $\sum_{i=1}^I log(x_i/y_i)/I = \alpha + \beta X$). </p>
<p>Having estimated $\alpha$ (i.e., the mean response) I would like to express $\frac{\bar{x}}{\bar{x}+\bar{y}}$ as a function of $\hat\alpha$. </p>
<p>So what I have is:
$\hat\alpha = \sum_{i=1}^I log(x_i/y_i)/I = log(\frac{\prod_{i=1}^I{x_i}}{\prod_{i=1}^I{y_i}})$</p>
<p>Therefore, $e^{I\alpha} = \frac{\prod_{i=1}^I{x_i}}{\prod_{i=1}^I{y_i}}$</p>
<p>Adding 1 to both sides and skipping a few steps gets me that:</p>
<p>$\frac{e^{I\alpha}}{1 + e^{I\alpha}} = \frac{\prod_{i=1}^I{x_i}}{\prod_{i=1}^I{x_i}+\prod_{i=1}^I{y_i}}$</p>
<p>So basically I'm trying to either get an intuition of how $\frac{\prod_{i=1}^I{x_i}}{\prod_{i=1}^I{x_i}+\prod_{i=1}^I{y_i}}$ relates to $\frac{\bar{x}}{\bar{x}+\bar{y}}$ or some function to connect the two.</p> | 28,559 | [
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<p>Deas,</p>
<p>One assumption for Spearman's correlation is that data have to be monotonic .. I tried to do scatterplot in spss, but I get the following graph that I couldn't decide if it is monotonic .. What is the solution please ? Note: my data are ordinal variables.</p>
<p><img src="http://i.stack.imgur.com/RZOUG.jpg" alt="Scatterplot gragh"></p> | 35,970 | [
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<p>This might be a question in general: due to computational burden, I have to use a subset of my complete data (say, 1,000 out of the complete 10,000 observations) to get a p-value of a test. The test itself is from Monte Carlo simulations. My question is, is there a way to quantify the uncertainty of the p-value <strong>due to the use of a subset of the 1,000 observations instead of using the complete dataset?</strong> Thanks!</p> | 73,572 | [
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<p>I'm doing a statistics paper in which I have to correlate the clinical diagnosis with the hystopathological one. So, basically I have one group of people with 2 different diagnosis:
1. clinical
2. hystopathological
Normally these two must be identical, but due to some clinical errors sometimes they aren't. My job is to calculate how many have the same diagnosis , how many have different diagnosis. What statistical test do I need to use?
Thank you ! </p> | 73,573 | [
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<p>I have a homework assignment about implementing the adaboost algorithm.<br>
We are following this <a href="https://www.cs.princeton.edu/picasso/mats/schapire02boosting_schapire.pdf" rel="nofollow">article by Schapire</a>. I'm not exactly sure what they mean in the assignment by :</p>
<blockquote>
<p>Let the training patterns and their labels in dataset D be denoted xi
and yi, respectively and let Wk(i) be the kth (discrete) distribution
over all these training samples. We initialize these weights Wk to to
be uniform in iteration k=1.</p>
</blockquote>
<p>Do I just make an array of ones?</p>
<p><code>Wk = ones((size(H(x)))</code> </p>
<p>Then updated it accordingly on each iteration.</p> | 35,972 | [
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<p>From Berger&Casella, Chapter 8, Problem 8.5, page 402 for 2nd edition.</p>
<p>The problem is:
A random sample, $X_{i}$ is drawn from a Pareto population with pdf
$$
f(x|\theta,v)=\frac{\theta v^{\theta}}{x^{\theta+1}}I_{[v,\infty)}(x),\theta>0,v>0
$$
Show that the LRT of
$$
H_{0}:\theta=1; H_{1}:\theta\not=1,
$$ with $v$ unknown has critical region of the form
$$
\{x:T(x)\le c_{1}; T(x)\ge c_{2}\}
$$
where $0<c_{1}<c_{2}$ and
$$
T=\log[\frac{\prod^{n}_{i=1}X_{i}}{(\min_{i}X_{i})^{n}}]
$$
Show that, under $H_{0}$, $2T$ has a chi squared distribution, and find the number of degrees of freedom. (Hint: Obtain the joint distribution of the $n-1$ non-trivial terms $\frac{X_{i}}{\min_{i} X_{i}}$ conditional on $\min_{i}X_{i}$. Put these $n-1$ terms together, and notice that the distribution of $T$ given $\min_{i}X_{i}$ does not depend on $\min_{i}X_{i}$, so it is the unconditional distribution of $T$).</p>
<p>After finishing the first part, I tried to compute the distribution of $T$, but after several hours of efforts I plainly got to nowhere. Here is what I did:</p>
<p>Since we know
$$
T=\log[\frac{\prod^{n}_{i=1}X_{i}}{(\min_{i}X_{i})^{n}}]
$$
We have to use Jacobian method. We want to compute the distribution of $A=\sum \log[X_{i}]$ and $B=n\log[X_{(1)}]$. Now, if $X$ has distribution $f_{X}[x]$, then $\log[X]$ has distribution $$
f_{Y}(y)=f_{X}(e^{y})*e^{y}=\frac{v}{e^{y}}I_{[\log[v],\infty)}(y)=ve^{-y}I_{[\log[v],\infty)}(y)
$$
On the other hand
$$
Z=\log[X_{(1)}]
$$
has distribution coming from $X_{(1)}$. We know $X_{(1)}$'s pdf is
$$
nf_{X}(x)(1-F_{X}(x))^{n-1}
$$
We know that
$$
f_{X}(x)=\frac{v}{x^{2}}I_{(v,\infty)}(x)
$$
Therefore
$$
F_{X}(x)=\int^{x}_{v}\frac{v}{t^{2}}dt=v(-\frac{1}{t})^{x}_{v}=v(-\frac{1}{x}+\frac{1}{v})=1-\frac{v}{x}
$$
Therefore
$$
f_{X_{(1)}}(x)=n\frac{v}{x^{2}}I_{(v,\infty)}\frac{v^{n-1}}{x^{n-1}}=\frac{nv^{n}}{x^{n+1}}I_{(v,\infty)}
$$
and if $\log[X_{(1)}]=z$, then we have
$$
f_{Z}(z)=\frac{nv^{n}}{e^{(n+1)z}}e^{z}I_{(\log[v],\infty)}=\frac{nv^{n}}{e^{nz}}I_{(\log[v],\infty)}
$$
Now it is suffice to compute the distribution of $$2(\sum_{i=1}^{n}\log[X_{i}]-n\log[Z])$$</p>
<p>Since $\log[X_{i}],\log[X_{j}]$ are independent, by Jacobian method we have
$$
Z=X+Y, W=Y; f_{Z,W}(z,w)=f_{X,Y}(Z-W,W)=f_{X}(Z-W)f_{Y}(W)
$$
So we have
$$
f_{(Z,W)}=ve^{-(z-w)}ve^{-w}=v^{2}e^{-z}
$$
where we have
$$
w\ge \log[v], z-w\ge \log[v]\rightarrow z\ge 2\log[v]
$$
Therefore
$$
\log[X_{i}]+\log[X_{j}]\sim v^{2}e^{-z}I_{(2\log[v],\infty)}
$$
Proceed inductively we have
$$
f_{\sum^{n}_{i=1}\log[X_{i}]}(s)=v^{n}e^{-s}I_{(n\log[v],\infty)}
$$
On the other hand we want to compute the pdf of $f_{n\log[X_{(1)}]}$. From the cdf we have:
$$
\log[X_{(1)}]\le t\leftrightarrow n\log[X_{(1)}]\le nt
$$
Therefore
$$
F_{n\log[X_{(1)}]}(t)=F_{\log[X_{i}]}(\frac{t}{n})
$$
This implies
$$
f_{n\log[X_{(1)}]}(t)=f_{\log[X_{i}]}(\frac{t}{n})*\frac{1}{n}=\frac{v^{n}}{e^{t}}I_{(n\log[v],\infty)}=v^{n}e^{-t}I_{(n\log[v],\infty)}
$$
Let $S=\sum^{n}_{i=1}\log[X_{i}],T=n\log[X_{(1)}]$. Then $S,T$ are independent because of Basu's theorem. But I cannot separate $f_{S,T}$ due to the restriction of the domain, as I have $S+T\ge n\log[v],T\ge n\log[v]$ does not imply $S\in (0,\infty)$. Also the resulting distribution is not $\chi^{2}$. So I think I need some help. </p> | 73,574 | [
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<p>In R, is it true that: </p>
<p><code>pt(q,df=Inf)</code> $=$ <code>pnorm(q)</code>? </p>
<p>Or in words, can I supply <code>df=Inf</code> in the cummulative t-distribution to get to the standard normal distribution function in R, as I should in theory?</p> | 35,974 | [
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] |
<p>I am not clear about how exactly the p values and t values are used. I know how to calculate them but I am not clear about the basic concept behind them.</p>
<p>Any help ?</p> | 49,294 | [
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<p>A typical approach to solving a classification problem is to identify a class of candidate models, and then perform model selection using some procedure like cross validation. Typically one selects the model with the highest accuracy, or some related function that encodes problem specific information, like <a href="http://en.wikipedia.org/wiki/F1_score" rel="nofollow">$\text{F}_\beta$</a>. </p>
<p>Assuming the end goal is to produce an accurate classifier (where the definition of accuracy is again, problem dependent), in what situations is it better to perform model selection using a <a href="http://en.wikipedia.org/wiki/Scoring_rule" rel="nofollow">proper scoring rule</a> as opposed to something improper, like accuracy, precision, recall, etc? Furthermore, let's ignore issues of model complexity and assume a priori we consider all the models equally likely.</p>
<p>Previously I would have said never. We know, in a formal sense, classification is an easier problem than regression [1], [2] and we can derive tighter bounds for the former than the later ($*$). Furthermore, there are cases when trying to accurately match probabilities can result in incorrect <a href="http://stats.stackexchange.com/a/50325/6248">decision boundaries</a> or <a href="http://yaroslavvb.blogspot.com/2007/06/log-loss-or-hinge-loss.html" rel="nofollow">overfitting</a>. However, based on the conversation <a href="http://stats.stackexchange.com/a/90705/6248">here</a> and the voting pattern of the community in regards to such issues, I've been questioning this view.</p>
<ol>
<li>Devroye, Luc. <a href="http://www.szit.bme.hu/~gyorfi/pbook.pdf" rel="nofollow">A probabilistic theory of pattern recognition.</a> Vol. 31. springer, 1996., Section 6.7</li>
<li>Kearns, Michael J., and Robert E. Schapire. <a href="http://www.cis.upenn.edu/~mkearns/papers/pconcepts.pdf" rel="nofollow">Efficient distribution-free learning of probabilistic concepts.</a> Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on. IEEE, 1990.</li>
</ol>
<p>$(*)$ This statement might be a little sloppy. I specifically mean that given labeled data of the form $S = \{(x_1, y_1), \ldots, (x_n, y_n)\}$ with $x_i \in \mathcal{X}$ and $y_i \in \{1, \ldots, K\}$, it seems to be easier to estimate a decision boundary than accurately estimate conditional probabilities.</p> | 28,579 | [
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<p>I have data that looks like this </p>
<p><img src="http://i.stack.imgur.com/2CdaI.png" alt="enter image description here"></p>
<p>I just want to predict the rest of 2013. What is the best way of doing this. </p>
<p>I used Excel's Forecast formula which uses linear regression. </p>
<p>I only have minor/very little knowledge in stats.</p> | 140 | [
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<p>I have a situation where I want to test for differences between rows and columns in my data frame:</p>
<pre><code> ID Cond1A Cond1B Cond2A Cond2B
Sample1 2000 2311 2323 2324
Sample2 2424 1313 2324 4546
Sample3 1213 3413 2342 3353
</code></pre>
<p>How do I test if Sample1 is different from Sample 2 and 3 across Condition 1 and 2 (where A and B are replicates). I prefer to do it in R, but really don't find a good way.</p> | 73,575 | [
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<p>While studying relevance feedback(Pseudo relevance feedback), I have learn that the model can go horribly wrong for some queries. Can anyone give reasons why this is?</p> | 73,576 | [
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<p>I am using WEKA to do classification. I wish to compare with an existing paper that uses the Precision and Recall Break-Even performance as their results.</p>
<p>Can someone please help in order for me to determine this Break-Even performance? More preferably - if possible - with WEKA. If not, a formula I can use.</p> | 73,577 | [
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<p>I wonder , if someone know the open source anomaly detection algorithm on computer log ?
For an example , computer log look like as mentioned below :</p>
<p>"value UL-CCCH-Message ::= { integrityCheckInfo { messageAuthenticationCode 0, rrc-MessageSequenceNumber 0 }, message cellUpdate : { u-RNTI { srnc-Identity 232, s-RNTI 178710 }, startList { { cn-DomainIdentity ps-domain, start-Value 58 } }, am-RLC-ErrorIndicationRb2-3or4 FALSE, am-RLC-ErrorIndicationRb5orAbove FALSE, cellUpdateCause cellReselection, rb-timer-indicator { t314-expired FALSE, t315-expired FALSE }, measuredResultsOnRACH { currentCell { modeSpecificInfo fdd : { measurementQuantity cpich-Ec-N0 : 24 } } } }"</p>
<p>there is way , i could extract features from logs and apply anomaly detection on time series data but there is also one other way is that i will discover general pattern from log automatically and make rule/ criteria , if future log belong to general pattern , if not not then its anomaly . I want to know any algoritm to find automatic pattern discovery in logs . </p>
<p>Please share your ideas, if you have expertise in it.</p>
<p>Thanks Kind Regard, Sash</p> | 35,983 | [
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<p>I have general statistical question concerning the analysis of two-factorial experiments. I would like to compare several response variables (biovolume, richness etc.) of two experiments (homogeneous and heterogeneous) with the same factorial combination (dispersal x disturbance).</p>
<p>I guess a ANOVA or nested ANOVA is not the appropriate method, since both experiments are conducted seperately from each other... </p>
<p>Has somebody an idea what would be the best way to realize it in R? </p>
<p>Thanks in advance,</p>
<p>Nils </p> | 73,578 | [
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<p>I am conducting a study for analyzing male involvement in Family planning. It is of interest to develop an index for the involvement of male. I am currently having difficulties fitting an IRT model since my data has a lot of missing values and answers like "don't know" and some confusing answers. Note that my variables are all dichotomous in nature. </p>
<p>Any suggestions for handling this kind of problem regarding missing values in IRT model?Thanks</p> | 35,985 | [
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<p>I have a single time series which will be divided on the date of the policy change before and after.
I want to compare the variances between the two time sections and I am told to do an F test of residual sum of squares.
How do I construct the F stat?
Do I run ARMA and get the residual sum of squares of the two time sections and get an F stat of the ratio of the two residual sum of squares or a ratio of two error variances?
Or do I do simple F test of the equality of variance of data in each of the two time sections?
Thanks a lot.</p> | 73,579 | [
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<p>What's the entropy of the following generalized probability distributions?</p>
<p>$P_1(x) = \delta(x)$</p>
<p>$P_2(x,y) = \delta(x+y)$, for $0\le x\le 1$, and $P_2(x,y)=0$ otherwise.</p>
<p>Integrals of the type $-\int \delta(x) \ln\delta(x) \mathrm{d}x$ seem to diverge to $-\infty$ (see <a href="http://math.stackexchange.com/q/625966/10063">here</a>). But entropy is supposed to be positive. What's going on here? How can I compute the entropy of these distributions? Is there a way to define entropy for these distributions?</p> | 73,580 | [
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<p>I am studying the Rocchio alorithm. I understand how it works. And typically we set Positive feedback is more valuable than negative feedback (so, set β < γ; e.g. γ = 0.25, β = 0.75).
<img src="http://i.stack.imgur.com/cIH7x.jpg" alt="enter image description here">
And many sytems allow only positive feedback which means γ = 0.0. My question why those systems don't consider the non-relevant documents? And what are those systems? Thanks.</p> | 35,987 | [
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<p>To avoid reinventing the wheel, which approaches are known to extract product / service data from previously unknown webpages via machine-learning?</p>
<p>Which keywords in a search engine might give me better results about this topic?</p> | 35,988 | [
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<p>This one is bothering me for a while, and a great dispute was held around it. In psychology (as well as in other social sciences), we deal with different ways of dealing with numbers :-) i.e. <strong>the levels of measurement</strong>. It's also common practice in psychology to standardize some questionnaire, hence transform the data into percentile scores (in order to assess a respondent's position within the representative sample).</p>
<p>Long story short, if you have a variable that holds the data expressed in percentile scores, how should you treat it? As an ordinal, interval, or even ratio variable?!</p>
<p>It's not ratio, cause there no real 0 (0<sup>th</sup> percentile doesn't imply absence of measured property, but the variable's smallest value). I advocate the view that percentile scores are ordinal, since P<sub>70</sub> - P<sub>50</sub> is not equal to P<sub>50</sub> - P<sub>30</sub>, while the other side says it's interval. </p>
<p>Please gentlemen, cut the cord. Ordinal or interval?</p> | 73,581 | [
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<p>I have a large set of data points received by experimentation and each point has n ,or let's say 8 in this case, independent variables and one output/dependent variable.</p>
<p>(x1, x2, x3, ...., xn) = y --> some measured output </p>
<p>How would I fit a "curve" through these points or interpolate with the data points? I am not too sure where to start with this. If there area any general formulas or keywords that anyone can pass along to get me started in the right direction, that would be much appreciated. </p> | 73,582 | [
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<p>I'm designing a multi class classifier (for 4 classes) using <strong>Discrete HMMs with States N and Symbols M</strong> for each of the HMM.</p>
<p>However, I found that recognition performance(i.e highest log likelihood) of a specific class depends upon the <strong>N and M</strong> with which it was trained.</p>
<p>Most of the literature I found on the web suggests <strong>use of same of number of states, symbols across all the HMMs</strong>. </p>
<p><strong>I was wondering if it need not be so?</strong> i.e. HMM1 can have 4 states, 18 symbols and HMM2 can have 5 states, 20 symbols............</p> | 73,583 | [
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<p>I am looking for a good and modern Kernel Regression package in R, which has the following features:</p>
<ol>
<li>It has cross-validation </li>
<li>It can automatically choose the "optimal" bandwidth</li>
<li>It doesn't have random effect - i.e. if I run the function at different times on the same data-set, the results should be exactly the same...</li>
</ol>
<p>I am trying "np", but I am seeing:</p>
<pre><code>Multistart 1 of 1 |
Multistart 1 of 1 |
...
</code></pre>
<p>It looks like in order to do the optimization, it's doing multiple-random-start optimization ... Am I right?</p>
<p>Could you please give me some pointers?</p>
<p>I did some google search but there are so many packages that do this... I just wanted to find the best/modern one to use...</p> | 35,990 | [
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<p>I was wondering if there under some circumstances is possible for ANN's to perform better if you prune away some connections on them as for example:</p>
<p>Constructing one ANN by taking two multi-layered ANN's A and B in parallel (same input and output nodes) adding a few "communication" connections between the hidden layers of A and B?</p>
<p>Could one get better generalization results?</p>
<p>Is this somehow used in practice or does one always just use multi-layered fully connected networks?</p> | 73,584 | [
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<p>I need to conduct an experiment and use one-way ANOVA test to analyze the data. I know what the test is, but I am having difficulty figuring out an experiment to perform. Any suggestions on an experiment? Something simple I can do at home.</p>
<p>Thank you</p> | 29,538 | [
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<p>Looking at the Heritage Health Prize, the data is structured as follows:</p>
<blockquote>
<p>Each of the Data Sets will be comprised of tables as follows:</p>
</blockquote>
<pre><code>a. Members Table, which will include:
i. MemberID (a unique member ID)
ii. AgeAtFirstClaim (member's age when first claim was made in the Data Set period)
iii. Sex
b. Claims Table, which will include:
i. MemberID
ii. ProviderID (the ID of the doctor or specialist providing the service)
iii. Vendor (the company that issues the bill)
iv. PCP (member's primary care physician)
v. Year (the year of the claim, Y1, Y2, Y3)
vi. Specialty
vii. PlaceSvc (place where the member was treated)
viii. PayDelay (the delay between the claim and the day the claim was paid for)
ix. LengthOfStay
x. DSFS (days since first service that year)
xi. PrimaryConditionGroup (a generalization of the primary diagnosis codes)
xii. CharlsonIndex (a generalization of the diagnosis codes in the form of a categorized comorbidity score)
xiii. ProcedureGroup (a generalization of the CPT code or treatment code)
xiv. SupLOS (a flag that indicates if LengthOfStay is null because it has been suppressed)
c. Labs Table, which will contain certain details of lab tests provided to members.
d. RX Table, which will contain certain details of prescriptions filled by members.
e. DaysInHospital Tables - Y2 and Y3, which will contain the number of days of hospitalization for each eligible member during Y2 and Y3 and will include:
i. MemberID;
ii. ClaimsTruncated (a flag for members who have had claims suppressed. If the flag is 1 for member xxx in DaysInHospital_Y2, some claims for member xxx will have been suppressed in Y1).
iii. DaysInHospital (the number of days in hospital Y2 or Y3, as applicable).
These two Tables are intended for use by Entrants to train and validate their algorithms. DaysInHospital Tables are based on the Claims Table with admissions in Y2 or Y3, as applicable. As a privacy measure, any member who spent more than two weeks in hospital is grouped; they are treated as though they spent 15 days in hospital.
f. Target - is "DaysInHospital_Y4" but doesn't include DaysInHospital. DaysInHospital data for Y4 are to be filled in by Entrants to produce entries. Seem SampleEntry.csv as an example.
</code></pre>
<p>So the goal is to predict the values in table f.</p>
<p>Clearly the first thing to do is to do a JOIN across the tables using member IDs. </p>
<p>Some of these appear to be standard discrete variables (age, sex etc) but others are much more tricky to deal with. For example, the labs index and RX index point to other tables. Also, there are many categorical types. Some of the categorical types (such as PrimaryConditionGroup) have many possible values. Normally for categorical data, these can be made into features using a binary flag for each category (simply using the index will fail as there is no ordering). However when there are many thousands of these this seems like an inefficient approach. Are there any better ways to deal with the mixture of numerical and categorical data such as this?</p> | 47,432 | [
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] |
<p>I know <code>glmnet(x,y)</code> generates $\lambda$ but I am very curious to know the actual formula that is behind this, generating $\lambda$. </p> | 35,994 | [
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<p>I am trying to approximate a certain factor in my graph.
<img src="http://i.stack.imgur.com/mvX9h.jpg" alt="enter image description here"></p>
<p>Following Tom Minka's tutorial what I have to do is as follows:
$$
\prod_{i=1}^3 q_{w_i}(\pi_2)\approx \int p(\pi_2|w_1)q_{\pi_1}(w_1)\prod_{i=1}^3 q_{D_i}(w_1)\, dw_1\,\, q_{w_2}(\pi_2)q_{w_3}(\pi_2)
$$
where $q_x(y)$ is the message that $y$ receives from $x$, $p(\pi_2|w_1)\propto \pi_2 (1-\pi_2)^{{w_1}^2}$ and the messages $q_{w_i}(\pi_2)$ are all also beta distributed and finally, $q_x(w_1)$ are all gaussians.</p>
<p>My question is with the right hand side of the approximation can I do the following:</p>
<p>$$
RHS \propto \int \pi(1-\pi)^{{w_1}^2}\exp\left(-\frac{1}{2\sigma^2}(w_1-\mu)^2\right)\times C_1\, dw_1\,\, C_2 \pi^{\alpha_0-1}(1-\pi)^{\beta_0-1}
$$
where I've let $\pi\equiv \pi_2$, $C_1,C_2$ are constants that do not depend on $\pi$ and $\alpha_0, \beta_0$ is obtained by combining the messages from $q_{w_2}, q_{w_3}$. Similarly I've combined the Gaussians, thus giving the exponential term.</p>
<p>So basically I end up with:
$$
RHS \propto \pi^{\alpha_0}(1-\pi)^{\beta_0-1}\left(\frac{1}{\sigma^2}-2\ln(1-\pi)\right)
$$</p>
<p><strong>Can I simply moment match this with a Beta distribution</strong> to get the approximation, in this case the moments being $E(\ln \pi), E(\ln (1-\pi))$ (my interpretation of the proj operation) and divide by the LHS messages or is there more to do.</p>
<p>Finally trying to find these messages for each factor seems tedious and no closed form for moments as above. <strong>Are there any tips or comments on making inference on this easier/ faster</strong>.</p> | 73,585 | [
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<p>I have a regression model with 3 different variables. One variable is only significant if I square it, and not include the linear version i.e. $Y = \beta_1X_1 + \beta_2X_2 + \beta_3X_3^2$</p>
<p>How should I interpret the $X_3^2$ variable?</p> | 49,477 | [
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<p>The may sound like a strange question but I was wondering if a Pseudoinverse of a matrix could be found using SVD whether there was a graphical modelling equivalent that could be used to estimate the inverse of a matrix. </p> | 73,586 | [
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<p>A question from a harried doctoral candidate working on dissertation revisions.</p>
<p>What is the current opinion about how to include and interpret 3-way interactions in linear models? If the p-value for a 3-way interaction is not meaningful to interpret, how does one address--both substantively and empirically--important intersections of three key variables using experimental data?</p>
<p>References on already on hand: </p>
<pre><code>Kam, Cindy D. and Robert J. Franzese, Jr. 2007. Modeling and Interpreting Interactive Hypotheses in Regression Analysis. Ann Arbor, MI: University of Michigan Press.
Braumoeller,Bear F. Hypothesis Testing and Multiplicative Interaction Terms International Organization , Vol. 58, No. 4 (Autumn, 2004), pp. 807-820 Article Stable URL: http://www.jstor.org/stable/3877804
</code></pre> | 73,587 | [
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<p>Suppose we have n by n covariance matrix of n normal random variables. Is there program where we plug in sample values for k of the n variables and the program, by using the covariance matrix and these k values, generates samples for the remaining n-k variables ?</p> | 35,997 | [
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<p>I Have a variable (time series) which is nonstationary. I found that from the graph which seems to have a stochastic trend and the correlogram has a typical nonstationary pattern.
After that, I've been asked to find the parsimonious ARMA model (using information criteria), and by following the Box-Jenkins methodology, go down till forecasting.
Notice that at this point, I havent been asked to make a unit root test to check if the series is stationary or not...</p>
<p>Is it possible a nonstationary series to produce a stationary ARMA model, which will give us reliable forecasting?</p>
<p>After the forecasting, they ask me to make a unit root test (where of course I find that the variable is non stationary). As a next step I take the variable in 1st differences, and follow the same procedure to find an ARMA model with best information criteria. I also make some comparissons with the model that I found at the 1st step, but after that they dont ask me to follow the Box-Jenkins methodology again and make a forecast from the ARMA model produced by the differenced variable...</p>
<p>So what I’m confused about is:</p>
<p>Why did they ask me to find an ARMA from a non stationary variable? Can a nonstat. Variable produce stationary ARMA model and reliable forecasting?</p>
<p>Wouldn’t it be normal to ask me to follow the Box Jenkins methodology in the second model (where the variable is differenced) and make a forecast from there?</p>
<p>Any answer would be helpful!</p> | 17,811 | [
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<p><strong>A definition before I start:</strong> </p>
<p>A trajectory $t$ of length $n$ is here defined as a series of 2D coordinates $$\{(x_1,y_1), (x_2, y_2),..., (x_n, y_n)\}$$
Now I have a set comprised of such trajectories denoted by set $T=\{t_1, t_2,...,t_n\}$. </p>
<hr>
<p>Take $t_1$ and $t_2$ as an example: let's say a segment of $t_1$ denoted by $S_{t_1}$ overlaps with a part of $t_2$ denoted by $S_{t_2}$. But for the other parts, they do not overlap.</p>
<p>All the trajectories may have such a "partially-overlapping" relationship with the others.</p>
<p>Obviously in this case, we cannot treat every trajectory as a whole, find a distance metric and finally cluster them. <strong>We have to sort of cluster the trajectories by part instead of as a whole.</strong></p>
<p>Is there any already-existed clustering technique that does the job? </p>
<p>Any suggestions or pointers are very much welcomed.</p> | 73,588 | [
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<p>I'm developing prediction model based on PLS (partial least squares) analysis.
I'm looking into variation captured by X(R square X), R square Y, number of components and accuracy after using cross validation, in the process of model development.</p>
<p>Can anybody suggest me what should be the approach to trade off among R2X, number of components and accuracy percentage of the model.</p>
<p>As a best practice, What are accepted levels of R2X and accuracy, in the industry? </p>
<p>Thanks a lot for your help.</p> | 73,589 | [
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<p>I am fitting curves to my data to extract one parameter. However, I am unsure what the certainty of that parameter is and how I would calculate / express its $95$% confidence interval.</p>
<p>Say for a dataset containing data that exponentially decays, I fit a curve to each dataset. Then the information I want to extract is the exponent $b$. I know the values of $t$ and the value of $a$ I am not interested in (thats a variable that comes from the population, not the process Im trying to model).</p>
<p>I use non-linear regression to fit these parameters. However I dont know how to compute the $95$% confidence interval for any method, so broader answers are welcome as well.</p>
<p>$$f= a\cdot e^{-bt}$$
<img src="http://www.mathworks.nl/help/releases/R2013b/matlab/math/expfun.gif" alt="example data and fit"></p>
<p>Once I have my value for $b$, how do I calculate its $95$% confidence interval? Thanks in advance!</p> | 25,042 | [
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<p>I'm new in the page and pretty new in statistics and R. I'm working on a project for college with the objective of finding the correlation between rain and water flow level in rivers. Once the correlation is proved I want to forecast/predict it.</p>
<p><strong>The data</strong>
I have a set of data of several years(taken every 5 minutes) for a particular rivers containing: </p>
<ul>
<li>Rainfall in millimetres</li>
<li>River flow in cubic meters per second</li>
</ul>
<p>This river doesn't have snow, so the model is just based on rain and time. There are occasionally freezing temperatures, but I'm thinking on removing those periods out of the data as outliers as that situation is out of scope for my project.</p>
<p><strong>Examples</strong>
Here you have a couple of plots of sample data the from a rain and the rise of water a few hours later.</p>
<p><img src="http://i.stack.imgur.com/ssmtM.jpg" alt="Bigger example a few days"></p>
<p><img src="http://i.stack.imgur.com/XSkvv.jpg" alt="Shorter example just one rainfall period"></p>
<p>The red line is the river flow. The orange is the rain. You can see it always rains before water raises in river. There is some rain starting again at the end of the time series, but it will affect the river flow later.</p>
<p>The correlation is there. Here is what I've done in R to prove the correlation using ccf in R: </p>
<ul>
<li>the cross-correlation</li>
<li>the leading variable</li>
<li>the lag</li>
</ul>
<p>This is my R line used for the second example (one rainfall period):</p>
<pre><code>ccf(arnoiaex1$Caudal, arnoiaex1$Precip, lag.max=1000, plot=TRUE, main="Flow & Rain")
</code></pre>
<p><img src="http://i.stack.imgur.com/IT62e.jpg" alt="ccf result for small example 2"></p>
<p>My interpretation is: </p>
<ul>
<li>that the rain leads (happens first),</li>
<li>there is a significant correlation that peaks at a lag of $\approx 450$ (I can check the exact number, I know that part). </li>
<li>I don't know how to find out the time that correlation affects the river flow, I think the name is “retention”. What I see is the graph follows the same shape of the first graph, when the river losing the water after the rain. I don't if based on that I can say the retention lasts from $\approx 450$ when it peaks to $\approx 800$ (I can check this in the object created in the dataframe returned by <code>ccf</code> and see when the water level comes back to the value of “before rain”. Is that right? Is there a better way to find the retention?</li>
</ul>
<p>Am I right?</p>
<p><strong>About the time series</strong>.
This time series doesn't have periodicity or seasonality. Rain can come any time and cause an effect. It does reduce in summer, but it still happens, it's an area with a lot of rain all year around.</p>
<p><strong>Model and forecast.</strong>
I don't know how to create a model to be able to do a forecast that tells me how much is a river going to increase the volume after a period of rain. I've been trying some <code>arima</code>, <code>auto arima</code> but haven't been very successful. Should I use <code>Arima</code>, <code>vars</code> or other different multivariate model? Any link to a example would be of great help.</p>
<p>Please, let me know if you know the best way to create this prediction, what model should I use. There are a few other things I'm considering doing but taken them out of this explanation for simplicity.
I can share some data if required.</p> | 73,590 | [
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<p>I have 17 patients who underwent cervical plating, I have calculated </p>
<ol>
<li>preoperative cervical measurements like angles at certain level to be operated, total cervical angles </li>
<li>same parameters calculated after operation </li>
<li>same parameters calculated after 2 years of operation</li>
</ol>
<p>Now, which test should I use, and in case it is a paired t-test whether to use equal variance or unequal variance? What happens to degrees of freedom in each case?</p> | 73,591 | [
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<p>I have the following data for 10 subjects based on before and after measurements:</p>
<pre><code>x <- c(12.9, 13.5, 12.8, 15.6, 17.2, 19.2, 12.6, 15.3, 14.4, 11.3)
y <- c(12.7, 13.6, 12.0, 15.2, 16.8, 20.0, 12.0, 15.9, 16.0, 11.1)
</code></pre>
<p>and would like to perform a permutation test.</p>
<p>I used the <code>permTS(...)</code> to perform a two-sided t-test and obtained a value of <code>0.982</code>.</p>
<p>But I would like to use the <code>expand.grid(rep(x,k ))</code> by interchanging the before and after tags. Does anyone know how this can be achieved?</p>
<p>Given the numbers of possible permutations it is not possible to stimulate every permutation, hence why I would like to use expand.grid to check if the same result is obtained.</p>
<p>Thanks</p> | 73,592 | [
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<p>I have a dataset collected using an accelerometer. I am extracting the magnitudes from the signal to find the difference in running pattern between two different running surfaces. Will normalization help to improve my classification accuracy? In general, when should we normalize time series data? Any suggestions?</p> | 73,593 | [
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<p>Is there a term(s) for calculating the variance or standard deviation from a value other than the mean?</p>
<p><strong>Example:</strong></p>
<p>If I have a set of estimates for jelly beans in a jar, calculating standard deviation and variance is an operation on the mean of those estimates. However, I also want to find the "standard deviation" of the distribution from the <em>actual</em> number of jelly beans in the jar. Is this still called <strong>standard deviation</strong> even though it's not the deviation from the mean? Is there a term or specific subject area for analysis of a distribution against an independent value like this?</p> | 41,344 | [
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] |
<p>Objective is to predict a car's retail value based on the characteristics of mileage, make, model, engine size, interior style, and cruise control.</p>
<hr>
<h2>Price</h2>
<p>The price of the vehicle </p>
<hr>
<h2>Cylinder</h2>
<p>4,6,8</p>
<hr>
<h2>Liter</h2>
<p>How many liters the engine is</p>
<hr>
<h2>Doors</h2>
<p>4 or 2 door</p>
<hr>
<h2>Cruise control</h2>
<p>1 if the car has, 0 if it does not</p>
<hr>
<h2>Leather</h2>
<p>1 if the car has, 0 if it does not</p>
<hr>
<h2>Make</h2>
<p>Buick,
Cadillac,
Chevrolet,
Pontiac,
SAAB,
Saturn</p>
<hr>
<h2>Model</h2>
<p>Century,
Lacrosse,
Lesabre,
Park Avenue,
CST-V,
CTS,
Deville,
STS-V6,
STS-V8,
XLR-V8,
AVEO,
Cavalier,
Classic,
Cobalt,
Corvette,
Impala,
Malibu,
Monte Carlo,
Bonneville,
G6,
Grand Am,
Grand Prix,
GTO,
Sunfire,
Vibe,
9_3,
9_3 HO,
9_5,
9_5 HO,
9-2X AWD,
Ion,
L Series</p>
<hr>
<h2>Trim</h2>
<p>Sedan 4D,
CX Sedan 4D,
CXL Sedan 4D,
CXS Sedan 4D,
Custom Sedan 4D,
Limited Sedan 4D,
Special Ed Ultra 4D,
DHS Sedan 4D,
DTS Sedan 4D,
Hardtop Conv 2D,
LS Hatchback 4D,
LS Sedan 4D,
LT Hatchback 4D,
LT Sedan 4D,
SVM Hatchback 4D,
SVM Sedan 4D,
Coupe 2D,
LS Coupe 2D,
LS Sport Coupe 2D,
LS Sport Sedan 4D,
Conv 2D,
SS Sedan 4D,
LS MAXX Hback 4D,
LT MAXX Hback 4D,
MAXX Hback 4D,
LT Coupe 2D,
SS Coupe 2D,
GXP Sedan 4D,
SE Sedan 4D,
SLE Sedan 4D,
GT Sedan 4D,
GT Coupe 2D,
GTP Sedan 4D,
AWD Sportwagon 4D,
GT Sportwagon,
Sportwagon 4D,
Linear Conv 2D,
Linear Sedan 4D,
Aero Conv 2D,
Aero Sedan 4D,
Arc Conv 2D,
Arc Sedan 4D,
Arc Wagon 4D,
Linear Wagon 4D,
Aero Wagon 4D,
Quad Coupe 2D,
L300 Sedan 4D</p>
<hr>
<h2>Type</h2>
<p>Sedan,
Convertible,
Hatchback,
Coupe,
Wagon</p>
<hr>
<p>I was just wondering how I should set up a model for this data set?</p>
<p>This is what I was thinking so far:</p>
<p>Model (dummy variables for Doors, Cylinder, Make, Model, Trim, and Type)</p>
<p>Price = Mileage + Make_bui + Make_Cad+Make_Che + Make_Pon + Make_Saa + Make_Sat + Cylinder + Liter + Doors + Cruise + Leather</p> | 37,451 | [
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<p>I'm looking for resources (articles, books or just suggestions) on cyclostationary time series parameter estimation. I have a 2D signal of that sort, which forms a grid. </p> | 36,002 | [
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<p>I am using the <code>mda</code> package and in particular the <code>fda</code> routine to classify in term of gear a set of 20 trips. I preformed a flexible discriminant analysis (FDA) using a set of 151 trips. </p>
<pre><code>FDAT1 <- fda(as.factor(gear) ~ . , data =matrizR)
</code></pre>
<p>A total of 22 predictors were considered. 20 of the predictors are "numeric" and 2 are "factors/discrete" (with several levels, e.g. A,B,C and D). </p>
<p>The resulting FDA rule was applied to the 20 trips and for the same predictors in order to predict the type of gear used. However a consistent error is occurring: </p>
<pre><code>Error in polybasis(newdata, degree, monomial) %*% object$coef :
non-conformable arguments
</code></pre>
<ul>
<li>Is it possible that the problem is related to the 2 predictors that are discrete variables?</li>
<li>Are there already available some routines to perform a discrete discriminant analysis considering continuous and discrete variables?</li>
</ul>
<hr>
<p>When I 'traceback' this is the result: </p>
<pre><code>6: predict.polyreg(object$fit, newdata)
5: predict(object$fit, newdata)
4: predict(object$fit, newdata)
3: predict.fda(FDAT1, matriz_vr)
2: predict(FDAT1, matriz_vr)
1: predict(FDAT1, matriz_vr)
</code></pre>
<p>Unfortunately, I'm not able to interprrt this...
It is some specific error of my commands?</p> | 73,594 | [
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<p>Suppose you had a bag with $n$ tiles, each with a letter on it. There are $n_A$ tiles with letter 'A', $n_B$ with 'B', and so on, and $n_*$ 'wildcard' tiles (we have $n = n_A + n_B + \ldots + n_Z + n_*$). Suppose you had a dictionary with a finite number of words.</p>
<p>You pick $k$ tiles from the bag without replacement.</p>
<p><strong>How would you compute (or estimate) the probability that you can form a given word, of length $l$ (with 1 < $l$ =< $k$) from the dictionary given the $k$ tiles selected?</strong></p>
<p>For those not familiar with Scrabble (TM), the wildcard character can be used to match any letter. Thus the word 'BOOT' could be 'spelled' with the tiles 'B', '*', 'O', 'T'. The order in which the letters are drawn does not matter.</p>
<p>Suggestion: in order to simplify the writing of answers, it might be better to just answer the question: what is the probability of having the word 'BOOT' among your possible moves after drawing 7 letters from a fresh bag.</p>
<p><em>(the problem's introduction has been copied from this similar <a href="http://stats.stackexchange.com/questions/3779/probability-of-not-drawing-a-word-from-a-bag-of-letters-in-scrabble">question</a>)</em></p> | 73,595 | [
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<p>What would be the MLE (Maximum Likelihood Estimator) of the given data:</p>
<p>Assuming the i-th point $ {p}_{i} = ({x}_{i}, {y}_{i}) $.<br>
We have the following measurements of this specific point - $ (x, y, r = \sqrt{{x}_{i}^{2} + {y}_{i}^{2}}) $.<br>
Namely, we have a measurement of the x, y coordinate and the norm of the point with the following distributions:</p>
<p>$$ x \sim N(0, {\sigma}_{x}), y \sim N(0, {\sigma}_{y}), r \sim N(0, {\sigma}_{r}) $$</p>
<p>Given this measurement vector, what would be the MLE of $ {p}_{i} $ coordinates (${x}_{i}, {y}_{i})$? </p>
<p>I know the brute force to calculate it, is there any efficient method for this specific formulation?</p>
<p>What about given a set of points and their corresponding measurements?<br>
Assuming the model of the set of points is linear (polynomial of degree N, etc...).</p>
<p>Has anyone encountered an article or anything about this problem?<br>
What about the Cramer-Rao lower bound, or better bound?</p>
<p>Thank You.</p> | 73,596 | [
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<p>I am analyzing rna-seq data in the format of counts. There is batch effect revealed by PCA. </p>
<p>One method I tried called <strong>RUVseq</strong>, it estimated the variation basing on control genes, and then added it to design matrix. I don't know why simply adding a continuous variable will work.</p>
<p>Here is an example, the experiment setting is:</p>
<pre><code>samples groups
A 1
B 1
C 1
D 2
E 2
F 2
G 3
H 3
I 3
</code></pre>
<p>Then <strong>RUVseq</strong> methods could estimate the unwanted variation, for example, </p>
<pre><code>B <- (-0.37670272 , 2.44136463, -0.79533912, -0.05487747, 0.25014132, 0.61824329 -0.17262350 ,-2.22390027, -1.26361438)
</code></pre>
<p>Then combine them together:</p>
<pre><code>samples groups B
A 1 -0.37670272
B 1 2.44136463
C 1 -0.79533912
D 2 -0.05487747
E 2 0.25014132
F 2 0.61824329
G 3 -0.17262350
H 3 -2.22390027
I 3 -1.26361438
</code></pre>
<p>Make a design matrix to fit glm, an example of design is </p>
<pre><code>model.matrix( ~ 0 + groups + B)
</code></pre>
<p>My question is why this works? The coefficient of continuous variable means how many the read counts of genes will change of one unit of B changed, right? Then why B could be used for correcting the batch effect? </p> | 73,597 | [
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<p>This has been originally asked at <a href="http://math.stackexchange.com/questions/480703/most-common-subgroups?noredirect=1#comment1035243_480703">math.stackexchange</a> where it was suggested that I try Cross Validated.</p>
<p>My background is not mathematics. This is an IT problem that may have a mathematical solution:</p>
<p>I have a large amount of sets of variables. The variables belong to a distinct set of data (for example, each variable may be the age of a person in years, in which case, my potential range has about 100 values). Each set has an unspecified number of data points (for the sake of making things easier to understand, lets assume that each set is the ages of people in a restaurant table): </p>
<ul>
<li>set 1: {1, 100, 2, 3}</li>
<li>set 2: {3, 4, 45, 2 ,1 ,34, 65, 33, 59, 32}</li>
<li>set 3: {40} </li>
<li>etc. etc. </li>
</ul>
<p>I need to identify common subsets within the data. So ideally I would like to look at a few millions of sets and determine that in 20% of these sets you can find the subset (30, 45, 50) for example. Which would then suggest that if you see a 50 year old and a 45 year old, then there is a good chance that a 30 year old will join them - or something similar)</p>
<p>Can anyone provide a few pointers?</p>
<hr>
<p>Some more information which should make this easier to understand:</p>
<p>Again, thanks to @whuber for the guidance.</p>
<ol>
<li>The values are not numbers. I think that the best description I can provide is something like product names. Therefore there is no error in their measurement and there is a finite number of them which is known beforehand.</li>
<li>There is a possibility to associate the values to each other, i.e. 'A' may be closely related to 'B' where as 'A' may be completely unrelated to 'C'. In any case, I may not be able to quantify the relationship, so for now I have to assume that there is no connection between the values. </li>
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<p>I am performing 5-fold cross-validation on a relatively large data set and I have noticed that the validation error for each of the 5 training sets are very similar. So I guess, in this case, cross-validation is not very useful (it would be about the same as just using one training and test set). So I was wondering if I am working with a special case, or is this the case for all large data sets. I'm thinking that perhaps if you have enough training examples, the average cross-validation score would not be very different than the score for one training and test set. Is this intuitition correct?</p> | 73,599 | [
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<p>I'm reading Newey & McFadden - Large sample estimation and hypothesis testing (in the <a href="http://rads.stackoverflow.com/amzn/click/0444887660" rel="nofollow">Handbook of Econometrics, Volume 4</a>, 1994, page 2178).</p>
<p>My model which I'm interested in has some former estimation done before the estimation of the primary model will take place. Hence the primary model (2nd-step) includes some estimated regressors from the former step (the 1st-step).</p>
<p>In order to account for the first-step variance I followed the approach from Newey & McFadden here the joint GMM-conditions are defined as $\widetilde{g}\left(z,\theta,\gamma\right) = \left[g\left(z,\theta,\gamma\right),m\left(z,\gamma\right)\right]$ where $g\left(z,\theta,\gamma\right)$ are the 2nd-step conditions and $m\left(z,\gamma\right)$ are the 1st-step ones.</p>
<p>The usual asymptotic GMM-variance of the GMM-estimator $\widehat{\theta}$ will look like</p>
<p>$Var\left(\widehat{\theta}\right)$ = $\left(G^TWG\right)^{-1}\mathbb{E}\left(g(z,\theta)g(z,\theta)^T\right)\left(G^TWG\right)^{-1}$</p>
<p>where $g(z,\theta)$ are some moment conditions, $G = \frac{\partial\mathbb{E}\left(g(z,\theta_0)\right)}{\partial \theta^T}$ and $W$ is some weight matrix.</p>
<p>In order to calculate the GMM-variance for the two-step approach I need to derive $G$, $W$ and the derivative of $\mathbb{E}$.</p>
<p><strong>My Question: I struggle to understand why this is true.</strong></p>
<p>Let $G_\theta = \frac{\partial\mathbb{E}\left(\widetilde{g}(z,\theta_0, \gamma_0)\right)}{\partial \theta^T}$, $G_\gamma = \frac{\partial\mathbb{E}\left(\widetilde{g}(z,\theta_0, \gamma_0)\right)}{\partial \gamma^T}$, $M = \frac{\partial\mathbb{E}\left(m(z, \gamma_0)\right)}{\partial \gamma^T}$</p>
<p>Then $\widetilde{G} = \frac{\partial\mathbb{E}\left(\widetilde{g}(z,\theta_0, \gamma_0)\right)}{\partial \left(\theta^T, \gamma^T\right)^T} = \begin{Bmatrix} G_\theta & G_\gamma\\ 0 & M\end{Bmatrix}$</p>
<p>This is easy to see but why is the inverse of $G$ this</p>
<p>$\widetilde{G}^{-1} = \begin{Bmatrix} G_\theta^{-1} & -G_\theta^{-1}G_\gamma M^{-1}\\ 0 & M^{-1}\end{Bmatrix}$</p>
<p>I know that the derivative of a 2x2-Jacobian-matrix is if $J = \begin{Bmatrix} x_u & x_v\\ y_u & y_v \end{Bmatrix}$ where $x_u,...$ are the partial derivatives then $J^{-1} = \frac{1}{det\left(J\right)}\begin{Bmatrix} y_v & -x_v\\ -y_u & x_u \end{Bmatrix}$</p>
<p>But then $\widetilde{G}^{-1} = M^{-1}G_\theta^{-1}\begin{Bmatrix} M & -G_\gamma\\ 0 & G_\theta \end{Bmatrix} = \begin{Bmatrix} M^{-1}G_\theta^{-1}M & -M^{-1}G_\theta^{-1}G_\gamma\\ 0 & M^{-1} \end{Bmatrix}$</p>
<p>But this is not the same. <strong>What have I done wrong?</strong></p> | 31,131 | [
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<p>Let X, Y, Z be i.i.d. distributed Gamma random variables. What could the mode of the vector $(X, X+Y, X+Y+Z)$ be?</p>
<p>Does the mode of a random vector equal the combination of the marginal modes?</p> | 36,008 | [
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<p>I'm trying to understand the properties of Mahalanobis distance of multivariate random points (my final goal is to use Mahalanobis distance for outlier detection). My calculations are in python.
I miss some basics here and will be glad if someone will explain me my mistake.</p>
<p>Here is my code:</p>
<p>First I generate 1000 multivariate (2D) random points around the origin</p>
<pre><code>#imports and definitions
import numpy as np
import scipy.stats as stats
import scipy.spatial.distance as distance
chi2 = stats.chi2
np.random.seed(111)
#covariance matrix: X and Y are normally distributed with std of 1
#and are independent one of another
covCircle = array([[1, 0.], [0., 1.]])
circle = multivariate_normal([0, 0], covCircle, 1000) #1000 points around [0, 0]
mahalanobis = lambda p: distance.mahalanobis(p, [0, 0], covCircle.T)
d = np.array(map(mahalanobis, circle)) #Mahalanobis distance values for the 1000 points
d2 = d ** 2 #MD squared
</code></pre>
<p>At this point I expect that <code>d2</code> (Mahalanobis distance values squared) to follow chi2 distribution with 2 degrees of freedom.
I would like to calculate the corresponding probability distribution function (PDF) values and use them for outlier detection.
To illustrate my problem, plot the calculated PDF values for values between 0 and 10 for chi-squared distribution with 2
degrees of freedom and with location and scale values equal to the mean and standard deviation of <code>d2</code>, respectively:</p>
<pre><code>theMean = mean(d2)
theStd = std(d2)
x = linspace(0, 10)
pdf = chi2.pdf(x, 2, theMean, theStd)
plot(x, pdf, '-b')
</code></pre>
<p>Here is the plot:</p>
<p><img src="http://i.stack.imgur.com/hLDyO.png" alt="enter image description here">
As we can see, the PDF values that correspond to <code>X</code> values below ~2.04 are neglegible low, suggesting that there are no points that are closer than 2.04 from the origin, which is obviously not true. What am I missing?</p>
<p><strong>EDIT</strong>
Why am I using <code>theMean</code> and <code>theStd</code> as arguments to the pdf function? As far as I understand, the distances are expected to have chi2 distribution with the given mean and stdev. That is why I use these as arguments to PDF. From a comment to this question I understand that this is not true. So what are the correct parameters of the theoretical chi2 distribution?</p> | 73,600 | [
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<p>I have an explanatory variable that is transformed as suggested in footnote 25 of <a href="http://www.federalreserve.gov/pubs/feds/2013/201353/201353pap.pdf" rel="nofollow">the article</a> as follows (the explanatory variable is continuous and can take negative, zero or positive values)
\begin{equation}
y=\text{sign}(x)\,\log{(|x|+1)}
\end{equation}
where $\text{sign}(x)$ takes a value of $1$ if $x>0$, $0$ if $x=0$, and $-1$ if $x<0$. Let's suppose my dependent variable is $y$ (not transformed). Now I need to take the derivative of $y$ with respect to $x$ to find out the marginal effect and elasticity. </p>
<p>I did this as follows:
\begin{equation}
\frac{\partial y}{\partial x}=\text{sign}(x)\,\frac {1}{(|x|+1)}\,\frac{\partial|x|}{\partial x}
\end{equation}</p>
<p>As far as I understand, $\frac{\partial|x|}{\partial x} =1$ if $x>0$ and $\frac{\partial|x|}{\partial x} =-1$ if $x<0$, and is not defined for $x=0$. My question is how I compute the marginal effect for the observation with the value $x=0$ with this transformation. </p> | 36,010 | [
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<p>I'm trying to understand how I might best model a variable where over time I've obtained increasingly detailed predictors. For example, consider modeling recovery rates on defaulted loans. Suppose we have a dataset with 20 years of data, and in the first 15 of those years we only know whether the loan was collateralized or not, but nothing about the characteristics of that collateral. For the last five years, however, we can break the collateral into a range of categories which are expected to be a good predictor of the recovery rate.</p>
<p>Given this setup I want to fit a model to the data, determine measures such as the statistical significance of the predictors, and then forecast with the model.</p>
<p>What missing data framework does this fit into? Are there any special considerations related to the fact that the more detailed explanatory variables only become available after a given point in time, as opposed to being scattered throughout the historical sample? </p> | 73,601 | [
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<p>I have a specific question about random selection, representativeness and inference. It is well known that it's necessary to use random selection to get representative samples from the population of interest. But what happens with non-random samples? </p>
<p>I am working with an intentional sample. I compared the means of some of the main variables of my database with official data. And I can conclude that there aren't important differences between them. So I have a quite representative sample. However, I am confused about how to deal with this sample. When I estimate the means of some variables or the correlations between them, do I need to compute CI or p-values for this estimations? </p>
<p>I guess that it's not necessary, since I got the sample without random selection. Therefore, I have no sampling error, and I can't know how my estimations differ from the population. However, I have read some papers where the authors work with non-random samples and they make estimations (they use CI's and p-values). Moreover, it's difficult to use multivariate techniques (like ANOVA or regression) without the help of statistical significance.</p>
<p>Can anybody help me? I am very confused with this matter.</p> | 25,048 | [
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<p>When estimating power calculations for a case control study, what does exposure of controls really mean? </p>
<p>I find lots of information how how to calculate exposure for cases when you already know exposure in controls - but nothing on how the actual number is calculated for controls. </p>
<p>Please enlighten me! </p> | 73,602 | [
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<p>this is my first post, so I hope everything is in the right format. I have some problems with glmer and don't know how to fix it, so I hope somebody can help me out with this. I could not find an answer to this anywhere.</p>
<p>My experiment: The experimental setup has eight sites, each site has a central area in which organisms get marked. From the central area, organisms have the choice to go into three different areas which are equipped with traps. The three areas are crossed with sites. The recapture rates in these three areas are very low and the sampling effort (trap days) is very high. Due to external influences, traps got destroyed to different degrees in all the sampling areas. In addition to the marked organisms, unmarked organisms of the same species get caught, too. </p>
<p>The dataset looks something like this:</p>
<pre><code>library(lme4)
data <- structure(list(site = c("A", "A", "A", "B", "B", "B", "C", "C", "C", "D", "D", "D", "E", "E", "E", "F", "F", "F", "G", "G", "G", "H", "H", "H"), area = c("I", "II", "III", "I", "II", "III", "I", "II", "III", "I", "II", "III", "I", "II", "III", "I", "II", "III", "I", "II", "III", "I", "II", "III"), marked = c(2, 6, 3, 5, 3, 9, 0, 8, 1, 1, 1, 18, 3, 0, 0, 1, 5, 6, 3, 0, 2, 2, 4, 5), unmarked = c(38, 78, 104, 1, 6, 10, 1, 13, 0, 13,7, 85, 7, 1, 0, 9, 4, 36, 3, 4, 3, 10, 20, 29), sampl_effort = c(9300, 9100, 8700, 9900, 9600, 8600, 9800, 9400, 10800, 11600, 11000, 13950, 10300, 9700, 9800, 10450, 10100, 10800, 9600, 9900, 9300, 11800, 11250, 9450)), .Names = c("site", "area", "marked", "unmarked", "sampl_effort"), row.names = c(NA,-24), class = "data.frame")
</code></pre>
<p>Now I wanted to fit a glmer. Because the amount of marked organisms may be related to the total amount of organisms, I chose to take a binomial approach, with cbind(marked, unmarked). I use area with the three treatments as explanatory variable, and site as random factor. Because the sampling effort differs between the different areas, I want to include it as an offset. The code looks like this:</p>
<pre><code>mod.glmer1= glmer(cbind(marked,unmarked) ~ area + (1 | site) + offset(sampl_effort), family=binomial, data=data)
</code></pre>
<p>Then, I get the error:</p>
<blockquote>
<p>Error: (maxstephalfit) PIRLS step-halvings failed to reduce deviance in pwrssUpdate</p>
</blockquote>
<p>If I try the glmer without the offset, everything works out fine:</p>
<pre><code>mod.glmer2= glmer(cbind(marked,unmarked) ~ area + (1 | site), family=binomial, data=data)
</code></pre>
<p>Out of interest, I tried a glm without the random factor and with offset:</p>
<pre><code>mod.glm1= glm(cbind(marked,unmarked) ~ area + offset(sampl_effort), family=binomial, data=data)
</code></pre>
<p>and I get the following Warnings:</p>
<blockquote>
<p>Warning messages:
1: glm.fit: fitted probabilities numerically 0 or 1 occurred
2: glm.fit: fitted probabilities numerically 0 or 1 occurred</p>
</blockquote>
<p>By looking at the fitted values</p>
<pre><code>round(cbind(data[,3:5],fits=fitted(mod.glm1)),8)
</code></pre>
<p>I can see that all the fitted values are 0.
I now thought that the offset is just too large to get reasonable fitted value. As the sampling effort is arbitrary (instead of days, I could have taken sampling hours, weeks, etc.), I decided to divide the sampling effort by 1000.
By doing this, the glm now works:</p>
<pre><code>mod.glm2= glm(cbind(marked,unmarked) ~ area + offset(sampl_effort/1000), family=binomial, data=data)
</code></pre>
<p>However, for the glmer</p>
<pre><code>mod.glmer3= glmer(cbind(marked,unmarked) ~ area + (1 | site) + offset(sampl_effort/1000), family=binomial, data=data)
</code></pre>
<p>I still get</p>
<blockquote>
<p>Error: (maxstephalfit) PIRLS step-halvings failed to reduce deviance in pwrssUpdate</p>
</blockquote>
<p>So my questions are basically:</p>
<ol>
<li><p>Am I allowed to divide the sampling effort by 1000? In my mind it should lead to the same results, as the relative differences in sampling effort stay the same (and sampling days are a arbitrary measurement). However, I of course tried it with e.g. dividing by 10000 and I get different results. </p></li>
<li><p>How can I include the sampling effort in the glmer? The sampling effort is just too important to keep it out, however, I also need the sites as random factor. Is the offset the right approach and if yes, why doesn't it work.</p></li>
</ol>
<p>P.S.: My session information:</p>
<blockquote>
<p>R version 3.0.2 (2013-09-25)
Platform: i386-w64-mingw32/i386 (32-bit)</p>
<p>locale:
[1] LC_COLLATE=English_United States.1252 LC_CTYPE=English_United States.1252<br>
[3] LC_MONETARY=English_United States.1252 LC_NUMERIC=C<br>
[5] LC_TIME=English_United States.1252 </p>
<p>attached base packages:
[1] stats graphics grDevices utils datasets methods base </p>
<p>other attached packages:
[1] car_2.0-19 lme4_1.0-6 Matrix_1.1-2 lattice_0.20-23</p>
<p>loaded via a namespace (and not attached):
[1] grid_3.0.2 MASS_7.3-29 minqa_1.2.3 nlme_3.1-111 nnet_7.3-7<br>
[6] Rcpp_0.11.0 splines_3.0.2 tools_3.0.2</p>
</blockquote>
<p>Thanks in advance!
John</p> | 73,603 | [
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<p>I'm doing a Support Vector Regression with about 70k samples with 500 features each. I'm using sklearn implementation of SVR and my input for the train set is a sparse matrix.
But, for my surprise, the training with Linear Kernel is taking a huge amount of time, a lot more than the training with RBF Kernel.</p>
<p>Does someone have a hint of why that happens? For me it seems that this is against intuition, given that RBF Kernel distances seem more expensive to calculate. </p> | 73,604 | [
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<p>Let $X_1, \cdots, X_n$ be $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. How to find $E[\Phi(\bar X)]$, where $\bar X:=\frac{\sum_{i=1}^nX_i}{n}$, please? I guess the answer should be $\Phi(\mu)$. Here is how I started. Note that $Y:= \bar X$ is also normal with mean $\mu$ and variance $\frac{\sigma^2}{n}$.</p>
<p>$$E[\Phi(\bar X)] = \int_{-\infty}^\infty \Phi(y) f_Y(y) dy,$$</p>
<p>where $\Phi(\cdot)$ is the cdf for standard normal random variable. I could not proceed any further from here. Does anyone know what to do, please? Thank you!</p> | 49,565 | [
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<p>Recently, randomly browsing questions triggered a memory of on off-hand comment from one of my professors a few years back warning about the usage of ratios in regression models. So I started reading up on this, leading eventually to Kronmal 1993.</p>
<p>I want to make sure that I'm correctly interpreting his suggestions on how to model these.</p>
<ol>
<li><p>For a model with a ratio with the same denominator on both the dependent and independent side:<br>
$ Z^{-1}Y = Z^{-1}1_n\beta_0 + Z^{-1}X\beta_X + \beta_Z + Z^{-1}\epsilon $</p>
<ul>
<li>Regress dependent ratio on the (inverse) denominator variable in addition to the other ratios</li>
<li>Weight by the (inverse) denominator variable</li>
</ul></li>
<li><p>For a model with dependent variable as a ratio:<br>
$ Y = \beta_0 + \beta_XX + Z1_n\alpha_0 + ZX\alpha_X + Z^{-1}\epsilon $</p>
<ul>
<li>Regress numerator by original variables, denominator, and denominator times original variables [what about categorical variables?]</li>
<li>Weight by (inverse) denominator</li>
</ul></li>
<li><p>For model with only independent variable ratios:
$ Y = \beta_0 + X\beta_X + Z^{-1}1_n\beta_{Z^{-1}} + W\beta_W + Z^{-1}W\beta_{Z^{-1}W} + \epsilon $</p>
<ul>
<li>Include numerator and (inverse) denominator as main effects, ratio as interaction term.</li>
</ul></li>
</ol>
<p>Are my interpretations here correct? Thanks.</p> | 41,280 | [
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<p>I am reading a book on linear regression and have some trouble understanding the variance-covariance matrix of $\mathbf{b}$:</p>
<p><img src="http://i.stack.imgur.com/RIgsS.png" alt="enter image description here"></p>
<p>The diagonal items are easy enough, but the off-diagonal ones are a bit more difficult, what puzzles me is that
$$
\sigma(b_0, b_1) = E(b_0 b_1) - E(b_0)E(b_1) = E(b_0 b_1) - \beta_0 \beta_1
$$</p>
<p>but there is no trace of $\beta_0$ and $\beta_1$ here.</p> | 73,605 | [
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<p>I'm trying to reproduce a derivation of the cokriging equations that involves the cross covariance and the cross variogram. This derivation is included in the Multivariate Geostatistics by Wackernagel. However, I can't reproduce the following result:</p>
<p>\begin{align}{}\sigma_{E}^{2} &= \sum_{i=1}^{N} \sum_{j=1}^{N} \sum_{\alpha=0}^{n_{i}}\sum_{\beta=0}^{n_{j}} w_{\alpha}^{i}w_{\beta}^{j}C_{ij}^{I}(x_{\alpha},x_{\beta})\\
&=2\sum_{i=1}^{N}\sum_{\alpha=1}^{n_{i}}w_{\alpha}^{i}\gamma_{ii_{0}}(x_{\alpha} -x_{0})-\gamma_{i_{0}i_{0}}(x_{0} -x_{0})\\
&-\sum_{i=1}^{N} \sum_{j=1}^{N} \sum_{\alpha=1}^{n_{i}}\sum_{\beta=1}^{n_{j}} w_{\alpha}^{i}w_{\beta}^{j}\gamma_{ij}(x_{\alpha} -x_{\beta})\end{align}</p>
<p>The cross covariance of the increments is defined as</p>
<p>$$C_{ij}^{I}= E[(Z_{i}(x_{\alpha})-Z_{i}(0))(Z_{j}(x_{\beta})-Z_{j}(0))]$$</p>
<p>and the cross variogram is defined as</p>
<p>$$\gamma_{ij}(h) = \frac{1}{2}E[(Z_{i}(x+h)-Z_{i}(x))(Z_{j}(x+h)-Z_{j}(x))]$$</p>
<p>There are a couple of conditions for the weights $w_{\alpha}^{i}$:</p>
<p>\begin{equation}
\sum_{\alpha=1}^{n_{i}}w_{\alpha}^{i} = \delta_{ii_{0}}\\
w_{0}^{i} = -\delta_{ii_{0}}
\end{equation}</p>
<p>where $\delta_{ii_{0}} = 1$ if $i=i_{0}$ and $\delta_{ii_{0}} = 0$ otherwise.</p>
<p>I think I can get to this expression that is extremely similar to the required equality</p>
<p>\begin{align}{}\sigma_{E}^{2}
&=2\sum_{i=1}^{N}\sum_{\alpha=1}^{n_{i}}w_{\alpha}^{i}C_{ii_{0}}(x_{\alpha},x_{0})-C_{i_{0}i_{0}}(x_{0},x_{0})\\
&-\sum_{i=1}^{N} \sum_{j=1}^{N} \sum_{\alpha=1}^{n_{i}}\sum_{\beta=1}^{n_{j}} w_{\alpha}^{i}w_{\beta}^{j}C_{ij}(x_{\alpha},x_{\beta})\end{align}</p>
<p>Next, I tried to obtain an expression for the cross covariances of the increments $C_{ij}^{I}$ in terms of the cross variogram $\gamma_{ij}(h)$ using the trick of adding 0 (with a random variable $Z(0)$) to the products of the variogram:</p>
<p>$$\gamma_{ij}(h) = \frac{1}{2}E[(Z_{i}(x+h)-Z_{i}(x)+Z_{i}(0)-Z_{i}(0))(Z_{j}(x+h)-Z_{j}(x)+Z_{j}(0)-Z_{j}(0))]$$</p>
<p>which produces the following:</p>
<p>$$C_{ij}^{I}(x, x+h)=-\gamma_{ij}(h)+\gamma_{ij}(x+h)+\gamma_{ij}(x)$$</p>
<p>However, when I substitute this in the equation for $\sigma_{E}^{2}$, there are extra terms that should not be there. So most likely I'm making a mistake.</p>
<p>I would appreciate any help.</p> | 73,606 | [
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<p>What statistical analysis would I use when testing 35 participants. pre & post test and checking for changes after 1 intervention and a control group with no intervention?
I have one mediating variable, 3 independent variables and two dependent variables.</p> | 73,607 | [
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<p>I am playing with <code>scikit-learn</code> to find the <code>tf-idf</code> values. </p>
<p>I have a set of <code>documents</code> like:</p>
<pre><code>D1 = "The sky is blue."
D2 = "The sun is bright."
D3 = "The sun in the sky is bright."
</code></pre>
<p>I want to create a matrix like this:</p>
<pre><code> Docs blue bright sky sun
D1 tf-idf 0.0000000 tf-idf 0.0000000
D2 0.0000000 tf-idf 0.0000000 tf-idf
D3 0.0000000 tf-idf tf-idf tf-idf
</code></pre>
<p>So, my code in <code>Python</code> is:</p>
<pre><code>import nltk
import string
from sklearn.feature_extraction.text import TfidfVectorizer
from nltk.corpus import stopwords
train_set = ["sky is blue", "sun is bright", "sun in the sky is bright"]
stop_words = stopwords.words('english')
transformer = TfidfVectorizer(stop_words=stop_words)
t1 = transformer.fit_transform(train_set).todense()
print t1
</code></pre>
<p>The result matrix I get is:</p>
<pre><code>[[ 0.79596054 0. 0.60534851 0. ]
[ 0. 0.4472136 0. 0.89442719]
[ 0. 0.57735027 0.57735027 0.57735027]]
</code></pre>
<p>If I do a hand calculation then the matrix should be:</p>
<pre><code> Docs blue bright sky sun
D1 0.2385 0.0000000 0.0880 0.0000000
D2 0.0000000 0.0880 0.0000000 0.0880
D3 0.0000000 0.058 0.058 0.058
</code></pre>
<p>I am calculating like say <code>blue</code> as <code>tf</code> = <code>1/2 = 0.5</code> and <code>idf</code> as <code>log(3/1) = 0.477121255</code>. Therefore <code>tf-idf = tf*idf = 0.5*0.477 = 0.2385</code>. In this way, I am calculating the other <code>tf-idf</code> values. Now, I am wondering, why I am getting different results in the matrix of hand calculation and in the matrix of Python? Which gives the correct results? Am I doing something wrong in hand calculation or is there something wrong in my Python code?</p> | 36,015 | [
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<p>I'm working on a project and I would like to know if the following strategy is good/correct. Sorry if this is a basic/stupid idea (I'm new to this).</p>
<p>The input is a dataset with 2.500 features and 1.000 instances. I have to apply a feature selection on this set. In the beginning I randomly make a learning set (which contains 70% of the original data) and a test set (with the remaining 30%). </p>
<p>Now I split the learning set in 10 folds (to do cross-validation), so we've split the learning set into 10 (more or less equal) parts and now we take 9 parts out of 10 to do a feature ranking (using chi squared), then we take the best 10 features (highest ranking) to do a model evaluation (using SVM) on the remaining part of the 'splitted learning set' (and we do this 10 times, so each part will be used as a 'test set'). </p>
<p>Doing this '10 folds' argument gives us 10 feature rankings (with 10 model evaluation/accuracies). Out of these 10 results we can take the best feature ranking (according to it's accuracy based on the model or to take the sum of the feature rankings) and then finally test it on the 30% test set from the beginning.</p>
<p>Please give your opinion whether this is good or really stupid...</p> | 73,608 | [
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<p>I wonder how to compute precision and recall using a confusion matrix for a multi-class classification problem. In specific, one observation can only be assigned with most probable class / label. </p>
<p>I would like to compute </p>
<ul>
<li>Precision = TP / (TP+FP) </li>
<li>Recall = TP / (TP+FN)</li>
</ul>
<p>for each class, and then compute the micro-averaged F-Measure.</p> | 73,609 | [
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<p>I conducted a panel OLS regression using gretl, on a balanced panel with around 25 zones and 10 time periods. I would like to use the model to forecast future values of y (dependant variable), given known future changes of x (explanatory variable).</p>
<p>I would like to ask: how to deal with (a) the constant, (b) the unit dummies; (c) the time dummies when it comes to forecasting. Do i have to include any/all of the above in computing y-hat, or simply use the coefficient for the x term?</p> | 73,610 | [
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<p>Suppose I have $X,Y$, which are independent random variables. </p>
<p>Why is it that $E(\frac{X}{Y}) = E(X)E(\frac{1}{Y})$? </p>
<p>Also, why is it that $E(X^2Y^2)=E(X^2)E(Y^2)$? How is it that the square of an independent random variable is also independent in relation to $Y$ or $Y^2$? Thanks! </p> | 73,611 | [
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<p>I have two sets of a longitudinal data that I hypothesize to measure same latent construct.I am trying to test this hypothesis using Structural equation modelling technique. Basically, I am trying to use confirmatory factor analysis in longitudinal data using SEM. My models are relatively simple CFA models.</p>
<p>The problem is that my sample size is small in term of SEM standards. We collected data only from 20 individuals for nearly two years (number of time points nearly equals 18) because the data collection is very expensive.</p>
<p>I am new to SEM and I was not able to decide if SEM is right choice for me. My question is given such a small sample size, is it a good idea to use SEM based techniques to model the data and test my hypothesis? Based on my researches, I concluded that I should not fit SEM using least squares, weighted least squares or Maximum likelihood since they require large samples which I do not have. It seems that they recommend fitting SEM using Partial least squares or Baysian approaches for small sample size but the problem is my sample size is very small. Are there any articles that discuss fitting SEM using very small sample size? </p>
<p>I am looking for suggestions, literatures or recomendations to fit such data with small sample size.</p> | 73,612 | [
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<p>I am working with R package "muhaz" to get the kernel estimate of the hazard function. If we have the following example which is the sample of uncensored data:</p>
<pre><code> x =c(0.001409591, 0.008451294, 0.019719245, 0.046626865 ,0.244813744,
0.410032119, 0.489286425, 0.697342699, 0.731541987, 0.954493031)
fit = muhaz(x, rep(1, length(x)), bw.method="local", b.cor="both", n.min.grid=51,
n.est.grid=101, kern="epanechnikov")
</code></pre>
<p>I have two questions:
1) How can I write the estimated hazard function as a formula in terms of the result of fit?
How can we get the estimated hazard function for any desire value not just for grid points (fit$est.grid points)?</p>
<p>2) Is it possible to get the estimated density function? because as far as I know the result does not provide it. How can we get the estimated density function for any desire value? not just for grid points (fit$est.grid points).</p> | 73,613 | [
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<p>Almost in all texts which are discussing theorems of statistical learning, they assume analyzing arbitrary unknown distribution (the worst case). But in practice different problems (different data) have different levels of hardness, for example linear separable data is easier to learn than the data that are less (or not) separable by hyperplanes.
Are there any works on formalizing the hardness of data (similar to the stuff done in complexity theory)</p>
<p>Note: I am not sure if the same as analyzing VC-dimension or not, but I think it is not. </p> | 73,614 | [
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<p>With respect to post <a href="http://stats.stackexchange.com/questions/7977/how-to-generate-uniformly-distributed-points-on-the-surface-of-the-3-d-unit-sphe">(1)</a> and post <a href="http://math.stackexchange.com/questions/87230/picking-random-points-in-the-volume-of-sphere-with-uniform-probability">(2)</a>, I generated a large number of uniformly distributed points inside the ball of radius $R$ using $\frac{R_s U^{1/3}}{\sqrt{X_1^2 + X_2^2 + X_3^2}} (X_1, X_2, X_3)$, where $U$ is uniformly distributed between 0 and 1, and $X_1, X_2, X_3$ are independent normal random variables with mean 0 and variance 1.
The following figure shows a uniform spherical distribution obtained by this method using 10000 independent draws in a sphere of radius 10.
<img src="http://i.stack.imgur.com/kzBDU.png" alt="enter image description here"></p>
<p>By computing the nearest neighbour distance $d_i$ of every point, I observed that the diagnostic plot of nearest neighbour distances does not follow a uniform distribution. Does this non-uniform distribution mean that one can cluster the points? Does it mean points dont have spatial randomness? If so, then how can I generate random points with uniform nearest neighbour distances.</p>
<hr>
<p>Temporary images for @Anony-Mousse consideration:
<img src="http://i.stack.imgur.com/txV1K.png" alt="enter image description here">
<img src="http://i.stack.imgur.com/1wLfB.png" alt="enter image description here"></p> | 73,615 | [
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<p>For example, the standard deviation is known to have bias that depends on the number of samples observed. If I wanted to do Bayesian inference on the SD of samples from two populations, and have unequal sample sizes from each, would the resulting inference take this bias into account?</p>
<p>(Yes, I'm aware that the bias in the standard deviation doesn't really manifest discernibly unless you have very small N, but I'm more interested in the theoretical question. Plus, I do have hierarchical data with lots of units of observation, but few observations within each unit and a systematic bias for more observations in one particular experimental condition than another, causing me trepidation in my desired analysis of the SD within each level of unit-crossed-with-condition.)</p> | 25,055 | [
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<p>This is a real small business problem, not a homework problem.</p>
<p>A monthly production goal is 60000. We want to be sure that say 95% of months meet this goal.
The goal is a minimum, so we want to be sure we aren't under this, but we don't care how much we are over.</p>
<p>We want to use daily production as a check on whether we are likely to meet the monthly production goal. We don't really care how low one day's production is as long as other days make up for it, but if we are too low one day it should be a signal that there will be a problem in meeting the monthly production goal.</p>
<p>How do we determine what is too low for daily production?</p>
<p>The simplest approach is to just divide the monthly production by the number of workdays (say 16) to get daily production goals. This works, in the sense of if you meet each daily production goal you will meet the monthly production goal, but it ignores the natural variability in daily production. It is too severe. We should be able to have daily production that is some amount less than this without having to be worried yet.</p>
<p>You can assume a normal distribution of daily production with errors that are independent from day to day, and that we have a history of daily production from which we can estimate the variance in daily production.</p>
<p>Thanks for your help.</p> | 73,616 | [
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<p>Suppose $a_1 = b + c_1$ and $a_2 = 2b + c_2$ where $b, c_1, c_2$ are all $N(0,1)$</p>
<p>Find $E[b|a_1,a_2]$</p>
<p>My attempt:
As $E[b] = 0$, I assume $E[b|a_1, a_2] = 0$. Is this a logical assumption?</p> | 73,617 | [
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<p>I'm reproducing an example from <a href="http://rads.stackoverflow.com/amzn/click/0470073713" rel="nofollow">Generalized, Linear, and Mixed Models</a>. My MWE is below:</p>
<pre><code>Dilution <- c(1/128, 1/64, 1/32, 1/16, 1/8, 1/4, 1/2, 1, 2, 4)
NoofPlates <- rep(x=5, times=10)
NoPositive <- c(0, 0, 2, 2, 3, 4, 5, 5, 5, 5)
Data <- data.frame(Dilution, NoofPlates, NoPositive)
fm1 <- glm(formula=NoPositive/NoofPlates~log(Dilution), family=binomial("logit"), data=Data)
summary(object=fm1)
</code></pre>
<hr>
<p><strong>Output</strong></p>
<hr>
<pre><code>Call:
glm(formula = NoPositive/NoofPlates ~ log(Dilution), family = binomial("logit"),
data = Data)
Deviance Residuals:
Min 1Q Median 3Q Max
-0.38326 -0.20019 0.00871 0.15607 0.48505
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 4.174 2.800 1.491 0.136
log(Dilution) 1.623 1.022 1.587 0.112
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 8.24241 on 9 degrees of freedom
Residual deviance: 0.64658 on 8 degrees of freedom
AIC: 6.8563
Number of Fisher Scoring iterations: 6
</code></pre>
<hr>
<p><strong>Code</strong></p>
<hr>
<pre><code>anova(object=fm1, test="Chisq")
</code></pre>
<hr>
<p><strong>Output</strong></p>
<hr>
<pre><code>Analysis of Deviance Table
Model: binomial, link: logit
Response: NoPositive/NoofPlates
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev Pr(>Chi)
NULL 9 8.2424
log(Dilution) 1 7.5958 8 0.6466 0.00585 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
</code></pre>
<hr>
<p><strong>Code</strong></p>
<hr>
<pre><code>library(aod)
wald.test(b=coef(object=fm1), Sigma=vcov(object=fm1), Terms=2)
</code></pre>
<hr>
<p><strong>Output</strong></p>
<hr>
<pre><code>Wald test:
----------
Chi-squared test:
X2 = 2.5, df = 1, P(> X2) = 0.11
</code></pre>
<p>Estimated coefficients are perfectly matching with the results given in the book but SE's are far apart. Based on LRT test the slope is significant but based on Wald and Z-test slope coefficient is insignificant. I wonder if I miss something basic. Thanks in advance for your help.</p> | 73,618 | [
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<p>I own a url shortening service. I want to deliver only legitimate statistics to my clients. There are possible scenarios that a particular user writes a script to automatically open the shortened URL, thus making the statistics look bad. What are the approaches one can follow to detect if a click is legitimate or not? Please point me to appropriate resources like tools or research papers. The very basic approach that I can think of is to monitor the IP address of the user and block if the number of requests exceed a threshold.</p> | 73,619 | [
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<p>From <a href="http://cran.r-project.org/web/packages/epiR/epiR.pdf" rel="nofollow">epiR manual</a> for <strong><em>epi.2by2</em></strong> function:</p>
<blockquote>
<p>units: multiplier for prevalence and incidence estimates.</p>
</blockquote>
<p>In examples it is set to 100, <code>units=100</code>. What does it mean?</p>
<p><strong>Note:</strong> I am working on ~5K case ~5K control data, no stratas, outcome vs exposure - 2by2 tables.</p>
<p><strong>Update:</strong> This <a href="http://www.medicalbiostatistics.com/prevalenceincidence.pdf" rel="nofollow">PDF</a> might be relevant.</p> | 73,620 | [
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<p>I know this might be a little ropey, statistically, but this is my problem.</p>
<p>I have a lot of range data, that is to say the minimum, maximum and sample size of a variable. For some of these data I also have a mean, but not many. I want to compare these ranges to each other to quantify the variability of each range, and also to compare the means. I have a good reason to assume that the distribution is symmetrical around the mean, and that the data will have a Gaussian distribution. For this reason I am thinking I can justify using the mid-point of the distribution as a proxy for the mean, when it is absent.</p>
<p>What I want to do is reconstruct a distribution for each range, and then use that to provide a standard deviation or standard error for that distribution. The only information I have is the max and min observed from a sample, and the mid-point as a proxy for the mean.</p>
<p>In this way I hope to be able to calculate weighted means for each group, and also to work out the coefficient of variation for each group as well, based on the range data I have and my assumptions (of a symmetrical and normal distribution).</p>
<p>I plan to use R to do this, so any code help would be appreciated as well.</p> | 73,621 | [
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<p>Give the following table</p>
<pre><code> Male | Female | Total
Likes 10 | 3 | 13
Indifferent 20 | 20 | 40
Dislikes 10 | 9 | 19
Total 40 32 72
</code></pre>
<p>Here the independent variable is gender which is nominal. The dependent variable which is ordinal is either likers, indifferent, or dislikes. I can use the Chi Square test or Exact Fisher to determine if a H0 such as <em>men and women are equally likely to be likers</em> should be rejected.</p>
<p>However, the question I would like to answer after this is: <strong>which gender is significantly more likely than the other to be a liker? And, by how much?</strong></p>
<p>For example: Men are x% significantly more likely than Women to be likers. </p>
<p>I thought that the next step would be to perform a test between liker and each group: i.e. males and likers. But I got stumped because I thought I should be comparing males to females with respect to likers. </p>
<p>Is there any standard way of approaching this question?</p> | 73,622 | [
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<p>I'm trying to understand the assumptions of different parameterizations in a Markov network. In this case, I'm trying to understand the assumptions (and effects) that result from parameterizing pairwise potentials with pairwise frequencies (i.e., $\psi_{(a, b)} = p(a, b)$), which the authors use in:</p>
<p>Mitrofanova, A. et al. Prediction of Protein Functions with Gene Ontology and Interspecies Protein Homology Data. IEEE/ACM Trans. Comput. Biol. and Bioinf., 8, 775–784.</p>
<p>Let's say we have a 3 node network (a)-(b)-(c), where the RV are Bernoulli. By chain rule of probability (and independence), in all cases, $p(a, b, c) = p(a) p(b|a) p(c|b)$.</p>
<p>First, if we define $p(a, b, c) = \psi_{a, b}(a,b) \psi_{b, c}(b,c)$, where $\psi_{a, b} = p(a, b)$ and $\psi_{b, c} = p(b, c)$, then it appears that $p(b)$ must be equal to 1 (or 0). What would happen in larger, loopy networks?</p>
<p>Second, if we use the same number of free parameters, constrained so each local potential sums to one, and learned the parameters (but the potentials need not be marginals), then I calculate that $p(b) = \psi_1(b) \psi_2(b)$, where $\psi_1(b) = \Sigma_a\psi_{b,c}$. As long as I don't assume $\psi_1(b) = \psi_2(b) = p(b)$, then is it a valid, useful parameterization? Are there any assumptions?</p>
<p>Third, let's now remove the constraint on the local potentials, and add in a global normalization constant instead. If I compare $p(a) p(b|a) p(c|b)$ to $\frac{1}{Z} p(a,b) p(c,b)$, I can see that $Z = p(b)$. That's good.</p>
<p>If, however, I try to calculate $Z$ by summing over all combinations of $a, b, c$, then I get that Z equals $\Sigma_b{p(b)p(b)}$. What happened?</p> | 73,623 | [
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<p>I am a little new to neural network. I have two questions:
1. Can we use neural network in the small $n$ large $p$ situation?
2. Any regularization methods designed for parameter elimination? (Shrink some $\omega$ to 0). For example to add $L_1$ regularizer as Lasso? I know some methods for avoiding over-fitting, (weight decay, early stopping, ect.) but none of them shrink parameters to 0. </p> | 73,624 | [
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<p>I have two columns of data, and would like to show their differences through the visualization approach. The current issue is that these two columns are in-fact very close to each other. In other words, I would like to zoom in the minor differences that are existed using a graphical way. Are there any suggestions? Thanks.</p>
<p>Here is part of the data set. The third column is the difference between the first column and second column.</p>
<pre><code>0.999998295 0.999998331 -3.56E-08
0.99999381 0.999986678 7.13E-06
0.999997682 0.999998494 -8.12E-07
0.999989429 0.999989062 3.67E-07
0.999998877 0.999998505 3.72E-07
0.999995327 0.999996709 -1.38E-06
0.999995779 0.999995146 6.33E-07
0.999997484 0.999996701 7.83E-07
0.999998829 0.999998076 7.53E-07
0.999997523 0.999998836 -1.31E-06
0.999998287 0.999996964 1.32E-06
0.999983543 0.999989135 -5.59E-06
0.999992705 0.999994249 -1.54E-06
0.999995663 0.999998168 -2.50E-06
0.999997263 0.999999195 -1.93E-06
0.999994474 0.999996713 -2.24E-06
0.999993058 0.999994063 -1.00E-06
0.999998295 0.999996262 2.03E-06
0.99999736 0.999998185 -8.25E-07
0.999983507 0.996875105 0.003108402
0.999996339 0.999996597 -2.58E-07
</code></pre> | 36,029 | [
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<p>How can I fit a copula for a bivariate vector of negative binomial and Bernoulli margins?</p>
<p>I would prefer a Frank or Clayton copula.</p> | 39,251 | [
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<p>I have a HLM model with significant variance in the level 1 intercepts across groups but no significant variance in the level 1 slopes across groups and find significant cross-level moderation effects. Does it make sense to interpret these or are random slopes a necessary condition for probing cross-level interaction effects? </p> | 73,625 | [
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<p>Suppose I have to estimate 3 models:</p>
<p>$y_1 = y_2\beta_1 + y_3\beta_2 + X\beta_3 + u_1$</p>
<p>$y_2 = y_1\alpha_1 + y_3\alpha_2 + X\alpha_3 + u_2$</p>
<p>$y_3 = y_1\gamma_1 + y_2\gamma_2 + X\gamma_3 + u_3$</p>
<p>I account for the endogeneity of $y_1, y_2$ and $y_3$ with a approach which creates a additional variable which allows me to not exlude any other regressors.</p>
<p>Suppose these additional regressors are $c_1$ for $y_1$, $c_2$ for $y_2$ and $c_3$ for $y_3$. Now the models will be:</p>
<p>$y_1 = y_2\beta_1 + y_3\beta_2 + X\beta_3 + c_2\beta_4 + c_3\beta_5 + u_1$</p>
<p>$y_2 = y_1\alpha_1 + y_3\alpha_2 + X\alpha_3 + c_1\alpha_4 + c_3\alpha_5 + u_2$</p>
<p>$y_3 = y_1\gamma_1 + y_2\gamma_2 + X\gamma_3 + c_1\gamma_4 + c_2\gamma_5 + u_2$</p>
<p>Given that the influence of the $c_i$ in each equation is significant, identification is achieved. The reason is that in each equation one $c_i$ is missing (for example $c_1$ in the $y_1$ equation).</p>
<p>It turns out that I cannot leave $c_1$ out of the equation regarding $y_1$ (same for all the other equations). If I would, the model would be seriously biased. </p>
<p>My questions are: When I estimate the full model, i.e., all $c_i$ are included in every equation and some of the regressors contained in $X$ are not signficant and these regressors are all different for each equation, is identification achieved?</p>
<p>I would say yes. But this approach does only achieve identification through the insignificance of a regressor. I'm not quite sure if this is the right angle to it.</p> | 36,030 | [
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