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Algae | 633 | Numbers | The most recent estimate suggests 72,500 algal species worldwide. |
Algae | 633 | Distribution | The distribution of algal species has been fairly well studied since the founding of phytogeography in the mid-19th century. Algae spread mainly by the dispersal of spores analogously to the dispersal of Plantae by seeds and spores. This dispersal can be accomplished by air, water, or other organisms. Due to this, spores can be found in a variety of environments: fresh and marine waters, air, soil, and in or on other organisms. Whether a spore is to grow into an organism depends on the combination of the species and the environmental conditions where the spore lands. |
Algae | 633 | Distribution | The spores of freshwater algae are dispersed mainly by running water and wind, as well as by living carriers. However, not all bodies of water can carry all species of algae, as the chemical composition of certain water bodies limits the algae that can survive within them. Marine spores are often spread by ocean currents. Ocean water presents many vastly different habitats based on temperature and nutrient availability, resulting in phytogeographic zones, regions, and provinces. |
Algae | 633 | Distribution | To some degree, the distribution of algae is subject to floristic discontinuities caused by geographical features, such as Antarctica, long distances of ocean or general land masses. It is, therefore, possible to identify species occurring by locality, such as "Pacific algae" or "North Sea algae". When they occur out of their localities, hypothesizing a transport mechanism is usually possible, such as the hulls of ships. For example, Ulva reticulata and U. fasciata travelled from the mainland to Hawaii in this manner. |
Algae | 633 | Distribution | Mapping is possible for select species only: "there are many valid examples of confined distribution patterns." For example, Clathromorphum is an arctic genus and is not mapped far south of there. However, scientists regard the overall data as insufficient due to the "difficulties of undertaking such studies." |
Algae | 633 | Ecology | Algae are prominent in bodies of water, common in terrestrial environments, and are found in unusual environments, such as on snow and ice. Seaweeds grow mostly in shallow marine waters, under 100 m (330 ft) deep; however, some such as Navicula pennata have been recorded to a depth of 360 m (1,180 ft). A type of algae, Ancylonema nordenskioeldii, was found in Greenland in areas known as the 'Dark Zone', which caused an increase in the rate of melting ice sheet. Same algae was found in the Italian Alps, after pink ice appeared on parts of the Presena glacier. |
Algae | 633 | Ecology | The various sorts of algae play significant roles in aquatic ecology. Microscopic forms that live suspended in the water column (phytoplankton) provide the food base for most marine food chains. In very high densities (algal blooms), these algae may discolor the water and outcompete, poison, or asphyxiate other life forms. |
Algae | 633 | Ecology | Algae can be used as indicator organisms to monitor pollution in various aquatic systems. In many cases, algal metabolism is sensitive to various pollutants. Due to this, the species composition of algal populations may shift in the presence of chemical pollutants. To detect these changes, algae can be sampled from the environment and maintained in laboratories with relative ease. |
Algae | 633 | Ecology | On the basis of their habitat, algae can be categorized as: aquatic (planktonic, benthic, marine, freshwater, lentic, lotic), terrestrial, aerial (subaerial), lithophytic, halophytic (or euryhaline), psammon, thermophilic, cryophilic, epibiont (epiphytic, epizoic), endosymbiont (endophytic, endozoic), parasitic, calcifilic or lichenic (phycobiont). |
Algae | 633 | Cultural associations | In classical Chinese, the word 藻 is used both for "algae" and (in the modest tradition of the imperial scholars) for "literary talent". The third island in Kunming Lake beside the Summer Palace in Beijing is known as the Zaojian Tang Dao (藻鑒堂島), which thus simultaneously means "Island of the Algae-Viewing Hall" and "Island of the Hall for Reflecting on Literary Talent". |
Algae | 633 | Cultivation | Algaculture is a form of aquaculture involving the farming of species of algae. |
Algae | 633 | Cultivation | The majority of algae that are intentionally cultivated fall into the category of microalgae (also referred to as phytoplankton, microphytes, or planktonic algae). Macroalgae, commonly known as seaweed, also have many commercial and industrial uses, but due to their size and the specific requirements of the environment in which they need to grow, they do not lend themselves as readily to cultivation (this may change, however, with the advent of newer seaweed cultivators, which are basically algae scrubbers using upflowing air bubbles in small containers). |
Algae | 633 | Cultivation | Commercial and industrial algae cultivation has numerous uses, including production of nutraceuticals such as omega-3 fatty acids (as algal oil) or natural food colorants and dyes, food, fertilizers, bioplastics, chemical feedstock (raw material), protein-rich animal/aquaculture feed, pharmaceuticals, and algal fuel, and can also be used as a means of pollution control and natural carbon sequestration. |
Algae | 633 | Cultivation | Seaweed farming or kelp farming is the practice of cultivating and harvesting seaweed. In its simplest form farmers gather from natural beds, while at the other extreme farmers fully control the crop's life cycle. |
Algae | 633 | Cultivation | The seven most cultivated taxa are Eucheuma spp., Kappaphycus alvarezii, Gracilaria spp., Saccharina japonica, Undaria pinnatifida, Pyropia spp., and Sargassum fusiforme. Eucheuma and K. alvarezii are attractive for carrageenan (a gelling agent); Gracilaria is farmed for agar; the rest are eaten after limited processing. Seaweeds are different from mangroves and seagrasses, as they are photosynthetic algal organisms and are non-flowering. |
Algae | 633 | Cultivation | The largest seaweed-producing countries are China, Indonesia, and the South Korea. Other notable producers include Philippines, North Korea, Japan, Malaysia, and Zanzibar (Tanzania). Seaweed farming has frequently been developed to improve economic conditions and to reduce fishing pressure. |
Algae | 633 | Cultivation | The Food and Agriculture Organization (FAO), reported that world production in 2019 was over 35 million tonnes. North America produced some 23,000 tonnes of wet seaweed. Alaska, Maine, France, and Norway each more than doubled their seaweed production since 2018. As of 2019, seaweed represented 30% of marine aquaculture. |
Algae | 633 | Uses | Agar, a gelatinous substance derived from red algae, has a number of commercial uses. It is a good medium on which to grow bacteria and fungi, as most microorganisms cannot digest agar. |
Algae | 633 | Uses | Alginic acid, or alginate, is extracted from brown algae. Its uses range from gelling agents in food, to medical dressings. Alginic acid also has been used in the field of biotechnology as a biocompatible medium for cell encapsulation and cell immobilization. Molecular cuisine is also a user of the substance for its gelling properties, by which it becomes a delivery vehicle for flavours. |
Algae | 633 | Uses | Between 100,000 and 170,000 wet tons of Macrocystis are harvested annually in New Mexico for alginate extraction and abalone feed. |
Algae | 633 | Uses | To be competitive and independent from fluctuating support from (local) policy on the long run, biofuels should equal or beat the cost level of fossil fuels. Here, algae-based fuels hold great promise, directly related to the potential to produce more biomass per unit area in a year than any other form of biomass. The break-even point for algae-based biofuels is estimated to occur by 2025. |
Algae | 633 | Uses | For centuries, seaweed has been used as a fertilizer; George Owen of Henllys writing in the 16th century referring to drift weed in South Wales: |
Algae | 633 | Uses | This kind of ore they often gather and lay on great heapes, where it heteth and rotteth, and will have a strong and loathsome smell; when being so rotten they cast on the land, as they do their muck, and thereof springeth good corn, especially barley ... After spring-tydes or great rigs of the sea, they fetch it in sacks on horse backes, and carie the same three, four, or five miles, and cast it on the lande, which doth very much better the ground for corn and grass. |
Algae | 633 | Uses | Today, algae are used by humans in many ways; for example, as fertilizers, soil conditioners, and livestock feed. Aquatic and microscopic species are cultured in clear tanks or ponds and are either harvested or used to treat effluents pumped through the ponds. Algaculture on a large scale is an important type of aquaculture in some places. Maerl is commonly used as a soil conditioner. |
Algae | 633 | Uses | Naturally growing seaweeds are an important source of food, especially in Asia, leading some to label them as superfoods. They provide many vitamins including: A, B1, B2, B6, niacin, and C, and are rich in iodine, potassium, iron, magnesium, and calcium. In addition, commercially cultivated microalgae, including both algae and cyanobacteria, are marketed as nutritional supplements, such as spirulina, Chlorella and the vitamin-C supplement from Dunaliella, high in beta-carotene. |
Algae | 633 | Uses | Algae are national foods of many nations: China consumes more than 70 species, including fat choy, a cyanobacterium considered a vegetable; Japan, over 20 species such as nori and aonori; Ireland, dulse; Chile, cochayuyo. Laver is used to make laverbread in Wales, where it is known as bara lawr. In Korea, green laver is used to make gim. It is also used along the west coast of North America from California to British Columbia, in Hawaii and by the Māori of New Zealand. Sea lettuce and badderlocks are salad ingredients in Scotland, Ireland, Greenland, and Iceland. Algae is being considered a potential solution for world hunger problem. |
Algae | 633 | Uses | Two popular forms of algae are used in cuisine: |
Algae | 633 | Uses | Furthermore, it contains all nine of the essential amino acids the body does not produce on its own |
Algae | 633 | Uses | The oils from some algae have high levels of unsaturated fatty acids. For example, Parietochloris incisa is high in arachidonic acid, where it reaches up to 47% of the triglyceride pool. Some varieties of algae favored by vegetarianism and veganism contain the long-chain, essential omega-3 fatty acids, docosahexaenoic acid (DHA) and eicosapentaenoic acid (EPA). Fish oil contains the omega-3 fatty acids, but the original source is algae (microalgae in particular), which are eaten by marine life such as copepods and are passed up the food chain. Algae have emerged in recent years as a popular source of omega-3 fatty acids for vegetarians who cannot get long-chain EPA and DHA from other vegetarian sources such as flaxseed oil, which only contains the short-chain alpha-linolenic acid (ALA). |
Algae | 633 | Uses | Agricultural Research Service scientists found that 60–90% of nitrogen runoff and 70–100% of phosphorus runoff can be captured from manure effluents using a horizontal algae scrubber, also called an algal turf scrubber (ATS). Scientists developed the ATS, which consists of shallow, 100-foot raceways of nylon netting where algae colonies can form, and studied its efficacy for three years. They found that algae can readily be used to reduce the nutrient runoff from agricultural fields and increase the quality of water flowing into rivers, streams, and oceans. Researchers collected and dried the nutrient-rich algae from the ATS and studied its potential as an organic fertilizer. They found that cucumber and corn seedlings grew just as well using ATS organic fertilizer as they did with commercial fertilizers. Algae scrubbers, using bubbling upflow or vertical waterfall versions, are now also being used to filter aquaria and ponds. |
Algae | 633 | Uses | Various polymers can be created from algae, which can be especially useful in the creation of bioplastics. These include hybrid plastics, cellulose-based plastics, poly-lactic acid, and bio-polyethylene. Several companies have begun to produce algae polymers commercially, including for use in flip-flops and in surf boards. |
Algae | 633 | Uses | The alga Stichococcus bacillaris has been seen to colonize silicone resins used at archaeological sites; biodegrading the synthetic substance. |
Algae | 633 | Uses | The natural pigments (carotenoids and chlorophylls) produced by algae can be used as alternatives to chemical dyes and coloring agents. The presence of some individual algal pigments, together with specific pigment concentration ratios, are taxon-specific: analysis of their concentrations with various analytical methods, particularly high-performance liquid chromatography, can therefore offer deep insight into the taxonomic composition and relative abundance of natural algae populations in sea water samples. |
Algae | 633 | Uses | Carrageenan, from the red alga Chondrus crispus, is used as a stabilizer in milk products. |
Analysis of variance | 634 | Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means. |
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Analysis of variance | 634 | History | While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler. These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis testing in the 1770s. Around 1800, Laplace and Gauss developed the least-squares method for combining observations, which improved upon methods then used in astronomy and geodesy. It also initiated much study of the contributions to sums of squares. Laplace knew how to estimate a variance from a residual (rather than a total) sum of squares. By 1827, Laplace was using least squares methods to address ANOVA problems regarding measurements of atmospheric tides. Before 1800, astronomers had isolated observational errors resulting from reaction times (the "personal equation") and had developed methods of reducing the errors. The experimental methods used in the study of the personal equation were later accepted by the emerging field of psychology which developed strong (full factorial) experimental methods to which randomization and blinding were soon added. An eloquent non-mathematical explanation of the additive effects model was available in 1885. |
Analysis of variance | 634 | History | Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article on theoretical population genetics,The Correlation Between Relatives on the Supposition of Mendelian Inheritance. His first application of the analysis of variance to data analysis was published in 1921, Studies in Crop Variation I, This divided the variation of a time series into components representing annual causes and slow deterioration. Fisher's next piece, Studies in Crop Variation II, written with Winifred Mackenzie and published in 1923, studied the variation in yield across plots sown with different varieties and subjected to different fertiliser treatments. Analysis of variance became widely known after being included in Fisher's 1925 book Statistical Methods for Research Workers. |
Analysis of variance | 634 | History | Randomization models were developed by several researchers. The first was published in Polish by Jerzy Neyman in 1923. |
Analysis of variance | 634 | Example | The analysis of variance can be used to describe otherwise complex relations among variables. A dog show provides an example. A dog show is not a random sampling of the breed: it is typically limited to dogs that are adult, pure-bred, and exemplary. A histogram of dog weights from a show might plausibly be rather complex, like the yellow-orange distribution shown in the illustrations. Suppose we wanted to predict the weight of a dog based on a certain set of characteristics of each dog. One way to do that is to explain the distribution of weights by dividing the dog population into groups based on those characteristics. A successful grouping will split dogs such that (a) each group has a low variance of dog weights (meaning the group is relatively homogeneous) and (b) the mean of each group is distinct (if two groups have the same mean, then it isn't reasonable to conclude that the groups are, in fact, separate in any meaningful way). |
Analysis of variance | 634 | Example | In the illustrations to the right, groups are identified as X1, X2, etc. In the first illustration, the dogs are divided according to the product (interaction) of two binary groupings: young vs old, and short-haired vs long-haired (e.g., group 1 is young, short-haired dogs, group 2 is young, long-haired dogs, etc.). Since the distributions of dog weight within each of the groups (shown in blue) has a relatively large variance, and since the means are very similar across groups, grouping dogs by these characteristics does not produce an effective way to explain the variation in dog weights: knowing which group a dog is in doesn't allow us to predict its weight much better than simply knowing the dog is in a dog show. Thus, this grouping fails to explain the variation in the overall distribution (yellow-orange). |
Analysis of variance | 634 | Example | An attempt to explain the weight distribution by grouping dogs as pet vs working breed and less athletic vs more athletic would probably be somewhat more successful (fair fit). The heaviest show dogs are likely to be big, strong, working breeds, while breeds kept as pets tend to be smaller and thus lighter. As shown by the second illustration, the distributions have variances that are considerably smaller than in the first case, and the means are more distinguishable. However, the significant overlap of distributions, for example, means that we cannot distinguish X1 and X2 reliably. Grouping dogs according to a coin flip might produce distributions that look similar. |
Analysis of variance | 634 | Example | An attempt to explain weight by breed is likely to produce a very good fit. All Chihuahuas are light and all St Bernards are heavy. The difference in weights between Setters and Pointers does not justify separate breeds. The analysis of variance provides the formal tools to justify these intuitive judgments. A common use of the method is the analysis of experimental data or the development of models. The method has some advantages over correlation: not all of the data must be numeric and one result of the method is a judgment in the confidence in an explanatory relationship. |
Analysis of variance | 634 | Classes of models | There are three classes of models used in the analysis of variance, and these are outlined here. |
Analysis of variance | 634 | Classes of models | The fixed-effects model (class I) of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see whether the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole. |
Analysis of variance | 634 | Classes of models | Random-effects model (class II) is used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are random variables, some assumptions and the method of contrasting the treatments (a multi-variable generalization of simple differences) differ from the fixed-effects model. |
Analysis of variance | 634 | Classes of models | A mixed-effects model (class III) contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types. |
Analysis of variance | 634 | Classes of models | Teaching experiments could be performed by a college or university department to find a good introductory textbook, with each text considered a treatment. The fixed-effects model would compare a list of candidate texts. The random-effects model would determine whether important differences exist among a list of randomly selected texts. The mixed-effects model would compare the (fixed) incumbent texts to randomly selected alternatives. |
Analysis of variance | 634 | Classes of models | Defining fixed and random effects has proven elusive, with multiple competing definitions. |
Analysis of variance | 634 | Assumptions | The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data. |
Analysis of variance | 634 | Assumptions | The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses: |
Analysis of variance | 634 | Assumptions | The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors ( ε {\displaystyle \varepsilon } ) are independent and |
Analysis of variance | 634 | Assumptions | In a randomized controlled experiment, the treatments are randomly assigned to experimental units, following the experimental protocol. This randomization is objective and declared before the experiment is carried out. The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald Fisher. This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University. Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox. |
Analysis of variance | 634 | Assumptions | In its simplest form, the assumption of unit-treatment additivity states that the observed response y i , j {\displaystyle y_{i,j}} from experimental unit i {\displaystyle i} when receiving treatment j {\displaystyle j} can be written as the sum of the unit's response y i {\displaystyle y_{i}} and the treatment-effect t j {\displaystyle t_{j}} , that is |
Analysis of variance | 634 | Assumptions | The assumption of unit-treatment additivity implies that, for every treatment j {\displaystyle j} , the j {\displaystyle j} th treatment has exactly the same effect t j {\displaystyle t_{j}} on every experiment unit. |
Analysis of variance | 634 | Assumptions | The assumption of unit treatment additivity usually cannot be directly falsified, according to Cox and Kempthorne. However, many consequences of treatment-unit additivity can be falsified. For a randomized experiment, the assumption of unit-treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit-treatment additivity is that the variance is constant. |
Analysis of variance | 634 | Assumptions | The use of unit treatment additivity and randomization is similar to the design-based inference that is standard in finite-population survey sampling. |
Analysis of variance | 634 | Assumptions | Kempthorne uses the randomization-distribution and the assumption of unit treatment additivity to produce a derived linear model, very similar to the textbook model discussed previously. The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies. However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations. In the randomization-based analysis, there is no assumption of a normal distribution and certainly no assumption of independence. On the contrary, the observations are dependent! |
Analysis of variance | 634 | Assumptions | The randomization-based analysis has the disadvantage that its exposition involves tedious algebra and extensive time. Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach. Few statisticians object to model-based analysis of balanced randomized experiments. |
Analysis of variance | 634 | Assumptions | However, when applied to data from non-randomized experiments or observational studies, model-based analysis lacks the warrant of randomization. For observational data, the derivation of confidence intervals must use subjective models, as emphasized by Ronald Fisher and his followers. In practice, the estimates of treatment-effects from observational studies generally are often inconsistent. In practice, "statistical models" and observational data are useful for suggesting hypotheses that should be treated very cautiously by the public. |
Analysis of variance | 634 | Assumptions | The normal-model based ANOVA analysis assumes the independence, normality, and homogeneity of variances of the residuals. The randomization-based analysis assumes only the homogeneity of the variances of the residuals (as a consequence of unit-treatment additivity) and uses the randomization procedure of the experiment. Both these analyses require homoscedasticity, as an assumption for the normal-model analysis and as a consequence of randomization and additivity for the randomization-based analysis. |
Analysis of variance | 634 | Assumptions | However, studies of processes that change variances rather than means (called dispersion effects) have been successfully conducted using ANOVA. There are no necessary assumptions for ANOVA in its full generality, but the F-test used for ANOVA hypothesis testing has assumptions and practical limitations which are of continuing interest. |
Analysis of variance | 634 | Assumptions | Problems which do not satisfy the assumptions of ANOVA can often be transformed to satisfy the assumptions. The property of unit-treatment additivity is not invariant under a "change of scale", so statisticians often use transformations to achieve unit-treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance. Also, a statistician may specify that logarithmic transforms be applied to the responses which are believed to follow a multiplicative model. According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition. |
Analysis of variance | 634 | Characteristics | ANOVA is used in the analysis of comparative experiments, those in which only the difference in outcomes is of interest. The statistical significance of the experiment is determined by a ratio of two variances. This ratio is independent of several possible alterations to the experimental observations: Adding a constant to all observations does not alter significance. Multiplying all observations by a constant does not alter significance. So ANOVA statistical significance result is independent of constant bias and scaling errors as well as the units used in expressing observations. In the era of mechanical calculation it was common to subtract a constant from all observations (when equivalent to dropping leading digits) to simplify data entry. This is an example of data coding. |
Analysis of variance | 634 | Algorithm | The calculations of ANOVA can be characterized as computing a number of means and variances, dividing two variances and comparing the ratio to a handbook value to determine statistical significance. Calculating a treatment effect is then trivial: "the effect of any treatment is estimated by taking the difference between the mean of the observations which receive the treatment and the general mean". |
Analysis of variance | 634 | Algorithm | ANOVA uses traditional standardized terminology. The definitional equation of sample variance is s 2 = 1 n − 1 ∑ i ( y i − y ¯ ) 2 {\textstyle s^{2}={\frac {1}{n-1}}\sum _{i}(y_{i}-{\bar {y}})^{2}} , where the divisor is called the degrees of freedom (DF), the summation is called the sum of squares (SS), the result is called the mean square (MS) and the squared terms are deviations from the sample mean. ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate treatment means, and a treatment variance. The treatment variance is based on the deviations of treatment means from the grand mean, the result being multiplied by the number of observations in each treatment to account for the difference between the variance of observations and the variance of means. |
Analysis of variance | 634 | Algorithm | The fundamental technique is a partitioning of the total sum of squares SS into components related to the effects used in the model. For example, the model for a simplified ANOVA with one type of treatment at different levels. |
Analysis of variance | 634 | Algorithm | The number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect. |
Analysis of variance | 634 | Algorithm | The F-test is used for comparing the factors of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic |
Analysis of variance | 634 | Algorithm | where MS is mean square, I {\displaystyle I} is the number of treatments and n T {\displaystyle n_{T}} is the total number of cases |
Analysis of variance | 634 | Algorithm | to the F-distribution with I − 1 {\displaystyle I-1} being the numerator degrees of freedom and n T − I {\displaystyle n_{T}-I} the denominator degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the ratio of two scaled sums of squares each of which follows a scaled chi-squared distribution. |
Analysis of variance | 634 | Algorithm | The expected value of F is 1 + n σ Treatment 2 / σ Error 2 {\displaystyle 1+{n\sigma _{\text{Treatment}}^{2}}/{\sigma _{\text{Error}}^{2}}} (where n {\displaystyle n} is the treatment sample size) which is 1 for no treatment effect. As values of F increase above 1, the evidence is increasingly inconsistent with the null hypothesis. Two apparent experimental methods of increasing F are increasing the sample size and reducing the error variance by tight experimental controls. |
Analysis of variance | 634 | Algorithm | There are two methods of concluding the ANOVA hypothesis test, both of which produce the same result: |
Analysis of variance | 634 | Algorithm | The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors (i.e. maximizing power for a fixed significance level). For example, to test the hypothesis that various medical treatments have exactly the same effect, the F-test's p-values closely approximate the permutation test's p-values: The approximation is particularly close when the design is balanced. Such permutation tests characterize tests with maximum power against all alternative hypotheses, as observed by Rosenbaum. The ANOVA F-test (of the null-hypothesis that all treatments have exactly the same effect) is recommended as a practical test, because of its robustness against many alternative distributions. |
Analysis of variance | 634 | Algorithm | ANOVA consists of separable parts; partitioning sources of variance and hypothesis testing can be used individually. ANOVA is used to support other statistical tools. Regression is first used to fit more complex models to data, then ANOVA is used to compare models with the objective of selecting simple(r) models that adequately describe the data. "Such models could be fit without any reference to ANOVA, but ANOVA tools could then be used to make some sense of the fitted models, and to test hypotheses about batches of coefficients." "[W]e think of the analysis of variance as a way of understanding and structuring multilevel models—not as an alternative to regression but as a tool for summarizing complex high-dimensional inferences ..." |
Analysis of variance | 634 | For a single factor | The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and Latin squares (and variants: Graeco-Latin squares, etc.). The more complex experiments share many of the complexities of multiple factors. A relatively complete discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available. |
Analysis of variance | 634 | For a single factor | There are some alternatives to conventional one-way analysis of variance, e.g.: Welch's heteroscedastic F test, Welch's heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, Alexander-Govern test, James second order test and Kruskal-Wallis test, available in onewaytests R |
Analysis of variance | 634 | For a single factor | It is useful to represent each data point in the following form, called a statistical model: |
Analysis of variance | 634 | For a single factor | where |
Analysis of variance | 634 | For a single factor | That is, we envision an additive model that says every data point can be represented by summing three quantities: the true mean, averaged over all factor levels being investigated, plus an incremental component associated with the particular column (factor level), plus a final component associated with everything else affecting that specific data value. |
Analysis of variance | 634 | For multiple factors | ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases. Consequently, factorial designs are heavily used. |
Analysis of variance | 634 | For multiple factors | The use of ANOVA to study the effects of multiple factors has a complication. In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for interactions (xy, xz, yz, xyz). All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance. Fortunately, experience says that high order interactions are rare. The ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results. |
Analysis of variance | 634 | For multiple factors | Caution is advised when encountering interactions; Test interaction terms first and expand the analysis beyond ANOVA if interactions are found. Texts vary in their recommendations regarding the continuation of the ANOVA procedure after encountering an interaction. Interactions complicate the interpretation of experimental data. Neither the calculations of significance nor the estimated treatment effects can be taken at face value. "A significant interaction will often mask the significance of main effects." Graphical methods are recommended to enhance understanding. Regression is often useful. A lengthy discussion of interactions is available in Cox (1958). Some interactions can be removed (by transformations) while others cannot. |
Analysis of variance | 634 | For multiple factors | A variety of techniques are used with multiple factor ANOVA to reduce expense. One technique used in factorial designs is to minimize replication (possibly no replication with support of analytical trickery) and to combine groups when effects are found to be statistically (or practically) insignificant. An experiment with many insignificant factors may collapse into one with a few factors supported by many replications. |
Analysis of variance | 634 | Associated analysis | Some analysis is required in support of the design of the experiment while other analysis is performed after changes in the factors are formally found to produce statistically significant changes in the responses. Because experimentation is iterative, the results of one experiment alter plans for following experiments. |
Analysis of variance | 634 | Associated analysis | In the design of an experiment, the number of experimental units is planned to satisfy the goals of the experiment. Experimentation is often sequential. |
Analysis of variance | 634 | Associated analysis | Early experiments are often designed to provide mean-unbiased estimates of treatment effects and of experimental error. Later experiments are often designed to test a hypothesis that a treatment effect has an important magnitude; in this case, the number of experimental units is chosen so that the experiment is within budget and has adequate power, among other goals. |
Analysis of variance | 634 | Associated analysis | Reporting sample size analysis is generally required in psychology. "Provide information on sample size and the process that led to sample size decisions." The analysis, which is written in the experimental protocol before the experiment is conducted, is examined in grant applications and administrative review boards. |
Analysis of variance | 634 | Associated analysis | Besides the power analysis, there are less formal methods for selecting the number of experimental units. These include graphical methods based on limiting the probability of false negative errors, graphical methods based on an expected variation increase (above the residuals) and methods based on achieving a desired confidence interval. |
Analysis of variance | 634 | Associated analysis | Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level. Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true. |
Analysis of variance | 634 | Associated analysis | Several standardized measures of effect have been proposed for ANOVA to summarize the strength of the association between a predictor(s) and the dependent variable or the overall standardized difference of the complete model. Standardized effect-size estimates facilitate comparison of findings across studies and disciplines. However, while standardized effect sizes are commonly used in much of the professional literature, a non-standardized measure of effect size that has immediately "meaningful" units may be preferable for reporting purposes. |
Analysis of variance | 634 | Associated analysis | Sometimes tests are conducted to determine whether the assumptions of ANOVA appear to be violated. Residuals are examined or analyzed to confirm homoscedasticity and gross normality. Residuals should have the appearance of (zero mean normal distribution) noise when plotted as a function of anything including time and modeled data values. Trends hint at interactions among factors or among observations. |
Analysis of variance | 634 | Associated analysis | A statistically significant effect in ANOVA is often followed by additional tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow-up tests are often distinguished in terms of whether they are "planned" (a priori) or "post hoc." Planned tests are determined before looking at the data, and post hoc tests are conceived only after looking at the data (though the term "post hoc" is inconsistently used). |
Analysis of variance | 634 | Associated analysis | The follow-up tests may be "simple" pairwise comparisons of individual group means or may be "compound" comparisons (e.g., comparing the mean pooling across groups A, B and C to the mean of group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels. Often the follow-up tests incorporate a method of adjusting for the multiple comparisons problem. |
Analysis of variance | 634 | Associated analysis | Follow-up tests to identify which specific groups, variables, or factors have statistically different means include the Tukey's range test, and Duncan's new multiple range test. In turn, these tests are often followed with a Compact Letter Display (CLD) methodology in order to render the output of the mentioned tests more transparent to a non-statistician audience. |
Analysis of variance | 634 | Study designs | There are several types of ANOVA. Many statisticians base ANOVA on the design of the experiment, especially on the protocol that specifies the random assignment of treatments to subjects; the protocol's description of the assignment mechanism should include a specification of the structure of the treatments and of any blocking. It is also common to apply ANOVA to observational data using an appropriate statistical model. |
Analysis of variance | 634 | Study designs | Some popular designs use the following types of ANOVA: |
Analysis of variance | 634 | Cautions | Balanced experiments (those with an equal sample size for each treatment) are relatively easy to interpret; unbalanced experiments offer more complexity. For single-factor (one-way) ANOVA, the adjustment for unbalanced data is easy, but the unbalanced analysis lacks both robustness and power. For more complex designs the lack of balance leads to further complications. "The orthogonality property of main effects and interactions present in balanced data does not carry over to the unbalanced case. This means that the usual analysis of variance techniques do not apply. Consequently, the analysis of unbalanced factorials is much more difficult than that for balanced designs." In the general case, "The analysis of variance can also be applied to unbalanced data, but then the sums of squares, mean squares, and F-ratios will depend on the order in which the sources of variation are considered." |
Analysis of variance | 634 | Cautions | ANOVA is (in part) a test of statistical significance. The American Psychological Association (and many other organisations) holds the view that simply reporting statistical significance is insufficient and that reporting confidence bounds is preferred. |
Analysis of variance | 634 | Generalizations | ANOVA is considered to be a special case of linear regression which in turn is a special case of the general linear model. All consider the observations to be the sum of a model (fit) and a residual (error) to be minimized. |
Analysis of variance | 634 | Generalizations | The Kruskal–Wallis test and the Friedman test are nonparametric tests which do not rely on an assumption of normality. |
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