Analysis of variance

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 group means in a sample.  ANOVA was developed by statistician and evolutionary biologist Ronald Fisher.  The 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. 

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.  The development of least-squares methods by Laplace and Gauss circa 1800 provided an improved method of combining observations (over the existing practices 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. 

Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article The Correlation Between Relatives on the Supposition of Mendelian Inheritance.   His first application of the analysis of variance was published in 1921.  Analysis of variance became widely known after being included in Fisher's 1925 book Statistical Methods for Research Workers. 

Randomization models were developed by several researchers.   The first was published in Polish by Jerzy Neyman in 1923. 

One of the attributes of ANOVA that ensured its early popularity was computational elegance.   The structure of the additive model allows solution for the additive coefficients by simple algebra rather than by matrix calculations.   In the era of mechanical calculators this simplicity was critical.   The determination of statistical significance also required access to tables of the F function which were supplied by early statistics texts. 

Example
The analysis of variance can be used as an exploratory tool to explain observations.   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). 
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). 

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. 

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. 

Background and terminology
ANOVA is a form of statistical hypothesis testing heavily used in the analysis of experimental data.   A test result (calculated from the null hypothesis and the sample) is called statistically significant if it is deemed unlikely to have occurred by chance, assuming the truth of the null hypothesis.  A statistically significant result, when a probability (p-value) is less than a pre-specified threshold (significance level), justifies the rejection of the null hypothesis, but only if the a priori probability of the null hypothesis is not high. 

In the typical application of ANOVA, the null hypothesis is that all groups are random samples from the same population.  For example, when studying the effect of different treatments on similar samples of patients, the null hypothesis would be that all treatments have the same effect (perhaps none).  Rejecting the null hypothesis is taken to mean that the differences in observed effects between treatment groups are unlikely to be due to random chance. 

By construction, hypothesis testing limits the rate of Type I errors (false positives) to a significance level.   Experimenters also wish to limit Type II errors (false negatives).   
The rate of Type II errors depends largely on sample size (the rate is larger for smaller samples), significance 
level (when the standard of proof is high, the chances of overlooking 
a discovery are also high) and effect size (a smaller effect size is more prone to Type II error). 

The terminology of ANOVA is largely from the statistical 
design of experiments.   The experimenter adjusts factors and 
measures responses in an attempt to determine an effect.   Factors are 
assigned to experimental units by a combination of randomization and 
blocking to ensure the validity of the results.   Blinding keeps the
weighing impartial.   Responses show a variability that is partially 
the result of the effect and is partially random error. 

ANOVA is the synthesis of several ideas and it is used for multiple 
purposes.   As a consequence, it is difficult to define concisely or precisely. 

"Classical" ANOVA for balanced data does three things at once:

In short, ANOVA is a statistical tool used in several ways to develop and confirm an explanation for the observed data. 

Additionally:

As a result:
ANOVA "has long enjoyed the status of being the most used (some would 
say abused) statistical technique in psychological research." 
ANOVA "is probably the most useful technique in the field of 
statistical inference." 

ANOVA is difficult to teach, particularly for complex experiments, with split-plot designs being notorious.   In some cases the proper 
application of the method is best determined by problem pattern recognition 
followed by the consultation of a classic authoritative test. 

Design-of-experiments terms
(Condensed from the "NIST Engineering Statistics Handbook": Section 5.7.  A 
Glossary of DOE Terminology.) 

 Balanced design An experimental design where all cells (i.e. treatment combinations) have the same number of observations. 
 Blocking A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental results due to a known change in raw materials, operators, machines, etc.,  become concentrated in the levels of the blocking variable.  The reason for blocking is to isolate a systematic effect and prevent it from obscuring the main effects.  Blocking is achieved by restricting randomization. 
 Design A set of experimental runs which allows the fit of a particular model and the estimate of effects. 
 DOE Design of experiments.   An approach to problem solving involving collection of data that will support valid, defensible, and supportable conclusions. 
 Effect How changing the settings of a factor changes the response.  The effect of a single factor is also called a main effect. 
 Error Unexplained variation in a collection of observations.  DOE's typically require understanding of both random error and lack of fit error. 
 Experimental unit The entity to which a specific treatment combination is applied. 
 Factors Process inputs that an investigator manipulates to cause a change in the output. 
 Lack-of-fit error Error that occurs when the analysis omits one or more important terms or factors from the process model.  Including replication in a DOE allows separation of experimental error into its components: lack of fit and random (pure) error. 
 Model Mathematical relationship which relates changes in a given response to changes in one or more factors. 
 Random error Error that occurs due to natural variation in the process.  Random error is typically assumed to be normally distributed with zero mean and a constant variance.  Random error is also called experimental error. 
 Randomization A schedule for allocating treatment material and for conducting treatment combinations in a DOE such that the conditions in one run neither depend on the conditions of the previous run nor predict the conditions in the subsequent runs. 
 Replication Performing the same treatment combination more than once.  Including replication allows an estimate of the random error independent of any lack of fit error. 
 Responses The output(s) of a process.  Sometimes called dependent variable(s). 
 Treatment A treatment is a specific combination of factor levels whose effect is to be compared with other treatments. 

Classes of models 
There are three classes of models used in the analysis of variance, and these are outlined here. 

Fixed-effects 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. 

Random-effects 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. 

Mixed-effects 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. 

Example:
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. 

Defining fixed and random effects has proven elusive, with competing 
definitions arguably leading toward a linguistic quagmire. 

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. 

Textbook analysis using a normal distribution
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:
 Independence of observations – this is an assumption of the model that simplifies the statistical analysis. 
 Normality – the distributions of the residuals are normal. 
 Equality (or "homogeneity") of variances, called homoscedasticity — the variance of data in groups should be the same. 

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 () are independent and

Randomization-based analysis

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. 

Unit-treatment additivity
In its simplest form, the assumption of unit-treatment additivity states that the observed response  from experimental unit  when receiving treatment  can be written as the sum of the unit's response  and the treatment-effect , that is 
 
The assumption of unit-treatment additivity implies that, for every treatment , the th treatment has  exactly the same effect  on every experiment unit. 

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. 

The use of unit treatment additivity and randomization is similar to the design-based inference that is standard in finite-population survey sampling. 

Derived linear model
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! 

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. 

Statistical models for observational data
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. 

Summary of 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. 

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. 

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. 

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. 

Logic
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". 

Partitioning of the sum of squares

ANOVA uses traditional standardized terminology.   The definitional 
equation of sample variance is
, 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. 

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. 

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. 

See also Lack-of-fit sum of squares. 

The F-test

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

where  MS is mean square,   = number of treatments and 
 = total number of cases

to the F-distribution with ,   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. 

The expected value of F is  (where  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. 

There are two methods of concluding the ANOVA hypothesis test, both of which produce the same result:
 The textbook method is to compare the observed value of F with the critical value of F determined from tables.  The critical value of F is a function of the degrees of freedom of the numerator and the denominator and the significance level (α).   If F ≥ FCritical, the null hypothesis is rejected. 
 The computer method calculates the probability (p-value) of a value of F greater than or equal to the observed value.   The null hypothesis is rejected if this probability is less than or equal to the significance level (α). 
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. 

Extended logic
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 ..."

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. 

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. 

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. 

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. 

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. 

Worked numeric examples
Numerous fully worked numerical examples are available in standard textbooks and online.   A 
simple case uses one-way (a single factor) analysis. 

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. 

Preparatory analysis

The number of experimental units

In the design of an experiment, the number of experimental units is planned to satisfy the goals of the experiment.  Experimentation is often sequential. 

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. 

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. 

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 confident interval. 

Power 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. 

Effect size

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. 

Follow-up analysis
It is always appropriate to carefully consider outliers.   They have a disproportionate impact on statistical conclusions and are often the result of errors. 

Model confirmation
It is prudent to verify that the assumptions of ANOVA have been met.  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.   One rule of thumb: "If the largest standard deviation is less than twice the smallest standard deviation, we can use methods based on the assumption of equal standard deviations and our results 
will still be approximately correct." 

Follow-up tests
A statistically significant effect in ANOVA is often followed up with one or more different follow-up 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 performed after looking at the data. 

Often one of the "treatments" is none, so the treatment group can act as a control.  Dunnett's test (a modification of the t-test) tests whether each of the other treatment groups has the same mean as the control. 

Post hoc tests such as Tukey's range test most commonly compare every group mean with every other group mean and typically incorporate some method of controlling for Type I errors.  Comparisons, which are most commonly planned, can be either simple or compound.  Simple comparisons compare one group mean with one other group mean.  Compound comparisons typically compare two sets of groups means where one set has two or more groups (e.g., compare average group means of group A, B and C with group D).  Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels. 

Following ANOVA with pair-wise multiple-comparison tests has been criticized on several grounds.  There are many such tests (10 in one table) and recommendations regarding their use are vague or conflicting. 

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. 

Some popular designs use the following types of ANOVA:
One-way ANOVA is used to test for differences among two or more independent groups (means), e.g. different levels of urea application in a crop, or different levels of antibiotic action on several different bacterial species, or different levels of effect of some medicine on groups of patients.  However, should these groups not be independent, and there is an order in the groups (such as mild, moderate and severe disease), or in the dose of a drug (such as 5 mg/mL, 10 mg/mL, 20 mg/mL) given to the same group of patients, then a linear trend estimation should be used.  Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test.  When there are only two means to compare, the t-test and the ANOVA F-test are equivalent; the relation between ANOVA and t is given by F = t2. 
Factorial ANOVA is used when the experimenter wants to study the interaction effects among the treatments. 
Repeated measures ANOVA is used when the same subjects are used for each treatment (e.g., in a longitudinal study). 
Multivariate analysis of variance (MANOVA) is used when there is more than one response variable. 

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."   The simplest techniques for handling unbalanced data restore balance by either throwing out data or by synthesizing missing data.   More complex techniques use regression. 

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. 

While ANOVA is conservative (in maintaining a significance level) against multiple comparisons in one dimension, it is not conservative against comparisons in multiple dimensions. 

A common mistake is to use an ANOVA (or Kruskal–Wallis) for analysis of ordered groups, e.g. in time sequence (changes over months), in disease severity (mild, moderate, severe), or in distance from a set point (10 km, 25 km, 50 km).  Data in three or more ordered groups that are defined by the researcher should be analysed by Linear Trend Estimation. 

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. 

The Kruskal–Wallis test and the Friedman test are nonparametric tests, which  do not rely on an assumption of normality. 

Connection to linear regression
Below we make clear the connection between multi-way ANOVA and linear regression. 

Linearly re-order the data so that  observation is associated with a response  and factors  where  denotes the different factors and  is the total number of factors.  In one-way ANOVA  and in two-way ANOVA .  Furthermore, we assume the  factor has  levels, namely .  Now, we can one-hot encode the factors into the  dimensional vector . 

The one-hot encoding function  is defined such that the  entry of  is

The vector  is the concatenation of all of the above vectors for all .  Thus, .  In order to obtain a fully general -way interaction ANOVA we must also concatenate every additional interaction term in the vector  and then add an intercept term.   Let that vector be . 

With this notation in place, we now have the exact connection with linear regression.  We simply regress response  against the vector .  However, there is a concern about identifiability.  In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero.  From here, one can use F-statistics or other methods to determine the relevance of the individual factors. 

Example
We can consider the 2-way interaction example where we assume that the first factor has 2 levels and the second factor has 3 levels. 

Define  if  and  if , i.e.  is the one-hot encoding of the first factor and  is the one-hot encoding of the second factor. 

With that,

where the last term is an intercept term.  For a more concrete example suppose that

Then,

See also

One-way analysis of variance (one-way ANOVA)
Two-way analysis of variance (two-way ANOVA)
ANOVA on ranks
ANOVA-simultaneous component analysis
Analysis of covariance (ANCOVA)
Multivariate analysis of variance (MANOVA)
Multivariate analysis of covariance (MANCOVA)
Analysis of molecular variance (AMOVA)
Analysis of rhythmic variance (ANORVA)
Explained variation
Mixed-design analysis of variance
Permutational analysis of variance
Repeated measures ANOVA
Variance decomposition
Linear trend estimation

Footnotes

Notes

References
 
  Pre-publication chapters are available on-line. 
 
 
 Cohen, Jacob (1988).  Statistical power analysis for the behavior sciences (2nd ed.).  Routledge 
 
Cox, David R. (1958).  Planning of experiments.   Reprinted as 

 Freedman, David A.(2005).  Statistical Models: Theory and Practice, Cambridge University Press.   
 

 

 Lehmann, E.L. (1959) Testing Statistical Hypotheses.   John Wiley & Sons. 
 
 Moore, David S. & McCabe, George P. (2003).   Introduction to the Practice of Statistics (4e).   W H Freeman & Co.  
 Rosenbaum, Paul R. (2002).  Observational Studies (2nd ed.).  New York: Springer-Verlag. 

Further reading

 
 
 
 
 
Cox, David R. & Reid, Nancy M. (2000).  The theory of design of experiments.   (Chapman & Hall/CRC).   
 
 Freedman, David A.; Pisani, Robert; Purves, Roger (2007) Statistics, 4th edition.  W.W. Norton & Company 
 
 
 Tabachnick, Barbara G. & Fidell, Linda S. (2007).  Using Multivariate Statistics (5th ed.).  Boston: Pearson International Edition. 

External links

 SOCR ANOVA Activity and interactive applet. 
  Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R (University of Southampton)
 NIST/SEMATECH e-Handbook of Statistical Methods, section 7.4.3: "Are the means equal?" 
Analysis of variance: Introduction
One Way & Two Way ANOVA Calculator

 
Category:Design of experiments
Category:Statistical tests
Category:Parametric statistics