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Analysis of variance
In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their
associated procedures, in which the observed variance is partitioned into components due to
different sources of variation. In its simplest form ANOVA provides a statistical test of whether
or not the means of several groups are all equal, and therefore generalizes Student's two-sample
t-test to more than two groups. ANOVAs are helpful because they possess a certain advantage
over a two-sample t-test. Doing multiple two-sample t-tests would result in a largely increased
chance of committing a type I error. For this reason, ANOVAs are useful in comparing three or
more means.
Overview
There are three conceptual classes of such models:
1. Fixed-effects models assume that the data came from normal populations which may
differ only in their means. (Model 1)
2. Random effects models assume that the data describe a hierarchy of different populations
whose differences are constrained by the hierarchy. (Model 2)
3. Mixed-effect models describe the situations where both fixed and random effects are
present. (Model 3)
In practice, there are several types of ANOVA depending on the number of treatments and the
way they are applied to the subjects in the experiment are:
One-way ANOVA is used to test for differences among two or more independent groups.
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 (Gosset, 1908). When
there are only two means to compare, the t-test and the F-test are equivalent; the relation
between ANOVA and t is given by F = t
2
.
Factorial ANOVA is used when the experimenter wants to study the effects of two or
more treatment variables. The most commonly used type of factorial ANOVA is the 2
2
(read "two by two") design, where there are two independent variables and each variable
has two levels or distinct values. However, such use of ANOVA for analysis of 2
k
factorial designs and fractional factorial designs is "confusing and makes little sense";
instead it is suggested to refer the value of the effect divided by its standard error to a t-
table.
[1]
Factorial ANOVA can also be multi-level such as 3
3
, etc. or higher order such as
2×2×2, etc. but analyses with higher numbers of factors are rarely done by hand because
the calculations are lengthy. However, since the introduction of data analytic software,
the utilization of higher order designs and analyses has become quite common.
Repeated measures ANOVA is used when the same subjects are used for each treatment
(e.g., in a longitudinal study). Note that such within-subjects designs can be subject to
carry-over effects.

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Mixed-design ANOVA. When one wishes to test two or more independent groups
subjecting the subjects to repeated measures, one may perform a factorial mixed-design
ANOVA, in which one factor is a between-subjects variable and the other is within-
subjects variable. This is a type of mixed-effect model.
Multivariate analysis of variance (MANOVA) is used when there is more than one
dependent variable.
PERMANOVA which tests the simultaneous responses of one or more variables to one
or more factors in an ANOVA experimental design on the basis of any distance measure,
using permutation methods.
Models
Fixed-effects models (Model 1)
Main article: Fixed effects model
The fixed-effects model of analysis of variance applies to situations in which the experimenter
applies several treatments to the subjects of the experiment to see if 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 (Model 2)
Main article: Random effects model
Random effects models are used when the treatments are not fixed. This occurs when the various
treatments (also known as factor levels) are sampled from a larger population. Because the
treatments themselves are random variables, some assumptions and the method of contrasting the
treatments differ from ANOVA model 1.
Most random-effects or mixed-effects models are not concerned with making inferences
concerning the particular sampled factors. For example, consider a large manufacturing plant in
which many machines produce the same product. The statistician studying this plant would have
very little interest in comparing the three particular machines to each other. Rather, inferences
that can be made for all machines are of interest, such as their variability and the mean.
However, if one is interested in the realized value of the random effect best linear unbiased
prediction can be used to obtain a "prediction" for the value.
Assumptions of ANOVA
There are several approaches to the analysis of variance.
A model often presented in textbooks
Many textbooks present the analysis of variance in terms of a linear model, which makes the
following assumptions:

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Analysis of variance In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different sources of variation. In its simplest form ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes Student's two-sample t-test to more than two groups. ANOVAs are helpful because they possess a certain advantage over a two-sample t-test. Doing multiple two-sample t-tests would result in a largely increased chance of committing a type I error. For this reason, ANOVAs are useful in comparing three or more means. Overview There are three conceptual classes of such models: 1. Fixed-effects models assume that the data came from normal populations which may differ only in their means. (Model 1) 2. Random effects models assume that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. (Model 2) 3. Mixed-effect models describe the situations where both fixed and random effects are present. (Model 3) In practice, there are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment are: One-way ANOVA is used to test for differences among two or more independent groups. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case ...
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