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ANALYSIS OF VARIANCE
Analysis of Variance, also known as ANOVA, is a collection of
statistical models, and their associated procedures, in which the observed
variance is partitioned into components due to different explanatory
variables. ANOVA is a statistical procedure for testing whether the observed
differences are significant.
The initial techniques of the analysis of variance were developed by
the statistician and geneticist R. A. Fisher in the 1920s and 1930s, and are
sometimes known as Fisher's ANOVA or Fisher's analysis of variance, due to
the use of Fisher's F-distribution as part of the test of statistical significance.
MODELS OF ANOVA:
There are three conceptual classes of such models:
1. Fixed-effects model assumes 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 effects models describe situations where both fixed and random
effects are present. (Model 3)
TYPES OF ANOVA:
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:
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 three or more groups, with the two-group
case relegated to the t-test (Gossett, 1908), which is a special case of
the ANOVA. The relation between ANOVA and t is given as F = t2.

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One-way ANOVA for repeated measures is used when the subjects are
subjected to repeated measures; this means that the same subjects are
used for each treatment. Note that this method can be subject to
carryover effects.
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. 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 because the calculations are lengthy
and the results are hard to interpret.
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 independent and the other is
repeated measures. This is a type of mixed effect model.
Multivariate analysis of variance (MANOVA) is used when there is
more than one dependent variable.
ASSUMPTIONS FOR ANOVA:
Independence of cases - this is a requirement of the design.
Normality - the distributions in each of the groups are normal (use the
Kolmogorov-Smirnov and Shapiro-Wilk normality tests to test it).
Some say that the F-test is extremely non-robust to deviations from
normality (Lindman, 1974) while others claim otherwise (Ferguson &
Takane 2005: 261-2). The Kruskal-Wallis test is a nonparametric
alternative which does not rely on an assumption of normality.
Homogeneity of variances - the variance of data in groups should be
the same (use Levene's test for homogeneity of variances).
These together form the common assumption that the error residuals are
independently, identically, and normally distributed for fixed effects models,
or:

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ANALYSIS OF VARIANCE Analysis of Variance, also known as ANOVA, is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables. ANOVA is a statistical procedure for testing whether the observed differences are significant. The initial techniques of the analysis of variance were developed by the statistician and geneticist R. A. Fisher in the 1920s and 1930s, and are sometimes known as Fisher's ANOVA or Fisher's analysis of variance, due to the use of Fisher's F-distribution as part of the test of statistical significance. MODELS OF ANOVA: There are three conceptual classes of such models: 1. Fixed-effects model assumes 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 effects models describe situations where both fixed and random effects are present. (Model 3) TYPES OF ANOVA: 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: 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 three or more groups, with the two-group case relegated to the t-test ...
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