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Concept# Multivariate analysis of variance

Summary

In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests involving individual dependent variables separately.
Without relation to the image, the dependent variables may be k life satisfactions scores measured at sequential time points and p job satisfaction scores measured at sequential time points. In this case there are k+p dependent variables whose linear combination follows a multivariate normal distribution, multivariate variance-covariance matrix homogeneity, and linear relationship, no multicollinearity, and each without outliers.
Assume -dimensional observations, where the ’th observation is assigned to the group and is distributed around the group center with Multivariate Gaussian noise: where is the covariance matrix. Then we formulate our null hypothesis as
MANOVA is a generalized form of univariate analysis of variance (ANOVA), although, unlike univariate ANOVA, it uses the covariance between outcome variables in testing the statistical significance of the mean differences.
Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution.
First, define the following matrices:
where the -th row is equal to
where the -th row is the best prediction given the group membership . That is the mean over all observation in group : .
where the -th row is the best prediction given no information. That is the empirical mean over all observations
Then the matrix is a generalization of the sum of squares explained by the group, and is a generalization of the residual sum of squares.

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