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    Multivariate ANOVA

    Multivariate analysis of variance, MANOVA, is a statistical test procedure for comparing multivariate means of several groups. Unlike ANOVA, it uses the variance-covariance between variables in testing the statistical significance of the mean differences. It is a generalized form of univariate analysis of variance. It is used when there are two or more dependent variables. Multivariate ANOVA takes scores from the multiple dependent variable and creates a single dependent variable giving the ability to test for the above effects.

    Multivariate analysis of variance for certain positive-definite matrices appear where sums of squares in univariate analysis of variance. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution.  

    MANOVA is based on the product of model variance matrix and inverse of the error variance matrix. The hypothesis implies that the product is equivalent to the inverse. Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product. There is no unique choice owning to the multi-dimensional nature of the alternative hypothesis. 

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