Multivariate analysis of variance

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 typically followed by significance tests involving individual dependent variables separately. It helps to answer

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.

MANOVA is based on the product of model variance matrix, ${\displaystyle \Sigma _{model}}$ and inverse of the error variance matrix, ${\displaystyle \Sigma _{res}^{-1}}$, or ${\displaystyle A=\Sigma _{model}\times \Sigma _{res}^{-1}}$. The hypothesis that ${\displaystyle \Sigma _{model}=\Sigma _{residual}}$ implies that the product ${\displaystyle A\sim I}$. Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, but there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis.

...
Wikipedia