Multivariate analysis of variance or multiple analysis of variance (MANOVA) is a statistical test procedure for comparing multivariate (population) means of several groups. As a multivariate procedure, it is used when there are two or more dependent variables, although statistical reports provide individual p-values for each dependent variable in order to test for statistical significance.
MANOVA is a generalized form of univariate analysis of variance (ANOVA), although, unlike univariate ANOVA, it uses the variance-covariance between 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.
Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution.
Analogous to ANOVA, MANOVA is based on the product of model variance matrix, and inverse of the error variance matrix, , or .
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.
S. Bartlett trace, the Lawley-Hotelling trace, Roy's greatest root (also called Roy's largest root),
Discussion continues over the merits of each, though the greatest root leads only to a bound on significance which is not generally of practical interest. A further complication is that the distribution of these statistics under the null hypothesis is not straightforward and can only be approximated except in a few low-dimensional cases.
In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square.