MANOVA

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Multivariate analysis of variance (MANOVA) is an extension of analysis of variance (ANOVA) methods to cover cases where there is more than one dependent variable and where the dependent variables cannot simply be combined. As well as identifying whether changes in the independent variables have a significant effect on the dependent variables, the technique also seeks to identify the interactions among the independent variables and the association between dependent variables, if any.

Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices (see matrix (mathematics)) 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.

Analogous to ANOVA, MANOVA is based on the product of model variance matrix and error variance matrix inverse. 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.

The most common statistics are

Samuel Stanley Wilks' lambda (Λ),
the Pillai-M. S. Bartlett trace (see trace of a matrix),
the Lawley-Hotelling trace and
Roy's greatest 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. The best-known approximation for Wilks' lambda was derived by C. R. Rao.

In the case of two groups all the statistics are equivalent and the test reduces to Hotelling's T-square.