Very brief description:

“Homogeneity of variance-covariance matrices” is the multivariate version of the univariate assumption of Homogeneity of variance and the bivariate assumption of Homoscedasticity. Refer to the post “Homogeneity of variance” for a discussion of equality of variances. In short, homogeneity of variance-covariance matrices concerns the variance-covariance matrices of the multiple dependent measures (such as in MANOVA) for each group. For example, if you have five dependent variables, it tests for five correlations and ten covariances for equality across the groups. So the more dependent variables in your equation, the higher the likelihood of non-equality of variances across the groups.

Who cares
Multivariate procedures such as MANOVA, Profile Analysis (GLM repeated models), Discriminant Function Analysis, Multivariate Regression (not Multiple  Regression), etc.

Why it is important
Refer to the post “Homoscedasticity”.

How to Test
Homogeneity of variance-covariance matrices for e.g. MANOVA is tested with the Box’s M test which should be non-significant (Box’s M tests the statistical hypothesis that the variance-covariance matrices are equal). In fact, you can first check the univariate homogeneity of variance with Levene’s test for each individual dependent variable, which should be non-significanct for all. However, Levene’s test does not take account of covariances, so variance-covariance matrices should be compared between groups using Box’s M-test. Note some of Box’s M limitations such as how it is affected by the size of the covariance matrix (these won’t be discussed here). Importantly, Box’s M test is sensitive to non-normality so do check for multivariate-normality. Note also that if the sample size across the independent variable groups are equal (in particular for 2-groups), Box’s M becomes unstable but we can then assume that Hotelling’s T² and Pillai’s stats are robust.

An easy eye-ball technique (for Discriminant Function Analysis as example) is to use the “split-data” function (compare groups) by the categorical dependent variable and then to create a scatterplot matrix for all the independent variables. Compare the scatterplots across the groups for the same independent variables. If no obvious differences, then the assumption has been met (variance-covariance matrices between/across groups are similar).

How to fix the problem
Problems related to the homogeneity of variance-covariance matrices violations can most often be attributed to the violation of multivariate normality. Applying remedies to correct for univariate normality (such as an increase in sample and/or transformations) are likely to be also a remedy to multivariate normality, and therefore a fix for the violation of the assumption of homogeneity of variance-covariance matrices.

Note: Where covariates are introduced into the model (e.g. ANCOVA and MANCOVA) the assumption of Homogeniety of regression slopes (test of parallelism) applies.
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