Assumptions and Effects of Violating Assumptions - Homogeneity of Variances and Covariances

Assumptions. In multivariate designs, with multiple dependent measures, the homogeneity of variances assumption described earlier (see Homogeneity of Variances) also applies. However, since there are multiple dependent variables, it is also required that their intercorrelations (covariances) are homogeneous across the cells of the design. ANOVA/MANOVA offers various specific tests of this assumption.

Effects of violations. The multivariate equivalent of the F-test is Wilks' Lambda. Not much is known about the robustness of Wilks' Lambda to violations of this assumption. However, because the interpretation of MANOVA results usually rests on the interpretation of significant univariate effects (after the overall test is significant), the discussion concerning univariate ANOVA (see Homogeneity of Variances) basically applies, and important significant univariate effects should be carefully scrutinized.

Special case: ANCOVA. A special serious violation of the homogeneity of variances/covariances assumption may occur when covariates are involved in the design. Specifically, if the correlations of the covariates with the dependent measure(s) are very different in different cells of the design, gross misinterpretations of results may occur. Remember that in ANCOVA, we in essence perform a regression analysis within each cell to partition out the variance component due to the covariates. The homogeneity of variances/covariances assumption implies that we perform this regression analysis subject to the constraint that all regression equations (slopes) across the cells of the design are the same. If this is not the case, serious biases may occur. GLM provides specific tests of this assumption, and it is advisable to look at those tests to ensure that the regression equations in different cells are approximately the same. Note that ANCOVA designs can be analyzed in the GLM module but not in the ANOVA/MANOVA module.