The effect of nonnormality on some multivariate tests and robustness to nonnormality in the linear model

The effect of nonnormality on multivariate regression tests, on the one-way multivariate analysis of variance and on tests of equality of covariance matrices is studied following the approach of Box & Watson (1962). In the nonnormal case, an approximation to the distribution of a generalized Mahalanobis distance type of statistic for the multivariate regression problem is derived. It is shown that sensitivity to nonnormality in the multivariate observations is determined by the extent of nonnormality of the regressors. The randomization distribution of the generalized Mahalanobis distance is deduced. The multivariate analysis of variance is found to be robust to nonnormality whereas the tests for equality of covariance matrices are found to be sensitive to nonnormality. An explanation for this varying degree of sensitivity to nonnormality is given.