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Biometrika 1971 58(1):105-121; doi:10.1093/biomet/58.1.105
© 1971 by Biometrika Trust
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The effect of nonnormality on some multivariate tests and robustness to nonnormality in the linear model

K. V. MARDIA

University of Hull

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.

Key Words: Contingency-type distributions • Nonnormality and the multivariate linear model • Tests on covariance matrices • Hotelling's T2 and generalized Mahalanobis distance • Mixtures of multivariate normal distributions • Multivariate Pareto distribution • Multivariate regression • Randomization distribution of multivariate statistics • Robustness of tests on mean vectors and on covariance matrices


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