On the robustness of the T02 test in multivariate analysis of variance when variance-covariance matrices are not equal

Current interest in multivariate analysis in biometrics and other areas of applied statistics must inevitably lead to a scrutiny of the robustness of the procedure used. One widely used test is the T02 test in multivariate analysis of variance (MANOVA). Among the assumptions underlying MANOVA are: (1) muitivariate normal distributions, (2) equality of variance covariance matrices, and (3) serially uncorrelated observations. The purpose of the present paper is to explore, in a preliminary fashion, the consequence to this test of violation of the assumption of equality of variance-covariance matrices. It is shown that in the case of two samples of equal size, the test is well behaved when there are variations of variance covariance matrices if the samples are very large; further that in the case of two samples of nearly equal size or k samples of equal size, the test is not affected seriously by moderate inequality of variance-covariance matrices as long as samples are very large. However, if some or all of k samples are of unequal size, quite a large effect occurs on the level of signi ficance and the power of the test from even moderate variations. It is to be noted that among numerous works in univariate analysis on the effect of heteroscedasticity of variances (see, e.g. David Gronow, 1951; Horsnell, 1953; Box, 1954; Scheffé, 1959, pp. 334–58), the present investigation is related to and has extended the work by Scheffé and Box as far as the effect on the level of significance of the test is concerned.