On Testing Homogeneity of Covariance Matrices with Box’s M and the Approximate Tests for Multivariate Data

Abstract

Homogeneity of covariance matrices, or equal covariance matrices across groups, is one of the most important assumptions in multivariate analysis of variance (MANOVA) and in discriminant analysis. Box’s M test, an exact test, is a generally accepted method used to check the violation of this assumption. The Box’s statistic M can be transformed to the statistics serving as the approximate tests based on chi-squared and F distributions. This study aims to assess the performance of Box’s M test compared with the approximate test using chi-squared distribution for normal data. When the data are non-normal, the performance of the approximate test and the nonparametric test using the bootstrap method are examined. The results showed that under normality with equal sample sizes and certain conditions to conduct the exact test, the Box’s M and the approximate chi-squared tests perform well whereas the performance of the approximate test is slightly better for detecting heterogeneity of covariance matrices. In the case of unequal sample sizes, in which the data do not meet the conditions to conduct Box’s M test, the approximate chi-squared test also performs well so it is applicable for such a case. When dealing with non-normal data, such as multivariate t-distributed data, the performance of the approximate test as well as the nonparametric test using the bootstrap method is unsatisfactory in the situations studied so it needs to be further developed.