Tests for Equality of Covariance Matrices

Abstract

Homogeneity of covariance matrices is one of the important assumptions of the multivariate analysis of variance (or MANOVA) and also the discriminant analysis. Box’M test is a commonly used method to check whether the covariance matrices across groups are equal. This test is included in many statistical packages such as SPSS. The Box’M statistic can be transformed to the test statistics and their distributions are approximated using chi-squared and F distributions. This study aims to evaluate the performance of Box’M test and compare to the other two related tests under multivariate normality. In the case of non-normality, the nonparametric method using bootstrap is studied.

The results were as follows: 1. Under multivariate normality and the sample sizes of all groups are quite the same, the Box’M test and also the other two tests using chi-squared and F  distributions performs well in the situations studied. Moreover, the number of populations, the number of variables, and the sample size affect the performance of all of the three tests. 2. Under non-normality, the bootstrap method used in this study and the two tests based on Box’M statistic including chi-squared and F distributions are unsatisfactory. 3. To apply Box’M tests in SPSS, the significance level of 0.001 is recommended and the distribution of data and the sample size should be taken into account.