The comparison of sample covariance matrices using likelihood ratio tests

SUMMARY The standard method for the comparison of two or more sample covariance matrices is the likelihood ratio test. The purpose of the present paper is to show how this test can be made more informative by hierarchically partitioning the test statistic into three components. Confronted with the comparison of several sample covariance matrices, most statisticians will probably consider using the likelihood ratio test (Seber, 1984, p. 449). The only reservation that needs to be made about the use of the test is its known sensitivity to nonnormality in the data (Layard, 1974; Olson, 1974; Manly, 1986). In the present paper we assume that the data are consistent with multivariate normality. The purpose of the present paper is to show that the test statistic for the standard likelihood ratio test can be partitioned into three components. These reflect differences between the covariance matrices due to: (a) matrices being proportional, (b) changing variances, and (c) changing correlations. The components can be tested for significance