On a Comprehensive Test for the Homogeneity of Variances and Covariances in Multivariate Problems

Abstract

The use of the Neyman and Pearson likelihood ratio method has enabled S. S. Wilks (Biometrika, 24, 471-94, 1932) to obtain a comprehensive criterion for testing the hypothesis that the corresponding variances and covariances in k samples drawn from normal multivariate populations are equal. The true sampling distribution of this criterion, l1, when the hypothesis tested is true, being unknown, methods of approximating to it are considered in the case where each of the samples is of the same size n. It is found that a Pearson Type I curve in the form p(l 1) = unspecified B([image] 1, [image] 2) unspecified-1 l 1ml-11)m2-1 will give an adequate approximation to the distribution of l1, if the parameters m1 and m2 are chosen to obtain agreement between the 1st and 2d moments of the true and approximate distributions. Thus the significance levels for l1 may be obtained either, directly, from the Tables of the Incomplete Beta-Function or, by means of a transformation, from Fisher''s z-tables. When n is not too small, empirical relations giving, with sufficient accuracy, m1 and m2 in terms of n, k and q, the number of variates, are obtained so that the labor of computing the moments may be avoided. The limiting form of the distribution of l1 is considered in the general case where the samples are large but not necessarily of the same size. An example illustrating the use of the test is worked out in full.