Power of the likelihood-ratio test of the general linear hypothesis in multivariate analysis

Abstract

1-1. General background. In a basic paper on multivariate tests of linear hypotheses, Wilks (1932) applied the method of the likelihood ratio to the problem of testing the equality of mean vectors in k multivariate populations. The general approach of this early paper has been extended and expanded in the literature until, now, a procedure is available for testing multivariate linear hypotheses in a completely general form (see, for example, Anderson, 1958). For the problem of computing the power of these tests, however, no such general procedure exists. This is due to the fact that the noncentral distribution of the test criterion has not been expressed in a numerically feasible form. A first step towards the derivation of the non-central distribution, however, was taken when Anderson & Girshick (1944) attacked the problem of finding the distribution of the Wishart matrix in the non-central case. The result led Anderson (1946) to the derivation of the moments of the criterion for testing the hypothesis of the general multivariate regression problem (or rather the equivalent Wilks-Lawley hypothesis). Anderson's results were valid for a certain class of alternatives, which he called linear and planar. The moments for the linear alternative involve an infinite sum of expressions containing gamma functions and for the planar alternative, involve a triple infinite sum of the same type of expressions. This appears to be the extent of the progress on the problem of obtaining the power of the multivariate likelihood-ratio test of the general linear hypothesis.t