A comparison of tests of the Wilks-Lawley hypothesis in multivariate analysis

Abstract

Three different tests have been proposed for the Wilks-Lawley hypothesis of the homogeneity of group means in multivariate analysis (i.e. of the equality of np-variate normal means). These tests are: the likelihood ratio test (Wilks, 1932); the analysis of variance analogue (Lawley, 1938) and the generalized Mahalanobis distance statistic (Pillai, 1954)*. We will call these test statistics W, L and D, respectively; their expressions as functions of the observations are given in ? 1-2. below. Under normal theory these tests are asymptotically equivalent. That is, suitably scaled, they have the same asymptotic distribution both when the null and the alternative hypotheses are true. This extends to non-normal theory, on central limit theorem considerations, provided the homoscedasticity and independence assumptions are retained. They are of course completely equivalent for all sample sizes in the univariate case, and also in the two-sample case. There is little to choose between them in respect of ease of computation and whilst intuitively it may be felt that W would be more seriously affected by departures from normality in small samples, relatively little is known of the robustness of the tests. (Barton & David, 1962 have studied the randomization theory of D and their results are available for the investigation of the effects of nonnormality on the first kind of error entailed by its use.) The null hypothesis percentage points of W were tabulated by Pearson & Wilks (1933) and Pillai (1954), gives those of D to a high degree of approximation. Both tabulations are for the usual percentage points, taken over a small range of values of N and p in the case of Pearson & Wilks, and a wide range of N andp in the case of Pillai; here N is the total sample size. The upshot of these considerations is that the distinction between the tests rests almost entirely on the comparison of their powers in small samples. Such comparison is made in the present paper for selected values of the several parameters concerned. In particular, the calculations have been restricted to the bivariate case p = 2, although the methods have been designed with the aim of application when p has any value. The overall aim has been to develop simple approximate methods of evaluating powers, checking the accuracy of approximation by comparison with exact values at points where these are obtainable by means of the theorems of Kshirsagar (1961) or the Fisher-Hsu-Roy theorem (cf. Hsu, 1940,