ON A COMPREHENSIVE TEST FOR THE HOMOGENEITY OF VARIANCES AND COVARIANCES IN MULTIVARIATE PROBLEMS

Now that satisfactory and probably final solutions have been obtained for a wide variety of statistical problems concerned with a single normally distributed variable, more and more attention has recently been given to the solution of multivariate problems. The multiple correlation methods of the old large sample theory have been replaced in many instances by others for which " studentized " test criteria are available, often having sampling distributions that are already familiar in univariate problems. In a recent paper on " The statistical utilization of multiple measurements", R. A. Fisher (1938a) has shown the connexion between certain of these methods: the D2-statistic work of Mahalanobis, the discriminant function methods of the Galton Laboratory and the generalized " Student's " ratio of Hotelling. A similar very general problem was dealt with some time ago by S. S. Wilks (1932), while mention may also be made of two papers by D. G. Lawley (1938 a, b) and a paper by P. L. Hsu (1938). The purpose of the methods put forward is to obtain information regarding the mean values of a number, say q, of correlated variables in one or more, say k, populations from which random samples have been drawn. If we denote by x8 a value of the sth variable (s = 1, 2, ..., q), then in all this work it has been assumed not only that x8 is normally distributed, but that it has the same variance o2 in every population sampled. Further, it is assumed that if x. is a second variable the correlation coefficients psu between x8 and x. is the same in all populations. The estimates of variance and covariance required in order to "studentize" the function of the sample means are therefore obtained by pooling together the sums of squares and sums of products from all samples. While it is true that even if oa, and psu are not the same in all populations the error involved may not be very large, it is however important to have available some means of testing the basic hypothesis which assumes homogeneity throughout the populations. Such a test has been derived by S. S. Wilks (1932) by an extension of Neyman & Pearson's likelihood ratio method of approach. Hitherto the somewhat lengthy computations required to obtain the moments of the sampling distribution of the test criterion have probably discouraged its use. The objects of the present paper are as follows: (a) In the simple but commonly met case, where the k samples are of the same