A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION

Abstract

There are several circumstances in which we may wish to test the homogeneity of the two sets of marginal probabilities in a two-way classification. For example, a sample from a bivariate distribution (say height of father, height of son) may be classified into a two-way table with identical (height) groupings in each margin. Or a similar classification may be possible for a non-measurable variable (say strength of right hand - strength of left hand). Again, in surveys of the same sample (a 'panel') on two different occasions, the interrelation of the results on the two occasions may be displayed in a two-way table, with one margin corresponding to each occasion. In all these cases, the question may arise: are the two sets of marginal probabilities identical? If the variable is measurable, we may test the difference between the means of the two marginal distributions by a large-sample standard-error test. However, we may be interested in the overall distributions, rather than only in their means. For the more stringent hypothesis of homogeneity, a test exists if we have two completely independent samples. when an ordinary x2 test of homogeneity may be applied (Cramer, 1946, p. 445). This test does not meet the essentially bivariate situations described above, where non-independence of the marginal distributions is a fundamental feature of the problem. When the classification is a double dichotomy, the problem of testing marginal homogeneity is simple, and its solution is a special case of the large-sample solution of the more general 2K classification problem given by Cochran (1950). Bowker (1948) gave a largesample test for complete symmetry in a two-way classification, a more restrictive hypothesis which is concerned with the entire set of probabilities in the classification, and not only with the marginal probabilities as we are here. In the present paper, a large-sample test for marginal homogeneity is derived and illustrated.