Testing Homogeneity Against Ordered Alternatives

In a one-way analysis of variance situation in which the populations are ordered under the alternative hypothesis, one desires a test that, unlike the usual normal theory F test, concentrates its power on the ordered alternatives, not on any alternatives. In this paper, two contributions are made. First, under the usual normal assumptions, work by Bartholomew 4, 7 on the likelihood ratio test, when the ordering is complete under the alternative hypothesis, is extended. By suitable characterization of the partition which the likelihood ratio induces on the sample space, the likelihood ratio test is shown to depend on incomplete Beta functions and certain probabilities of the above partitions of the sample space. The major contribution in this paper is for the case of equal sample sizes, where explicit expressions for these probabilities are obtained by indicating their relationship to Sparre Andersen's 1, 2 results. Second, under the analogous nonparametric assumptions and for equal sample sizes, a parallel test based on ranks is proposed and discussed for stochastic ordering of the populations. The asymptotic Pitman efficiency of the nonparametric test relative to the test in the normal case is derived.