A Test of Homogeneity of Means Under Restricted Alternatives

Abstract

SUMMARY: Suppose that observations x1, x2, …, xk have been obtained from normal populations with unknown means μ1, μ2, …, μk and known standard deviations σ1, σ2, …, σk. The paper discusses tests of the hypothesis μ 1, μ 2, … = μk when prior information is available about the μ‘s under the alternative hypothesis. A generalized form of the χ 2-test, denoted by x2¯, has been developed for use when this information takes the form of order restrictions between some or all of the μ‘s. The distribution is shown to depend on that of χ 2 and certain probabilities which have been determined for many sets of order restrictions. The power function has been obtained in certain cases and used to compare the x2¯-test with χ 2, which assumes no prior information. This shows that the utilization of order information often leads to a considerable gain in power. Power comparisons with other versions of x2¯ and with tests based on scores suggest that no worthwhile improvement over x2¯ is possible. The paper is an extension and generalization of work contained in two earlier papers (Bartholomew, 1959a, b).