A One-sided Studentized Range Test for lesting against a Simple Ordered Alternative

Abstract

Consider the usual balanced one-way fixed effects analysis of variance (ANOVA) model X ij = μ i + ε ij (1 ≤ j ≤ n ; 1 ≤ i ≤ k ), where the ε ij are independent N (0, [sgrave] 2 ) random variables and μ i is the mean of the i th treatment (1 ≤ i ≤ k ). The μ i and [sgrave] 2 are unknown parameters. Let X i be the sample mean of the i th treatment based on n observations (1 ≤ i ≤ k ), and let S 2 be an unbiased estimate of [sgrave] 2 , which is distributed independently of the X i as a σ 2 χ 2 v / v random variable. Usually, the ANOVA mean squared error with v = k ( n − 1) df is used as the estimate S 2 . A problem that has received considerable attention is that of testing the null hypothesis H 0 : μ 1 , = … = μ k against the simple ordered alternative hypothesis H A : μ 1 ≤ … ≤ μ k with at least one strict inequality [see, e.g., Barlow, Bartholomew, Bremner, and Brunk (1972) and Robertson, Wright, and Dykstra (1988)]. Various test procedures have been proposed for this problem, such as the likelihood ratio test (LRT) of Bartholomew (1961), which are particularly sensitive to configurations of the treatment means satisfying the order restriction. Many of the test procedures, however, suffer from the disadvantage of not having a convenient inversion to a set of simultaneous confidence intervals for useful contrasts of the treatment means. In this article the following simple multiple contrast test procedure is considered, which may be thought of as a “one-sided” studentized range test (OSRT). The OSRT rejects the null hypothesis H 0 iff the statistic max 1≤ i < j ≤ k ( X j − X j )/(S/√ n ) exceeds a suitable critical point. A considerable advantage of this test procedure is that it allows the simple construction of exact simultaneous one-sided confidence intervals of the form ( d ij , ∞) for all of the ordered pairwise differences of the treatment means μ j ) − μ i (1 ≤ i < j ≤ k ). These confidence intervals will necessarily be the shortest possible confidence intervals of this kind. In Section 1 it is shown how these simultaneous confidence intervals may also be extended to produce exact simultaneous confidence intervals for all “nonnegative” contrasts of the treatment means ∑ i = 1 k c i μ i , where ∑ i = 1 k c i = 0 and ∑ i = 1 j c i ≤ 0 (1 ≤ j ≤ k − 1). The critical points of the usual (two-sided) studentized range test (SRT) are easily obtained by solving a two-dimensional integral equation, regardless of the number of treatments k . The calculation of the exact critical points for the OSRT is more difficult, however, with the complexity increasing as k increases. Nevertheless, in Section 2, it is shown how for 3 ≤ k ≤ 9 the critical points may be obtained by solving no more than a three-dimensional integral equation, and some tables of the critical points are given for these cases. In Section 3, the power properties of the OSRT are discussed and power comparisons are made with other test procedures. Modifications of the OSRT for an unbalanced ANOVA model are discussed in Section 4. The natural modification which is known to give a conservative test procedure when applied to the SRT, is shown to give a possibly liberal test if applied to the OSRT. Finally, Section 5 contains a summary.