On the isotonic range statistic for testing against an ordered alternative

Consider the balanced one-way layout for comparing k treatment effects mu(i), 1 less than or equal to i less than or equal to k. Marcus (Biometrika 63 (1976) 177) considers a test procedure for testing the null hypothesis H-0: mu(1) = mu(2) = (...) = mu(k) against the simple ordered alternative H-A: mu(1) less than or equal to mu(2) less than or equal to (...) less than or equal to mu(k) with at least one strict inequality based upon the range (μ) over cap (k) - (μ) over cap (1) of the isotonic estimates of the treatment effects. This test statistic was originally proposed in Williams (Biometrics 27 (1971) 103) and is sometimes referred to as the modified Williams statistic. It has many useful applications in biostatistical studies. In this article, we present an algorithm for calculating the distribution function of this isotonic range statistic for a general number of treatments k using recursive integration techniques and we provide tables of critical points for this test procedure. This allows the full implementation of this test procedure. We also show how this procedure can be used to construct a set of simultaneous confidence intervals for a set of monotone contrasts of the treatment effects and compare it with other multiple comparisons procedures. (C) 2001 Elsevier Science B.V. All rights reserved.