On Step-up Tests for Comparing Several Treatments

Consider a one-way layout in which independent observations Xi have distributions F (x− μi) with μi unknown, 1 ≤ i ≤ k. The basic problem is to test the family of subset hypotheses about μi. The step-up test scheme of Welsch (1977) is studied in detail. A general result is given on how to determine the critical values so that the type I familywise error rate is strongly controlled at a preassigned level α, and a particular set of critical values is recommended. The interrelationship between the step-up tests and the closed tests of Marcus, Peritz and Gabriel (1976) is illuminated. This enables us to construct a closed test which is uniformly more powerful than a step-up test. Monte Carlo simulation is carried out to compare the power of the two new tests with that of Welsch’s (1977) test and several other multiple tests. The two new tests turn out to be preferable. The one-way ANOVA setting is given special attention, and the critical values of the new step-up test are