A generalized step-up-down multiple test procedure

A generalization of step-up and step-down multiple test procedures is proposed. This step-up-down procedure is useful when the objective is to reject a specified minimum number, q, out of a family of k hypotheses. If this basic objective is met at the first step, then it proceeds in a step-down manner to see if more than q hypotheses can be rejected. Otherwise it proceeds in a step-up manner to see if some number less than q hypotheses can be rejected. The usual step-down procedure is the special case where q = 1, and the usual step-up procedure is the special case where q = k. Analytical and numerical comparisons between the powers of the step-up-down procedures with different choices of q are made to see how these powers depend on the actual number of false hypotheses. Examples of application include comparing the efficacy of a treatment to a control for multiple endpoints and testing the sensitivity of a clinical trial for comparing the efficacy of a new treatment with a set of standard treatments.