Optimal design for pp-value consistent step-up procedures for multiple comparisons with a control in direction-mixed families

It is common in clinical studies for several treatments to be compared to a control. Most of the related statistical techniques have been developed to accommodate inferential families in which all hypotheses are either one- or two-sided such that the familywise error rate is controlled at a specified level. Several multiple testing procedures were recently proposed to perform multiple comparisons with a control in direction-mixed families that contain a mixture of one- and two-sided inferences. Of these procedures, the pp-value consistent step-up procedure is found to be superior in terms of its power and pp-value consistent property. In this paper, we examine the techniques for computing the all-pairs power of this testing procedure. We also provide the means to obtain the optimal design when a desired level of all-pairs power is given. Compared to the conservative method of treating all hypotheses as two-sided, the proposed procedure requires a substantially smaller sample, as all useful information on the direction of the alternatives is utilized in an optimal way. The computation of the optimal design relies on an efficient algorithm, which is also discussed in this paper. A clinical study example is employed to illustrate the proposed procedure.