Multiple comparison procedures for optimally discriminating between good, equivalent, and bad treatments with respect to a control

Given several experimental treatments and an associated family of test problems in terms of two-sided comparisons with a control, a multiple test is proposed which simultaneously controls the probability of Type I and Type II familywise errors including incorrect directional decisions by means of one single probability requirement. The multiple test can alternatively be written as a partition procedure that keeps the probability of misclassifications in discriminating bad, equivalent, and good treatments with regard to the control below a prespecified level. Concerning the 0-1 loss function of indicating incorrect classifications, the proposed decision procedure is derived as a minimax rule among all so-called natural partition procedures. For such an optimum rule, least favorable parameter configuration results on the probability of correct partition are obtained under general distribution assumptions. These results are used for the usual one-way layout with normally distributed observations to determine smallest total sample sizes with optimum allocation among the treatments and the control subject to preassigned bounds on the probability of correct partition. Both the known and unknown common variance case are dealt with. For known variance asymptotic considerations reveal close relationships between the optimum sample size as well as corresponding critical value of the partition procedure here and that one suggested by Sobel and Tong (1971, Biometrika, 58, 177-181) for discriminating only good from bad treatments in considering one-sided comparisons with a control. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: primary 62F07, 62B15; secondary 60C20, 62H30.