OPTIMAL DESIGNS FOR MULTIPLE COMPARISONS WITH THE BEST

In comparing k treatments, it is not always necessary to compare all pairs of treatments. Often comparisons of each treatment with the best of the other treatments suffice. If the i-th treatment is better than the best of the other treatments, then it is the best. On the other hand, if the j-th treatment is worse than the best of the other treatments, then it can be safely discarded. Hsu (1981, 1984a,b) proposed the method of Multiple Comparisons with the Best (MCB) to provide simultaneous confidence intervals for the difference between each treatment and the best of the other treatments. We first extend the MCB method to the General Linear Model, then the problem of optimal design for MCB inference is considered. The traditional A-, D-, and E-optimal design criteria do not apply to multiple comparisons, since the parameters of interest in multiple comparisons are not invariant under orthogonal transformations. Thus new optimal design criteria are formulated for MCB. Then designs optimal within variance-balanced designs in the classes of block designs, row-column designs, and nested row-column designs are discussed. Finally, simulations of randomly generated designs indicate that these optimal variance-balanced designs perform well outside the class of variance-balanced designs as well.