Ordered multiple comparisons with the best and their applications to dose-response studies.

This article considers the problem of comparing several treatments (dose levels, interventions, etc.) with the best, where the best treatment is unknown and the treatments are ordered in some sense. Order relations among treatments often occur quite naturally in practice. They may be ordered according to increasing risks, such as tolerability or safety problems with increasing dose levels in a dose-response study, for example. We tackle the problem of constructing a lower confidence bound for the smallest index of all treatments being at most marginally less effective than the (best) treatment having the largest effect. Such a bound ensures at confidence level 1 -alpha that all treatments with lower indices are relevantly less effective than the best competitor. We derive a multiple testing strategy that results in sharp confidence bounds. The proposed lower confidence bound is compared with those derived from other testing strategies. We further derive closed-form expressions for power and sample size calculations. Finally, we investigate several real data sets to illustrate various applications of our methods.