Stepwise multiple comparisons with the best

Abstract

Multiple comparisons with the best (MCB) compares each treatment with the best of the other treatments. The parameters of interest in MCB are the differences between each treatment and the best of the other treatments. The simultaneous confidence intervals constructed in Hsu ( Ann. Statist. 9, 1981, Ann. Statist. 12, 1984) for these differences are closed intervals constrained to contain zero, which may seem undesirable to some users. This article derives a stepwise MCB method where the first step is Tukey's all-pairwise comparison method, and the last step is the original 1-step MCB method. At the same confidence level, stepwise MCB will always infer at least as many treatments to be strictly not the best as Tukey's method, often more. Open confidence intervals for the differences between each treatment and the best of the other treatments associated with the stepwise MCB are also derived. Thus, the common perception that stepwise procedures have no corresponding confidence sets (Lehmann, Testing Statistical Hypothesis, 1986, p.388) is not true for the MCB problem of this article.