Partitioning principle and selection of good treatments

In a recent paper, Finner and Straßburger [2002. The partitioning principle: a powerful tool in multiple decision theory. Ann. Statist. 30, 1194–1213.] introduced a general, weak and strong partitioning principle (SPP), respectively, for the construction of multiple decision procedures, e.g., multiple testing or selection procedures. Partitioning principles can be viewed as natural extensions of the well-known closure principle and may yield more powerful decision procedures. In this paper, we are concerned with the construction of a step-down procedure for selecting a subset of k⩾2k⩾2 treatments containing all good treatments by rigorously applying the weak partitioning principle (WPP). This results in some new least favourable parameter configuration (LFC) problems and an improved set of critical values. The new step-down procedure improves the step-down procedure based on the closure principle considerably and decreases the required sample size with respect to a pre-specified power criterion. Various procedures including a Newman–Keuls-type procedure which is related to the SPP and probably the best possible step-down procedure will be compared with respect to power and sample sizes. Finally, we reanalyse an experiment with 13 treatment means.