Within a one-way layout with normally distributed observations Tong (1969) proposed a multiple decision procedure for partitioning a given set of treatments into two subsets with the purpose of separating good and bad treatments. The qualities good and bad are defined in terms of the mean difference from a control. Tong's procedure is optimal in the sense that it maximizes the minimum probability of a correct partition (MPCP) within the class of so-called natural procedures (Giani and Straßburger, 1997). In this contribution we show that the optimality of Tong's procedure cannot be extended to a larger class of multiple decision rules. In fact, a non-natural partitioning procedure will be presented, which leads to a greater MPCP than Tong's procedure. Although the new procedure has a stepwise structure, it substantially differs from the well-known step-up and step-down selection procedures used for comparisons with a control. After a discussion of the theoretical results, the minimum total sample sizes necessary for Tong's and the new stepwise procedure to control the probability of correct partition at a preassigned confidence level P are compared with each other. It turns out that in practically relevant situations () Tong's procedure has nearly the same efficiency as the new decision rule.

References:

1. Tong, Y.L. (1969): On Partitioning a Set of Normal Populations by Their Locations with Respect to a Control. Annals of Mathematical Statistics 40, 1300-1324.
2. Giani, G., Straßburger, K. (1997): Optimum partition procedures for separating good and bad treatments. Journal of the American Statistical Association 92, 291-298.