A Generalization of the Partition Problem

Abstract

The problem of partitioning with respect to a control or a standard is an important statistical problem in the area of selection and ranking. In the last 60 + years a number of formulations have been proposed for this problem utilizing the two-stage sampling strategy of Stein (1945) and other multistage sampling strategies. One such formulation, which had generalized many other formulations, was proposed in Tong (1969) for the normally distributed populations. Since then the formulation in Tong (1969) has been used by a multitude of researchers and practitioners. The indifference zone in Tong's (1969) formulation is a region in which the experimenter is indifferent to differentiate any treatment mean from the control treatment mean. However, this indifference zone also serves the additional role of defining the boundaries for “bad” and “good” treatments compared to the control treatment. Though the formulation in Tong (1969) is straightforward and easy to implement, this additional role that the indifference zone plays could make the formulation somewhat undesirable to the practitioner. To illustrate, assume that in a clinical trial a drug with less than 10% improvement over the placebo is considered as a drug with insignificant improvement and the drug is termed a “bad treatment” and a drug with more than 30% improvement over the placebo is considered a drug with significant improvement and the drug is termed a “good treatment.” In such a scenario, Tong's formulation would define the indifference zone to be between 10% and 30% improvement over placebo, and any treatment with an effect size of between 10% and 30% improvement can be partitioned either a good treatment or a bad treatment without any penalty. Such a wide indifference zone makes the partition obtained to be rather misleading as the treatments termed good treatment or bad treatment could include treatments with effect size between 10% and 30%. One could potentially reduce the size of the indifference zone in Tong's formulation, but this would then alter the definition of the bad and good treatments, which are generally provided by the experts in the area. In this article, a generalization of Tong's formulation is proposed that will partition the treatments in the indifference zone as a separate identifiable group without altering the definition of the good and bad treatments. It is shown that the formulation in Tong (1969) is a limiting case of the generalized partition formulation proposed in this article. A fixed sample procedure is constructed for the case when the common population variance is known. In addition, a purely sequential procedure is proposed for the unknown variance case. The first-order and second-order asymptotic properties are derived and verified using Monte Carlo simulation studies. An example is provided to illustrate an application of the proposed generalization.