Stochastic Optimization and Subgroup Selection

Abstract

An important area in statistics is that of experimental or study design. The most typical problem is that of finding the required sample size that meets specific goals such as controlling type I and II error rates at given levels. In most experiments, additional constraints may be imposed from both practical and technical perspectives. From the practical perspective, it is expensive, both in monetary and time scales, or even impossible, to perform experiments over all possible values that the design variables can take. Some approaches have been developed to design an experiment to achieve maximum information given the restrictions in sample size, known as `optimal designs'. In an optimal design, the design points are selected to maximize a pre-selected optimality criterion, and it can be done using optimization methods. In this work, we propose a method for stochastic optimization called “forward slice” and evaluate its performance relative to other optimization methods. We also demonstrate the use of our method in design problems. Specifically, our simulation studies indicate that the “forward slice” selects the global optimum more often than other optimization methods. Further, when applying the “forward slice” to design problems, our method performs well when obtaining locally D-optimal design points and achieves a higher median overall D-optimality compared to the design points obtained using an alternative algorithm. In addition to experimental design, optimality considerations also arise post-design and post-experiment. One such example is in the problem of subgroup selection in a clinical trial. Most studies are designed to address only the primary inferential questions. However, in many studies, it is also of interest to assess differential associations in subpopulations. We propose a decision-theoretic approach to subgroup analysis consisting of two stages: model selection and subgroup reporting. We assess and compare the performance of our proposed method with some traditional approaches for subgroup selection under different scenarios. Our simulation studies show that the performance of the decision-theoretic method is similar to that of testing an interaction followed by stratified analyses based on the results of the interaction test. In the selection of the subgroups, the proposed method favors reporting subgroups that exhibit larger treatment effect, are larger, and are simpler (less complex). We also observe a trade-off where approaches that tend to have a larger power for detecting a subpopulation may perform more poorly when the effect is in fact homogeneous in the overall population. In addition, the proposed method allows for incorporation of prior information in both the model selection and subgroup reporting stages which may increase power while keeping the type-I error controlled.