Optimal Tests of Treatment Effects for the Overall Population and Two Subpopulations in Randomized Trials, Using Sparse Linear Programming

We propose new, optimal methods for analyzing randomized trials, when it is suspected that treatment effects may differ in two predefined subpopulations. Such subpopulations could be defined by a biomarker or risk factor measured at baseline. The goal is to simultaneously learn which subpopulations benefit from an experimental treatment, while providing strong control of the familywise Type I error rate. We formalize this as a multiple testing problem and show it is computationally infeasible to solve using existing techniques. Our solution involves a novel approach, in which we first transform the original multiple testing problem into a large, sparse linear program. We then solve this problem using advanced optimization techniques. This general method can solve a variety of multiple testing problems and decision theory problems related to optimal trial design, for which no solution was previously available. In particular, we construct new multiple testing procedures that satisfy minimax and Bayes optimality criteria. For a given optimality criterion, our new approach yields the optimal tradeoff between power to detect an effect in the overall population versus power to detect effects in subpopulations. We demonstrate our approach in examples motivated by two randomized trials of new treatments for HIV. Supplementary materials for this article are available online.

针对怀疑治疗效应在两个预设亚组中存在差异的随机试验，本文提出了全新的最优分析方法。此类亚组可通过基线测量的生物标志物或风险因子进行定义。研究目标为同时识别出可从试验性治疗中获益的亚组，同时严格控制家族式I类错误率（familywise Type I error rate）。本文将该问题形式化为多重检验问题，并证明现有技术无法在计算层面解决该问题。本文提出的解决方案采用一种全新思路：首先将原始多重检验问题转化为大规模稀疏线性规划问题，随后借助先进优化技术对其求解。该通用方法可解决此前尚无有效解决方案的、各类与最优试验设计相关的多重检验问题与决策理论问题。具体而言，本文构建了满足极小极大（minimax）与贝叶斯（Bayes）最优性准则的全新多重检验流程。针对给定的最优性准则，本文新方法可实现总体人群效应检出效力与亚组效应检出效力间的最优权衡。本文结合两项针对人类免疫缺陷病毒（HIV，human immunodeficiency virus）新型治疗方案的随机试验实例，验证了所提方法的有效性。本文补充材料可在线获取。