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.

我们提出了全新的最优分析方法，用于当怀疑存在两种预定义亚群的治疗效应存在差异时，对随机对照试验进行数据分析。此类亚群可通过基线阶段测量的生物标志物（biomarker）或风险因子进行定义。本研究的目标是同时明确可从试验性治疗中获益的亚群，同时严格控制家族式I类错误率（familywise Type I error rate）。我们将该问题形式化为多重检验问题（multiple testing problem），并证明现有技术无法在合理计算开销下完成求解。我们的解决方案采用了一种新颖的思路：首先将原始多重检验问题转换为大规模稀疏线性规划（linear program）问题，随后借助先进的优化技术完成求解。该通用方法可解决多种与最优试验设计相关的多重检验问题与决策理论（decision theory）问题，而此前针对这类问题尚无可行的解决方案。具体而言，我们构建了满足极小极大与贝叶斯最优准则（minimax and Bayes optimality criteria）的新型多重检验流程。针对给定的最优准则，我们的新方法可在总体人群效应检测功效与亚群效应检测功效之间实现最优的效能权衡。我们通过两项针对人类免疫缺陷病毒（HIV）新型治疗药物的随机对照试验案例，验证了所提方法的有效性与实用性。本文的补充材料可通过在线渠道获取。