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We propose new, optimal methods for analyzing randomized trials, when it issuspected that treatment effects may differ in two predefined subpopulations.Such sub-populations could be defined by a biomarker or risk factor measured atbaseline. The goal is to simultaneously learn which subpopulations benefit froman experimental treatment, while providing strong control of the familywiseType I error rate. We formalize this as a multiple testing problem and show itis computationally infeasible to solve using existing techniques. Our solutioninvolves a novel approach, in which we first transform the original multipletesting problem into a large, sparse linear program. We then solve this problemusing advanced optimization techniques. This general method can solve a varietyof multiple testing problems and decision theory problems related to optimaltrial design, for which no solution was previously available. In particular, weconstruct new multiple testing procedures that satisfy minimax and Bayesoptimality criteria. For a given optimality criterion, our new approach yieldsthe optimal tradeoff? between power to detect an effect in the overallpopulation versus power to detect effects in subpopulations. We demonstrate ourapproach in examples motivated by two randomized trials of new treatments forHIV.

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