Uniformly most powerful tests for simultaneously detecting a treatment effect in the overall population and at least one subpopulation

We take the perspective of a researcher planning a randomized trial of a new treatment, where it is suspected that certain subpopulations may benefit more than others. These subpopulations could be defined by a risk factor or biomarker measured at baseline. We focus on situations where the overall population is partitioned into two, predefined subpopulations. When the true average treatment effect for the overall population is positive, it logically follows that it must be positive for at least one subpopulation. Our goal is to construct multiple testing procedures that maximize power for simultaneously rejecting the overall population null hypothesis and at least one subpopulation null hypothesis. We show that uniformly most powerful tests exist for this problem, in the case where outcomes are normally distributed. We construct new multiple testing procedures that, to the best of our knowledge, are the first to have this property. These procedures have the advantage of not requiring any sacrifice for detecting a treatment effect in the overall population, compared to the uniformly most powerful test of the overall population null hypothesis.