On a Holm-related MTP for rejecting at least k hypotheses: general validity, optimality property, confidence regions, and applications

This article concerns p-value-based multiple testing procedures (MTPs) that can be used in a confirmatory clinical study under minimal assumptions in case the requirement for study-success is that at least k out of m primary/important hypotheses become rejected. Recently, a simple, generally valid Holm-type MTP was discussed that can be used for such a requirement for any k from one to m. It can only reject at least k (or zero) hypotheses, but this increases the power to reject k or more hypotheses compared to Holm’s step-down MTP. The present article provides a simple formulation and proof of strong family-wise error rate (FWER) control for a stepwise MTP that is sharper in that for any k strictly between one and m it: (a) always rejects at least as much, and (b) can potentially reject fewer than k hypotheses. This sharper MTP too is generally valid, without any assumption about logical or stochastic relationships. It has a gatekeeping step, followed by m steps where ordered primary p-values are compared to critical constants and rejections are made in a step-down manner. These constants have the optimality property that under a natural monotonicity restriction, they cannot be increased without losing the general strong FWER control. Confidence regions like those for Holm’s MTP are provided. Applications are discussed in connection with three interesting approaches proposed earlier for confirmatory studies: (a) the Superiority-Noninferiority approach; (b) Fallback tests for co-primary endpoints; and (c) Multistage gatekeeping MTPs that utilize so-called k-truncated Holm MTPs in some stages.

本文聚焦于基于p值的多重检验程序（multiple testing procedures, MTPs），适用于在最小假设条件下开展的确证性临床试验，此时试验成功的判定标准为：在m个主要/重要假设中，至少需拒绝k个。近期有研究提出了一种简单且普适有效的霍尔姆型MTP，可针对1到m间任意k值满足该判定标准。该程序仅能拒绝至少k个（或0个）假设，相较霍尔姆逐步下降型MTP，其拒绝k个及以上假设的检验效能有所提升。本文针对某逐步型MTP给出了简洁的形式化表述与强家族整体错误率（family-wise error rate, FWER）控制的证明，该MTP具有更优的特性：当k严格介于1与m之间时，(a) 其拒绝的假设数始终不少于同类程序，且(b) 理论上可拒绝少于k个的假设。该更优MTP同样具备普适有效性，无需任何逻辑或随机关系的假设。它包含一个守门步骤，随后是m个步骤：将按序排列的主要p值与临界常数进行比较，并以逐步下降的方式完成假设拒绝。这些临界常数具有最优性性质：在自然单调性约束下，若提升临界常数，则会丧失普适强FWER控制特性。本文还给出了与霍尔姆MTP类似的置信区域。最后讨论了其与此前提出的三种适用于确证性研究的方法的结合应用：(a) 优效-非劣效性方法；(b) 共同主要终点的回落检验；(c) 在部分阶段采用所谓k截断霍尔姆MTP的多阶段守门多重检验程序。