Mielke et al. (2021) proposed a stepwise multiple testing procedure (MTP) based on marginal p-values for rejecting at least k out of m null hypotheses. Briefly, the MTP cannot reject less than k hypotheses, but this property improves the power to reject k or more hypotheses relative to the stepdown MTP of Holm (1979). Mielke et al. discussed why such an MTP is of interest in the context of biosimilarity developments. This article describes how a slight modification of Holm's simple direct arguments can be used as an alternative to the closed-testing arguments of Mielke et al. to show the strong control of the family-wise error rate. This modification is based on a Holm-type formulation of the rejection algorithm where each step involves a single ordered p-value. With k equal to one, the stepwise MTP reduces to Holm's stepdown MTP. The MTP is valid quite generally. A version of the MTP with weights for hypotheses is also described and discussed. As in the case without weights: (a) a modification of Holm's arguments can be used; (b) with k equal to one, the MTP reduces to Holm's MTP with weights; and (c) the MTP is valid quite generally.

中文翻译：

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关于拒绝 m 个假设中的至少 k 个的逐步过程的注意事项：一个简单的 Holm 型公式和证明

米尔克等人。(2021) 提出了一种基于边际p值的逐步多重检验程序 (MTP)，用于拒绝m个零假设中的至少k个。简而言之，MTP 不能拒绝少于k个假设，但是这个属性提高了拒绝k个假设的能力或更多与 Holm (1979) 的降级 MTP 相关的假设。米尔克等人。讨论了为什么在生物相似性发展的背景下对这样的 MTP 感兴趣。本文描述了如何对 Holm 的简单直接论点进行轻微修改，以替代 Mielke 等人的封闭测试论点。以显示对家庭错误率的强大控制。此修改基于拒绝算法的 Holm 型公式，其中每个步骤都涉及单个有序p值。与k等于 1，逐步 MTP 减少为 Holm 的降压 MTP。MTP 非常普遍有效。还描述和讨论了具有假设权重的 MTP 版本。与没有权重的情况一样： (a) 可以使用对 Holm 论证的修改；(b) 当k等于 1 时，MTP 减少到带权重的 Holm 的 MTP；(c) MTP 相当普遍有效。