In multiple testing several criteria to control for type I errors exist. The false discovery rate, which evaluates the expected proportion of false discoveries among the rejected null hypotheses, has become the standard approach in this setting. However, false discovery rate control may be too conservative when the effects are weak. In this paper we alternatively propose to control the number of significant effects, where 'significant' refers to a pre-specified threshold $\gamma$. This means that a $(1-\alpha)$-lower confidence bound $L$ for the number of non-true null hypothesis with p-values below $\gamma$ is provided. When one rejects the nulls corresponding to the $L$ smallest p-values, the probability that the number of false positives exceeds the number of false negatives among the significant effects is bounded by $\alpha$. Relative merits of the proposed criterion are discussed. Procedures to control for the number of significant effects in practice are introduced and investigated both theoretically and through simulations. Illustrative real data applications are given.

翻译：在多重检验中，存在多种控制第一类错误的准则。错误发现率（评估被拒绝的原假设中错误发现比例的期望值）已成为该场景下的标准方法。然而，当效应较弱时，错误发现率控制可能过于保守。本文提出了另一种方法——控制显著效应数量，其中“显著”指预先指定的阈值$\gamma$。这意味着我们为非真原假设中p值低于$\gamma$的数量提供了一个$(1-\alpha)$水平的下置信界$L$。当拒绝对应$L$个最小p值的原假设时，显著效应中假阳性数量超过假阴性数量的概率被控制在$\alpha$以内。本文讨论了所提准则的相对优势，通过理论分析和模拟研究介绍了实际中控制显著效应数量的具体流程，并给出了实际数据应用的示例。