Testing Procedures for Claiming Success on at Least <i>k</i> Out of <i>m</i> Hypotheses with an Application to Biosimilar Development

Multiplicity is a common issue in clinical drug development and there exists many proposals for the handling of multiple testing in clinical trials. However, the literature on testing procedures for claiming success on at least k out of m tests and the operating characteristics of these procedures is still sparse. Such testing is very relevant in biosimilar drug development, for example, where products have gained regulatory approval in the past, even though not all hypotheses could be rejected. Obviously, simple adjustments for multiplicity like the Bonferroni-adjustment or the Holm-procedure are valid as well for this testing problem, but can be conservative. In this article, we propose simple testing procedures for claiming success on at least k out of m tests which are more powerful than standard procedures while still providing strong control of the family-wise error rate. We illustrate their applicability in practice using an example from biosimilar drug development. In the supplementary materials, we provide proofs of the properties of our testing procedures and demonstrate the superiority of the proposed methodologies using simulations.多重性问题是临床药物研发中的常见议题，当前已有诸多针对临床试验多重检验的处理方案。然而，现有文献中针对「至少通过m项检验中的k项以判定研发成功」的检验方法，以及这类方法的操作特性的相关研究仍较为匮乏。这类检验在生物类似药（biosimilar）研发中极具现实意义，例如过往已有产品即便未拒绝全部原假设，也成功获得监管批准的案例。显然，诸如Bonferroni校正法（Bonferroni-adjustment）或Holm法（Holm-procedure）这类经典的多重性校正方法，同样适用于该检验场景，但往往过于保守。本文提出了适用于「至少通过m项检验中的k项以判定研发成功」的简便检验流程，其检验效能优于标准流程，同时仍可严格控制家族错误率（family-wise error rate）。我们通过一则生物类似药研发的实例，阐释了该方法的实际应用价值。在补充材料中，我们给出了所提检验方法的性质证明，并通过模拟实验验证了所提方法的优越性。