When simultaneously testing multiple hypotheses, the usual approach in the context of confirmatory clinical trials is to control the familywise error rate (FWER), which bounds the probability of making at least one false rejection. In many trial settings, these hypotheses will additionally have a hierarchical structure that reflects the relative importance and links between different clinical objectives. The graphical approach of Bretz et al (2009) is a flexible and easily communicable way of controlling the FWER while respecting complex trial objectives and multiple structured hypotheses. However, the FWER can be a very stringent criterion that leads to procedures with low power, and may not be appropriate in exploratory trial settings. This motivates controlling generalized error rates, particularly when the number of hypotheses tested is no longer small. We consider the generalized familywise error rate (k‐FWER), which is the probability of making k or more false rejections, as well as the tail probability of the false discovery proportion (FDP), which is the probability that the proportion of false rejections is greater than some threshold. We also consider asymptotic control of the false discovery rate, which is the expectation of the FDP. In this article, we show how to control these generalized error rates when using the graphical approach and its extensions. We demonstrate the utility of the resulting graphical procedures on three clinical trial case studies.

中文翻译：

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控制广义错误率的图形方法。

当同时测试多个假设时，在验证性临床试验的背景下，通常的方法是控制家庭错误率 (FWER)，它限制了至少做出一次错误拒绝的概率。在许多试验环境中，这些假设还具有反映不同临床目标之间的相对重要性和联系的层次结构。Bretz 等人 (2009) 的图形方法是一种灵活且易于交流的方法，可以在尊重复杂试验目标和多个结构化假设的同时控制 FWER。然而，FWER 可能是一个非常严格的标准，导致程序的功效较低，并且可能不适用于探索性试验设置。这促使控制广义错误率，特别是当测试的假设数量不再少时。我们考虑广义的家庭错误率（k ‐FWER)，即做出 k 个或更多错误拒绝的概率，以及错误发现比例 (FDP) 的尾部概率，即错误拒绝的比例大于某个阈值的概率。我们还考虑了对错误发现率的渐近控制，这是 FDP 的期望。在本文中，我们将展示如何在使用图形方法及其扩展时控制这些广义错误率。我们在三个临床试验案例研究中展示了由此产生的图形程序的实用性。