广义族差错率的分步控制
On Stepwise Control of the Generalized Familywise Error Rate
摘要：
处理多重检验问题的经典方法是将注意力限制在控制族差错率（FWER）的程序上，即至少有一个错误拒绝的概率。在许多应用中，人们可能愿意容忍一个以上的错误拒绝，只要这种情况的数量得到控制，从而增加了该过程检测错误零假设的能力。这建议通过控制$k$或更多错误拒绝的概率来取代对FWER的控制，这被称为$k$ -FWER。本文提出了一种推导$k$ -FWER控制程序的统一方法。我们首先概括了众所周知的封闭原则的上下文中的FWER的情况下，控制的$k$ -FWER。然后，我们讨论了如何推导出$k$ -FWER控制逐步（stepdown或stepup）程序的基础上，边际$p$ -值使用这一原则。证明了在一定条件下，广义闭检验过程可归结为逐步检验过程，且任何逐步检验过程都等价于广义闭检验过程。最后，我们将著名的Hommel检验过程在两个方向上进行了推广，并证明了任何推广的Hommel检验过程都等价于一个具有相同临界值的推广的封闭检验过程.
英文摘要：
A classical approach for dealing with the multiple testing problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of at least one false rejection. In many applications, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of $k$ or more false rejections, which is called the $k$-FWER. In this article, a unified approach is presented for deriving the $k$-FWER controlling procedures. We first generalize the well-known closure principle in the context of the FWER to the case of controlling the $k$-FWER. Then, we discuss how to derive the $k$-FWER controlling stepwise (stepdown or stepup) procedures based on marginal $p$-values using this principle. We show that, under certain conditions, generalized closed testing procedures can be reduced to stepwise procedures, and any stepwise procedure is equivalent to a generalized closed testing procedure. Finally, we generalize the well-known Hommel procedure in two directions, and show that any generalized Hommel procedure is equivalent to a generalized closed testing procedure with the same critical values.
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