Generalizing Simes’ test and Hochberg’s stepup procedure
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推广 Simes 检验与 Hochberg 逐步上升程序
- 研究问题
- 当允许少量错误拒绝时，如何以控制至少发生指定数量错误拒绝的概率为目标，改进多重检验发现错误原假设的能力？
- 方法与思路
- 作者利用各个同维子集具有相同联合零分布的条件，推广 Simes 检验及逐步程序，并在特定正依赖条件下建立错误率控制。
- 材料陈述的贡献
- 摘要称，推广程序分别实现弱或强错误率控制；在独立检验概率值的情形下，部分推广检验比所述原有程序更有检验力。
- 阅读关注点
- 应核对弱控制与强控制的区别、子集联合分布相同和正依赖的条件，以及独立情形下检验力比较针对哪些备择配置。
- 阅读边界
- 仅据版本未注明的索引摘要；错误率控制和检验力优势均有条件，摘要不能支持在任意依赖结构下直接使用这些程序。
原文摘要
In a multiple testing problem where one is willing to tolerate a few false rejections, procedure controlling the familywise error rate (FWER) can potentially be improved in terms of its ability to detect false null hypotheses by generalizing it to control the k-FWER, the probability of falsely rejecting at least k null hypotheses, for some fixed k>1. Simes’ test for testing the intersection null hypothesis is generalized to control the k-FWER weakly, that is, under the intersection null hypothesis, and Hochberg’s stepup procedure for simultaneous testing of the individual null hypotheses is generalized to control the k-FWER strongly, that is, under any configuration of the true and false null hypotheses. The proposed generalizations are developed utilizing joint null distributions of the k-dimensional subsets of the p-values, assumed to be identical. The generalized Simes’ test is proved to control the k-FWER weakly under the multivariate totally positive of order two (MTP2) condition [J. Multivariate Analysis 10 (1980) 467–498] of the joint null distribution of the p-values by generalizing the original Simes’ inequality. It is more powerful to detect k or more false null hypotheses than the original Simes’ test when the p-values are independent. A stepdown procedure strongly controlling the k-FWER, a version of generalized Holm’s procedure that is different from and more powerful than [Ann. Statist. 33 (2005) 1138–1154] with independent p-values, is derived before proposing the generalized Hochberg’s procedure. The strong control of the k-FWER for the generalized Hochberg’s procedure is established in situations where the generalized Simes’ test is known to control its k-FWER weakly.