A novel class of methods for combining $p$-values to perform aggregate hypothesis tests has emerged that exploit the properties of heavy-tailed Stable distributions. These methods offer important practical advantages including robustness to dependence and better-than-Bonferroni scaleability, and they reveal theoretical connections between Bayesian and classical hypothesis tests. The harmonic mean $p$-value (HMP) procedure is based on the convergence of summed inverse $p$-values to the Landau distribution, while the Cauchy combination test (CCT) is based on the self-similarity of summed Cauchy-transformed $p$-values. The CCT has the advantage that it is analytic and exact. The HMP has the advantage that it emulates a model-averaged Bayes factor, is insensitive to $p$-values near 1, and offers multilevel testing via a closed testing procedure. Here I investigate whether other Stable combination tests can combine these benefits, and identify a new method, the L\'evy combination test (LCT). The LCT exploits the self-similarity of sums of L\'evy random variables transformed from $p$-values. Under arbitrary dependence, the LCT possesses better robustness than the CCT and HMP, with two-fold worst-case inflation at small significance thresholds. It controls the strong-sense familywise error rate through a multilevel test uniformly more powerful than Bonferroni. Simulations show that the LCT behaves like Simes' test in some respects, with power intermediate between the HMP and Bonferroni. The LCT represents an interesting and attractive addition to combined testing methods based on heavy-tailed distributions.
翻译:结合美元价值以进行总体假设测试的新型方法类别已经出现,它利用了重尾调稳定分布的特性。这些方法提供了重要的实际优势,包括依赖性强,比Bonferroni的可缩放性更好,它们揭示了巴伊西亚和古典假设测试之间的理论联系。调和平均美元价值(HMP)程序的基础是对美元价值与Landau分布的反正数值的趋同。而Couch组合测试(CCT)则以粗尾调变换的美元价值的自我相似性为基础。CCT的中间性功能优势在于它具有分析性和准确性。HMP的优势在于它模仿模型平均湾值与古典假设值之间的理论联系,它通过封闭测试程序提供多级测试。在这里,我调查其他稳定的混合测试能否结合这些好处,并找出一种新的方法,即LCCT变换值值值值值值值。LCT利用了最弱的自我-C-Cylanceral 和最强性硬性货币价值的LMvyral 测试方法,它展示了比Oral-ral-ral-lalalalalalalalalalal-lation roal rolal rol) 的较重的自我缩缩缩缩检验法。