The partial conjunction null hypothesis is tested in order to discover a signal that is present in multiple studies. The standard approach of carrying out a multiple test procedure on the partial conjunction (PC) $p$-values can be extremely conservative. We suggest alleviating this conservativeness, by eliminating many of the conservative PC $p$-values prior to the application of a multiple test procedure. This leads to the following two step procedure: first, select the set with PC $p$-values below a selection threshold; second, within the selected set only, apply a family-wise error rate or false discovery rate controlling procedure on the conditional PC $p$-values. The conditional PC $p$-values are valid if the null p-values are uniform and the combining method is Fisher. The proof of their validity is based on a novel inequality in hazard rate order of partial sums of order statistics which may be of independent interest. We also provide the conditions for which the false discovery rate controlling procedures considered will be below the nominal level. We demonstrate the potential usefulness of our novel method, CoFilter (conditional testing after filtering), for analyzing multiple genome wide association studies of Crohn's disease.

翻译：部分合取零假设的检验旨在发现存在于多项研究中的信号。对部分合取（PC）$p$值执行多重检验程序的标准方法可能极为保守。我们建议通过在应用多重检验程序前剔除大量保守的PC $p$值来缓解这种保守性。这引出了以下两步程序：首先，筛选出PC $p$值低于选择阈值的集合；其次，仅在筛选后的集合中，对条件PC $p$值应用控制族错误率或错误发现率的程序。当零假设$p$值服从均匀分布且合并方法为Fisher时，条件PC $p$值是有效的。其有效性证明基于顺序统计量部分和在风险率序上的一个新不等式，该不等式可能具有独立的理论价值。我们还提供了所考虑的几种错误发现率控制程序低于名义水平的条件。通过分析多个克罗恩病全基因组关联研究，我们证明了新方法CoFilter（过滤后条件检验）的潜在实用价值。