Cauchy combination test: a powerful test with analytic <i>p</i>-value calculation under arbitrary dependency structures^*

Combining individual <i>p</i>-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual <i>p</i>-values. We prove a non-asymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the <i>p</i>-value calculation of our proposed test is not only accurate, but also as simple as the classic <i>z</i>-test or <i>t</i>-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to <i>p</i>-value calculations, especially for very small <i>p</i>-values. The proposed test has also been applied to a genome-wide association study of Crohn's disease and compared with several existing tests.

将单个p值（p-value）进行组合以整合多个微小效应，是统计学领域长期以来备受关注的研究课题，其历史可追溯至经典的费希尔组合检验（Fisher's combination test）。在现代大规模数据分析中，变量间相关性与稀疏性是两类常见特征，而高效计算则是处理海量数据的必要前提。为应对上述挑战，我们提出了一种依托柯西分布（Cauchy distribution）的新型检验方法。该检验统计量形式简洁，被定义为单个p值经柯西变换后的加权和。我们证明了一项非渐近结论：在任意相依结构下，所提检验统计量的零分布尾部可通过柯西分布实现良好近似。基于这一理论结果，所提检验的p值计算不仅精度可靠，还具备与经典z检验（z-test）、t检验（t-test）相当的简便性，使其非常适用于海量数据分析场景。我们进一步证明，在强稀疏性设定下，所提检验的统计功效具有渐近最优性。大量仿真实验表明，所提检验在针对稀疏备择假设时兼具强劲功效，且p值计算精度优异，尤其针对极小p值时表现突出。此外，我们将所提检验应用于一项克罗恩病（Crohn's disease）的全基因组关联研究（genome-wide association study），并与多款现有检验方法开展了对比分析。