Cauchy Combination Test: A Powerful Test With Analytic <i>p</i>-Value Calculation Under Arbitrary Dependency Structures

<b><i>Abstract–</i>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 nonasymptotic 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. Supplementary materials for this article are available online.</b>