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Combining individual p-values to aggregate multiple small effects has along-standing interest in statistics, dating back to the classic Fisher'scombination test. In modern large-scale data analysis, correlation and sparsityare common features and efficient computation is a necessary requirement fordealing with massive data. To overcome these challenges, we propose a new testthat takes advantage of the Cauchy distribution. Our test statistic has a verysimple form and is defined as a weighted sum of Cauchy transformation ofindividual p-values. We prove a non-asymptotic result that the tail of the nulldistribution of our proposed test statistic can be well approximated by aCauchy distribution under arbitrary dependency structures. Based on thistheoretical result, the p-value calculation of our proposed test is not onlyaccurate, but also as simple as the classic z-test or t-test, making our testwell suited for analyzing massive data. We further show that the power of theproposed test is asymptotically optimal in a strong sparsity setting. Extensivesimulations demonstrate that the proposed test has both strong power againstsparse alternatives and a good accuracy with respect to p-value calculations,especially for very small p-values. The proposed test has also been applied toa genome-wide association study of Crohn's disease and compared with severalexisting tests.

Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures