  Combining individual p-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 very
simple form and is defined as a weighted sum of Cauchy transformation of
individual p-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 p-value calculation of our proposed test is not only
accurate, but also as simple as the classic z-test or t-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 p-value calculations,
especially for very small p-values. The proposed test has also been applied to
a genome-wide association study of Crohn's disease and compared with several
existing tests.
