The Cauchy Combination Test under Arbitrary Dependence Structures

Combining individual <i>p</i>-values to perform an overall test is often encountered in statistical applications. The Cauchy combination test (CCT) (<i>Journal of the American Statistical Association</i>, 2020, 115, 393–402) is a powerful and computationally efficient approach to integrate individual <i>p</i>-values under arbitrary dependence structures for sparse signals. We revisit this test to additionally show that (i) the tail probability of the CCT can be approximated just as well when more relaxed assumptions are imposed on individual <i>p</i>-values compared to those of the original test statistics; (ii) such assumptions are satisfied by six popular copula distributions; and (iii) the power of the CCT is no less than that of the minimum <i>p</i>-value test when the number of <i>p</i>-values goes to infinity under some regularity conditions. These findings are confirmed by both simulations and applications in two real datasets, thus, further broadening the theory and applications of the CCT.