The Cauchy Combination Test under Arbitrary Dependence Structures

Abstract

Combining individual p-values to perform an overall test is often encountered in statistical applications. The Cauchy combination test (CCT) (Journal of the American Statistical Association, 2020, 115, 393–402) is a powerful and computationally efficient approach to integrate individual p-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 p-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 p-value test when the number of p-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.

Keywords:

• Cauchy combination test
• Cauchy distribution
• Copula
• p-value combination
• Sparse signals

Supplementary Materials

The Supplementary Materials contain three parts. In Part 1, we provide the technical details of theorems and Corollary in Section 2 of the main text. Part 2 includes the additional simulation results for Models 2 and 4 in Section 3. Part 3 gives the R codes for the numerical studies (Model 1 in the simulation study and real data analysis for Air Quality study).

Acknowledgments

We would like to thank the editor, the associate editor, and two anonymous reviewers for their insightful comments that have greatly improved the article. We also have to thank David Martin and Kai Yu for their valuable help.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Qizhai Li was supported in part by Beijing Natural Science Foundation (Z180006).