Effective Positive Cauchy Combination Test

TL;DR

This paper introduces the positive Cauchy combination test (PCCT), a new method for combining p-values that improves control over type I errors under various dependence structures, validated through simulations and genetic data analysis.

Contribution

The paper proposes the PCCT, a novel p-value combination method that addresses limitations of the original CCT under weak dependence and provides robust error control.

Findings

PCCT effectively controls type I errors under weak dependence.

Simulation studies show PCCT outperforms existing methods.

Application to genetic data demonstrates practical utility.

Abstract

In the field of multiple hypothesis testing, combining p-values represents a fundamental statistical method. The Cauchy combination test (CCT) (Liu and Xie, 2020) excels among numerous methods for combining p-values with powerful and computationally efficient performance. However, large p-values may diminish the significance of testing, even extremely small p-values exist. We propose a novel approach named the positive Cauchy combination test (PCCT) to surmount this flaw. Building on the relationship between the PCCT and CCT methods, we obtain critical values by applying the Cauchy distribution to the PCCT statistic. We find, however, that the PCCT tends to be effective only when the significance level is substantially small or the test statistics are strongly correlated. Otherwise, it becomes challenging to control type I errors, a problem that also pertains to the CCT. Thanks to the theories of stable distributions and the generalized central limit theorem, we have demonstrated critical values under weak dependence, which effectively controls type I errors for any given significance level. For more general scenarios, we correct the test statistic using the generalized mean method, which can control the size under any dependence structure and cannot be further optimized. Our method exhibits excellent performance, as demonstrated through comprehensive simulation studies. We further validate the effectiveness of our proposed method by applying it to a genetic dataset.