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

Abstract

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 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.

Keywords: Cauchy distribution; Correlation matrix; Global hypothesis testing; High dimensional data; Non-asymptotic approximation; Sparse alternative.

Figure 1:

The ratio of empirical size to significance level for Model 1–4, summarized by boxplots. The x-axis is the significance level at α = 10⁻¹, 10⁻², 10⁻³, 10⁻⁴, 10⁻⁵.

Figure 2:

The ratio of empirical size to significance level (dashed lines) for Model 5. The straight line in each plot is the reference line. The plot on the right is a zoom-in image of the plot on the left. Note that the non-smoothness and fluctuation of the dashed curve in the right plot is due to the Monte Carlo errors.

Figure 3:

Power comparison of CCT, MinP, HC and BJ. The x-axis is the correlation strength ρ. The columns from left to right correspond to the dimension d = 20, 40, 60. The rows from top to bottom correspond to the signal percentage 5%, 10% and 20%. The signal strength is chosen to make the power in every setting comparable.