The Chi-Square Test of Distance Correlation

Abstract

Distance correlation has gained much recent attention in the data science community: the sample statistic is straightforward to compute and asymptotically equals zero if and only if independence, making it an ideal choice to discover any type of dependency structure given sufficient sample size. One major bottleneck is the testing process: because the null distribution of distance correlation depends on the underlying random variables and metric choice, it typically requires a permutation test to estimate the null and compute the p-value, which is very costly for large amount of data. To overcome the difficulty, in this article, we propose a chi-squared test for distance correlation. Method-wise, the chi-squared test is nonparametric, extremely fast, and applicable to bias-corrected distance correlation using any strong negative type metric or characteristic kernel. The test exhibits a similar testing power as the standard permutation test, and can be used for K-sample and partial testing. Theory-wise, we show that the underlying chi-squared distribution well approximates and dominates the limiting null distribution in upper tail, prove the chi-squared test can be valid and universally consistent for testing independence, and establish a testing power inequality with respect to the permutation test. Supplementary files for this article are available online.

Keywords:

• Centered chi-squared distribution
• Nonparametric test
• Testing independence
• Unbiased distance covariance

Supplementary Materials

The appendix (supplementary material) includes detailed background information, all theorem proofs and intermediate results, and detailed simulation functions.

Acknowledgments

The authors thank to Dr. Austin Brockmeier, Mr. Ronak Mehta, Dr. Yuexiao Dong, Dr. Carey Priebe for helpful discussions and suggestions. We also thank to the editors and reviewers for the valuable suggestions that greatly improved the article in organization, expositions, and experiments.

Notes

1 https://github.com/neurodata/mgc-matlab.

2 https://github.com/neurodata/mgc.

Additional information

Funding

This work was supported by the National Science Foundation award DMS-1921310, the National Institute of Health award R01MH120482, and DARPA L2M program FA8650-18-2-7834.