Asymptotic Distribution-Free Independence Test for High-Dimension Data

Abstract

Test of independence is of fundamental importance in modern data analysis, with broad applications in variable selection, graphical models, and causal inference. When the data is high dimensional and the potential dependence signal is sparse, independence testing becomes very challenging without distributional or structural assumptions. In this article, we propose a general framework for independence testing by first fitting a classifier that distinguishes the joint and product distributions, and then testing the significance of the fitted classifier. This framework allows us to borrow the strength of the most advanced classification algorithms developed from the modern machine learning community, making it applicable to high dimensional, complex data. By combining a sample split and a fixed permutation, our test statistic has a universal, fixed Gaussian null distribution that is independent of the underlying data distribution. Extensive simulations demonstrate the advantages of the newly proposed test compared with existing methods. We further apply the new test to a single cell dataset to test the independence between two types of single cell sequencing measurements, whose high dimensionality and sparsity make existing methods hard to apply. Supplementary materials for this article are available online.

Keywords:

• Permutation
• Rank sum test
• Sample splitting
• Test of independence

Supplementary Materials

The supplementary materials contain the technical proof and the code for replicating the numerical results.

Acknowledgments

This work used the Bridges-2 system, which is supported by NSF award OAC-1928147 at the Pittsburgh Supercomputing Center. This project was funded by National Institute of Mental Health (NIMH) grant R01MH123184 and NSF DMS-2015492.