Comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering

TL;DR

This paper introduces a fast, assumption-free statistical test for comparing large covariance matrices in genomics, along with a gene clustering method, validated on asthma gene expression data.

Contribution

It develops a robust, computationally efficient test for high-dimensional covariance matrices without structural assumptions, and proposes a new gene clustering algorithm for genomics.

Findings

Test effectively compares covariance matrices across biological states.

Gene clustering reveals differences in gene patterns between disease and control groups.

Methods are implemented in an R package available on CRAN.

Abstract

Comparing large covariance matrices has important applications in modern genomics, where scientists are often interested in understanding whether relationships (e.g., dependencies or co-regulations) among a large number of genes vary between different biological states. We propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes. A distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices. Hence the test is robust with respect to various complex dependence structures that frequently arise in genomics. We prove that the proposed procedure is asymptotically valid under weak moment conditions. As an interesting application, we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high-dimensional genomics data. Using an asthma gene expression dataset, we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets/pathways between the disease group and the control group, and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups. The proposed methods have been implemented in an R-package HDtest and is available on CRAN.