Hypothesis Testing for the Covariance Matrix in High-Dimensional Transposable Data with Kronecker Product Dependence Structure

TL;DR

This paper proposes robust hypothesis tests for the covariance matrices in high-dimensional transposable data modeled by a Kronecker product structure, with applications in genomics and neuroscience.

Contribution

It introduces novel tests for row and column covariance matrices that are robust to normality deviations in high-dimensional matrix-variate data.

Findings

Tests maintain nominal levels in simulations

Tests are powerful against alternatives

Applications demonstrate practical utility in genomics and neuroscience

Abstract

The matrix-variate normal distribution is a popular model for high-dimensional transposable data because it decomposes the dependence structure of the random matrix into the Kronecker product of two covariance matrices: one for each of the row and column variables. We develop tests for assessing the form of the row (column) covariance matrix in high-dimensional settings while treating the column (row) dependence structure as a nuisance. Our tests are robust to normality departures provided that the Kronecker product dependence structure holds. In simulations, we observe that the proposed tests maintain the nominal level and are powerful against the alternative hypotheses tested. We illustrate the utility of our approach by examining whether genes associated with a given signalling network show correlated patterns of expression in different tissues and by studying correlation patterns within measurements of brain activity collected using electroencephalography.