A robust covariance testing approach for high-throughput data

The problem of testing changes in covariance has received increasing attention in recent years, especially in the context of high-dimensional testing. A number of approaches have been proposed, all limited to the two-sample problem and involving varying statistics and assumptions on the number of features $p$ vs. the sample size $n$. There are no general approaches to test association of covariances with a continuous outcome. We propose a uniform framework for testing association of covariances with an experimental variable, whether discrete or continuous. The approach is not limited by the data dimensions. Our test procedure (i) does not rely on parametric assumptions, (ii) works well for a range of $p$ and $n$ (e.g., does not require $n > p$), (iii) provides correct type I error control, and (iv) includes four different statistics, to ensure power and flexibility under various settings, including a new "connectivity" statistic that is sensitive to changes in overall covariance magnitude. We demonstrate that, for the two-sample special case, the proposed statistics are permutationally equivalent or similar to existing proposed statistics. We demonstrate the power and utility of our approaches via simulation and analysis of real data. The approach is implemented in an $R$ package.