  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.
