  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.
