  Having observed an $m\times n$ matrix $X$ whose rows are possibly correlated,
we wish to test the hypothesis that the columns are independent of each other.
Our motivation comes from microarray studies, where the rows of $X$ record
expression levels for $m$ different genes, often highly correlated, while the
columns represent $n$ individual microarrays, presumably obtained
independently. The presumption of independence underlies all the familiar
permutation, cross-validation and bootstrap methods for microarray analysis, so
it is important to know when independence fails. We develop nonparametric and
normal-theory testing methods. The row and column correlations of $X$ interact
with each other in a way that complicates test procedures, essentially by
reducing the accuracy of the relevant estimators.
