  Many inference techniques for multivariate data analysis assume that the rows
of the data matrix are realizations of independent and identically distributed
random vectors. Such an assumption will be met, for example, if the rows of the
data matrix are multivariate measurements on a set of independently sampled
units. In the absence of an independent random sample, a relevant question is
whether or not a statistical model that assumes such row exchangeability is
plausible. One method for assessing this plausibility is a statistical test of
row covariation. Maintenance of a constant type I error rate regardless of the
column covariance or matrix mean can be accomplished with a test that is
invariant under an appropriate group of transformations. In the context of a
class of elliptically contoured matrix regression models (such as matrix normal
models), I show that there are no non-trivial invariant tests if the number of
rows is not sufficiently larger than the number of columns. Furthermore, I show
that even if the number of rows is large, there are no non-trivial invariant
tests that have power to detect arbitrary row covariance in the presence of
arbitrary column covariance. However, we can construct biased tests that have
power to detect certain types of row covariance that may be encountered in
practice.
