Limitations on detecting row covariance in the presence of column covariance

TL;DR

This paper investigates the limitations of detecting row covariance in multivariate data when column covariance is present, showing that invariant tests often lack power unless specific conditions are met, but biased tests can be effective in practice.

Contribution

It demonstrates fundamental limitations of invariant tests for row covariance detection under certain data conditions and proposes biased tests as a practical alternative.

Findings

No non-trivial invariant tests exist if rows are not sufficiently numerous.

Invariant tests lack power to detect arbitrary row covariance with arbitrary column covariance.

Biased tests can detect certain practical types of row covariance.

Abstract

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.