  A common problem in genetics is that of testing whether a set of highly
dependent gene expressions differ between two populations, typically in a
high-dimensional setting where the data dimension is larger than the sample
size. Most high-dimensional tests for the equality of two mean vectors rely on
naive diagonal or trace estimators of the covariance matrix, ignoring
dependencies between variables. A test recently proposed by Lopes et al. (2012)
implicitly incorporates dependencies by using random pseudo-projections to a
lower-dimensional space. Their test offers higher power when the variables are
dependent, but lacks desirable invariance properties and relies on asymptotic
p-values that are too conservative. We illustrate how a permutation approach
can be used to obtain p-values for the Lopes et al. test and how modifying the
test using random subspaces leads to a test statistic that is invariant under
linear transformations of the marginal distributions. The resulting test does
not rely on assumptions about normality or the structure of the covariance
matrix. We show by simulation that the new test has higher power than competing
tests in realistic settings motivated by microarray gene expression data. We
also discuss the computational aspects of high-dimensional permutation tests
and provide an efficient R implementation of the proposed test.
