A high-dimensional two-sample test for the mean using random subspaces

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 dependences between variables. A test using random subspaces is proposed, which offers higher power when the variables are dependent and is invariant under linear transformations of the marginal distributions. The p-values for the test are obtained using permutations. The test does not rely on assumptions about normality or the structure of the covariance matrix. It is shown by simulation that the new test has higher power than competing tests in realistic settings motivated by microarray gene expression data. Computational aspects of high-dimensional permutation tests are also discussed and an efficient R implementation of the proposed test is provided.