  Scientists routinely compare gene expression levels in cases versus controls
in part to determine genes associated with a disease. Similarly, detecting
case-control differences in co-expression among genes can be critical to
understanding complex human diseases; however statistical methods have been
limited by the high dimensional nature of this problem. In this paper, we
construct a sparse-Leading-Eigenvalue-Driven (sLED) test for comparing two
high-dimensional covariance matrices. By focusing on the spectrum of the
differential matrix, sLED provides a novel perspective that accommodates what
we assume to be common, namely sparse and weak signals in gene expression data,
and it is closely related with Sparse Principal Component Analysis. We prove
that sLED achieves full power asymptotically under mild assumptions, and
simulation studies verify that it outperforms other existing procedures under
many biologically plausible scenarios. Applying sLED to the largest
gene-expression dataset obtained from post-mortem brain tissue from
Schizophrenia patients and controls, we provide a novel list of genes
implicated in Schizophrenia and reveal intriguing patterns in gene
co-expression change for Schizophrenia subjects. We also illustrate that sLED
can be generalized to compare other gene-gene "relationship" matrices that are
of practical interest, such as the weighted adjacency matrices.
