Making accurate inference for gene regulatory networks, including inferring about pathway by pathway interactions, is an important and difficult task. Motivated by such genomic applications, we consider multiple testing for conditional dependence between subgroups of variables. Under a Gaussian graphical model framework, the problem is translated into simultaneous testing for a collection of submatrices of a high-dimensional precision matrix with each submatrix summarizing the dependence structure between two subgroups of variables. A novel multiple testing procedure is proposed and both theoretical and numerical properties of the procedure are investigated. Asymptotic null distribution of the test statistic for an individual hypothesis is established and the proposed multiple testing procedure is shown to asymptotically control the false discovery rate (FDR) and false discovery proportion (FDP) at the pre-specified level under regularity conditions. Simulations show that the procedure works well in controlling the FDR and has good power in detecting the true interactions. The procedure is applied to a breast cancer gene expression study to identify between pathway interactions.
Keywords: Between pathway interactions; Gaussian graphical model; conditional dependence; covariance structure; false discovery proportion; false discovery rate; multiple testing; precision matrix; testing submatrices.