Multiple Testing of Submatrices of a Precision Matrix With Applications to Identification of Between Pathway Interactions

ABSTRACT

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 prespecified 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. Supplementary materials for this article are available online.

KEYWORDS:

• Between pathway interactions
• Conditional dependence
• Covariance structure
• False discovery proportion
• False discovery rate
• Gaussian graphical model
• Multiple testing
• Precision matrix
• Testing submatrices

Supplement Materials

In the supplement we collect a few technical lemmas, provide detailed proofs of Theorems 1–3 and Corollary 1 and summarize the numerical comparisons with Bonferroni correction procedure.

Funding

The research of Yin Xia was supported in part by “The Recruitment Program of Global Experts” Youth Project from China, the startup fund from Fudan University and NSF Grant DMS-1612906. The research of Tianxi Cai was supported in part by NIH Grants R01 GM079330, P50 MH106933, and U54 HG007963. The research of Tony Cai was supported in part by NSF Grants DMS-1208982 and DMS-1403708, and NIH Grant R01 CA127334.