High-Dimensional Gaussian Graphical Regression Models with Covariates

• Published in: Journal of the American Statistical Association Vol. 118; no. 543; pp. 2088 - 2100
• Main Authors: Zhang, Jingfei, Li, Yi
• Format: Journal Article
• Language: English
• Published: United States Taylor & Francis 03.07.2023; Taylor & Francis Ltd
• Subjects: Brain cancer; brain neoplasms; Co-expression QTL; Consistency; Convergence; Efficacy; Gaussian graphical model with covariates; genes; Graphical models; Nonasymptotic convergence rate; normal distribution; Parameters; quantitative trait loci; regression analysis; Regression models; Simulation; Sparse group lasso; Sparsity; Statistics; Subject-specific Gaussian graphical model; Variants; co-expression QTL; subject-specific Gaussian graphical model; non-asymptotic convergence rate; sparse group lasso; Gaussian graphical model with covariates
• ISSN: 0162-1459; 1537-274X; 1537-274X
• DOI: 10.1080/01621459.2022.2034632

Though Gaussian graphical models have been widely used in many scientific fields, relatively limited progress has been made to link graph structures to external covariates. We propose a Gaussian graphical regression model, which regresses both the mean and the precision matrix of a Gaussian graphical model on covariates. In the context of co-expression quantitative trait locus (QTL) studies, our method can determine how genetic variants and clinical conditions modulate the subject-level network structures, and recover both the population-level and subject-level gene networks. Our framework encourages sparsity of covariate effects on both the mean and the precision matrix. In particular for the precision matrix, we stipulate simultaneous sparsity, that is, group sparsity and element-wise sparsity, on effective covariates and their effects on network edges, respectively. We establish variable selection consistency first under the case with known mean parameters and then a more challenging case with unknown means depending on external covariates, and establish in both cases the convergence rates and the selection consistency of the estimated precision parameters. The utility and efficacy of our proposed method is demonstrated through simulation studies and an application to a co-expression QTL study with brain cancer patients. Supplementary materials for this article are available online.