Covariate-Adjusted Precision Matrix Estimation with an Application in Genetical Genomics.
Motivated by analysis of genetical genomics data, we introduce a sparse high dimensional multivariate regression model for studying conditional independence relationships among a set of genes adjusting for possible genetic effects. The precision matrix in the model specifies a covariate-adjusted Gaussian graph, which presents the conditional dependence structure of gene expression after the confounding genetic effects on gene expression are taken into account. We present a covariate-adjusted precision matrix estimation method using a constrained ℓ1minimization, which can be easily implemented by linear programming. Asymptotic convergence rates in various matrix norms and sign consistency are established for the estimators of the regression coefficients and the precision matrix, allowing both the number of genes and the number of the genetic variants to diverge. Simulation shows that the proposed method results in significant improvements in both precision matrix estimation and graphical structure selection when compared to the standard Gaussian graphical model assuming constant means. The proposed method is also applied to analyze a yeast genetical genomics data for the identification of the gene network among a set of genes in the mitogen-activated protein kinase pathway.
Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
T Tony
TT
Cai
Department of Biostatistics and Epidemiology, University of Pennsylvania Perelman School of Medicine, Philadelphia, Pennsylvania 19104, USA.
Hongzhe
H
Li
Department of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China.
Weidong
W
Liu
Department of Statistics, Fox School of Business, Temple University, Philadelphia, Pennsylvania 19122, USA.
Jichun
J
Xie
Constrained ℓ1 penalization; Gaussian graphical model; high-dimensionality; multivariate regression