Covariate Adjusted Precision Matrix Estimation with an Application in Genetical Genomics
Tony Cai, Hongzhe Li, Weidong Liu, and Jichun Xie
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Abstract:
Motivated by the analysis of eQTL data, we introduce a
sparse high dimensional multivariate regression model for
studying the conditional independent relationships among a
set of genes adjusting for possible genetic effects, as
well as the genetic architecture that influences the gene
expressions. The precision matrix in the model specifies a
covariate-adjusted Gaussian graph, which presents the
conditional dependency structure of gene expression after
the confounding genetic effects on gene expressions are
taken into account. We present a covariate-adjusted
precision matrix estimation (CAPME) method using
constrained l1 minimization, which can be easily
implemented by linear programming. Asymptotic convergence
rates 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. Numerical performance of
the estimators is investigated using both simulated and
real data sets. Simulation results show that CAPME
results in significant improvements in both precision
matrix estimation and in graphical structure selection when
compared to the standard Gaussian graphical model assuming
constant means. CAPME is also applied to analyze a yeast
eQTL data for the identification of the gene network among
a set of genes in the Mitogen-activated protein kinase
(MAPK) pathway.
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