  Genetical genomics experiments have now been routinely conducted to measure
both the genetic markers and gene expression data on the same subjects. The
gene expression levels are often treated as quantitative traits and are subject
to standard genetic analysis in order to identify the gene expression
quantitative loci (eQTL). However, the genetic architecture for many gene
expressions may be complex, and poorly estimated genetic architecture may
compromise the inferences of the dependency structures of the genes at the
transcriptional level. In this paper we introduce a sparse conditional Gaussian
graphical model for studying the conditional independent relationships among a
set of gene expressions adjusting for possible genetic effects where the gene
expressions are modeled with seemingly unrelated regressions. We present an
efficient coordinate descent algorithm to obtain the penalized estimation of
both the regression coefficients and the sparse concentration matrix. The
corresponding graph can be used to determine the conditional independence among
a group of genes while adjusting for shared genetic effects. Simulation
experiments and asymptotic convergence rates and sparsistency are used to
justify our proposed methods. By sparsistency, we mean the property that all
parameters that are zero are actually estimated as zero with probability
tending to one. We apply our methods to the analysis of a yeast eQTL data set
and demonstrate that the conditional Gaussian graphical model leads to a more
interpretable gene network than a standard Gaussian graphical model based on
gene expression data alone.
