Variable Selection and Joint Estimation of Mean and Covariance Models with an Application to eQTL Data

In genomic data analysis, it is commonplace that underlying regulatory relationship over multiple genes is hardly ascertained due to unknown genetic complexity and epigenetic regulations. In this paper, we consider a joint mean and constant covariance model (JMCCM) that elucidates conditional dependent structures of genes with controlling for potential genotype perturbations. To this end, the modified Cholesky decomposition is utilized to parametrize entries of a precision matrix. The JMCCM maximizes the likelihood function to estimate parameters involved in the model. We also develop a variable selection algorithm that selects explanatory variables and Cholesky factors by exploiting the combination of the GCV and BIC as benchmarks, together with Rao and Wald statistics. Importantly, we notice that sparse estimation of a precision matrix (or equivalently gene network) is effectively achieved via the proposed variable selection scheme and contributes to exploring significant hub genes shown to be concordant to a priori biological evidence. In simulation studies, we confirm that our model selection efficiently identifies the true underlying networks. With an application to miRNA and SNPs data from yeast (a.k.a. eQTL data), we demonstrate that constructed gene networks reproduce validated biological and clinical knowledge with regard to various pathways including the cell cycle pathway.