  The Genotype-Tissue Expression (GTEx) project collects samples from multiple
human tissues to study the relationship between genetic variation or single
nucleotide polymorphisms (SNPs) and gene expression in each tissue. However,
most existing eQTL analyses only focus on single tissue information. In this
paper, we develop a multi-tissue eQTL analysis that improves the single tissue
cis-SNP gene expression association analysis by borrowing information across
tissues. Specifically, we propose an empirical Bayes regression model for
SNP-expression association analysis using data across multiple tissues. To
allow the effects of SNPs to vary greatly among tissues, we use a mixture
distribution as the prior, which is a mixture of a multivariate Gaussian
distribution and a Dirac mass at zero. The model allows us to assess the
cis-SNP gene expression association in each tissue by calculating the Bayes
factors. We show that the proposed estimator of the cis-SNP effects on gene
expression achieves the minimum Bayes risk among all estimators. Analyses of
the GTEx data show that our proposed method is superior to traditional simple
regression methods in terms of predicting accuracy for gene expression levels
using cis-SNPs in testing data sets. Moreover, we find that although genetic
effects on expression are extensively shared among tissues, effect sizes still
vary greatly across tissues.
