  Expression quantitative trait loci (eQTL) analyses, which identify genetic
markers associated with the expression of a gene, are an important tool in the
understanding of diseases in human and other populations. While most eQTL
studies to date consider the connection between genetic variation and
expression in a single tissue, complex, multi-tissue data sets are now being
generated by the GTEx initiative. These data sets have the potential to improve
the findings of single tissue analyses by borrowing strength across tissues,
and the potential to elucidate the genotypic basis of differences between
tissues.
  In this paper we introduce and study a multivariate hierarchical Bayesian
model (MT-eQTL) for multi-tissue eQTL analysis. MT-eQTL directly models the
vector of correlations between expression and genotype across tissues. It
explicitly captures patterns of variation in the presence or absence of eQTLs,
as well as the heterogeneity of effect sizes across tissues. Moreover, the
model is applicable to complex designs in which the set of donors can (i) vary
from tissue to tissue, and (ii) exhibit incomplete overlap between tissues. The
MT-eQTL model is marginally consistent, in the sense that the model for a
subset of tissues can be obtained from the full model via marginalization.
Fitting of the MT-eQTL model is carried out via empirical Bayes, using an
approximate EM algorithm. Inferences concerning eQTL detection and the
configuration of eQTLs across tissues are derived from adaptive thresholding of
local false discovery rates, and maximum a-posteriori estimation, respectively.
We investigate the MT-eQTL model through a simulation study, and rigorously
establish the FDR control of the local FDR testing procedure under mild
assumptions appropriate for dependent data.

Methodology