Sparse variable and covariance selection for high-dimensional seemingly unrelated Bayesian regression

High-throughput technology for molecular biomarkers is increasingly producing multivariate phenotype data exhibiting strong correlation structures. Existing approaches for combining such data with genetic variants for multivariate Quantitative Trait Loci analysis generally either ignore correlation structure or make other restrictive assumptions about the associations between phenotypes and genetic loci. We present a Bayesian Variable Selection (BVS) model with sparse variable and covariance selection for high-dimensional seemingly unrelated regressions. The model includes a matrix of binary variable selection indicators for multivariate regression, thus allowing different phenotype responses to be associated with different genetic predictors (a seemingly unrelated regressions framework). A general covariance structure is allowed for the residuals relating to the conditional dependencies between phenotype variables. The covariance structure may be dense (unrestricted) or sparse, with a graphical modelling prior. The graphical structure amongst the multivariate responses can be estimated as part of the model. To achieve feasible computation of the large and complex model space, we exploit a factorisation of the covariance matrix parameter to enable faster computation using Markov Chain Monte Carlo (MCMC) methods. We are able to infer associations with thousands of candidate predictors multivariately on hundreds of responses. We illustrate the model using a dataset of 158 NMR spectroscopy measured metabolites and over 9000 Single Nucleotide Polymorphisms on chromosome 16, measured in a cohort of more than 5000 people.