A Computationally Efficient Bayesian Seemingly Unrelated Regressions Model for High-Dimensional Quantitative Trait Loci Discovery

Abstract

Our work is motivated by the search for metabolite quantitative trait loci (QTL) in a cohort of more than 5000 people. There are 158 metabolites measured by NMR spectroscopy in the 31‐year follow‐up of the Northern Finland Birth Cohort 1966 (NFBC66). These metabolites, as with many multivariate phenotypes produced by high‐throughput biomarker technology, exhibit strong correlation structures. Existing approaches for combining such data with genetic variants for multivariate QTL analysis generally ignore phenotypic correlations or make restrictive assumptions about the associations between phenotypes and genetic loci. We present a computationally efficient Bayesian seemingly unrelated regressions model for high‐dimensional data, with cell‐sparse variable selection and sparse graphical structure for covariance selection. Cell sparsity allows different phenotype responses to be associated with different genetic predictors and the graphical structure is used to represent the conditional dependencies between phenotype variables. To achieve feasible computation of the large model space, we exploit a factorisation of the covariance matrix. Applying the model to the NFBC66 data with 9000 directly genotyped single nucleotide polymorphisms, we are able to simultaneously estimate genotype–phenotype associations and the residual dependence structure among the metabolites. The R package BayesSUR with full documentation is available at https://cran.r-project.org/web/packages/BayesSUR/

BayesSUR: An R Package for High-Dimensional Multivariate Bayesian Variable and Covariance Selection in Linear Regression

“…The BayesSUR package fits a Bayesian seemingly unrelated regression model with a number of options for variable selection, and where the covariance matrix structure is allowed to be diagonal, dense or sparse. It encompasses three classes of Bayesian multi-response linear regression models: hierarchical related regressions (HRR, Richardson et al 2011), dense and sparse seemingly unrelated regressions (dSUR and SSUR, Bottolo et al 2021), and the structured seemingly unrelated regression, which makes use of a Markov random field (MRF) prior (Zhao et al 2021).…”

“…It is possible to estimate a full covariance matrix by specifying an inverse Wishart prior, i.e., C ∼ IW(ν, τ I s ). To avoid estimating the dense and large covariance matrix directly, Bottolo et al (2021) exploited a factorization of the dense covariance matrix to transform the parameter space (ν, τ ) of the inverse Wishart distribution to space…”

“…|S q | and |R q | denote the number of variables in these components. For more technical details, please refer to Bottolo et al (2021).…”

“…In the SSUR models, another important parameter is G in the hyper-inverse Wishart prior for the covariance matrix C. It is updated by the junction tree sampler (Green and Thomas 2013;Bottolo et al 2021) jointly with the corresponding proposal for σ 2 qt and ρ qt |σ 2 qt in (10). At each MCMC iteration we then extract the adjacency matrix G (t) (t = 1, • • • , L), from which we derive posterior mean estimators of the edge inclusion probabilities as Ĝ…”

“…However, these approaches are not computationally feasible if the number of input variables is very large. Bottolo, Banterle, Richardson, Ala-Korpela, Järvelin, and Lewin (2021) recently developed a Bayesian variable selection model which employs the hotspot prior for variable selection, learns a structured covariance matrix and implements the ESS algorithm in the SUR framework to further improve computational efficiency.…”