Bayesian LASSO for Quantitative Trait Loci Mapping

Abstract

The mapping of quantitative trait loci (QTL) is to identify molecular markers or genomic loci that influence the variation of complex traits. The problem is complicated by the facts that QTL data usually contain a large number of markers across the entire genome and most of them have little or no effect on the phenotype. In this article, we propose several Bayesian hierarchical models for mapping multiple QTL that simultaneously fit and estimate all possible genetic effects associated with all markers. The proposed models use prior distributions for the genetic effects that are scale mixtures of normal distributions with mean zero and variances distributed to give each effect a high probability of being near zero. We consider two types of priors for the variances, exponential and scaled inverse-x 2 distributions, which result in a Bayesian version of the popular least absolute shrinkage and selection operator (LASSO) model and the well-known Student's t model, respectively. Unlike most applications where fixed values are preset for hyperparameters in the priors, we treat all hyperparameters as unknowns and estimate them along with other parameters. Markov chain Monte Carlo (MCMC) algorithms are developed to simulate the parameters from the posteriors. The methods are illustrated using well-known barley data.

Group Variable Selection Methods and Their Applications in Analysis of Genomic Data

“…. , β p are equivalent to Ridge when we assume normal prior distributions for β's, and are equivalent to Lasso when we assume Laplace prior distributions [11,21]. Bayesian approach offers an alternative framework of variable selection.…”

Mapping quantitative trait loci using the MCMC procedure in SAS

“…The scaled inverse w 2 distribution for the prior of a variance component is not very complicated, but one can easily handle hierarchical models by assigning prior distributions to the degree of belief and scale parameter in the scaled inverse w 2 distribution. Details of the Mapping QTL using PROC MCMC S Xu and Z Hu hierarchical Bayesian model for QTL mapping can be found in the study by Yi and Xu (2008).…”

“…This program is not problem specific and can handle models with high level of complexity. WinBUGS has been applied to QTL mapping in many studies (Sillanpää and Bhattacharjee, 2005;Yi and Xu, 2008). Another general purpose of Bayesian analysis program is the MCMC procedure in SAS (Chen, 2009;SAS Institute Inc, 2009).…”

“…We are sure that the MCMC procedure in SAS has never been applied to QTL mapping. As Bayesian QTL mapping has been widely accepted by researchers and will become the main stream method of QTL mapping in the foreseeable future (Satagopan et al, 1996;Sillanpää and Arjas, 1998;Wang et al, 2005;Yi and Xu, 2008), PROC MCMC will become one of the most popular QTL mapping programs. This is particularly evident because many clinic traits have complicated densities beyond the normal distribution (Baima et al, 1999;Balmain, 2002;Rankinen et al, 2006) and Bayesian QTL mapping for these traits is difficult to conduct using other programs.…”

Multiple loci in silico mapping in inbred lines

“…There are several methods to estimate the parameters in model (2), that is, the PML and Bayesian LASSO (Park and Casella, 2008;Yi and Xu, 2008). If residual variance is heterozygous, the method proposed by Yi and Banerjee (2009) is available as well.…”