For the linear mixed model (LMM): y = Xβ + Zb + e; y is the vector of phenotype value; β the vector of fixed effects, b is the vector of additive genetic effects. Distributions: b ~ N(0,G), e ~ N(0,R), y ~ N(Xβ, V), where V=ZGZ′ + R. Var(G)=Aσ2, where A is the matrix of an additive genetic relationship constructed based on the pedigree (BLUP) or the genomic marker information provided by the SNPs (GBLUP).We can get Mixed Model Equations for β and b using Restricted Maximum Likelihood (REML).

The module is y = Xβ + Zs + e where and s the sum of the vector of SNP effects derived from different assumed distributions. BayesB assumes that most of the genetic markers have zero effect, which can be described as a mixture prior of a scaled t-distribution with probability π and a point mass at 0 with probability 1−π. BayesCπ assumes that SNP effects have a mixture prior of a normal distribution that has mean 0 and variance σ2 with probability π and null effect markers with probability 1−π. BayesN is the nested BayesCπ model, where the SNPs within a 0.2 Mb non-overlapping genomic region are collectively considered as a window. BayesS is similar to BayesCπ but the variance of SNP effects (for SNPs with non-zero effects) is related to MAF (pi) through a parameter S (σi^2=[2pi(1-pi)]^S*σ^2). BayesR assumes that SNP effects follow a mixture of four normal distributions N(0, γk*σk^2), the γk are 0, 0.01, 0.1 and 1 with probability π1, π2, π3 and π4, respectively, and π1+π2+π3+π4=1. The unknown parameters and SNP effects of Bayesian models were obtained from a Gibbs scheme based on the Markov chain Monte Carlo (MCMC) iterations.