Bayes factor for linear mixed model in genetic association studies

Bayes factor has advantages over p-value as test statistics for association, particularly when comparing multiple non-nested alternative models. An efficient method to compute Bayes factor for linear mixed model in the context of genetic association studies is lacking. In this study, we transform the standard linear mixed model to a Bayesian linear regression by substituting the random effect with fixed effects, where the covariates of the fixed effects are eigenvectors of the genetic relatedness matrix and their respective prior effect sizes are proportional to the corresponding eigenvalues. Using conjugate normal inverse gamma priors on regression parameters, Bayes factors can be computed in a closed form. We demonstrate numerically the known relationship between Bayes factors and p-values for the linear mixed model. We then show that predictions based on the transformed Bayesian linear regression are identical to those of the best linear unbiased prediction (BLUP) of the standard linear mixed model. Our results provided a new perspective and derivation to a known connection between BLUP and Bayesian estimates. Methods described in this note are implemented in the software IDUL as two new functionalities: computing Bayes factors and residuals for the linear mixed model. IDUL and its source code are freely available at https://github.com/haplotype/idul.