The linear mixed model (LMM) has emerged as a powerful statistical tool in genetic association studies, enabling researchers to control for population structure and relatedness among study participants to reduce false positive findings. In the standard LMM framework, a hyperparameter η quantifies the relative contribution of random genetic effects versus random noise. While classical approaches treat η as known, genome-wide association studies require its estimation, typically via maximum likelihood (ML) or restricted maximum likelihood (REML) methods. Although likelihood ratio tests and Wald tests provide p-values for association testing under the LMM, Bayes factors offer distinct advantages—particularly when comparing multiple non-nested alternative models. However, no efficient method has been available for computing Bayes factors under the LMM framework in the context of genetic association studies.

Recently, Yongtao Guan and Daniel Levy from the Framingham Heart Study at the National Heart, Lung, and Blood Institute published a technical note, "Bayes factor for linear mixed model in genetic association studies" in the journal Quantitative Biology. The team developed an efficient method to compute Bayes factors for the LMM by transforming the standard LMM into a Bayesian linear regression through substitution of 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, the authors derived a closed-form expression for Bayes factors that can be evaluated efficiently without requiring sampling or numerical integration. They further demonstrated numerically that the Bayesian prediction from the transformed model is identical to the best linear unbiased prediction (BLUP) of the standard LMM, providing a new perspective and derivation for this known connection. The methods described in this study are implemented in the software package IDUL as two new functionalities: computing Bayes factors and residuals for the LMM.

Figure 1 illustrates the comparison between p-values and Bayes factors for the linear mixed model. The analysis was performed using real data consisting of 5,757 whole genome sequenced samples with a high level of relatedness and 79 protein assay phenotypes. The left panel shows log₁₀BF versus −log₁₀P, revealing that large Bayes factors correspond to small p-values, while also demonstrating that different Bayes factors may yield the same p-values and vice versa. The stratification by minor allele frequencies (MAF) shows that SNPs with larger MAF tend to have larger Bayes factors for the same p-values, because Bayes factors penalize SNPs with small MAF due to their lower information contents through the prior effect size. The right panel displays χ² versus 2logₑBF, reflecting the mathematical relationship between Bayes factors and their corresponding chi-square values.
DOI：10.1002/qub2.70036