Polygenic modeling with bayesian sparse linear mixed models

Abstract

Both linear mixed models (LMMs) and sparse regression models are widely used in genetics applications, including, recently, polygenic modeling in genome-wide association studies. These two approaches make very different assumptions, so are expected to perform well in different situations. However, in practice, for a given dataset one typically does not know which assumptions will be more accurate. Motivated by this, we consider a hybrid of the two, which we refer to as a "Bayesian sparse linear mixed model" (BSLMM) that includes both these models as special cases. We address several key computational and statistical issues that arise when applying BSLMM, including appropriate prior specification for the hyper-parameters and a novel Markov chain Monte Carlo algorithm for posterior inference. We apply BSLMM and compare it with other methods for two polygenic modeling applications: estimating the proportion of variance in phenotypes explained (PVE) by available genotypes, and phenotype (or breeding value) prediction. For PVE estimation, we demonstrate that BSLMM combines the advantages of both standard LMMs and sparse regression modeling. For phenotype prediction it considerably outperforms either of the other two methods, as well as several other large-scale regression methods previously suggested for this problem. Software implementing our method is freely available from http://stephenslab.uchicago.edu/software.html.

Figure 1. Comparison of PVE estimates from LMM (blue), BVSR (red), and BSLMM (purple) in two simulation scenarios.

The x-axis show the number of causal SNPs (Scenario I) or the number of medium/small effect SNPs (Scenario II). Results are based on 20 replicates in each case. (A) (true PVE = 0.2) and (C) (true PVE = 0.6) show RMSE of PVE estimates. (B) (true PVE = 0.2) and (D) (true PVE = 0.6) show boxplots of PVE estimates, where the true PVE is shown as a horizontal line. Notice a break point on the y-axis in (C).

Figure 2. Comparison of prediction performance of LMM (blue), BVSR (red), and BSLMM (purple) in two simulation scenarios, where all causal SNPs are included in the data.

Performance is measured by Relative Predictive Gain (RPG). True PVE = 0.6. Means and standard deviations (error bars) are based on 20 replicates. The x-axis show the number of causal SNPs (Scenario I) or the number of medium/small effect SNPs (Scenario II).

Figure 3. Comparison of prediction performance of LMM (blue), BVSR (red), and BSLMM (purple) for seven diseases in the WTCCC dataset.

Performance is measured by area under the curve (AUC), where a higher value indicates better performance. The order of the diseases is based on the performance of BSLMM. The mean and standard deviation of AUC scores for BSLMM in the seven diseases are 0.60 (0.02) for HT, 0.60 (0.03) for CAD, 0.61 (0.03) for T2D, 0.65 (0.02) for BD, 0.68 (0.02) for CD, 0.72 (0.01) for RA, 0.88 (0.01) for T1D.

Figure 4. Comparison of prediction performance of several models with BSLMM for three traits in the heterogenous stock mouse dataset.

Performance is measured by RMSE difference with respect to BSLMM, where a positive value indicates worse performance than BSLMM. The x-axis shows two different ways to split the data into a training set and a test set, each with 20 replicates. The mean RMSE of BSLMM for the six cases are 0.70, 0.80, 0.79, 0.90, 0.98 and 0.99, respectively.