An Empirical Bayes Mixture Model for Effect Size Distributions in Genome-Wide Association Studies

Characterizing the distribution of effects from genome-wide genotyping data is crucial for understanding important aspects of the genetic architecture of complex traits, such as number or proportion of non-null loci, average proportion of phenotypic variance explained per non-null effect, power for discovery, and polygenic risk prediction. To this end, previous work has used effect-size models based on various distributions, including the normal and normal mixture distributions, among others. In this paper we propose a scale mixture of two normals model for effect size distributions of genome-wide association study (GWAS) test statistics. Test statistics corresponding to null associations are modeled as random draws from a normal distribution with zero mean; test statistics corresponding to non-null associations are also modeled as normal with zero mean, but with larger variance. The model is fit via minimizing discrepancies between the parametric mixture model and resampling-based nonparametric estimates of replication effect sizes and variances. We describe in detail the implications of this model for estimation of the non-null proportion, the probability of replication in de novo samples, the local false discovery rate, and power for discovery of a specified proportion of phenotypic variance explained from additive effects of loci surpassing a given significance threshold. We also examine the crucial issue of the impact of linkage disequilibrium (LD) on effect sizes and parameter estimates, both analytically and in simulations. We apply this approach to meta-analysis test statistics from two large GWAS, one for Crohn’s disease (CD) and the other for schizophrenia (SZ). A scale mixture of two normals distribution provides an excellent fit to the SZ nonparametric replication effect size estimates. While capturing the general behavior of the data, this mixture model underestimates the tails of the CD effect size distribution. We discuss the implications of pervasive small but replicating effects in CD and SZ on genomic control and power. Finally, we conclude that, despite having very similar estimates of variance explained by genotyped SNPs, CD and SZ have a broadly dissimilar genetic architecture, due to differing mean effect size and proportion of non-null loci.Author Summary: We describe in detail the implications of a particular mixture model (a scale mixture of two normals) for effect size distributions from genome-wide genotyping data. Parameters from this model can be used for estimation of the non-null proportion, the probability of replication in de novo samples, the local false discovery rate, power for detecting non-null loci, and proportion of variance explained from additive effects. Here, we fit this model by minimizing discrepancies with nonparametric estimates from a resampling-based algorithm. We examine the effects of linkage disequilibrium (LD) on effect sizes and parameter estimates, both analytically and in simulations. We validate this approach using meta-analysis test statistics (“z-scores”) from two large GWAS, one for Crohn’s disease and the other for schizophrenia. We demonstrate that for these studies a scale mixture of two normal distributions generally fits empirical replication effect sizes well, providing an excellent fit for the schizophrenia effect sizes but underestimating the tails of the distribution for Crohn’s disease.