Non-parametric Polygenic Risk Prediction via Partitioned GWAS Summary Statistics

Abstract

In complex trait genetics, the ability to predict phenotype from genotype is the ultimate measure of our understanding of genetic architecture underlying the heritability of a trait. A complete understanding of the genetic basis of a trait should allow for predictive methods with accuracies approaching the trait's heritability. The highly polygenic nature of quantitative traits and most common phenotypes has motivated the development of statistical strategies focused on combining myriad individually non-significant genetic effects. Now that predictive accuracies are improving, there is a growing interest in the practical utility of such methods for predicting risk of common diseases responsive to early therapeutic intervention. However, existing methods require individual-level genotypes or depend on accurately specifying the genetic architecture underlying each disease to be predicted. Here, we propose a polygenic risk prediction method that does not require explicitly modeling any underlying genetic architecture. We start with summary statistics in the form of SNP effect sizes from a large GWAS cohort. We then remove the correlation structure across summary statistics arising due to linkage disequilibrium and apply a piecewise linear interpolation on conditional mean effects. In both simulated and real datasets, this new non-parametric shrinkage (NPS) method can reliably allow for linkage disequilibrium in summary statistics of 5 million dense genome-wide markers and consistently improves prediction accuracy. We show that NPS improves the identification of groups at high risk for breast cancer, type 2 diabetes, inflammatory bowel disease, and coronary heart disease, all of which have available early intervention or prevention treatments.

Keywords: genome-wide association study; linkage disequilibrium; non-parametric prediction; phenotype prediction; polygenic score; prognosis; summary statistics.

Figure 1

Overview of Non-Parametric Shrinkage (NPS) (A) For unlinked markers, NPS partitions SNPs into K subgroups splitting the GWAS effect sizes (${\hat{\beta }}_j$) at cut-offs of $b_0,b_1,\dots ,b_K$. Partitioned risk scores $G_{ik}$ are calculated for each partition $k$ and individual $i$ using an independent genotype-level training cohort. The per-partition shrinkage weights ${\omega }_k$ are determined by the separation of $G_{ik}$ between training case subjects and control subjects. Estimating the per-partition shrinkage weights is a far easier problem than estimating per-SNP effects. The training sample size is small but still larger than the number of partitions, whereas for per-SNP effects, the GWAS sample size is considerably smaller than the number of markers in the genome. This procedure “shrinks” the estimated effect sizes not relying on any specific assumption about the distribution of true effect sizes. (B) For markers in LD, genotypes and estimated effects are decorrelated first by a linear projection $\mathcal{P}$ in non-overlapping windows of ∼2.5 Mb in length, and then NPS is applied to the data. The size of black dots indicates genotype frequencies in population. Before projection, genotypes at SNP 1 and 2 are correlated due to LD ($\mathbf{D}$), and thus sampling errors of estimated effects (${\hat{\beta }}_j|{\beta }_j$) are also correlated between adjacent SNPs. The projection $\mathcal{P}$ neutralizes both correlation structures. The axes of projection are marked by red dashed lines. ${\beta }_j$ denotes the true genetic effect at SNP $j$. $N_g$ is the sample size of GWAS cohort.

Figure 2

Per-Partition Shrinkage Weights Estimated by Non-Parametric Shrinkage (NPS) Approximate the Conditional Mean Effects in the Decorrelated Space (A) NPS shrinkage weights ${\omega }_k$ (red line) compared to the theoretical optimum (black line), ${\lambda }_{lj}/\left({\lambda }_{lj}+\frac{M}{N_g{h}^2}\right)$, under infinitesimal architecture. The partition of largest eigenvalues ${\mathcal{S}}_{10}$ is marked by gray box. (B) Conditional mean effects estimated by NPS (red line) in sub-partitions of ${\mathcal{S}}_{10}$ by $\left|{\hat{\eta }}_{lj}\right|$ under infinitesimal architecture. The theoretical line (black) is the average over all ${\lambda }_{lj}$ in ${\mathcal{S}}_{10}$. (C and D) Conditional mean effects estimated by NPS (red line) in sub-partitions of ${\mathcal{S}}_{10}$ (C) and ${\mathcal{S}}_2$ (D) on intervals of $\left|{\hat{\eta }}_{lj}\right|$ under non-infinitesimal architecture with the causal SNP fraction of 1%. The true conditional means (black) were estimated over 40 simulation runs. The mean NPS shrinkage weights (red line) and their 95% CIs (red shade) were estimated from five replicates. Grey vertical lines indicate partitioning cut-offs. No shrinkage line (green) indicates ${\omega }_k=1$. The number of markers $M$ is 101,296. The discovery GWAS size $N_g$ equals to $M$. The heritability $h^2$ is 0.5.