Polygenic risk prediction: why and when out-of-sample prediction R² can exceed SNP-based heritability

In polygenic score (PGS) analysis, the coefficient of determination (R²) is a key statistic to evaluate efficacy. R² is the proportion of phenotypic variance explained by the PGS, calculated in a cohort that is independent of the genome-wide association study (GWAS) that provided estimates of allelic effect sizes. The SNP-based heritability (h_SNP², the proportion of total phenotypic variances attributable to all common SNPs) is the theoretical upper limit of the out-of-sample prediction R². However, in real data analyses R² has been reported to exceed h_SNP², which occurs in parallel with the observation that h_SNP² estimates tend to decline as the number of cohorts being meta-analyzed increases. Here, we quantify why and when these observations are expected. Using theory and simulation, we show that if heterogeneities in cohort-specific h_SNP² exist, or if genetic correlations between cohorts are less than one, h_SNP² estimates can decrease as the number of cohorts being meta-analyzed increases. We derive conditions when the out-of-sample prediction R² will be greater than h_SNP² and show the validity of our derivations with real data from a binary trait (major depression) and a continuous trait (educational attainment). Our research calls for a better approach to integrating information from multiple cohorts to address issues of between-cohort heterogeneity.