Variance Explained (R²)

R² represents the proportion of phenotypic variance in the target dataset that is accounted for by the PRS. It is calculated as the squared Pearson correlation between the predicted genetic risk and the observed phenotype in a continuous trait setting.

• Scale: Ranges from 0 to 1 (or 0% to 100%)
• Interpretation: An R² of 0.05 means the PRS explains 5% of the variance in the trait
• Upper bound: Constrained by SNP heritability (h²_SNP), which is the maximum variance explainable by all common SNPs
• Binary traits: For case-control studies, a pseudo-R² such as Nagelkerke's R² is often reported on the liability scale

While R² measures variance explained, AUC-ROC measures discriminative ability. These metrics are mathematically related but capture distinct aspects of predictive performance.

• For binary traits, R² on the liability scale can be derived from AUC using the variance of the standard logistic distribution (π²/3)
• A PRS with a modest R² (e.g., 2-5%) can still achieve a clinically meaningful AUC when the trait has high heritability
• Key distinction: R² is sensitive to the scale of the phenotype, while AUC is rank-based and scale-invariant
• Both metrics should be reported together for a complete picture of model performance

R² estimates are highly sensitive to the sample size of the validation cohort and the discovery GWAS used to derive effect sizes.

• Winner's curse: R² in the discovery dataset is often inflated due to overfitting; out-of-sample validation is mandatory
• As GWAS sample sizes increase, the precision of effect size estimates improves, directly increasing the achievable R²
• Standard error: Confidence intervals around R² narrow with larger validation cohorts, enabling more precise comparisons between PRS methods
• Small validation samples can produce noisy, unreliable R² estimates that fail to generalize

R² is not a universal constant for a given PRS; it is population-specific and degrades when the validation cohort's ancestry diverges from the GWAS discovery population.

• Cross-ancestry portability: R² typically drops by 2-5 fold when applying a European-derived PRS to non-European populations
• Differences in linkage disequilibrium (LD) patterns and allele frequencies between populations reduce predictive accuracy
• Population stratification can artificially inflate R² if not corrected via principal components or linear mixed models
• Reporting ancestry-stratified R² values is essential for transparent evaluation of PRS equity

The incremental R² quantifies the additional variance explained when a PRS is added to an existing clinical model, directly measuring its practical value.

• ΔR² = R²(clinical + PRS) - R²(clinical only)
• Even a small incremental R² can translate to meaningful net reclassification improvement (NRI) in risk categories
• For traits with established non-genetic risk factors (e.g., cardiovascular disease), the incremental R² over models like the Framingham Risk Score is the key metric
• Absolute risk models combine baseline incidence rates with PRS-derived relative risks to produce clinically actionable probabilities

R² should be interpreted alongside complementary metrics to avoid over-reliance on a single summary statistic.

• Mean squared error (MSE): Measures absolute prediction error; R² is a scaled, relative metric
• Correlation coefficient (r): R² is simply r² for continuous traits, but r retains directional information
• Odds ratio per standard deviation: For binary traits, this quantifies risk discrimination in epidemiological terms
• Calibration slope: Assesses whether predicted risks match observed risks across the probability spectrum; a model can have high R² but poor calibration
• Report multiple metrics to satisfy both statistical rigor and clinical interpretability requirements