From Nagelkerke's R2 to Liability-Scale Variance Explained for Polygenic Scores

It is desirable to quantify the prediction accuracy of polygenic scores (PGS) for disease on the scale of liability and adjusted for case-control ascertainment in the test sample, because that allows comparison across prevalence and ascertainment. Previous expressions have focused in their derivation and implementation on linear regression on the observed 0-1 scale followed by a transformation of the coefficient of determination (R2) to the scale of liability, adjusted for ascertainment. Yet most statistical analyses with empirical data use logistic regression. The differences in scale have led to confusion and incorrect transformations in the literature. Here we provide a new derivation and simple equation, validated by simulation, that allows a direct transformation from the empirical results from logistic regression to the scale of liability.