the polygenic score is normally distributed, the expected AUC can be calculated as [21] (12)In this expression, is formed from effects on estimated on the observed scale whereas the conditional means and variances are conveniently calculated on the liability scale for . There is a linear transformation between effects on and those on , defined by their bivariate normal distribution, and equation 11 gives another linear transformation between effects on and those on . Equation 12 may therefore be equivalently written in terms of effects on with the corresponding score denoted . The conditional means and variances are functions of the variance in explained by [21], [40], which is , givingandSimilarlyand
