Let yi be a quantitative trait measured on the ith participant, i = 1,…, n, and xi a k × 1 vector of covariates such as confounders. Let g1i,…,gPi be allelic counts or dosages of P independent variants associated with yi, and α1,…, αP their effect sizes, so that the additive linear model holds: (1)yi=xiβ+∑p=1Pgpiαp+ϵi,i=1,…,n where ϵi are residual errors. An optimal PRS for yi is Giopt=∑p=1Pgpiαp, the weighted sum with the causal genotypes and their true effects.
