From Eq 2, we deduce that (i) as the effective number of SNPs S increases, the R2 of the PGS deteriorates (since every SNP-effect estimate contains noise, owing to imperfect inferences in finite samples), (ii) given the effective number of SNPs, under a polygenic architecture, the precise fraction of effective SNPs that is associated does not affect the R2, (iii) R2 is quadratically proportional to ρG, implying a strong sensitivity to CGR, and (iv) as the sample size of the discovery study grows, the upper limit of the R2 is given by hSNP2ρG2, implying that the full SNP heritability in the hold-out sample cannot be entirely captured as long as CGR is imperfect.
