Since the population regression cannot be run, the vector γ is unknown, so gi cannot be constructed empirically. What can be constructed empirically is a “polygenic index (PGI),” g^i, which is a standardized, weighted sum of allele counts using some other weight vector γ^ calculated from GWAS summary statistics: g^i≡xi′γ^sd(xi′γ^). In general, γ^ will not be equal to γ because γ^ is calculated from GWAS summary statistics that are estimated in a finite sample. The key observation for our framework is that when γ^ is calculated using standard methods (that include all the SNPs in xi), such as LDpred30 and PRS-CS31, the resulting PGI can be expressed as g^i=(gi+ei)ρ, where ei is mean-zero estimation error that is uncorrelated with gi, and ρ≡sdxi′γ^/sdxi′γ is a scaling factor that standardizes g^i. In words, the PGI is a standardized, noisy measure of the additive SNP factor, where the noise is classical measurement error.
