One way to characterize the amount of measurement error is the value ρ. In Methods, we show that ρ2=1+Varei=hSNP2R2≥1, where hSNP2 is the SNP heritability (the predictive power of gi) and R2 is the fraction of variance explained in a regression of the phenotype yi on the PGI g^i (the predictive power of g^i). The ratio hSNP2/R2 is greater than or equal to one because the weights that define gi maximize the variance explained in yi, and therefore any other weights—including those used to construct the PGI—explain at most hSNP2 of the variation. Furthermore, the amount of measurement error ρ would achieve its minimum value of one only if the PGI weights were based on GWAS summary statistics from an infinite sample.
