The “theoretical regression” is what we call a regression with the (unobserved) standardized additive SNP factor as an explanatory variable. Consider an OLS regression of a phenotype ϕi on the standardized additive SNP factor gi, a vector of covariates zi, and a vector wi of interactions between gi and a subset of the regressors in zi (possibly all of them): (1)ϕi=giβg+ziζg+wiδg+ϵg,i, where the g subscripts indicate that these are parameters from the theoretical regression. (Note that the phenotype ϕi need not be the same phenotype yi for which the standardized additive SNP factor is the best linear predictor. For example, some papers have studied the relationship between the PGI for educational attainment and test scores at younger ages14. Note also that the covariates in zi may be measured with error; equation (1) represents whatever regression is run by a researcher except that gi is measured without error.) The “feasible regression” is what we call the regression using the PGI g^i in place of gi: (2)ϕi=g^iβg^+ziζg^+w^iδg^+ϵg^,i, where w^i is the vector of interactions with g^i in place of gi. We denote the vectors of coefficients from the theoretical and feasible regressions by αg≡βg,ζg,δg′ and αg^≡βg^,ζg^,δg^′, respectively.
