The idea underlying our “corrected” estimator follows immediately from equation (3) by inverting the bias term: (4)αcorr=Vg−1(Vg+Ω)P−1αg^=αg. This expression is called a regression-disattenuation estimator. It cannot be implemented directly, however, because Vg involves the variance and covariances of the unobserved standardized additive SNP factor gi. However, the variance and covariances involving gi differ from analogous terms involving g^i only due to measurement error, and the amount of measurement error is given by ρ. Therefore, the variance and covariances involving gi can be inferred from estimable quantities. In the Supplementary Methods, we derive an expression for αcorr in terms of ρ and population parameters that can be estimated consistently using the observed data. That expression is stated in Methods. We implement that version of the estimator. In the Supplementary Methods, we also derive standard errors for the regression coefficients, under the assumption that ρ is known.
