The usual assumptions of linear regression apply to the estimation of b. The residuals, which are the mean-centred prognosis effects, should be uncorrelated. When marginal single-SNP effects are considered, as is usual in GWAS, correlation can arise through LD, and we therefore fit Eq. (3) to a pruned set of approximately independent SNPs. Even with pruned SNPs, LD can lead to heteroscedasticity, since a SNP in a region of high LD is expected to have greater variance in its marginal effect on both incidence and prognosis37. Furthermore, allele frequency has also been observed to relate to effect size variance38, again creating potential heteroscedasticity. Residual heteroscedasticity does not affect the bias of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b^ \ast$$\end{document}b^* but its standard error is needed for our SIMEX adjustment, and so we calculate a heteroscedasticity robust estimate of that standard error (Supplementary Note 1).
