We performed 1000 simulations and compared type-1 error at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P \, < \, 0.05$$\end{document}P<0.05 for the unadjusted estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _{GY}^\prime$$\end{document}β^GY′ to our adjusted estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _{GY}$$\end{document}β^GY, using the Hedges–Olkin estimator to correct for regression dilution. Type-1 error rates vary among SNPs, since the index event bias is proportional to the effect on incidence and the rejection rate for a non-zero bias is greater for allele frequencies nearer 0.5. Firstly, we estimated the mean type-1 error over all SNPs with no effect on prognosis. As this is dominated by the large number of SNPs without effects on incidence, and therefore no index event bias, we also estimated the mean type-1 error over SNPs with effects on incidence but not on prognosis. To assess lower error rates, we estimated the family-wise type-1 error over the same SNPs, as the proportion of simulations in which at least one SNP had \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}
