However, as has been discussed in the context of Mendelian randomisation23, this approach can have large variance, and can lead to implausible negative adjustments for regression dilution, as we found in our IPF data. Therefore we follow Bowden et al.23 in proposing simulation extrapolation24 (SIMEX) for the analysis of real datasets. Briefly, this approach simulates new estimates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _{GX}$$\end{document}β^GX with increasing degrees of measurement error, by adding Gaussian noise with variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \sigma _{GX}^2$$\end{document}λσGX2 to the given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _{GX}$$\end{document}β^GX, for various values of λ. The linear regression of Eq. (3) is repeated for each simulated dataset, and the estimator of its slope \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^* considered as a function of λ. Standard applications of SIMEX, including that of Bowden et al.23, fit a linear or quadratic model relating \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^* to λ,
