Mendelian randomization is a method that uses genetic variants as instrumental variables to test for causative association between an exposure and an outcome9. Let z be a genetic variant (e.g., SNP), x be the exposure (e.g., health risk factor) and y be the outcome (e.g., disease). If z is significantly associated with x, the effect of x on y can be estimated using a two-step least squares (2SLS) approach51\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy} = \hat b_{zy}/\hat b_{zx}\;{\mathrm{with}}\;{\mathrm{var}}\left( {\hat b_{xy}} \right) = {\mathrm{var}}(y)(1 - R_{xy}^2)/\left[ {n{\mathrm{var}}\left( x \right)R_{zx}^2} \right],$$\end{document}b^xy=b^zy∕b^zxwithvarb^xy=var(y)(1-Rxy2)∕nvarxRzx2,where n is the sample size, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{xy}^2$$\end{document}Rxy2 is the variance in y explained by x, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{zx}^2$$\end{document}Rzx2 is the variance in x explained z. This analysis requires individual-level data so that the statistical power could be limited if bxy is small. We have previously proposed an approach that only requires summary-level data to estimate bxy so that the power can be greatly improved if bzx and bzy are estimated
