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 from independent studies of large sample size17, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy} = \hat b_{zy}/\hat b_{zx}$$\end{document}b^xy=b^zy∕b^zx with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}(\hat b_{xy}) \approx \frac{{b_{zy}^2}}{{b_{zx}^2}}\left[\frac{{{\mathrm{var}}\left( {\hat b_{zx}} \right)}}{{b_{zx}^2}} + \frac{{{\mathrm{var}}\left( {\hat b_{zy}} \right)}}{{b_{zy}^2}}\right]$$\end{document}var(b^xy)≈bzy2bzx2varb^zxbzx2+varb^zybzy2. We called this approach a summary data-based Mendelian randomization (SMR) analysis17. We have also shown previously that a SMR analysis using a single genetic variant is unable to distinguish between causality (the effect of SNP on outcome is mediated by exposure) and pleiotropy (the SNP has distinct effects on exposure and outcome). Here, we extend the SMR method to use all the top associated SNPs at a genome-wide significance level for the exposure as instrumental variables to test for causality. We call this method a generalized SMR (GSMR) analysis. The basic idea of GSMR is that if x is causal for y, any SNP associated with x will have an effect on y, and the
