and bzy from independent samples17. If there are multiple independent (or nearly independent) SNPs associated with x and the effect of x on y is causal, then all the x-associated SNPs will have an effect on y through x (Fig. 1a). In this case, bxy at any of the x-associated SNPs is expected to be identical in the absence of pleiotropy13,16,22 as all the SNP effects on y are mediated by x (Fig. 1b). Therefore, increased statistical power can be achieved by integrating the estimates of bxy from all the x-associated SNPs using a generalized least squares (GLS) approach (Methods). The GSMR method essentially implements SMR analysis for each SNP instrument individually, and then integrates the bxy estimates of all the SNP instruments by GLS, accounting for the sampling variance in both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zx}$$\end{document}b^zx and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zy}$$\end{document}b^zy for each SNP and the LD among SNPs. It is important to note that in accordance with one of the basic assumptions for MR9, only the SNPs that are strongly associated with the risk factor should be used as the instruments for MR analyses including
