1b) and that the estimate of bxy is unbiased under the alternative (Supplementary Table 1). In comparison with the existing methods that use summary data to make causal inference12,13,16,18, GSMR is more powerful as demonstrated by simulation (Supplementary Fig. 3) because GSMR accounts 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 while the other approaches assume that bzx is estimated without error.
