b_{xy(i)}^2\left[\frac{{{\mathrm{var}}\left( {\hat b_{zx\left( i \right)}} \right)}}{{b_{zx\left( i \right)}^2}} + \frac{{{\mathrm{var}}\left( {\hat b_{zy\left( i \right)}} \right)}}{{b_{zy\left( i \right)}^2}} - \frac{{{\mathrm{var}^2}\left( {\hat b_{zx\left( i \right)}} \right)}}{{b_{zx\left( i \right)}^4}} \right]$$\end{document}varb^xyi=bxy(i)2varb^zxibzxi2+varb^zyibzyi2-var2b^zxibzxi4. Therefore, we can estimate bxy from all the instruments using the generalized least squares approach as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy} = ({\mathbf 1}^{\prime}{\mathbf{V}}^{ - 1}{\mathbf 1})^{ - 1}{\mathbf 1}^{\prime}{\mathbf{V}}^{ - 1}{\hat{\mathbf b}}_{xy}$$\end{document}b^xy=(1′V -11)-11′V -1b^xy with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}(\hat b_{xy}) = ({\mathbf 1}^{\prime}{\mathbf{V}}^{ - 1}{\mathbf 1})^{ - 1}$$\end{document}var(b^xy)=(1′V -11)-1. The statistical significance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy}$$\end{document}b^xy can be tested by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{GSMR} = \hat b_{xy}^2/{\mathrm{var}}(\hat b_{xy})$$\end{document}TGSMR=b^xy2∕var(b^xy) which follows a χ2 distribution with 1 degree of freedom. Note that because logOR is free of ascertainment bias (i.e., the bias due to a higher proportion of cases in the sample than in the general population), the method can be applied to disease data from case–control studies, and the estimate of bxy should be interpreted as that of the general population.
