Let y be the liability of a disease on the logit scale, x be a risk factor in standard deviation (SD) units and z be the genotype of a SNP (coded as 0, 1, or 2). The MR estimate of the causal effect of risk factor on disease9 is \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, where bzy is the effect of z on y on the logit scale (logarithm of odds ratio, logOR), bzx is the effect of z on x, and bxy is the effect of x on y free of confounding from non-genetic factors (note that bxy can be approximately interpreted as logOR; see below). SMR is a flexible and powerful MR approach that is able to estimate and test the significance of bxy using the estimates of bzx 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.
