To test whether the effect of a risk factor (x0) on a disease (y) depends on other risk factors (x = {x1, x2,…, xi}), we usually perform a joint analysis based on the model below\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y = x_0b_0 + {\mathbf{xb}}_{xy} + e,$$\end{document}y=x0b0+xbxy+e,where b0 is the effect of x0 on y, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{b}}_{xy} = \{ b_{x_iy}\}$$\end{document}bxy={bxiy} is a t-length vector with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{x_iy}$$\end{document}bxiy being the effect of a covariate xi on y, and e is the residual. Such an analysis is equivalent to a two-step analysis with the first step to adjust both x0 and y by x and the second step to estimate the effect of adjusted x0 on adjusted y. We therefore can estimate the effect size of x0 on y accounting for x by a GSMR analysis using SNP effects on x0 and y conditioning on x.
