b_{zy}|{\hat{\mathbf b}}_{xy} = \hat b_{zy} - {\hat{\mathbf b}}_{zx}^t{\hat{\bf b}}_{xy},$$\end{document}b^zy∣b^xy=b^zy-b^zxtb^xy,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zy}$$\end{document}b^zy is the SNP effect on the disease on the logit scale (i.e., logOR), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathbf b}}_{xy}$$\end{document}b^xy is a t-length vector with the i-th element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{x_iy}$$\end{document}b^xiy being the effect of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{x_i}$$\end{document}gxi on the disease when all the covariates are fitted jointly, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\rm b}}_{zx}$$\end{document}b^zx is a t-length vector of SNP effects on x. For the ease of derivation, we assume each covariate has been standardized with mean 0 and variance 1 (note that the method can be applied to data on the original scale without standardization). We know from previous studies53 that the joint effects of gx on y (bxy) can be transformed from the marginal effects (βxy), i.e.,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{b}}_{xy} = {\mathbf{D}}^{ - \frac{1}{2}}{\mathbf{R}}_x^{ - 1}{\mathbf{D}}^{\frac{1}{2}}{\boldsymbol{\beta}}_{xy},$$\end{document}bxy=D-12Rx-1D12βxy,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{R}}_x = \{ r_{g(x_i,x_j)}\}$$\end{document}Rx={rg(xi,xj)} is a t×t matrix with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}
