of estimation errors (if there is a sample overlap), i.e.,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}( {\hat b_i,\hat b_j}) = {\mathrm{cov}}( {b_i,b_j} ) + {\mathrm{cov}}( {e_i,e_j}) = {\mathrm{cov}}( {b_i,b_j}) + r_e\sqrt {{\mathrm{var}}( {e_i} ){\mathrm{var}}( {e_j} )}$$\end{document}cov(b^i,b^j)=cov(bi,bj)+cov(ei,ej)=cov(bi,bj)+revar(ei)var(ej)where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {e_i} \right)$$\end{document}varei and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}( {e_j} )$$\end{document}var(ej) are the variance of the estimation errors across genes in tissues i and j, respectively, and re is the correlation of estimation errors across genes between two tissues, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_e = {\mathrm{cor}}( {e_i,e_j} )$$\end{document}re=cor(ei,ej). We know from Bulik-Sullivan et al.41 and Zhu et al.44 that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\it \it r_e \approx r_{\mathrm{\it_p}}\rho$$\end{document}re≈rpρ, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho = \frac{{N_s}}{{\sqrt {N_iN_j} }}$$\end{document}ρ=NsNiNj measures the sample overlap with Ni and Nj being the sample sizes in tissues i and j, respectively, and Ns being the number of overlapping individuals, and rp is the correlation of gene expression levels between two tissues in the overlapping sample. If i = j, then re = 1 and \documentclass[12pt]{minimal} \usepackage{amsmath}
