Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b$$\end{document}b^ be the estimated effect at the top-associated cis-eQTL for a gene (i.e., one SNP per gene). We can model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b$$\end{document}b^ as1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b = b + e$$\end{document}b^=b+ewhere b is the true effect and e is the estimation error. We assume that b and e are random variables when interrogated across genes, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\sim N\left( {0,{\mathrm{var}}\left( b \right)} \right)$$\end{document}b~N0,varb and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e\sim N\left( {0,{\mathrm{var}}\left( e \right)} \right)$$\end{document}e~N0,vare. The covariance of the estimated cis-eQTL effects between tissues i and j across genes can be partitioned into the covariance of true cis-eQTL effects and the covariance 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}
