)$$\end{document}r^e=cov ^b^i,b^jvar ^eivar ^(ej)=cov ^b^i,b^jvar ^b^ivar ^b^j=cor^(b^i,b^j) for null SNPs, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {{\mathrm{cor}}}( {\hat b_i,\hat b_j} )$$\end{document}cor^(b^i,b^j) is the observed sample correlation between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_i$$\end{document}b^i and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_j$$\end{document}b^j in the set of genes. In practice, we computed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat r_e$$\end{document}r^e for each gene using “null” SNPs (PeQTL > 0.01) in the cis-region by a simple correlation approach and took the average across genes.
