Chunk #48 — Methods — Correlation of cis-eQTL effects between tissues

Source

Identifying gene targets for brain-related traits using transcriptomic and methylomic data from blood.

Embedded

yes

Text

\begin{document}$$\widehat {{\mathrm{var}}}( {e_j} )$$\end{document}var^(ej) and \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 are not observable. We know that SE2 of \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^ of a SNP is an estimate of the variance of e over repeated experiments for a gene. We therefore can approximate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {{\mathrm{var}}}( e )$$\end{document}var^(e) by the average of SE2 across genes (one SNP per gene). We also know from Eq. (2) that if bi = bj = 0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}( {\hat b_i,\hat b_j} ) = r_e\sqrt {{\mathrm{var}}\left( {e_i} \right){\mathrm{var}}(e_j)}$$\end{document}cov(b^i,b^j)=revareivar(ej). Hence, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat r_e = \frac{{\widehat {{\mathrm{cov}}}\left( {\hat b_i,\hat b_j} \right)}}{{\sqrt {\widehat {{\mathrm{var}}}\left( {e_i} \right)\widehat {{\mathrm{var}}}(e_j)} }} = \frac{{\widehat {{\mathrm{cov}}}\left( {\hat b_i,\hat b_j} \right)}}{{\sqrt {\widehat {{\mathrm{var}}}\left( {\hat b_i} \right)\widehat {{\mathrm{var}}}\left( {\hat b_j} \right)} }} = \widehat {{\mathrm{cor}}}( {\hat b_i,\hat b_j} )$$\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}