Chunk #19 — Methods — Proposed methods — The proposed efficient method

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An efficient genome-wide association test for multivariate phenotypes based on the Fisher combination function.

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To estimate δjk in Eq. (2) accurately, we have taken two steps to remove potential biases. First, since the sample correlation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat {\rho }_{\text {\textit {j,k}}}$\end{document}ρ^j,k is not an unbiased estimator of ρj,k [30], we estimate ρj,k by the bias-corrected sample correlation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat {r}_{\text {\textit {j,k}}}$\end{document}r^j,k: (3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \hat{r}_{j,k} = \hat{\rho}_{j,k}\left(1+ \frac{1-\hat{\rho}_{j,k}^{2}}{2(n-3)} \right), $$ \end{document}r^j,k=ρ^j,k1+1−ρ^j,k22(n−3),