Without loss of generality, we assume that the association test statistic for the jth phenotypic trait is zj where j=1,…,m. The corresponding two-sided p-value is defined as pj=2Φ(−|zj|), where Φ is the standard Gaussian distribution function. Under the null hypothesis of no association between a SNP and multivariate phenotypic traits, the distribution of T statistic is approximated by a Gaussian distribution with the mean of T as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mu = E[T] = 2m $$ \end{document}μ=E[T]=2m and the variance of T as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} \sigma^{2} &=& \text{Var}[T] \\ & = &\text{Var} \left\{\sum_{j=1}^{m} -2\log(p_{j})\right\}\\ & = & \sum_{j=1}^{m} \text{Var}\{-2\log(p_{j})\} \! +\! \sum_{j \ne k} \text{cov}\{-2\log(p_{j}), \! -2\log(p_{k}\!)\}\\ & = & 4m + \sum_{j \ne k} \text{cov}\{-2\log(p_{j}), -2\log(p_{k})\}. \end{array} $$ \end{document}σ2=Var[T]=Var∑j=1m−2log(pj)=∑j=1mVar{−2log(pj)}+∑j≠kcov{−2log(pj),−2log(pk)}=4m+∑j≠kcov{−2log(pj),−2log(pk)}.
