For correlated phenotypic traits, T is the sum of dependent chi-squared statistics. Brown [27] and Yang [28] have shown that, under the null hypothesis of no association between a SNP and multivariate phenotypic traits, the distribution of T statistic follows a scale chi-squared distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\gamma {\chi _{v}^{2}})$\end{document}(γχv2), or equivalently, a gamma distribution with the shape parameter v/2 and the scale parameter 2γ. Therefore, to calculate the global p-value of T statistic, we only need to estimate the parameters v and γ. Suppose that the mean of T is μ and the variance of T is σ2. Using the first and second moments of T, the values of v and γ can be calculated as v=2μ2/σ2 and γ=σ2/(2μ). The following are technical details of the derivation of the mean and variance of T statistic when the marginal p-values are based on two-sided tests (see Brown [27] and Yang [28] for the case of one-sided marginal tests):
