sum of sample sizes.Table 1(a) Multivariate test statistic in the meta-analysis of results based on overlapping samples; (b) expected value for the cross-trait-intercept; (c) effective sample size for a GWAMA.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{\rm{multi},j} = \frac{{\mathop {\sum }\nolimits_{i = 1}^P w_{ji}Z_{ji}}}{{\sqrt {\mathop {\sum }\nolimits_{i = 1}^P w_{ji}V_{ji} + \mathop {\sum }\nolimits_{i = 1}^P \mathop {\sum }\nolimits_{k = 1}^P \sqrt {w_{ji}w_{jk}} CTI_{ik}\,\rm{for}\,i \ne k} }}$$\end{document}Zmulti,j=∑i=1PwjiZji∑i=1PwjiVji+∑i=1P∑k=1PwjiwjkCTIikfori≠k(a) Multivariate test statistic for jth SNP. P is the number of GWASs across that we run the meta-analysis; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{ji} = \sqrt {N_{ji}h_{SNP,i}^2}$$\end{document}wji=NjihSNP,i2 is the weight given to the jth SNP in GWAS i, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{SNP,i}^2$$\end{document}hSNP,i2 being the SNP-heritability of the trait analyzed in GWAS i; and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{ji} = 1$$\end{document}Vji=1 represents the variance of the distribution of Zji under the null hypothesis of no effect.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CTI_{ik} = \frac{{N_sr_p}}{{\sqrt {N_{ji}N_{jk}} }}$$\end{document}CTIik=NsrpNjiNjk(b) Cross-trait intercept between GWAS i and k. Ns represents the sample overlap; rp indicates the phenotypic correlation; Nji and Njk are the sample sizes at SNP j
