and estimate the correlation of SNP effects (see ‘URLs’ section). The expected proportion, E[P], of sign-consistent SNP effects between discovery and replication was calculated using the quadrant probability of a standard bivariate Gaussian distribution with correlation E[ρb], denoting the expected correlation between estimated SNP effects in the discovery and replication sample:1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[P]=\frac{1}{2}+\frac{{\sin }^{-1}(E[{\rho }_{{\rm{b}}}t])}{\pi },$$\end{document}E[P]=12+sin−1(E[ρbt])π,where sin−1 denotes the inverse of the sine function and E[ρb] the expectation of the ρb statistic under the assumption that the true SNP effects are the same across discovery and replications cohorts. E[ρb] was calculated as2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[\,{\rho }_{{\rm{b}}}]=\,\frac{{\sigma }_{{\rm{b}}}^{2}}{\sqrt{\left({\sigma }_{{\rm{b}}}^{2}\,+\,[1-{\sigma }_{{\rm{b}}}^{2}{h}_{{\rm{d}}}]/({N}_{{\rm{d}}}{h}_{{\rm{d}}})\,\right)\left({\sigma }_{{\rm{b}}}^{2}\,+\,[1-{\sigma }_{{\rm{b}}}^{2}{h}_{{\rm{r}}}]/({N}_{{\rm{r}}}{h}_{{\rm{r}}})\right)}},$$\end{document}E[ρb]=σb2σb2+[1−σb2hd]/(Ndhd)σb2+[1−σb2hr]/(Nrhr),where Nd and Nr denote the sizes of the discovery and replication samples, respectively; hd and hr the average heterozygosity under Hardy–Weinberg equilibrium (that is, 2 × MAF × (1 − MAF)) across GWS SNPs in the discovery and replication samples, respectively; and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{\sigma }}}_{{\rm{b}}}^{2}$$\end{document}σb2 the mean per-SNP variance explained by GWS SNPs, which we calculated (as per ref. 60.) as the sample variance of estimated SNP effects in the discovery sample minus the median squared standard error.
