worth underlining here that direct attempts to predict \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_1^2$$\end{document}R12 or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_2^2$$\end{document}R22 are challenging as they require prior knowledge of the number of causal variants (MC). Unfortunately, no method to date can provide estimates of MC with high enough precision. However, under the assumption that causal variants are shared between ancestries, focusing on the ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_2^2/R_1^2$$\end{document}R22/R12 overcomes this limitation and therefore allows us to derive the approximate closed-form formula shown below in Eq. (1) (details of our derivations are given in Supplementary Note 1):1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_2^2/R_1^2 \approx \frac{{\rho _b^2h_2^2}}{{h_1^2}} \times \left( {\frac{{\mathop {\sum }\nolimits_{k = 1}^{M_{\mathrm{T}}} \sqrt {\frac{{p_{k,2}(1 - p_{k,2})}}{{p_{k,1}(1 - p_{k,1})}}} \left[ {\mathop {\sum }\nolimits_{j = 1}^{M_{\mathrm{C}}} r_{jk,1}r_{jk,2}} \right]}}{{\mathop {\sum }\nolimits_{k = 1}^{M_{\mathrm{T}}} \left( {\mathop {\sum }\nolimits_{j = 1}^{M_{\mathrm{C}}} r_{jk,1}^2} \right)}}} \right)^2 \times \frac{{{\mathrm{var}}({\mathrm{PGS}}_1)}}{{{\mathrm{var}}({\mathrm{PGS}}_2)}},$$\end{document}R22/R12≈ρb2h22h12×∑k=1MTpk,2(1−pk,2)pk,1(1−pk,1)∑j=1MCrjk,1rjk,2∑k=1MT∑j=1MCrjk,122×var(PGS1)var(PGS2),where MT denotes the number of GWS SNPs used to calculate PGSs. Note that a special case of Eq. (1) was derived in de Vlaming et al.18 to characterize the accuracy of PGS in the presence of causal effects heterogeneity (modelled in their
