Chunk #12 — Results — Expected relative accuracy of PGS in ancestry divergent populations

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Theoretical and empirical quantification of the accuracy of polygenic scores in ancestry divergent populations.

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quantities over all candidate causal variants, as shown below in Eq. (2):2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_2^2/R_1^2 \approx \frac{{\rho _b^2h_2^2}}{{h_1^2}} \times \left( {\frac{{\mathop {\sum }\nolimits_{k = 1}^{M_{\mathrm{T}}} \overline {r_{k,1}r_{k,2}} \sqrt {\frac{{p_{k,2}(1 - p_{k,2})}}{{p_{k,1}(1 - p_{k,1})}}} }}{{\mathop {\sum }\nolimits_{k = 1}^{M_{\mathrm{T}}} \overline {r_{k,1}^2} }}} \right)^2 \times \frac{{\mathop {\sum }\nolimits_{k = 1}^{M_{\mathrm{T}}} p_{k,1}(1 - p_{k,1})\hat \beta _k^2}}{{\mathop {\sum }\nolimits_{k = 1}^{M_{\mathrm{T}}} p_{k,2}(1 - p_{k,2})\hat \beta _k^2}},$$\end{document}R22/R12≈ρb2h22h12×∑k=1MTrk,1rk,2¯pk,2(1−pk,2)pk,1(1−pk,1)∑k=1MTrk,12¯2×∑k=1MTpk,1(1−pk,1)β^k2∑k=1MTpk,2(1−pk,2)β^k2,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _k$$\end{document}β^k represents the effect size of the kth PGS-SNP estimated in the discovery GWAS. We use the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline {r_{k,1}^2}$$\end{document}rk,12¯ to denote the mean squared correlation of allele counts between the kth PGS-SNP and all candidate causal SNPs within 100 kb. Similarly, we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline {r_{k,1}r_{k,2}}$$\end{document}rk,1rk,2¯ as the mean of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{jk,1}r_{jk,2}$$\end{document}rjk,1rjk,2’s between the kth PGS-SNP and all candidate causal SNPs within 100 kb.