A common12,28 definition of a raw PS for the individual j is the sum of its alleles weighted by their effect sizes:2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S}_{j}=\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{V}}}}{\beta }_{i}{x}_{ij},$$\end{document}Sj= ∑i=1NVβixij,where xij is the allelic state at site i, individual j, βi is the associated effect size and NV is the number of variants used to compute it. An equivalent alternative is to compute the mean, instead of the sum:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{x}}_{j}=\frac{1}{{N}_{{\rm{V}}}}\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{V}}}}{\beta }_{i}{x}_{ij}.$$\end{document}x¯j=1NV ∑i=1NVβixij.In order to compute partial PS we only consider a subset of variants of the genome, with size NS (NS < NV); we can therefore define a raw pPS as4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{x}}_{j}^{\prime}=\frac{1}{{N}_{{\rm{S}}}}\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{S}}}}{\beta }_{i}{x}_{ij}.$$\end{document}x¯j′=1NS ∑i=1NSβixij.We then devised a standardized pPS using a Z-score of the raw pPS, as is common:5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{pPS}}}_{j}=\frac{{\overline{x}}_{j}^{\prime}-{\mu }_{\overline{x}^{\prime} }}{{\sigma }_{\overline{x}^{\prime} }},$$\end{document}pPSj=x¯j′−μx¯′σx¯′,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu }_{\overline{x}^{\prime} }$$\end{document}μx¯′ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\overline{x}^{\prime} }$$\end{document}σx¯′ are,
