\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{PRS}\sim {\alpha}_0+{\sum}_{k=1}^5{\alpha}_k\times {\boldsymbol{PC}}_k,\kern2.5em \boldsymbol{\delta} \sim {\beta}_0+{\sum}_{k=1}^5{\beta}_k\times {\boldsymbol{PC}}_k,$$\end{document}PRS∼α0+∑k=15αk×PCk,δ∼β0+∑k=15βk×PCk, where δ is the residual variance of the first regression. Fitting the two regressions gives how individual-level PRS vary with ancestry captured by PCs. For any individual i projected into the same PC space with the raw score PRSi, raw and PC coordinates PCi, k, k = 1, 2, ⋯, 5, an ancestry adjusted PRS can then be calculated as:
