Adjustment for index event bias in genome-wide association studies of subsequent events.

Directed acyclic graph of association of SNP G with prognosis Y conditional on incidence X. U is a composite variable including all common causes of X and Y, and may include polygenic effects as well as non-genetic risk factors. In our examples, X is idiopathic pulmonary fibrosis or Crohn’s disease, and Y is survival or prognosis. Conditioning on X induces the moralised association between G with U shown by the dotted line. This creates association between G and Y via the path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G - U \to Y$$\end{document}G-U→Y in addition to the direct effect G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\to$$\end{document}→ Y

Directed acyclic graph of association of SNP G with risk factor X conditional on outcome Y. U is as in Fig. 1. For example, X may be body mass index, and Y may be type-2 diabetes, with the study design being case/control or a cohort depleted for cases, such as UK Biobank17. Conditioning on Y induces the moralised association between G and U shown by the dotted line. This creates association of G with X via the path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G - U \to X$$\end{document}G-U→X, in addition to the direct effect G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\to$$\end{document}→ X. The direct effect itself is biased by conditioning on Y, as shown by the additional dotted line connecting G and X. The resulting selection bias is not the focus of this paper

Directed acyclic graph of association of SNPs Gi with prognosis Y conditional on incidence X. U is as in Fig. 1. Conditioning on X induces the moralised associations shown by dotted lines. These create association of each Gi with Y via the path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_i - U \to Y$$\end{document}Gi-U→Y and all paths \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_i - G_j \to Y$$\end{document}Gi-Gj→Y where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \ne j$$\end{document}i≠j. Under a polygenic model in which individual SNPs explain little covariation between X and Y, the combined effect of U and all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{j \ne i}$$\end{document}Gj≠i is approximately constant across SNPs Gi. If a SNP Gk has a major effect on X and/or Y, the associations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{j \ne k}$$\end{document}Gj≠k can be conditioned on Gk to prevent the major gene contributing to index event bias

Association of SNP G with prognosis Y conditional on incidence X derived from trait S. U is a composite variable as in Fig. 1. For example, X may be a diagnosis of disease (e.g., Crohn’s disease), and S a subtype of disease (e.g., ileal, colonic, ileocolonic or healthy). Conditioning on X, which is a descendant of the collider S, induces the moralised association between G and U shown by the dotted line. This creates association of G with Y via the path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G - U \to Y$$\end{document}G-U→Y in addition to its direct effect via G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow$$\end{document}→ Y and its mediated effect via \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G \to S \to Y$$\end{document}G→S→Y. Further conditioning on S blocks the mediation path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G \to S \to Y$$\end{document}G→S→Y, but leaves open the path \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G - U \to Y$$\end{document}G-U→Y creating index event bias