Similarly, we used GenomicSEM16 to model the joint genetic architecture of the four subgroups. First, we ran a common factor model without individual SNP effects, following the tutorial ‘Models without individual SNP effects’ on the GenomicSEM GitHub website (Code availability). Second, we ran a multivariate GWAS of the common factor (see Supplementary Note 5 for details). We specified the model using unit variance identification, for which the latent factor variance is fixed to 1 and the loadings of the traits are estimated freely. This ensures that we capture how much of each subgroup contributes to the latent factor. GenomicSEM also generates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q}_{{\rm{SNP}}}$$\end{document}QSNP values, which indicate possible heterogeneous effects across the subgroups. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q}_{{\rm{SNP}}}$$\end{document}QSNP statistic is mathematically similar to the Q statistic from standard meta-analysis and is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ X}^{2}$$\end{document}X2-distributed test statistic, with larger values indexing a violation of the null hypothesis that the SNP acts entirely through the common factor.
