We evaluated evidence of effect size moderation by ancestry in the multi-ancestry model for each independent variant. To do so, we extended the MEMO model into a mixture model that separated variants with homogenous effects (models with only an intercept term) from those with possible heterogeneous effects (on at least one axis of genetic variation). We considered six sub-models including the null model, and the models in which the number of included components varied from 0 to 4.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(\,y)=\prod _{a}{p}_{a}^{{\rm{N}}{\rm{U}}{\rm{L}}{\rm{L}}}p({b}_{j}|{\rm{N}}{\rm{U}}{\rm{L}}{\rm{L}})+{{\rm{p}}}_{{\rm{a}}}^{{\rm{A}}{\rm{L}}{\rm{T}}}\sum _{{\rm{j}}\in {{\rm{S}}}_{{\rm{a}}}}[{{\rm{q}}}_{{\rm{j}}0}\,{\rm{p}}({{\rm{b}}}_{{\rm{j}}}|{\rm{M}}{{\rm{R}}}_{0}(\,{\rm{j}}))+\ldots +{{\rm{q}}}_{{\rm{j}}4}\,{\rm{p}}({{\rm{b}}}_{{\rm{j}}}|{\rm{M}}{{\rm{R}}}_{4}(\,{\rm{j}}))],$$\end{document}L(y)=∏apaNULLp(bj|NULL)+paALT∑j∈Sa[qj0p(bj|MR0(j))+…+qj4p(bj|MR4(j))],where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p({b}_{j}|{\rm{N}}{\rm{U}}{\rm{L}}{\rm{L}})$$\end{document}p(bj|NULL) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\left({b}_{j}| M{R}_{l}\right)$$\end{document}pbj∣MRl are the likelihoods of the variant j effect sizes under the null model and the meta-regression models with l axes of genetic variation, respectively; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{a}^{{\rm{NULL}}}$$\end{document}paNULL and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{a}^{{\rm{ALT}}}$$\end{document}paALT are the probabilities of locus a carrying zero or at least one causal variant, respectively. The term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}
