multiple correlated instruments17. Here, we extend this approach to detect heterogeneity in bxy estimated at m near-independent instruments (note that the method accounts for remaining LD not removed by clumping). The basic idea is to test where there is a significant difference between bxy estimated at an instrument i (i.e., bxy(i)) and bxy estimated at a target SNP that shows a strong association with the exposure. The power of detecting heterogeneity increases with the strength of association between the target SNP and exposure. However, we cannot simply choose the top exposure-associated SNP because sometimes when a SNP has an extremely strong effect on the exposure, it is also likely to be a pleiotropic outlier (e.g., the top LDL-associated SNP at the APOE locus shows a very strong pleiotropic effect on Alzheimer’s disease, as shown in Fig. 4). Therefore, to increase the robustness of the HEIDI-outlier test, we examine the distribution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy}$$\end{document}b^xy as a function of −log10(P-value) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zx}$$\end{document}b^zx and choose the SNP that shows the strongest association with the exposure in the third quintile of the distribution of \documentclass[12pt]{minimal}
