We have shown above that under a causal model the expected value 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 estimated at any of the SNP instruments is identical in the absence of pleiotropy. If there are SNPs that have pleiotropic effects on x and y, \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 estimated at these SNPs will deviate from the expected value under a causal model, and hence will present as outliers. There have been methods to assess the sensitivity of an MR analysis to detect pleiotropy52. These methods, however, do not account for possible LD between SNPs nor the sampling errors in the estimated effect sizes of the instruments on the exposures. We previously proposed an approach (heterogeneity in dependent instrument, HEIDI) to test for heterogeneity in bxy estimated at 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
