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. 2020 Sep 11;21:232. doi: 10.1186/s13059-020-02026-y

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

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Fig. 7 — Histograms of I² distribution for validating exclusion restriction from PTWAS scan signals. Panel a shows the histogram of the I² statistics computed from 2.09 million trait-gene-tissue combinations, where the gene has at least two independent eQTLs with individual signal-level PIP >0.50. For each trait-gene-tissue pair, the I² index measures the consistency of the causal effects estimated from each independent eQTL. Panels b and c show the histograms with at least 3 and 5 eligible independent eQTLs, respectively. The overall patterns of the distributions are the same in all panels, where majority of the test cases show strong consistency among independent instruments. However, the ability of identifying violations of ER improves with the increasing availability of independent eQTL instruments