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Flexible statistical methods for estimating and testing effects in genomic studies with multiple conditions


We introduce new statistical methods for analyzing genomic data sets that measure many effects in many conditions (for example, gene expression changes under many treatments). These new methods improve on existing methods by allowing for arbitrary correlations in effect sizes among conditions. This flexible approach increases power, improves effect estimates and allows for more quantitative assessments of effect-size heterogeneity compared to simple shared or condition-specific assessments. We illustrate these features through an analysis of locally acting variants associated with gene expression (cis expression quantitative trait loci (eQTLs)) in 44 human tissues. Our analysis identifies more eQTLs than existing approaches, consistent with improved power. We show that although genetic effects on expression are extensively shared among tissues, effect sizes can still vary greatly among tissues. Some shared eQTLs show stronger effects in subsets of biologically related tissues (for example, brain-related tissues), or in only one tissue (for example, testis). Our methods are widely applicable, computationally tractable for many conditions and available online.

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Fig. 1: Overview of fitting procedure in mash, which estimates multivariate distribution of effects present in the data.
Fig. 2: Comparison of methods on simulated data.
Fig. 3: Summary of primary patterns identified by mash in GTEx data.
Fig. 4: mash uses learned patterns of sharing to inform effect estimates in the GTEx data.
Fig. 5: Number of tissues shared by sign and magnitude.
Fig. 6: Pairwise sharing by magnitude of eQTLs among tissues.

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Data availability

The GTEx study data are available through dbGaP under accession phs000424.v6.p1. The GTEx summary statistics used in the mash analysis have been deposited in Zenodo (


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This work was supported by National Institutes of Health (NIH) grants MH090951 and HG02585 to M.S., and by a grant from the Gordon and Betty Moore Foundation (GBMF 4559) to M.S. S.M.U. was supported by NIH grant T32HD007009. Expression (GTEx) Project was supported by the Common Fund of the Office of the Director of the NIH. Additional funds were provided by the National Cancer Institute (NCI), National Human Genome Research Institute, National Heart, Lung, and Blood Institute, National Institute on Drug Abuse, National Institute of Mental Health, and National Institute of Neurological Disorders and Stroke. Donors were enrolled at Biospecimen Source Sites funded by NCI/SAIC-Frederick (SAIC-F) subcontracts to the National Disease Research Interchange (10XS170), Roswell Park Cancer Institute (10XS171), and Science Care (X10S172). The Laboratory, Data Analysis, and Coordinating Center was funded through a contract (HHSN268201000029C) to the Broad Institute. Biorepository operations were funded through an SAIC-F subcontract to Van Andel Institute (10ST1035). Additional data repository and project management were provided by SAIC-F (HHSN261200800001E). The Brain Bank was supported by a supplement to University of Miami grants DA006227 and DA033684 and to contract N01MH000028. Statistical methods development grants were made to the University of Geneva (MH090941 and MH101814), the University of Chicago (MH090951, MH090937, MH101820 and MH101825), the University of North Carolina at Chapel Hill (MH090936 and MH101819), Harvard University (MH090948), Stanford University (MH101782), Washington University in St. Louis (MH101810) and the University of Pennsylvania (MH101822). The data used for the analyses described in this manuscript were obtained from the GTEx Portal on 17 October 2015.

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Authors and Affiliations



S.M.U. and M.S. conceived of the project and developed the statistical methods. S.M.U. implemented the comparisons with simulated data. S.M.U. and G.W. performed the analyses of the GTEx data and additional analyses. S.M.U., G.W. and M.S. implemented the software, with contributions from P.C. S.M.U. and M.S. wrote the manuscript, with input from G.W. and P.C. P.C. and G.W. prepared the online code and data resources.

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Correspondence to Matthew Stephens.

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Urbut, S.M., Wang, G., Carbonetto, P. et al. Flexible statistical methods for estimating and testing effects in genomic studies with multiple conditions. Nat Genet 51, 187–195 (2019).

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