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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.

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 (


  1. 1.

    Blischak, J. D., Tailleux, L., Mitrano, A., Barreiro, L. B. & Gilad, Y. Mycobacterial infection induces a specific human innate immune response. Sci. Rep. 5, 16882 (2015).

    CAS  Article  Google Scholar 

  2. 2.

    Ferguson, J. P., Cho, J. H. & Zhao, H. A new approach for the joint analysis of multiple ChIP-Seq libraries with application to histone modification. Stat. Appl. Genet. Mol. Biol. 11, (2012).

  3. 3.

    Pickrell, J., Berisa, T., Ségurel, L., Tung, J. Y. & Hinds, D. Detection and interpretation of shared genetic influences on 40 human traits. Nat. Genet. 48, 709–717 (2016).

    CAS  Article  Google Scholar 

  4. 4.

    Dimas, A. S. et al. Common regulatory variation impacts gene expression in a cell type-dependent manner. Science 325, 1246–1250 (2009).

    CAS  Article  Google Scholar 

  5. 5.

    Flutre, T., Wen, X., Pritchard, J. & Stephens, M. A statistical framework for joint eQTL analysis in multiple tissues. PLoS Genet. 9, e1003486 (2013).

    CAS  Article  Google Scholar 

  6. 6.

    Li, G., Shabalin, A. A., Rusyn, I., Wright, F. A. & Nobel, A. B. An Empirical Bayes approach for multiple tissue eQTL Analysis. Biostatistics 19, 391–406 (2017).

    Article  Google Scholar 

  7. 7.

    Petretto, E. et al. New insights into the genetic control of gene expression using a Bayesian multi-tissue approach. PLoS Comput. Biol. 6, e1000737 (2010).

    Article  Google Scholar 

  8. 8.

    Wen, X. & Stephens, M. Using linear predictors to impute allele frequencies from summary of pooled genotype data. Ann. Appl. Stat. 4, 1158–1182 (2010).

    Article  Google Scholar 

  9. 9.

    Han, B. & Eskin, E. Random-effects model aimed at discovering associations in meta-analysis of genome-wide association studies. Am. J. Hum. Genet. 88, 586–598 (2011).

    CAS  Article  Google Scholar 

  10. 10.

    Stephens, M. Unified framework for association analysis with multiple related phenotypes. PLoS One 8, e65245 (2013).

    CAS  Article  Google Scholar 

  11. 11.

    Sul, J. H., Han, B., Ye, C., Choi, T. & Eskin, E. Effectively identifying eQTLs from multiple tissues by combining mixed model and meta-analytic approaches. PLoS Genet. 9, e1003491 (2013).

    CAS  Article  Google Scholar 

  12. 12.

    Wei, Y., Tenzen, T. & Ji, H. Joint analysis of differential gene expression in multiple studies using correlation motifs. Biostatistics 16, 31–46 (2015).

    Article  Google Scholar 

  13. 13.

    Zhou, X. & Stephens, M. Efficient multivariate linear mixed model algorithms for genome-wide association studies. Nat. Methods 11, 407–409 (2014).

    CAS  Article  Google Scholar 

  14. 14.

    Han, B. & Eskin, E. Interpreting meta-analyses of genome-wide association studies. PLoS Genet. 8, e1002555 (2012).

    CAS  Article  Google Scholar 

  15. 15.

    Lebrec, J. J., Stijnen, T. & van Houwelingen, H. C. Dealing with heterogeneity between cohorts in genomewide SNP association studies. Stat. Appl. Genet. Mol. Biol. 9, (2010).

  16. 16.

    Stephens, M. False discovery rates: a new deal. Biostatistics 18, 275–294 (2017).

    PubMed  Google Scholar 

  17. 17.

    GTEx Consortium. The Genotype-Tissue Expression (GTEx) pilot analysis: multitissue gene regulation in humans. Science 348, 648–660 (2015).

    Article  Google Scholar 

  18. 18.

    Nicolae, D. L. et al. Trait-associated SNPs are more likely to be eQTLs: annotation to enhance discovery from GWAS. PLoS Genet. 6, e1000888 (2010).

    Article  Google Scholar 

  19. 19.

    Shabalin, A. A. Matrix eQTL: ultra fast eQTL analysis via large matrix operations. Bioinformatics 28, 1353–1358 (2012).

    CAS  Article  Google Scholar 

  20. 20.

    Price, A. L. et al. Principal components analysis corrects for stratification in genome-wide association studies. Nat. Genet. 38, 904–909 (2006).

    CAS  Article  Google Scholar 

  21. 21.

    Engelhardt, B. E. & Stephens, M. Analysis of population structure: a unifying framework and novel methods based on sparse factor analysis. PLoS Genet. 6, e1001117 (2010).

    Article  Google Scholar 

  22. 22.

    Tversky, A. & Kahneman, D. Judgment under uncertainty: heuristics and biases. Science 185, 1124–1131 (1974).

    CAS  Article  Google Scholar 

  23. 23.

    Wen, X. & Stephens, M. Bayesian methods for genetic association analysis with heterogeneous subgroups: from meta-analyses to gene-environment interactions. Ann. Appl. Stat. 8, 176–203 (2014).

    Article  Google Scholar 

  24. 24.

    Servin, B. & Stephens, M. Imputation-based analysis of association studies: candidate regions and quantitative traits. PLoS Genet. 3, e114 (2007).

    Article  Google Scholar 

  25. 25.

    Veyrieras, J.-B. et al. High-resolution mapping of expression-QTLs yields insight into human gene regulation. PLoS Genet. 4, e1000214 (2008).

    Article  Google Scholar 

  26. 26.

    Hormozdiari, F., Kostem, E., Kang, E. Y., Pasaniuc, B. & Eskin, E. Identifying causal variants at loci with multiple signals of association. Genetics 198, 497–508 (2014).

    CAS  Article  Google Scholar 

  27. 27.

    Kichaev, G. et al. Integrating functional data to prioritize causal variants in statistical fine-mapping studies. PLoS Genet. 10, e1004722 (2014).

    Article  Google Scholar 

  28. 28.

    Chen, W. et al. Fine mapping causal variants with an approximate Bayesian method using marginal test statistics. Genetics 200, 719–736 (2015).

    Article  Google Scholar 

  29. 29.

    Moyerbrailean, G. A. et al. Which genetic variants in DNase-seq footprints are more likely to alter binding? PLoS Genet. 12, e1005875 (2016).

    Article  Google Scholar 

  30. 30.

    Giambartolomei, C. et al. Bayesian test for colocalisation between pairs of genetic association studies using summary statistics. PLoS Genet. 10, e1004383 (2014).

    Article  Google Scholar 

  31. 31.

    Fortune, M. D. et al. Statistical colocalization of genetic risk variants for related autoimmune diseases in the context of common controls. Nat. Genet. 47, 839–846 (2015).

    CAS  Article  Google Scholar 

  32. 32.

    Nica, A. C. et al. Candidate causal regulatory effects by integration of expression QTLs with complex trait genetic associations. PLoS Genet. 6, e1000895 (2010).

    Article  Google Scholar 

  33. 33.

    Benjamini, Y. & Hochberg, Y. Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. R. Stat. Soc. Series B Stat. Methodol. 57, 289–300 (1995).

    Google Scholar 

  34. 34.

    Storey, J. D. The positive false discovery rate: a Bayesian interpretation and the q-value. Ann. Stat. 31, 2013–2035 (2003).

    Article  Google Scholar 

  35. 35.

    Bovy, J., Hogg, D. W. & Roweis, S. T. Extreme Deconvolution: inferring complete distribution functions from noisy, heterogeneous and incomplete observations. Ann. Appl. Stat. 5, 1657–1677 (2011).

    Article  Google Scholar 

  36. 36.

    Larribe, F. & Fearnhead, P. Composite likelihood methods in statistical genetics. Stat. Sin. 21, 43–69 (2011).

    Google Scholar 

  37. 37.

    Dempster, A. P., Laird, N. M. & Rubin, D. B. Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Series B Stat. Methodol. 39, 1–38 (1977).

    Google Scholar 

  38. 38.

    Varadhan, R. & Roland, C. Simple and globally convergent methods for accelerating the convergence of any EM algorithm. Scand. J. Stat. 35, 335–353 (2008).

    Article  Google Scholar 

  39. 39.

    Efron, B. Microarrays empirical Bayes and the two-groups model. Stat. Sci. 23, 1–22 (2008).

    Article  Google Scholar 

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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.

Author information




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.

Corresponding author

Correspondence to Matthew Stephens.

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The authors declare no competing interests.

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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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