Research and Reporting Methods15 August 2017
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    Sensitivity analysis is useful in assessing how robust an association is to potential unmeasured or uncontrolled confounding. This article introduces a new measure called the “E-value,” which is related to the evidence for causality in observational studies that are potentially subject to confounding. The E-value is defined as the minimum strength of association, on the risk ratio scale, that an unmeasured confounder would need to have with both the treatment and the outcome to fully explain away a specific treatment–outcome association, conditional on the measured covariates. A large E-value implies that considerable unmeasured confounding would be needed to explain away an effect estimate. A small E-value implies little unmeasured confounding would be needed to explain away an effect estimate. The authors propose that in all observational studies intended to produce evidence for causality, the E-value be reported or some other sensitivity analysis be used. They suggest calculating the E-value for both the observed association estimate (after adjustments for measured confounders) and the limit of the confidence interval closest to the null. If this were to become standard practice, the ability of the scientific community to assess evidence from observational studies would improve considerably, and ultimately, science would be strengthened.

    References

    • 1. Wasserstein RL  and  Lazar NL The ASA's statement on p-values: context, process, and purpose. Am Stat2016;70:129-33. CrossrefGoogle Scholar
    • 2. Altman DG Machin D Bryant TN , and  Gardner MJ Statistics with Confidence. 2nd ed. London BMJ Books 2000. Google Scholar
    • 3. Rosner B Fundamentals of Biostatistics. 8th ed. Boston Cengage Learning 2015. Google Scholar
    • 4. Pagano M  and  Gavreau K Principles of Biostatistics. Belmont, CA Brooks/Cole 2000. Google Scholar
    • 5. Greenland S Senn SJ Rothman KJ Carlin JB Poole C Goodman SN et alStatistical tests, P values, confidence intervals, and power: a guide to misinterpretations. Eur J Epidemiol2016;31:337-50. [PMID: 27209009] doi:10.1007/s10654-016-0149-3 CrossrefMedlineGoogle Scholar
    • 6. Goodman S A dirty dozen: twelve p-value misconceptions. Semin Hematol2008;45:135-40. [PMID: 18582619] doi:10.1053/j.seminhematol.2008.04.003 CrossrefMedlineGoogle Scholar
    • 7. Greenland S Null misinterpretation in statistical testing and its impact on health risk assessment. Prev Med2011;53:225-8. [PMID: 21871481] doi:10.1016/j.ypmed.2011.08.010 CrossrefMedlineGoogle Scholar
    • 8. Greenland S  and  Poole C Problems in common interpretations of statistics in scientific articles, expert reports, and testimony. Jurimetrics2011;51:11329. Google Scholar
    • 9. Sterne JA  and  DaveySmith G Sifting the evidence—what's wrong with significance tests? BMJ2001;322:226-31. [PMID: 11159626] CrossrefMedlineGoogle Scholar
    • 10. Stang A Poole C , and  Kuss O The ongoing tyranny of statistical significance testing in biomedical research. Eur J Epidemiol2010;25:225-30. [PMID: 20339903] doi:10.1007/s10654-010-9440-x CrossrefMedlineGoogle Scholar
    • 11. Goodman SN Toward evidence-based medical statistics. 1: The P value fallacy. Ann Intern Med1999;130:995-1004 LinkGoogle Scholar
    • 12. Hill AB The environment and disease: association or causation? Proc R Soc Med1965;58:295-300. [PMID: 14283879] CrossrefMedlineGoogle Scholar
    • 13. Greenland S Randomization, statistics, and causal inference. Epidemiology1990;1:421-9. [PMID: 2090279] CrossrefMedlineGoogle Scholar
    • 14. Imbens GW  and  Rubin DB Sensitivity analysis and bounds. In: Causal Inference for Statistics, Social, and Biomedical Sciences. New York Cambridge Univ Pr 2015 496-512. Google Scholar
    • 15. Hernan MA, Robins JR. Confounding. In: Causal Inference. 11 September 2016. Accessed at https://cdn1.sph.harvard.edu/wp-content/uploads/sites/1268/2016/09/hernanrobins_v1.10.31.pdf on 22 May 2017. Google Scholar
    • 16. Rosenbaum PR Sensitivity to hidden bias. In: Observational Studies. 2nd ed. New York Springer 2002 105-70. Google Scholar
    • 17. Rosenbaum PR Design sensitivity. In: Design of Observational Studies. New York Springer 2010 269-71. Google Scholar
    • 18. Rosenbaum PR Design sensitivity and efficiency in observational studies. J Am Stat Assoc2010;105:692-702. CrossrefGoogle Scholar
    • 19. Greenland S Multiple-bias modeling for analysis of observational data. J R Stat Soc Ser A2005;168:267-308. CrossrefGoogle Scholar
    • 20. Lash TL Fox MP , and  Fink AK Applying Quantitative Bias Analysis to Epidemiologic Data. New York Springer 2009. Google Scholar
    • 21. Greenland S  and  Lash TL Bias analysis.. In: Rothman KJ, Greenland S, Lash TL, eds. Modern Epidemiology. Philadelphia Lippincott Williams & Wilkins 2008 345-80. Google Scholar
    • 22. Cornfield J Haenszel W Hammond EC Lilienfeld AM Shimkin MB , and  Wynder EL Smoking and lung cancer: recent evidence and a discussion of some questions. J Natl Cancer Inst1959;22:173-203. [PMID: 13621204] MedlineGoogle Scholar
    • 23. Victora CG Smith PG Vaughan JP Nobre LC Lombardi C Teixeira AM et alEvidence for protection by breast-feeding against infant deaths from infectious diseases in Brazil. Lancet1987;2:319-22. [PMID: 2886775] CrossrefMedlineGoogle Scholar
    • 24. Bross ID Spurious effects from an extraneous variable. J Chronic Dis1966;19:637-47. [PMID: 5966011] CrossrefMedlineGoogle Scholar
    • 25. Schlesselman JJ Assessing effects of confounding variables. Am J Epidemiol1978;108:3-8. [PMID: 685974] MedlineGoogle Scholar
    • 26. Rosenbaum PR  and  Rubin DB Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. J R Stat Soc Series B Stat Methodol1983;45:212-8. Google Scholar
    • 27. Lin DY Psaty BM , and  Kronmal RA Assessing the sensitivity of regression results to unmeasured confounders in observational studies. Biometrics1998;54:948-63. [PMID: 9750244] CrossrefMedlineGoogle Scholar
    • 28. Imbens GW Sensitivity to exogeneity assumptions in program evaluation. Am Econ Rev2003;93:126-32. CrossrefGoogle Scholar
    • 29. Vanderweele TJ  and  Arah OA Bias formulas for sensitivity analysis of unmeasured confounding for general outcomes, treatments, and confounders. Epidemiology2011;22:42-52. [PMID: 21052008] doi:10.1097/EDE.0b013e3181f74493 CrossrefMedlineGoogle Scholar
    • 30. Bross ID Pertinency of an extraneous variable. J Chronic Dis1967;20:487-95. [PMID: 6028268] CrossrefMedlineGoogle Scholar
    • 31. Lee WC Bounding the bias of unmeasured factors with confounding and effect modifying potentials. Stat Med2011;30:1007-17. CrossrefMedlineGoogle Scholar
    • 32. Robins J M Scharfstein D , and  Rotnitzky A Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models.. In: Halloran E, Berry D, eds. Statistical Models for Epidemiology, the Environment, and Clinical Trials. New York Springer-Verlag 2000 1-94. Google Scholar
    • 33. McCandless LC Gustafson P , and  Levy A Bayesian sensitivity analysis for unmeasured confounding in observational studies. Stat Med2007;26:2331-47. [PMID: 16998821] CrossrefMedlineGoogle Scholar
    • 34. Brumback BA Hernán MA Haneuse SJ , and  Robins JM Sensitivity analyses for unmeasured confounding assuming a marginal structural model for repeated measures. Stat Med2004;23:749-67. [PMID: 14981673] CrossrefMedlineGoogle Scholar
    • 35. VanderWeele TJ Sensitivity analysis for mediation. In: Explanation in Causal Inference: Methods for Mediation and Interaction. New York Oxford Univ Pr 2015 66-97. Google Scholar
    • 36. VanderWeele TJ Sensitivity analysis for contagion effects in social networks. Sociol Methods Res2011;40:240-55. [PMID: 25580037] CrossrefMedlineGoogle Scholar
    • 37. Ding P  and  VanderWeele TJ Sensitivity analysis without assumptions. Epidemiology2016;27:368-77. [PMID: 26841057] doi:10.1097/EDE.0000000000000457 CrossrefMedlineGoogle Scholar
    • 38. Gilbert PB Bosch RJ , and  Hudgens MG Sensitivity analysis for the assessment of causal vaccine effects on viral load in HIV vaccine trials. Biometrics2003;59:531-41. [PMID: 14601754] CrossrefMedlineGoogle Scholar
    • 39. Chiba Y  and  VanderWeele TJ A simple method for principal strata effects when the outcome has been truncated due to death. Am J Epidemiol2011;173:745-51. [PMID: 21354986] doi:10.1093/aje/kwq418 CrossrefMedlineGoogle Scholar
    • 40. Huang TH  and  Lee WC Bounding formulas for selection bias. Am J Epidemiol2015;182:868-72. [PMID: 26519426] doi:10.1093/aje/kwv130 CrossrefMedlineGoogle Scholar
    • 41. Rosenbaum PR Discussing hidden bias in observational studies. Ann Intern Med1991;115:901-5 LinkGoogle Scholar
    • 42. Ip S Chung M Raman G Chew P Magula N DeVine D et alBreastfeeding and Maternal and Infant Health Outcomes in Developed Countries. Evidence Report/Technology Assessment no. 153. (Prepared by Tufts-New England Medical Center Evidence-based Practice Center under contract no. 290-02-0022.) AHRQ publication no. 07-E007. Rockville Agency for Healthcare Research and Quality April 2007. Google Scholar
    • 43. Moorman PG Calingaert B Palmieri RT Iversen ES Bentley RC Halabi S et alHormonal risk factors for ovarian cancer in premenopausal and postmenopausal women. Am J Epidemiol2008;167:1059-69. [PMID: 18303003] doi:10.1093/aje/kwn006 CrossrefMedlineGoogle Scholar
    • 44. Taubes G Epidemiology faces its limits. Science1995;269:164-9. [PMID: 7618077] CrossrefMedlineGoogle Scholar
    • 45. VanderWeele TJ. On a square-root transformation of the odds ratio for a common outcome. Epidemiology. 2017. [Forthcoming]. Google Scholar
    • 46. Borenstein M Hedges LV Higgins JPT , and  Rothstein HR Converting among effect sizes. In: Introduction to Meta-Analysis. Hoboken, NJ Wiley 2009 45-51. Google Scholar
    • 47. Hasselblad V  and  Hedges LV Meta-analysis of screening and diagnostic tests. Psychol Bull1995;117:167-78. [PMID: 7870860] CrossrefMedlineGoogle Scholar