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Bahadur Efficiency of Sensitivity Analyses in Observational Studies

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  • Paul R. Rosenbaum

Abstract

An observational study draws inferences about treatment effects when treatments are not randomly assigned, as they would be in a randomized experiment. The naive analysis of an observational study assumes that adjustments for measured covariates suffice to remove bias from nonrandom treatment assignment. A sensitivity analysis in an observational study determines the magnitude of bias from nonrandom treatment assignment that would need to be present to alter the qualitative conclusions of the naive analysis, say leading to the acceptance of a null hypothesis rejected in the naive analysis. Observational studies vary greatly in their sensitivity to unmeasured biases, but a poor choice of test statistic can lead to an exaggerated report of sensitivity to bias. The Bahadur efficiency of a sensitivity analysis is introduced, calculated, and connected to established concepts, such as the power of a sensitivity analysis and the design sensitivity. The Bahadur slope equals zero when the sensitivity parameter equals the design sensitivity, but the Bahadur slope permits more refined distinctions. Specifically, the Bahadur relative efficiency can also compare the relative performance of two test statistics at a value of the sensitivity parameter below the minimum of their design sensitivities. Adaptive procedures that combine several tests can achieve the best design sensitivity and the best Bahadur slope of their component tests. Ultimately, in sufficiently large sample sizes, design sensitivity is more important than efficiency for the power of a sensitivity analysis, and the exponential rate at which rate design sensitivity overtakes efficiency is characterized.

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  • Paul R. Rosenbaum, 2015. "Bahadur Efficiency of Sensitivity Analyses in Observational Studies," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 205-217, March.
    • Handle: RePEc:taf:jnlasa:v:110:y:2015:i:509:p:205-217
    DOI: 10.1080/01621459.2014.960968
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    References listed on IDEAS

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    1. Jesse Y. Hsu & Dylan S. Small, 2013. "Calibrating Sensitivity Analyses to Observed Covariates in Observational Studies," Biometrics, The International Biometric Society, vol. 69(4), pages 803-811, December.
    2. Guildo W. Imbens, 2003. "Sensitivity to Exogeneity Assumptions in Program Evaluation," American Economic Review, American Economic Association, vol. 93(2), pages 126-132, May.
    3. Rosenbaum, Paul R., 2010. "Design Sensitivity and Efficiency in Observational Studies," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 692-702.
    4. Small, Dylan S., 2007. "Sensitivity Analysis for Instrumental Variables Regression With Overidentifying Restrictions," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1049-1058, September.
    5. Dylan S. Small & Jing Cheng & M. Elizabeth Halloran & Paul R. Rosenbaum, 2013. "Case Definition and Design Sensitivity," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(504), pages 1457-1468, December.
    6. Paul R. Rosenbaum, 2007. "Sensitivity Analysis for m-Estimates, Tests, and Confidence Intervals in Matched Observational Studies," Biometrics, The International Biometric Society, vol. 63(2), pages 456-464, June.
    7. Peter B. Gilbert & Ronald J. Bosch & Michael G. Hudgens, 2003. "Sensitivity Analysis for the Assessment of Causal Vaccine Effects on Viral Load in HIV Vaccine Trials," Biometrics, The International Biometric Society, vol. 59(3), pages 531-541, September.
    8. Paul R. Rosenbaum, 2004. "Design sensitivity in observational studies," Biometrika, Biometrika Trust, vol. 91(1), pages 153-164, March.
    9. Rosenbaum, Paul R. & Silber, Jeffrey H., 2009. "Amplification of Sensitivity Analysis in Matched Observational Studies," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1398-1405.
    10. Jesse Y. Hsu & Dylan S. Small & Paul R. Rosenbaum, 2013. "Effect Modification and Design Sensitivity in Observational Studies," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 135-148, March.
    11. P. R. Rosenbaum, 2012. "Testing one hypothesis twice in observational studies," Biometrika, Biometrika Trust, vol. 99(4), pages 763-774.
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    Cited by:

    1. Siyu Heng & Dylan S. Small & Paul R. Rosenbaum, 2020. "Finding the strength in a weak instrument in a study of cognitive outcomes produced by Catholic high schools," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(3), pages 935-958, June.
    2. Paul R. Rosenbaum, 2015. "Some Counterclaims Undermine Themselves in Observational Studies," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1389-1398, December.
    3. Kwonsang Lee & Dylan S. Small & Paul R. Rosenbaum, 2018. "A powerful approach to the study of moderate effect modification in observational studies," Biometrics, The International Biometric Society, vol. 74(4), pages 1161-1170, December.
    4. Ruoqi Yu, 2021. "Evaluating and improving a matched comparison of antidepressants and bone density," Biometrics, The International Biometric Society, vol. 77(4), pages 1276-1288, December.
    5. Paul R. Rosenbaum, 2023. "A second evidence factor for a second control group," Biometrics, The International Biometric Society, vol. 79(4), pages 3968-3980, December.

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