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Bias Formulas for Sensitivity Analysis of Unmeasured Confounding for General Outcomes, Treatments, and Confounders

VanderWeele, Tyler J.a,b; Arah, Onyebuchi A.c,d

doi: 10.1097/EDE.0b013e3181f74493
Methods: Original Article

Uncontrolled confounding in observational studies gives rise to biased effect estimates. Sensitivity analysis techniques can be useful in assessing the magnitude of these biases. In this paper, we use the potential outcomes framework to derive a general class of sensitivity-analysis formulas for outcomes, treatments, and measured and unmeasured confounding variables that may be categorical or continuous. We give results for additive, risk-ratio and odds-ratio scales. We show that these results encompass a number of more specific sensitivity-analysis methods in the statistics and epidemiology literature. The applicability, usefulness, and limits of the bias-adjustment formulas are discussed. We illustrate the sensitivity-analysis techniques that follow from our results by applying them to 3 different studies. The bias formulas are particularly simple and easy to use in settings in which the unmeasured confounding variable is binary with constant effect on the outcome across treatment levels.


From the Departments ofaEpidemiology and bBiostatistics, Harvard School of Public Health, Boston, MA; cDepartment of Epidemiology, School of Public Health, University of California, Los Angeles (UCLA), Los Angeles, CA; and dDepartment of Public Health, Academic Medical Center, University of Amsterdam, Amsterdam, The Netherlands.

Submitted 4 March 2010; accepted 10 August 2010; posted 3 November 2010.

Supported by National Institutes of Health grant R03 HD060696-01A1 (to T.J.V.). Supported by a career grant (VENI number 916.96.059) from the Netherlands Organization for Scientific Research (NWO) (to O.A.A.).

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Correspondence: Tyler J. VanderWeele, Departments of Epidemiology and Biostatistics, Harvard School of Public Health, 677 Huntington Avenue, Boston, MA 02115. E-mail:

© 2011 Lippincott Williams & Wilkins, Inc.