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Bias Formulas for Sensitivity Analysis for Direct and Indirect Effects

VanderWeele, Tyler J.


VanderWeele TJ. Bias formulas for sensitivity analysis for direct and indirect effects. Epidemiology. 2010;21:540–551.

In Appendix 1 of the paper “Bias formulas for sensitivity analysis for direct and indirect effects,” the formula for the bias of the natural direct effect risk ratio was written as:

and ought to have been written as


To obtain the simpler formulas in Appendix 1 for the natural direct and indirect effect risk ratio bias terms:

an additional assumption that vm = 1 for all m is necessary ie, M has no effect on Y when the unmeasured confounder is absent. The more general results are given in the eAppendix of the original paper.

Epidemiology. 22(1):134, January 2011.

doi: 10.1097/EDE.0b013e3181df191c
Methods: Original Article

A key question in many studies is how to divide the total effect of an exposure into a component that acts directly on the outcome and a component that acts indirectly, ie, through some intermediate. For example, one might be interested in the extent to which the effect of diet on blood pressure is mediated through sodium intake and the extent to which it operates through other pathways. In the context of such mediation analysis, even if the effect of the exposure on the outcome is unconfounded, estimates of direct and indirect effects will be biased if control is not made for confounders of the mediator-outcome relationship. Often data are not collected on such mediator-outcome confounding variables; the results in this paper allow researchers to assess the sensitivity of their estimates of direct and indirect effects to the biases from such confounding. Specifically, the paper provides formulas for the bias in estimates of direct and indirect effects due to confounding of the exposure-mediator relationship and of the mediator-outcome relationship. Under some simplifying assumptions, the formulas are particularly easy to use in sensitivity analysis. The bias formulas are illustrated by examples in the literature concerning direct and indirect effects in which mediator-outcome confounding may be present.


From the Departments of Epidemiology and Biostatistics, Harvard University, Boston, MA.

Supported by the National Institute of Child Health and Human Development.

Submitted 8 October 2009; accepted 23 November 2009; posted 17 May 2010.

Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article (

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

© 2010 Lippincott Williams & Wilkins, Inc.