Statistical Science

Identification, Inference and Sensitivity Analysis for Causal Mediation Effects

Kosuke Imai, Luke Keele, and Teppei Yamamoto

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Causal mediation analysis is routinely conducted by applied researchers in a variety of disciplines. The goal of such an analysis is to investigate alternative causal mechanisms by examining the roles of intermediate variables that lie in the causal paths between the treatment and outcome variables. In this paper we first prove that under a particular version of sequential ignorability assumption, the average causal mediation effect (ACME) is nonparametrically identified. We compare our identification assumption with those proposed in the literature. Some practical implications of our identification result are also discussed. In particular, the popular estimator based on the linear structural equation model (LSEM) can be interpreted as an ACME estimator once additional parametric assumptions are made. We show that these assumptions can easily be relaxed within and outside of the LSEM framework and propose simple nonparametric estimation strategies. Second, and perhaps most importantly, we propose a new sensitivity analysis that can be easily implemented by applied researchers within the LSEM framework. Like the existing identifying assumptions, the proposed sequential ignorability assumption may be too strong in many applied settings. Thus, sensitivity analysis is essential in order to examine the robustness of empirical findings to the possible existence of an unmeasured confounder. Finally, we apply the proposed methods to a randomized experiment from political psychology. We also make easy-to-use software available to implement the proposed methods.

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Statist. Sci. Volume 25, Number 1 (2010), 51-71.

First available in Project Euclid: 3 August 2010

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Causal inference causal mediation analysis direct and indirect effects linear structural equation models sequential ignorability unmeasured confounders


Imai, Kosuke; Keele, Luke; Yamamoto, Teppei. Identification, Inference and Sensitivity Analysis for Causal Mediation Effects. Statist. Sci. 25 (2010), no. 1, 51--71. doi:10.1214/10-STS321.

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  • Albert, J. M. (2008). Mediation analysis via potential outcomes models. Stat. Med. 27 1282–1304.
  • Angrist, J. D., Imbens, G. W. and Rubin, D. B. (1996). Identification of causal effects using instrumental variables (with discussion). J. Amer. Statist. Assoc. 91 444–455.
  • Avin, C., Shpitser, I. and Pearl, J. (2005). Identifiability of path-specific effects. In Proceedings of the International Joint Conference on Artificial Intelligence. Morgan Kaufman, San Francisco, CA.
  • Baron, R. M. and Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology 51 1173–1182.
  • Cochran, W. G. (1957). Analysis of covariance: Its nature and uses. Biometrics 13 261–281.
  • Deaton, A. (2009). Instruments of development: Randomization in the tropics, and the search for the elusive keys to economic development. Proc. Br. Acad. 162 123–160.
  • Egleston, B., Scharfstein, D. O., Munoz, B. and West, S. (2006). Investigating mediation when counterfactuals are not metaphysical: Does sunlight UVB exposure mediate the effect of eyeglasses on cataracts? Working Paper 113, Dept. Biostatistics, Johns Hopkins Univ., Baltimore, MD.
  • Elliott, M. R., Raghunathan, T. E. and Li, Y. (2010). Bayesian inference for causal mediation effects using principal stratification with dichotomous mediators and outcomes. Biostatistics. 11 353–372.
  • Gallop, R., Small, D. S., Lin, J. Y., Elliot, M. R., Joffe, M. and Ten Have, T. R. (2009). Mediation analysis with principal stratification. Stat. Med. 28 1108–1130.
  • Geneletti, S. (2007). Identifying direct and indirect effects in a non-counterfactual framework. J. Roy. Statist. Soc. Ser. B 69 199–215.
  • Glynn, A. N. (2010). The product and difference fallacies for indirect effects. Unpublished manuscript, Dept. Government, Harvard Univ.
  • Goodman, L. A. (1960). On the exact variance of products. J. Amer. Statist. Assoc. 55 708–713.
  • Green, D. P., Ha, S. E. and Bullock, J. G. (2010). Yes, but what’s the mechanism? (don’t expect an easy answer). Journal of Personality and Social Psychology 98 550–558.
  • Hafeman, D. M. and Schwartz, S. (2009). Opening the black box: A motivation for the assessment of mediation. International Journal of Epidemiology 38 838–845.
  • Hafeman, D. M. and VanderWeele, T. J. (2010). Alternative assumptions for the identification of direct and indirect effects. Epidemiology 21. To appear.
  • Imai, K. and Yamamoto, T. (2010). Causal inference with differential measurement error: Nonparametric identification and sensitivity analysis. American Journal of Political Science 54 543–560.
  • Imai, K., Keele, L. and Tingley, D. (2009). A general approach to causal mediation analysis. Psychological Methods. To appear.
  • Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2010). Causal mediation analysis using R. In Advances in Social Science Research Using R (H. D. Vinod, ed.). Lecture Notes in Statist. 196 129–154. Springer, New York.
  • Imai, K., Keele, L. and Yamamoto, T. (2010). Replication data for: Identification, inference, and sensitivity analysis for causal mediation effects. Available at
  • Imai, K., Tingley, D. and Yamamoto, T. (2009). Experimental designs for identifying causal mechanisms. Technical report, Dept. Politics, Princeton Univ. Available at
  • Imbens, G. W. (2003). Sensitivity to exogeneity assumptions in program evaluation. American Economic Review 93 126–132.
  • Jo, B. (2008). Causal inference in randomized experiments with mediational processes. Psychological Methods 13 314–336.
  • Joffe, M. M., Small, D., Ten Have, T., Brunelli, S. and Feldman, H. I. (2008). Extended instrumental variables estimation for overall effects. Int. J. Biostat. 4 Article 4.
  • Joffe, M. M., Small, D. and Hsu, C.-Y. (2007). Defining and estimating intervention effects for groups that will develop an auxiliary outcome. Statist. Sci. 22 74–97.
  • Judd, C. M. and Kenny, D. A. (1981). Process analysis: Estimating mediation in treatment evaluations. Evaluation Review 5 602–619.
  • Kraemer, H. C., Kiernan, M., Essex, M. and Kupfer, D. J. (2008). How and why criteria definig moderators and mediators differ between the Baron & Kenny and MacArthur approaches. Health Psychology 27 S101–S108.
  • Kraemer, H. C., Wilson, T., Fairburn, C. G. and Agras, W. S. (2002). Mediators and moderators of treatment effects in randomized clinical trials. Archives of General Psychiatry 59 877–883.
  • MacKinnon, D. P. (2008). Introduction to Statistical Mediation Analysis. Taylor & Francis, New York.
  • Nelson, T. E., Clawson, R. A. and Oxley, Z. M. (1997). Media framing of a civil liberties conflict and its effect on tolerance. American Political Science Review 91 567–583.
  • Pearl, J. (2001). Direct and indirect effects. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (J. S. Breese and D. Koller, eds.) 411–420. Morgan Kaufman, San Francisco, CA.
  • Pearl, J. (2010). An introduction to causal inference. Int. J. Biostat. 6 Article 7.
  • Petersen, M. L., Sinisi, S. E. and van der Laan, M. J. (2006). Estimation of direct causal effects. Epidemiology 17 276–284.
  • Robins, J. (1999). Marginal structural models versus structural nested models as tools for causal inference. In Statistical Models in Epidemiology, the Environment and Clinical Trials (M. E. Halloran and D. A. Berry, eds.) 95–134. Springer, New York.
  • Robins, J. M. (2003). Semantics of causal DAG models and the identification of direct and indirect effects. In Highly Structured Stochastic Systems (P. J. Green, N. L. Hjort and S. Richardson, eds.) 70–81. Oxford Univ. Press, Oxford.
  • Robins, J. M. and Greenland, S. (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology 3 143–155.
  • Roy, J., Hogan, J. W. and Marcus, B. H. (2008). Principal stratification with predictors of compliance for randomized trials with 2 active treatments. Biostatistics 9 277–289.
  • Rubin, D. B. (2004). Direct and indirect causal effects via potential outcomes (with discussion). Scand. J. Statist. 31 161–170.
  • Rubin, D. B. (2005). Causal inference using potential outcomes: Design, modeling, decisions. J. Amer. Statist. Assoc. 100 322–331.
  • Sjölander, A. (2009). Bounds on natural direct effects in the presence of confounded intermediate variables. Stat. Med. 28 558–571.
  • Skrabanek, P. (1994). The emptiness of the black box. Epidemiology 5 5553–5555.
  • Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology 13 290–321.
  • Sobel, M. E. (2008). Identification of causal parameters in randomized studies with mediating variables. Journal of Educational and Behavioral Statistics 33 230–251.
  • Ten Have, T. R., Joffe, M. M., Lynch, K. G., Brown, G. K., Maisto, S. A. and Beck, A. T. (2007). Causal mediation analyses with rank preserving models. Biometrics 63 926–934.
  • VanderWeele, T. J. (2008). Simple relations between principal stratification and direct and indirect effects. Statist. Probab. Lett. 78 2957–2962.
  • VanderWeele, T. J. (2009). Marginal structural models for the estimation of direct and indirect effects. Epidemiology 20 18–26.
  • VanderWeele, T. J. (2010). Bias formulas for sensitivity analysis for direct and indirect effects. Epidemiology. To appear.
  • Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J. Amer. Statist. Assoc. 57 348–368.