Statistical Science

Interference and Sensitivity Analysis

Tyler J. VanderWeele, Eric J. Tchetgen Tchetgen, and M. Elizabeth Halloran

Full-text: Open access


Causal inference with interference is a rapidly growing area. The literature has begun to relax the “no-interference” assumption that the treatment received by one individual does not affect the outcomes of other individuals. In this paper we briefly review the literature on causal inference in the presence of interference when treatments have been randomized. We then consider settings in which causal effects in the presence of interference are not identified, either because randomization alone does not suffice for identification or because treatment is not randomized and there may be unmeasured confounders of the treatment–outcome relationship. We develop sensitivity analysis techniques for these settings. We describe several sensitivity analysis techniques for the infectiousness effect which, in a vaccine trial, captures the effect of the vaccine of one person on protecting a second person from infection even if the first is infected. We also develop two sensitivity analysis techniques for causal effects under interference in the presence of unmeasured confounding which generalize analogous techniques when interference is absent. These two techniques for unmeasured confounding are compared and contrasted.

Article information

Statist. Sci., Volume 29, Number 4 (2014), 687-706.

First available in Project Euclid: 15 January 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Causal inference infectiousness effect interference sensitivity analysis spillover effect stable unit treatment value assumption vaccine trial


VanderWeele, Tyler J.; Tchetgen Tchetgen, Eric J.; Halloran, M. Elizabeth. Interference and Sensitivity Analysis. Statist. Sci. 29 (2014), no. 4, 687--706. doi:10.1214/14-STS479.

Export citation


  • Ali, M., Emch, M., von Seidlein, L., Yunus, M., Sack, D. A., Rao, M., Holmgren, J. and Clemens, J. D. (2005). Herd immunity conferred by killed oral cholera vaccines in Bangladesh: A reanalysis. Lancet 366 44–49.
  • Aronow, P. M. and Samii, C. (2013). Estimating average causal effects under general interference. Available at arXiv:1305.6156v1.
  • Christakis, N. A. and Fowler, J. H. (2007). The spread of obesity in a large social network over 32 years. New England Journal of Medicine 357 370–379.
  • Cox, D. R. (1958). Planning of Experiments. Wiley, New York.
  • Frangakis, C. E. and Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics 58 21–29.
  • Gilbert, P. B., Bosch, R. J. and Hudgens, M. G. (2003). Sensitivity analysis for the assessment of causal vaccine effects on viral load in HIV vaccine trials. Biometrics 59 531–541.
  • Halloran, M. E. and Hudgens, M. G. (2012a). Causal inference for vaccine effects on infectiousness. Int. J. Biostat. 8 Art. 6, front matter + 40.
  • Halloran, M. E. and Hudgens, M. G. (2012b). Comparing bounds for vaccine effects on infectiousness. Epidemiology 23 931–932.
  • Halloran, M. E. and Struchiner, C. J. (1991). Study designs for dependent happenings. Epidemiology 2 331–338.
  • Halloran, M. E. and Struchiner, C. J. (1995). Causal inference in infectious diseases. Epidemiology 6 142–151.
  • Hong, G. and Raudenbush, S. W. (2006). Evaluating kindergarten retention policy: A case study of causal inference for multilevel observational data. J. Amer. Statist. Assoc. 101 901–910.
  • Hudgens, M. G. and Halloran, M. E. (2006). Causal vaccine effects on binary postinfection outcomes. J. Amer. Statist. Assoc. 101 51–64.
  • Hudgens, M. G. and Halloran, M. E. (2008). Toward causal inference with interference. J. Amer. Statist. Assoc. 103 832–842.
  • Jemiai, Y., Rotnitzky, A., Shepherd, B. E. and Gilbert, P. B. (2007). Semiparametric estimation of treatment effects given base-line covariates on an outcome measured after a post-randomization event occurs. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 879–901.
  • Kempton, R. A. (1997). Interference between plots. In Statistical Methods for Plant Variety Evaluation (R. A. Kempton and P. N. Fox, eds.) 101–116. Chapman & Hall, London.
  • Liu, L. and Hudgens, M. G. (2013). On inverse probability weighted estimators in the presence of interference. Technical report.
  • Liu, L. and Hudgens, M. G. (2014). Large sample randomization inference of causal effects in the presence of interference. J. Amer. Statist. Assoc. 109 288–301.
  • Luo, X., Small, D. S., Li, C.-S. R. and Rosenbaum, P. R. (2012). Inference with interference between units in an fMRI experiment of motor inhibition. J. Amer. Statist. Assoc. 107 530–541.
  • Manski, C. F. (2013). Identification of treatment response with social interactions. Econom. J. 16 S1–S23.
  • Ogburn, E. L. and VanderWeele, T. J. (2012). Causal diagrams for interference and contagion. Preprint.
  • Perez-Heydrich, C., Hudgens, M. G., Halloran, M. E., Clemens, J. D., Ali, M. and Emch, M. E. (2013). Assessing effects of cholera vaccination in the presence of interference. Preprint.
  • Préziosi, M.-P. and Halloran, M. E. (2003). Effects of pertussis vaccination on transmission: Vaccine efficacy for infectiousness. Vaccine 21 1853–1861.
  • Ramsay, M. E., Andrews, N. J., Trotter, C. L., Kaczmarski, E. B. and Miller, E. (2003). Herd immunity from meningococcal serogroup C conjugate vaccination in England: Database analysis. Br. Med. J. 326 365–366.
  • Robins, J. M., Rotnitzky, A. and Scharfstein, D. O. (2000). Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models. In Statistical Models in Epidemiology, the Environment, and Clinical Trials (Minneapolis, MN, 1997) (E. Halloran and D. Berry, eds.). IMA Vol. Math. Appl. 116 1–94. Springer, New York.
  • Rosenbaum, P. R. (2007). Interference between units in randomized experiments. J. Amer. Statist. Assoc. 102 191–200.
  • Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. Ann. Statist. 6 34–58.
  • Rubin, D. B. (1986). Which if’s have causal answers? J. Amer. Statist. Assoc. 81 961–962.
  • Rubin, D. B. (1990). Comment on J. Neyman and causal inference in experiments and observational studies: “On the application of probability theory to agricultural experiments. Essay on principles. Section 9” [Ann. Agric. Sci. 10 (1923), 1–51]. Statist. Sci. 5 472–480.
  • Scharfstein, D. O., Rotnitzky, A. and Robins, J. M. (1999). Adjusting for nonignorable drop-out using semiparametric nonresponse models. J. Amer. Statist. Assoc. 94 1096–1146.
  • Sobel, M. E. (2006). What do randomized studies of housing mobility demonstrate?: Causal inference in the face of interference. J. Amer. Statist. Assoc. 101 1398–1407.
  • Struchiner, C. J., Halloran, M. E., Robins, J. M. and Spielman, A. (1990). The behavior of common measures of association used to assess a vaccination program under complex disease transmission patterns—a computer simulation study of malaria vaccines. Int. J. Epidemiol. 19 187–196.
  • Tchetgen Tchetgen, E. J. (2011). On causal mediation analysis with a survival outcome. Int. J. Biostat. 7 Art. 33, 38.
  • Tchetgen Tchetgen, E. J. and Shpitser, I. (2012). Semiparametric theory for causal mediation analysis: Efficiency bounds, multiple robustness and sensitivity analysis. Ann. Statist. 40 1816–1845.
  • Tchetgen Tchetgen, E. J. and VanderWeele, T. J. (2012). On causal inference in the presence of interference. Stat. Methods Med. Res. 21 55–75.
  • VanderWeele, T. J. (2010a). Bias formulas for sensitivity analysis for direct and indirect effects. Epidemiology 21 540–551.
  • VanderWeele, T. J. (2010b). Direct and indirect effects for neighborhood-based clustered and longitudinal data. Sociol. Methods Res. 38 515–544.
  • VanderWeele, T. J. (2011). Sensitivity analysis for contagion effects in social networks. Sociol. Methods Res. 40 240–255.
  • VanderWeele, T. J. and Arah, O. A. (2011). Bias formulas for sensitivity analysis of unmeasured confounding for general outcomes, treatments, and confounders. Epidemiology 22 42–52.
  • VanderWeele, T. J. and Tchetgen Tchetgen, E. J. (2011a). Effect partitioning under interference in two-stage randomized vaccine trials. Statist. Probab. Lett. 81 861–869.
  • VanderWeele, T. J. and Tchetgen Tchetgen, E. J. (2011b). Bounding the infectiousness effect in vaccine trials. Epidemiology 22 686–693.
  • VanderWeele, T. J., Vandenbroucke, J. P., Tchetgen Tchetgen, E. J. and Robins, J. M. (2012). A mapping between interactions and interference: Implications for vaccine trials. Epidemiology 23 285–292.
  • VanderWeele, T. J., Hong, G., Jones, S. M. and Brown, J. L. (2013). Mediation and spillover effects in group-randomized trials: A case study of the 4Rs educational intervention. J. Amer. Statist. Assoc. 108 469–482.
  • Van der Laan, M. J. (2012). Causal inference for networks. Working Paper 300, Univ. California, Berkeley, Berkeley, CA.