Bayesian Analysis

Approximate Bayesian Inference for Doubly Robust Estimation

Daniel J. Graham, Emma J. McCoy, and David A. Stephens

Full-text: Open access

Abstract

Doubly robust estimators are typically constructed by combining outcome regression and propensity score models to satisfy moment restrictions that ensure consistent estimation of causal quantities provided at least one of the component models is correctly specified. Standard Bayesian methods are difficult to apply because restricted moment models do not imply fully specified likelihood functions. This paper proposes a Bayesian bootstrap approach to derive approximate posterior predictive distributions that are doubly robust for estimation of causal quantities. Simulations show that the approach performs well under various sources of misspecification of the outcome regression or propensity score models. The estimator is applied in a case study of the effect of area deprivation on the incidence of child pedestrian casualties in British cities.

Article information

Source
Bayesian Anal., Volume 11, Number 1 (2016), 47-69.

Dates
First available in Project Euclid: 4 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.ba/1423083639

Digital Object Identifier
doi:10.1214/14-BA928

Mathematical Reviews number (MathSciNet)
MR3447091

Zentralblatt MATH identifier
1357.62186

Keywords
approximate bayes doubly robust propensity score treatment effect

Citation

Graham, Daniel J.; McCoy, Emma J.; Stephens, David A. Approximate Bayesian Inference for Doubly Robust Estimation. Bayesian Anal. 11 (2016), no. 1, 47--69. doi:10.1214/14-BA928. https://projecteuclid.org/euclid.ba/1423083639


Export citation

References

  • Bang, H. and Robins, J. M. (2005). “Doubly robust estimation in missing data and causal inference models.” Biometrics, 61: 962–972.
  • Chamberlain, G. and Imbens, G. W. (2003). “Nonparametric Applications of Bayesian Inference.” Journal of Business & Economic Statistics, 21(1): 12–18.
  • Christie, N. (1995). “Social, economic and environmental factors in child pedestrian accidents: a research overview.” Technical Report 116, Transport Research Laboratory, Berkshire.
  • Flores, C. A., Flores-Lagunes, A., Gonzalez, A., and Neumann, T. C. (2012). “Estimating the Effects of Length of Exposure to Instruction in a Training Program: The Case of Job Corps.” The Review of Economics and Statistics, 94(1): 153–171.
  • Graham, D. J., McCoy, E. J., and Stephens, D. A. (2012). “Semiparametric double-robust estimation for continuous treatment effects.” Paper Presented at the 2012 Joint Statistical Meetings, San Diego.
  • — (2013). “Quantifying the effect of area deprivation on child pedestrian casualties using longitudinal mixed models to adjust for confounding, interference, and spatial dependence.” Journal of the Royal Statistical Society: Series A, 176(4): 931–950.
  • Graham, D. J. and Stephens, D. A. (2008). “Decomposing the impact of deprivation on child pedestrian casualties in England.” Accident Analysis & Prevention, 40: 1351–1364.
  • Gustafson, P. (2012). “Double-robust estimators: slightly more Bayesian than meets the eye.” The International Journal of Biostatistics, 8(2): 1–15.
  • Hirano, K. and Imbens, G. W. (2004). “The propensity score with continuous treatments.” In Gelman, A. and Meng, X. (eds.), Applied Bayesian modeling and causal inference from incomplete data perspectives, 73–84. New York: Wiley.
  • Horvitz, D. G. and Thompson, D. J. (1952). “A generalization of sampling without replacement from a finite universe.” Journal of the American Statistical Association, 47: 663–685.
  • Imbens, G. W. (1999). “The role of the propensity score in estimating dose–response functions.” NBER Working Paper, 237.
  • — (2000). “The role of the propensity score in estimating dose–response functions.” Biometrika, 87(3): 706–710.
  • Kang, J. D. Y. and Schafer, J. L. (2007). “Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data.” Statistical Science, 22(4): 523–539.
  • Kass, R. E. and Steffey, D. (1989). “Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models).” Journal of the American Statistical Association, 84(407): 717–726.
  • Lunceford, J. K. and Davidian, M. (2004). “Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study.” Statistics in Medicine, 23: 2937–2960.
  • McCandless, L. C., Richardson, S., and Best, N. (2012). “Adjustment for Missing Confounders Using External Validation Data and Propensity Scores.” Journal of the American Statistical Association, 107(497): 40–51.
  • Newton, M. A. and Raftery, A. E. (1994). “Approximate Bayesian Inference with the Weighted Likelihood Bootstrap (with discussion).” Journal of the Royal Statistical Society. Series B (Methodological), 56(1): pp. 3–48.
  • Pearl, J. (2009). Causality – models, reasoning and inference. Cambridge: Cambridge University Press, 2nd edition.
  • — (2010). “On a Class of Bias-Amplifying Variables that Endanger Effect Estimates.” In Proceeding of the 25th Conference on Uncertainty in Artificial Intelligence (UAI 2010), 425–432. Corvallis: Association for Uncertainty in Artificial Intelligence.
  • Robins, J. M. and Rotnitzky, A. (2001). “Comment on “Inference for semiparametric models: some questions and an answer”.” Statistica Sinica, 11: 920–936.
  • Rubin, D. B. (1981). “The Bayesian Bootstrap.” The Annals of Statistics, 9(1): 130–134.
  • Scharfstein, D. O., Rotnitzky, A., and Robins, J. M. (1999). “Adjusting for Nonignorable Drop-Out Using Semiparametric Nonresponse Models.” Journal of the American Statistical Association, 94(448): 1096–1120 (with rejoinder 1135–1146).
  • Tsiatis, A. A. (2006). Semiparametric theory and missing data. Berlin: Springer.
  • Tsiatis, A. A. and Davidian, M. (2007). “Comment: Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data.” Statistical Science, 22(4): 569–573.
  • van der Laan, M. and Robins, J. M. (2003). Unified methods for censored longitudinal data and causality. Berlin: Springer.