Statistical Science

Causal Inference Through Potential Outcomes and Principal Stratification: Application to Studies with “Censoring” Due to Death

Donald B. Rubin

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Causal inference is best understood using potential outcomes. This use is particularly important in more complex settings, that is, observational studies or randomized experiments with complications such as noncompliance. The topic of this lecture, the issue of estimating the causal effect of a treatment on a primary outcome that is “censored” by death, is another such complication. For example, suppose that we wish to estimate the effect of a new drug on Quality of Life (QOL) in a randomized experiment, where some of the patients die before the time designated for their QOL to be assessed. Another example with the same structure occurs with the evaluation of an educational program designed to increase final test scores, which are not defined for those who drop out of school before taking the test. A further application is to studies of the effect of job-training programs on wages, where wages are only defined for those who are employed. The analysis of examples like these is greatly clarified using potential outcomes to define causal effects, followed by principal stratification on the intermediated outcomes (e.g., survival).

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Statist. Sci., Volume 21, Number 3 (2006), 299-309.

First available in Project Euclid: 20 December 2006

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Missing data quality of life Rubin causal model truncation due to death


Rubin, Donald B. Causal Inference Through Potential Outcomes and Principal Stratification: Application to Studies with “Censoring” Due to Death. Statist. Sci. 21 (2006), no. 3, 299--309. doi:10.1214/088342306000000114.

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  • Aitkin, M. and Rubin, D. B. (1985). Estimation and hypothesis testing in finite mixture models. J. Roy. Statist. Soc. Ser. B 47 67--75.
  • 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--472.
  • Dempster, A. P., Laird, N. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1--38.
  • Frangakis, C. E. and Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics 58 21--29.
  • Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004). Bayesian Data Analysis, 2nd ed. Chapman and Hall/CRC, New York.
  • Gelman, A. E., Meng, X.-L. and Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies (with discussion). Statist. Sinica 6 733--807.
  • Holland, P. W. (1986). Statistics and causal inference (with discussion). J. Amer. Statist. Assoc. 81 945--970.
  • Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer, New York.
  • Rosenbaum, P. R. and Rubin, D. B. (1983). Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. J. Roy. Statist. Soc. Ser. B 45 212--218.
  • Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. J. Educational Psychology 66 688--701.
  • Rubin, D. B. (1975). Bayesian inference for causality: The importance of randomization. In Proc. Social Statistics Section 233--239. Amer. Statist. Assoc., Alexandria, VA.
  • Rubin, D. B. (1976). Inference and missing data (with discussion). Biometrika 63 581--592.
  • Rubin, D. B. (1977). Assignment to treatment group on the basis of a covariate. J. Educational Statistics 2 1--26.
  • Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. Ann. Statist. 6 34--58.
  • Rubin, D. B. (1979). Discussion of ``Conditional independence in statistical theory,'' by A. P. Dawid. J. Roy. Statist. Soc. Ser. B 41 27--28.
  • Rubin, D. B. (1980). Comment on ``Randomization analysis of experimental data: The Fisher randomization test,'' by D. Basu. J. Amer. Statist. Assoc. 75 591--593.
  • Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist. 12 1151--1172.
  • Rubin, D. B. (1990). Neyman (1923) and causal inference in experiments and observational studies. Statist. Sci. 5 472--480.
  • Rubin, D. B. (1998). More powerful randomization-based $p$-values in double-blind trials with noncompliance. Statistics in Medicine 17 371--385.
  • Rubin, D. B. (2000). Comment on ``Causal inference without counterfactuals,'' by A. P. Dawid. J. Amer. Statist. Assoc. 95 435--438.
  • Rubin, D. B. (2005). Causal inference using potential outcomes: Design, modeling, decisions. 2004 Fisher Lecture. J. Amer. Statist. Assoc. 100 322--331.
  • Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985). Statistical Analysis of Finite Mixture Distributions. Wiley, New York.
  • Zhang, J. L. (2002). Bayesian estimation of causal effects in the presence of truncation by death. Ph.D. dissertation, Harvard Univ.
  • Zhang, J. L. and Rubin, D. B. (2003). Estimation of causal effects via principal stratification when some outcomes are truncated by ``death.'' J. Educational and Behavioral Statistics 28 353--368.
  • Zhang, J. L., Rubin, D. B. and Mealli, F. (2005). Using the EM algorithm to estimate the effects of job training programs on wages. In Proc. 55th Session of the International Statistical Institute.
  • Zhang, J. L., Rubin, D. B. and Mealli, F. (2006). Evaluating the effects of training programs on wages through principal stratification. In Modelling and Evaluating Treatment Effects in Econometrics (D. Millimet, J. Smith and E. Vytlacil, eds.). Elsevier, Amsterdam. To appear.