The Annals of Statistics

Statistical modeling of causal effects in continuous time

Judith J. Lok

Full-text: Open access

Abstract

This article studies the estimation of the causal effect of a time-varying treatment on time-to-an-event or on some other continuously distributed outcome. The paper applies to the situation where treatment is repeatedly adapted to time-dependent patient characteristics. The treatment effect cannot be estimated by simply conditioning on these time-dependent patient characteristics, as they may themselves be indications of the treatment effect. This time-dependent confounding is common in observational studies. Robins [(1992) Biometrika 79 321–334, (1998b) Encyclopedia of Biostatistics 6 4372–4389] has proposed the so-called structural nested models to estimate treatment effects in the presence of time-dependent confounding. In this article we provide a conceptual framework and formalization for structural nested models in continuous time. We show that the resulting estimators are consistent and asymptotically normal. Moreover, as conjectured in Robins [(1998b) Encyclopedia of Biostatistics 6 4372–4389], a test for whether treatment affects the outcome of interest can be performed without specifying a model for treatment effect. We illustrate the ideas in this article with an example.

Article information

Source
Ann. Statist., Volume 36, Number 3 (2008), 1464-1507.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1211819571

Digital Object Identifier
doi:10.1214/009053607000000820

Mathematical Reviews number (MathSciNet)
MR2418664

Zentralblatt MATH identifier
1360.62511

Subjects
Primary: 62P10: Applications to biology and medical sciences
Secondary: 62M99: None of the above, but in this section

Keywords
Causality in continuous time counterfactuals longitudinal data observational studies

Citation

Lok, Judith J. Statistical modeling of causal effects in continuous time. Ann. Statist. 36 (2008), no. 3, 1464--1507. doi:10.1214/009053607000000820. https://projecteuclid.org/euclid.aos/1211819571


Export citation

References

  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Billingsley, P. (1986). Probability and Measure. Wiley, New York.
  • Collett, D. (1994). Modelling Survival Data in Medical Research. Chapman and Hall, London.
  • Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman and Hall, London.
  • Cox, D. R. and Oakes, D. (1984). Analysis of Survival Data. Chapman and Hall, London.
  • Duistermaat, J. J. and Eckhaus, W. (1995). Analyse van Gewone Differentiaalvergelijkingen. Epsilon, Utrecht.
  • Gill, R. D. and Robins, J. M. (2001). Causal inference for complex longitudinal data: The continuous case. Ann. Statist. 29 1785–1811.
  • Hernán, M. A., Cole, S. R., Margolick, J., Cohen, M. and Robins J. M. (2005). Structural accelerated failure time models for survival analysis in studies with time-varying treatments. Pharmacoepidemiology and Drug Safety 14 477–491.
  • Keiding, N. (1999). Event history analysis and inference from observational epidemiology. Statistics in Medicine 18 2353–2363.
  • Keiding, N., Filiberti, M., Esbjerg, S., Robins, J. M. and Jacobsen, N. (1999). The graft versus leukemia effect after bone marrow transplantation: A case study using structural nested failure time models. Biometrics 55 23–28.
  • Lok, J. J. (2001). Statistical modelling of causal effects in time. Ph.D. thesis, Dept. Mathematics, Free Univ. Amsterdam. Available at http://www.math.vu.nl/research/theses/pdf/lok.pdf.
  • Lok, J. J. (2004). Mimicking counterfactual outcomes for estimation of causal effects. Available at http://arXiv.org/abs/math.ST/0409045.
  • Lok, J. J. (2007). Structural nested models and standard software: A mathematical foundation through partial likelihood. Scand. J. Statist. 34 186–206.
  • Lok, J. J., Gill, R. D., van der Vaart, A. W. and Robins, J. M. (2004). Estimating the causal effect of a time-varying treatment on time-to-event using structural nested failure time models. Statist. Neerlandica 58 271–295.
  • Mark, S. D. and Robins, J. M. (1993). Estimating the causal effect of smoking cessation in the presence of confounding factors using a rank preserving structural failure time model. Statistics in Medicine 12 1605–1628.
  • Pearl, J. (2000). Causality. Models, Reasoning, and Inference. Cambridge Univ. Press.
  • Robins, J. M. (1989). The analysis of randomized and non-randomized AIDS treatment trials using a new approach to causal inference in longitudinal studies. In Health Service Research Methodology: A Focus on AIDS (L. Sechrest, H. Freeman and A. Bailey, eds.) 113–159. NCHSR, U.S. Public Health Service, Washington, DC.
  • Robins, J. M. (1992). Estimation of the time-dependent accelerated failure time model in the presence of confounding factors. Biometrika 79 321–334.
  • Robins, J. M. (1993). Analytic methods for HIV treatment and cofactor effects. In Methodological Issues of AIDS Behavioral Research (D. G. Ostrow and R. Kessler, eds.) 213–287. Plenum Press, New York.
  • Robins, J. M. (1997). Causal inference from complex longitudinal data. Latent Variable Modeling and Applications to Causality. Lecture Notes in Statist. 120 69–117. Springer, New York.
  • Robins, J. M. (1998a). Correcting for non-compliance in equivalence trials. Statistics in Medicine 17 269–302.
  • Robins, J. M. (1998b). Structural nested failure time models. In Survival Analysis (P. Armitage and T. Colton, eds.). Encyclopedia of Biostatistics 6 4372–4389. Wiley, Chichester.
  • Robins, J. M. (2000). Robust estimation in sequentially ignorable missing data and causal inference models. In Proceedings of the American Statistical Association Section on Bayesian Statistical Science 1999 6–10.
  • Robins, J. M., Blevins, D., Ritter, G. and Wulfsohn, M. (1992). G-estimation of the effect of prophylaxis therapy for pneumocystis carinii pneumonia on the survival of AIDS patients. Epidemiology 3 319–336.
  • 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 (M. E. Halloran and D. Berry, eds.) 1–94. Springer, New York.
  • Robins, J. M. and Wasserman, L. (1997). Estimation of effects of sequential treatments by reparameterizing directed acyclic graphs. In Proceedings of the Thirteenth Conference on Uncertainty in Artificial Intelligence (D. Geiger and P. Shenoy, eds.) 409–420. Morgan Kaufmann, San Francisco, CA.
  • Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales. Wiley, New York.
  • Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika 70 41–55.
  • Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.
  • Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • Witteman, J. C. M., D’Agostino, R. B., Stijnen, T., Kannel, W. B., Cobb, J. C., de Ridder, M. A. J., Hofman, A. and Robins, J. M. (1998). G-estimation of causal effects: Isolated systolic hypertension and cardiovascular death in the Framingham Study. American J. Epidemiology 148 390–401.