Bernoulli

A martingale approach to continuous-time marginal structural models

Kjetil Røysland

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Abstract

Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. The key point is that this can be understood in terms of Girsanov’s change of measure. This offers a mathematical interpretation of marginal structural models that has not been available before. We consider both a model of an observational study and a model of a hypothetical randomized trial. These models correspond to different martingale measures – the observational measure and the randomized trial measure – on some underlying space. We describe situations where the randomized trial measure is absolutely continuous with respect to the observational measure. The resulting continuous-time likelihood ratio process with respect to these two probability measures corresponds to the weights in discrete-time marginal structural models. In order to do inference for the hypothetical randomized trial, we can simulate samples using observational data weighted by this likelihood ratio.

Article information

Source
Bernoulli Volume 17, Number 3 (2011), 895-915.

Dates
First available in Project Euclid: 7 July 2011

Permanent link to this document
http://projecteuclid.org/euclid.bj/1310042849

Digital Object Identifier
doi:10.3150/10-BEJ303

Mathematical Reviews number (MathSciNet)
MR2817610

Citation

Røysland, Kjetil. A martingale approach to continuous-time marginal structural models. Bernoulli 17 (2011), no. 3, 895--915. doi:10.3150/10-BEJ303. http://projecteuclid.org/euclid.bj/1310042849.


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References

  • [1] Aalen, O., Borgan, Ø. and Gjessing, H. (2008). Survival and Event History Analysis: A Process Point of View. New York: Springer.
  • [2] Andersen, P.K., Borgan, Ø., Gill, R.D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. New York: Springer.
  • [3] Aalen, O.O., Borgan, Ø., Keiding, N. and Thormann, J. (1980). Interaction between life history events. Nonparametric analysis for prospective and retrospective data in the presence of censoring. Scand. J. Statist. 7 161–171.
  • [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. New York: Wiley.
  • [5] Brémaud, P. (1981). Point Processes and Queues. New York: Springer.
  • [6] Didelez, V. (2008). Graphical models for marked point processes based on local independence. J. Roy. Statist. Soc. Ser. B 70 245–264.
  • [7] Florens, J.-P. and Fougere, D. (1996). Noncausality in continuous time. Econometrica 64 1195–1212.
  • [8] Hernán, M.A., Brumback, B. and Robins, J.M. (2000). Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology 11 561–570.
  • [9] Hernán, M.A., Hernández-Díaz, S. and Robins, J.M. (2004). A structural approach to selection bias. Epidemiology 15 615.
  • [10] Jacod, J. (1974). Multivariate point processes: Predictable projection, Radon–Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253.
  • [11] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Berlin: Springer.
  • [12] Lépingle, D. and Mémin, J. (1978). Sur l’intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. Verw. Gebiete 42 175–203.
  • [13] Lok, J.J. (2008). Statistical modeling of causal effects in continuous time. Ann. Statist. 36 1464–1507.
  • [14] Protter, P.E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Berlin: Springer.
  • [15] Robins, J.M., Hernán, M.A. and Brumback, B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology 11 550–560.
  • [16] Robins, J. (1992). Estimation of the time-dependent accelerated failure time model in the presence of confounding factors. Biometrika 79 321–334.
  • [17] Robins, J.M. (1998). Structural nested failure time models. In The Encyclopedia of Biostatistics 4372–4389. Chichester: Wiley.
  • [18] Schweder, T. (1970). Composable Markov processes. J. Appl. Probab. 7 400–410.
  • [19] Sterne, J.A., Hernán, M.A., Ledergerber, B., Tilling, K., Weber, R., Sendi, P., Rickenbach, M., Robins, J.M. and Egger, M. (2005). Long-term effectiveness of potent antiretroviral therapy in preventing AIDS and death: A prospective cohort study. Lancet 366 378–384.