Skip to main content

State Occupation Probabilities in Non-Markov Models

Abstract

The consistency of the Aalen—Johansen-derived estimator of state occupation probabilities in non-Markov multi-state settings is studied and established via a new route. This new route is based on interval functions and relies on a close connection between additive and multiplicative transforms of interval functions, which is established. Under certain assumptions, the consistency follows from explicit expressions of the additive and multiplicative transforms related to the transition probabilities as interval functions, which are obtained, in combination with certain censoring and positivity assumptions

This is a preview of subscription content, access via your institution.

References

  1. O. O. Aalen and S. Johansen, “An Empirical Transition Matrix for Non-Homogeneous Markov Chains Based on Censored Observations”, Scand. J. Statist. 5 (3), 141–150 (1978).

    MathSciNet  MATH  Google Scholar 

  2. O. O. Aalen, Ø. Borgan, and H. Fekjær, “Covariate Adjustment of Event Histories Estimated from Markov Chains: The Additive Approach”, Biometrics 57 (4), 993–1001 (2001).

    MathSciNet  Article  Google Scholar 

  3. P. K. Andersen, Ø. Borgan, R. D. Gill, and N. Keiding, Statistical Models Based on Counting Processes (Springer-Verlag, New York, 1993).

    Book  Google Scholar 

  4. S. Datta and G. A. Satten, “Validity of the Aalen—Johansen Estimators of Stage Occupation Probabilities and Nelson—Aalen Estimators of Integrated Transition Hazards for non-Markov Models”, Statist. Probab. Lett. 55 (4), 403–411 (2001).

    MathSciNet  Article  Google Scholar 

  5. R. M. Dudley and R. Norvaiša, Concrete Functional Calculus (Springer, New York, 2011).

    Book  Google Scholar 

  6. R. D. Gill and S. Johansen, “A Survey of Product-Integration with a View toward Application in Survival Analysis”, Ann. Statist. 18 (4), 1501–1555 (1990).

    MathSciNet  Article  Google Scholar 

  7. D. V. Glidden, “Robust Inference for Event Probabilities with Non-Markov Event Data”, Biometrics, 58 (2), 361–368 (2002).

    MathSciNet  Article  Google Scholar 

  8. P. Jepsen, H. Vilstrup, and P. K. Andersen, “The Clinical Course of Cirrhosis: the Importance of Multistate Models and Competing Risks Analysis”, Hepatology 62 (1), 292–302 (2015).

    Article  Google Scholar 

  9. N. Keiding, J. P. Klein, and M. M. Horowitz, “Multi-State Models and Outcome Prediction in Bone Marrow Transplantation”, Statistics in Medicine 20 (12), 1871–1885 (2001).

    Article  Google Scholar 

  10. M. Overgaard, “Counting Processes in p-Variation with Applications to Recurrent Events”, https://arxiv.org/abs/1903.04296, unpublished manuscript (2019).

  11. M. Overgaard and S. N. Hansen, “On the Assumption of Independent Right Censoring”, https://arxiv.org/abs/1905.02508, unpublished manuscript (2019).

  12. H. Putter, M. Fiocco, and R. B. Geskus, “Tutorial in Biostatistics: Competing Risks and Multi-State Models”, Statistics in Medicine 26 (11), 2389–2430 (2007).

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

The author would like to thank Erik Thorlund Parner and Jan Pedersen for discussions and comments on drafts of this paper.

This research is supported by the Novo Nordisk Foundation, grant NNF17OC0028276.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Overgaard.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Overgaard, M. State Occupation Probabilities in Non-Markov Models. Math. Meth. Stat. 28, 279–290 (2019). https://doi.org/10.3103/S1066530719040033

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530719040033

Keywords

  • Aalen—Johansen estimator
  • interval function
  • additive transform
  • multiplicative transform
  • product integral

AMS 2010 Subject Classification

  • 62N02
  • 62N05
  • 62F12
  • 62F40