Skip to main content


Log in

Estimating stage occupation probabilities in non-Markov models

  • OriginalPaper
  • Published:
Lifetime Data Analysis Aims and scope Submit manuscript


We study non-Markov multistage models under dependent censoring regarding estimation of stage occupation probabilities. The individual transition and censoring mechanisms are linked together through covariate processes that affect both the transition intensities and the censoring hazard for the corresponding subjects. In order to adjust for the dependent censoring, an additive hazard regression model is applied to the censoring times, and all observed counting and “at risk” processes are subsequently given an inverse probability of censoring weighted form. We examine the bias of the Datta–Satten and Aalen–Johansen estimators of stage occupation probability, and also consider the variability of these estimators by studying their estimated standard errors and mean squared errors. Results from different simulation studies of frailty models indicate that the Datta–Satten estimator is approximately unbiased, whereas the Aalen–Johansen estimator either under- or overestimates the stage occupation probability due to the dependent nature of the censoring process. However, in our simulations, the mean squared error of the latter estimator tends to be slightly smaller than that of the former estimator. Studies on development of nephropathy among diabetics and on blood platelet recovery among bone marrow transplant patients are used as demonstrations on how the two estimation methods work in practice. Our analyses show that the Datta–Satten estimator performs well in estimating stage occupation probability, but that the censoring mechanism has to be quite selective before a deviation from the Aalen-Johansen estimator is of practical importance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Aalen OO (1988) Dynamic description of a Markov chain with random time scale. Math Scientist 13:90–103

    MATH  MathSciNet  Google Scholar 

  • Aalen OO (1989) A linear regression model for the analysis of life times. Stat Med 8:907–925

    Google Scholar 

  • Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer-Verlag, New York

    MATH  Google Scholar 

  • Borgan Ø (1998) Aalen–Johansen estimator. In Encyclopedia of biostatistics. Wiley, Chichester, pp 5–10

  • Datta S, Satten GA (2002) Estimation of integrated transition hazards and stage occupation probabilities for non-Markov systems under dependent censoring. Biometrics 58:792–802

    Article  MathSciNet  Google Scholar 

  • Efron B, Tibshirani R (1993) An introduction to the bootstrap. Chapman & Hall, New York

    MATH  Google Scholar 

  • Fosen J, Borgan Ø, Weedon-Fekjaer H, Aalen OO (2006) Dynamic analysis of recurrent event data using the additive hazard model. Biometrical J 48:381–398

    Article  MathSciNet  Google Scholar 

  • Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning: data mining, inference, and prediction. Springer-Verlag, New York

    MATH  Google Scholar 

  • Hougaard P (2000) Analysis of multivariate survival data. Springer-Verlag, New York

    MATH  Google Scholar 

  • Klein JP, Moeschberger ML (2003) Survival analysis: techniques for censored and truncated data, 2nd edn. Springer-Verlag, New York

    MATH  Google Scholar 

  • Robins JM, Finkelstein DM (2000) Correcting for noncompliance and dependent censoring in an AIDS clinical trial with inverse probability of censoring weighted (IPCW) log-rank tests. Biometrics 56:779–788

    Article  MATH  Google Scholar 

  • Satten GA, Datta S, Robins JM (2001) Estimating the marginal survival function in the presence of time dependent covariates. Stat Probab Lett 54:397–403

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nina Gunnes.

Additional information

N. Gunnes—Supported by a grant from the Norwegian Cancer Society.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gunnes, N., Borgan, Ø. & Aalen, O.O. Estimating stage occupation probabilities in non-Markov models. Lifetime Data Anal 13, 211–240 (2007).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: