Dependent Competing Risks with Time-Dependent Covariates

  • Eric V. Slud
  • Leonid Kopylev


This paper discusses two mechanisms with time-dependent covariates for dependence between competing-risk latent failure times under which the marginal survival functions are identifiable. The first is the finite-state nonhomogeneous Markov chain with two absorbing failure states, A and B, where “marginal survival function for A” is the probability that failure due to A does not occur before t when transitions to B are suppressed. Even in highly stratified models, the nonparametric survival estimator due to Aalen & Johansen (1978) is readily computable when transitions between each pair (i,j) states can occur only in one direction.


Markov Chain Model Chronological Time Crude Survival Nonparametric Maximum Likelihood Estimator Compete Risk Data 
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  1. Aalen, O. and Johansen, S. (1978), “An Empirical Transition Matrix for Nonhomogeneous Markov Chains Based on Censored Observations,” Scandinavian Journal of Statistics, 5, 141–150.Google Scholar
  2. Andersen, P. Borgan, O., Gill, R. and Keiding, N. (1993) Statistical Models Based on Counting Processes New York: Springer-Verlag.Google Scholar
  3. Cox, D.R. (1972), “Regression Models and Life Tables (with discussion),” Journal of the Royal Statistical Society, Series B 34, 187–220.Google Scholar
  4. Dambrosia, J.M. (1993), Personal communication.Google Scholar
  5. Davis, P.H., DambrosiaJ.M. et al. (1987), “Risk Factors for Ischemic Stroke: A Prospective study in Rochester, Minnesota,” Annals of Neurology, 22, 319–327.CrossRefGoogle Scholar
  6. Fleming, T. (1978), “Nonparametric Estimation for Nonhomogeneous Markov Processes in the Problem of Competing Risks,” Annals of Statistics, 6, 1057–1070.MathSciNetMATHCrossRefGoogle Scholar
  7. Keiding, N. (1991), “Age-specific Incidence and Prevalence: a Statistical Perspective,” Jour-nal of the Royal Statistical Society, Series A 154, 371–412.Google Scholar
  8. Kopylev, L. and Slud, E. (1994), “Aalen-Johansen Estimators for Marginal Survival Functions in Highly Stratified Competing Risk Data with Time-dependent Covariates,” Technical report in preparation.Google Scholar
  9. Murray, S. and Tsiatis, A. (1993), “A Nonparametric Approach to Incorporating Prognostic Longitudinal Covariate Information in Survival Estimation,” ENAR Invited Talk, April 1994.Google Scholar
  10. Prentice, R., Kalbfleisch, J., Peterson, A., Flournoy, N., Farewell, V. and Breslow, N. (1978), “The Analysis of Failure Times in the Presence of Competing Risks,” Biometrics, 34, 541–554.MATHCrossRefGoogle Scholar
  11. Schatzkin, A. and Slud, E. (1989), “ Competing Risks Bias Arising from an Omitted Risk Factor,” American Journal of Epidemiology, 129, 850–856.Google Scholar
  12. Slud, E. and Rubinstein, L. (1983), “ Dependent Competing Risks and Summary Survival Curves,” Biometrika, 78, 643–649.MathSciNetCrossRefGoogle Scholar
  13. Tsiatis, A. (1975), “ A Nonidentifiability Aspect of Competing Risks,” Proceedings of the National Academy of Sciences of the USA, 72, 20–22.MathSciNetMATHCrossRefGoogle Scholar
  14. Vardi, Y. (1982), “Nonparametric Estimation in the Presence of Length Bias,” Annals of Statistics, 10, 616–620.MathSciNetMATHCrossRefGoogle Scholar
  15. Wang, M. C. (1991), “Nonparametric Estimation from Cross-sectional Survey Data,” Journal of the American Statistical Association, 86, 130–143.MathSciNetMATHCrossRefGoogle Scholar
  16. Weldon, K. and Potvin, D. (1991), “Nonparametric Recovery of Duration Distributions from Cross-sectional Sample Surveys,” Communications in Statistics Theory and Methods, 20, 3943–3973.MathSciNetMATHCrossRefGoogle Scholar
  17. Yang, G. and He, S. (1994), “ Estimating Lifetime Distribution Under Different Sampling Plans,” Statistical Decision Theory and Related Topics, New York: Springer-Verlag, 73–87.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Eric V. Slud
    • 1
  • Leonid Kopylev
    • 1
  1. 1.Department of MathematicsUniversity of Maryland at College ParkCollege ParkUSA

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