Statistical Methods for Dependent Competing Risks

  • M. L. Moeschberger
  • John P. Klein


Many biological and medical studies have as a response of interest the time to occurrence of some event, X, such as the occurrence of cessation of smoking, conception, a particular symptom or disease, remission, relapse, death due to some specific disease, or simply death. Often it is impossible to measure X due to the occurrence of some other competing event, usually termed a competing risk. This competing event may be the withdrawal of the subject from the study (for whatever reason), death from some cause other than the one of interest, or any eventuality that precludes the main event of interest from occurring. Usually the assumption is made that all such censoring times and lifetimes are independent. In this case one uses either the Kaplan-Meier estimator or the Nelson-Aalen estimator to estimate the survival function. However, if the competing risk or censoring times are not independent of X, then there is no generally acceptable way to estimate the survival function. There has been considerable work devoted to this problem of dependent competing risks scattered throughout the statistical literature in the past several years and this paper presents a survey of such work.


Survival Function Cumulative Incidence Function Accelerate Failure Time Model Compete Risk Analysis Marginal Survival 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aalen, O. (1978), “Nonparametric estimation of partial transition probabilities in multiple decrement models,” Annals of Statistics, 6, 534–545.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Andersen, P.K., Borgan. O, Gill, R.D. and Keiding, N. (1993), Statistical Models Based on Counting Processes. Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  3. Basu, A.P. and Klein, J.P. (1982), “Some recent results in competing risks theory,” Survival analysis. J.Crowley and R.A. Johnson (eds.) Hayward, California, 216–229.Google Scholar
  4. Benichou, J. and Gail, M.H. (1990), “Estimates of absolute cause-specific risk in cohort studies,” Biometrics, 46, 813–826.CrossRefGoogle Scholar
  5. Berman, S.M. (1963), “Notes on extreme values, competing risks, and semi-Markov processes,” Annals of Mathematical Statistics, 34, 1104–06.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Chiang, C.L. (1968), Introduction to Stochastic Processes in Biostatistics. Wiley, New York.zbMATHGoogle Scholar
  7. Chiang, C.L. (1970), “Competing risks and conditional probabilities,” Biometrics, 26, 767–776.MathSciNetCrossRefGoogle Scholar
  8. Clayton, D.G. (1978), “A model for association on bivariate life tables and its applications in epidemiological studies of familial tendency in chronic disease incidence,” Biometrika, 65, 141–151.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Cox, D.R. (1959), “The analysis of exponentially distributed lifetimes with two types of failure,” Journal of the Royal Statistical Society Series B, 21, 411–421.zbMATHGoogle Scholar
  10. Cox, D.R. (1962), Renewal Theory. Methuen, London.zbMATHGoogle Scholar
  11. Cox, D.R. (1972), “Regression models and life tables (with discussion),” Journal of the Royal Statistical Society Series B, 34, 187–202.zbMATHGoogle Scholar
  12. Cox, D.R. and Oakes, D. (1984), Analysis of survival data. Chapman and Hall, London.Google Scholar
  13. Crowder, M. (1991), “On the identifiability crisis in competing risks analysis,” Scandinavian Journal of Statistics, 18, 223–233.MathSciNetzbMATHGoogle Scholar
  14. David, H.A. and Moeschberger, M.L. (1978), The Theory of Competing Risks. Griffin, High Wycombe.zbMATHGoogle Scholar
  15. Digram, J.J., Weissfeld, L.A., and Anderson, S.J. (1994), “Methods for bounding the marginal survival distribution,” Technical Report-Methods #15, Department of Biostatistics, University of Pittsburgh, Pittsburgh, PA.Google Scholar
  16. Fisher, L. and Kanarek, P. (1974), “Presenting censored survival data when censoring and survival times may not be independent,” Proschan and Serfling (eds.). Reliability and Biometry: Statistical Analysis of Lifelength, SIAM, Philadelphia, PA, 303–326.Google Scholar
  17. Gail, M. (1975), “A review and critique of some models used in competing risk analyses,”Google Scholar
  18. Biometrics,31, 209–222.Google Scholar
  19. Gaynor, J.J, Feuer, E.J., Tan, C.C., Wu, D.H., Little, C.R., Straus, D.J., Clarkson, B.D., and Brennan, M.F. (1993), “On the use of cause-specific failure and conditional failure probabilities: examples from clinical oncology data,” Journal of the American Statistical Association, 88, 400–409.zbMATHCrossRefGoogle Scholar
  20. Gray, R. J. (1988), “A class of k-sample tests for comparing the cumulative incidence of a competing risk,” The Annals of Statistics, 16, No. 3, 1141–1154.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Heckman, J.J. and Honore, B.E. (1989), “The identifiability of the competing risks model,” Biometrika, 76 (2), 325–330.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Hoover, D.R. and Guess, F.M. (1990), “Response linked censoring: modeling and estimation,” Biometrika, 77, 893–896.CrossRefGoogle Scholar
  23. Hougaard, P. (1986), “A class of multivariate failure time distributions,” Biometrika, 73, 671–678.MathSciNetzbMATHGoogle Scholar
  24. Kalbfleisch, J.D. and Prentice, R.L. (1980), The Statistical Analysis of Failure Time Data. Wiley, New York.zbMATHGoogle Scholar
  25. Kaplan, E.L. and Meier, P. (1958), “Nonparametric estimation from incomplete observa-Google Scholar
  26. tions,“ Journal of the American Statistical Association,53, 457–481.Google Scholar
  27. Kimball, A.W. (1958), “Disease incidence estimation in populations subject to multipleGoogle Scholar
  28. causes of death,“ Bull. Int. Inst. Statist.,36, 103–204.Google Scholar
  29. Kimball, A.W. (1969), “Models for the estimation of competing risks from grouped data,” Biometics, 25, 329–337.CrossRefGoogle Scholar
  30. Kimball, A.W. (1971), “Model I vs. Model II in competing risk theory,” Biometrics, 27, 462–465.Google Scholar
  31. Klein, J.P. and Moeschberger, M.L. (1984), “Asymptotic bias of the product limit estimator under dependent competing risks,” Indian Journal of Productivity, Reliability and Quality Control, 9, 1–7.MathSciNetzbMATHGoogle Scholar
  32. Klein, J.P. and Moeschberger, M.L. (1986), “Consequences of assuming independence in a bivariate exponential series system,” IEEE Transactions on Reliability, R-35, 330–335.Google Scholar
  33. Klein, J.P. and Moeschberger, M.L. (1987), “Independent or dependent competing risks: Does it make a difference?” Communications in Statistics-Computation and Simulation, 16 (2), 507–533.MathSciNetzbMATHCrossRefGoogle Scholar
  34. Klein, J.P. and Moeschberger, M.L. (1988), “Bounds on net survival probabilities for dependent competing risks,” Biometrics, 44, 529–538.MathSciNetzbMATHCrossRefGoogle Scholar
  35. Klein, J., Moeschberger, M., Li, Y., Wang, S. (1992), “Estimating random effects in the Framingham heart study,” J.P. Klein and P.K. Goel (eds.), Survival Analysis: State of the Art. Kluwer Academic Publishers, Boston, 99–120.Google Scholar
  36. Korn, E.L. and Dorey, F.J. (1992), “Applications of crude incidence curves,” Statistics in Medicine, 11, 813–829.CrossRefGoogle Scholar
  37. Lagakos, S.W. (1979), “General right-censoring and its impact on the analysis of survival data,” Biometrics, 35, 139–156.zbMATHCrossRefGoogle Scholar
  38. Lagakos, S.W. and Williams, J.S. (1978), “Models for censored survival analysis: A cone class of variable-sum models,” Biometrika, 65, 181–189.zbMATHCrossRefGoogle Scholar
  39. Link, W.A. (1989), “A model for informative censoring,” Journal of the American Statistical Association, 84, 749–752.CrossRefGoogle Scholar
  40. Moeschberger, M.L. (1974), “Life tests under dependent competing causes of failure,” Technometrics, 16, 39–47.MathSciNetzbMATHCrossRefGoogle Scholar
  41. Moeschberger, M.L. and Klein, J.P. (1984), “Consequences of departures from independence in exponential series systems,” Technometrics, 26, 277–284.MathSciNetCrossRefGoogle Scholar
  42. Nelson, W. (1972), “Theory and applications of hazard plotting for censored failure data,” Technometrics, 14, 945–966.CrossRefGoogle Scholar
  43. Oakes, D. (1982), “A concordance test for independence in the presence of censoring,” Biometrics, 38, 451–455.zbMATHCrossRefGoogle Scholar
  44. Pepe, M.S. (1991), “Inference for events with dependent risks in multiple endpoint studies,” Journal of the American Statistical Association, 86, 770–778.MathSciNetzbMATHCrossRefGoogle Scholar
  45. Pepe, M.S. and Mori M. (1993), “Kaplan-Meier, marginal or conditional probability curves in summarizing competing risks failure time data?” Statistics in Medicine, 12, 737–751.CrossRefGoogle Scholar
  46. Peterson, A.V. (1976), “Bounds for a joint distribution function with fixed sub-distribution functions: Applications to competing risks,” Proceedings of the National Academy of Sciences, 73, 11–13.MathSciNetzbMATHCrossRefGoogle Scholar
  47. Prentice, R.L., Kalbfleisch, J.D., Peterson, A.V., Flournoy, N., Farewell, V.T., and Breslow, N.E. (1978), “The analysis of failure time data in the presence of competing risks,” Biometrics, 34, 541–554.zbMATHCrossRefGoogle Scholar
  48. Robins, J. M. (1993), “Analytic methods for estimating HIV-treatment and cofactor effects,” In Methodological Issues in AIDS Research, (eds. D. G. Ostrow and R. C. Kessler) New York, Plenum, 213–290.Google Scholar
  49. Robins, J. M. (1992), “Estimation of the time-dependent accelerated failure time model in the presence of confounding factors,” Biometrika, 79, 321–34.MathSciNetzbMATHCrossRefGoogle Scholar
  50. Slud, E. (1992), “Nonparametric identifiability of marginal survival distributions in the presence of dependent competing risks and a prognostic covariate,” JP Klein and PK noel (eds.), Survival Analysis: State of the Art. Kluwer Academic Publishers, Boston, 355–368.Google Scholar
  51. Slud, E.V. and Rubinstein L.V. (1983), “Dependent competing risks and summary survival curves,” Biometrika, 70, 643–649.MathSciNetzbMATHCrossRefGoogle Scholar
  52. Slud, E.V. and Byar, D. (1988), “How dependent causes of death can make risk factors appear protective,” Biometrics, 44, 265–269.MathSciNetzbMATHCrossRefGoogle Scholar
  53. Slud, E.V., Byar, D., and Schatzkin, A. (1988), “Dependent competing risks and the latent-failure model,” Biometrics, 44, 1203–1205.CrossRefGoogle Scholar
  54. Tsiatis, A. (1975), “A nonidentifiability aspect of the problem of competing risks,” Proceedings of the National Academy of Sciences, USA, 72, 20–22.Google Scholar
  55. Williams, J.S. and Lagakos, S.W. (1977), “Models for censored survival analysis: Constant-sum and variable-sum models,” Biometrika, 64, 215–224.MathSciNetzbMATHGoogle Scholar
  56. Zheng, M. and Klein, J.P. (1994a), “Estimates of marginal survival for dependent competing risks based on an assumed copula,” Biometrika, (to appear).Google Scholar
  57. Zheng, M. and Klein, J.P. (1994b), “A self-consistent estimator of marginal survival funtions based on dependent competing risk data and an assumed copula,” Communication in Statistics, (to appear).Google Scholar
  58. Zheng, M. and Klein, J.P. (1994c), “Identifiability and estimation of marginal survival functions for dependent competing risks assuming the copula is known,” 1994 International Research Conference on Lifetime Data Models in Reliability and Survival Analysis. Boston, pp. to be supplied.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • M. L. Moeschberger
    • 1
  • John P. Klein
    • 2
  1. 1.104B Starling-Loving HallThe Ohio State UniversityColumbusUSA
  2. 2.Medical College of WisconsinMilwaukeeUSA

Personalised recommendations