Lifetime Data Analysis

, Volume 15, Issue 2, pp 241–255 | Cite as

On pseudo-values for regression analysis in competing risks models

  • Frederik Graw
  • Thomas A. Gerds
  • Martin Schumacher


For regression on state and transition probabilities in multi-state models Andersen et al. (Biometrika 90:15–27, 2003) propose a technique based on jackknife pseudo-values. In this article we analyze the pseudo-values suggested for competing risks models and prove some conjectures regarding their asymptotics (Klein and Andersen, Biometrics 61:223–229, 2005). The key is a second order von Mises expansion of the Aalen-Johansen estimator which yields an appropriate representation of the pseudo-values. The method is illustrated with data from a clinical study on total joint replacement. In the application we consider for comparison the estimates obtained with the Fine and Gray approach (J Am Stat Assoc 94:496–509, 1999) and also time-dependent solutions of pseudo-value regression equations.


Competing risks Generalized estimating equation Jackknife pseudo-values Regression models Survival analysis Von Mises expansion 


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  1. Aalen OO, Johansen S (1978) An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scand J Stat 5: 141–150zbMATHMathSciNetGoogle Scholar
  2. Andersen PK, Klein JP (2007) Regression analysis for multistate models based on a pseudo-value approach, with applications to bone marrow transplantation studies. Scand J Stat 34: 3–16zbMATHCrossRefMathSciNetGoogle Scholar
  3. Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkzbMATHGoogle Scholar
  4. Andersen PK, Klein JP, Rosthøj S (2003) Generalised linear models for correlated pseudo-observations, with applications to multi-state models. Biometrika 90(1): 15–27zbMATHCrossRefMathSciNetGoogle Scholar
  5. Fine JP (1999) Analysing competing risks data with transformation models. J R Stat Soc B 61(4): 817–830zbMATHCrossRefMathSciNetGoogle Scholar
  6. Fine JP, Gray RJ (1999) A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc 94(446): 496–509zbMATHCrossRefMathSciNetGoogle Scholar
  7. Gill RD (1989) Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. Scand J Stat 16(2): 97–128zbMATHMathSciNetGoogle Scholar
  8. Gill RD, Johansen S (1990) A survey of product-integration with a view toward application in survival analysis. Ann Stat 18(4): 1501–1555zbMATHCrossRefMathSciNetGoogle Scholar
  9. Halekoh U, Højsgaard S (2006) The R package geepack for generalized estimating equations. J Stat Softw 15(2): 1–11Google Scholar
  10. Hampel FR (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69: 1179–1186CrossRefMathSciNetGoogle Scholar
  11. Huber P (1977) Robust statistical procedures. SIAM, PhiladelphiazbMATHGoogle Scholar
  12. James LF (1997) A study of a class of weighted bootstrap for censored data. Ann Stat 25(4): 1595–1621zbMATHCrossRefGoogle Scholar
  13. Jewell NP, Lei X, Ghani AC, Donnelly CA, Leung GM, Ho LM, Cowling BJ, Hedley AJ (2007) Non- parametric estimation of the case fatality ratio with competing risks data: an application to Severe Acute Respiratory Syndrome (SARS). Stat Med 26(9): 1982–98CrossRefMathSciNetGoogle Scholar
  14. Klein JP, Andersen PK (2005) Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function. Biometrics 61(1): 223–229zbMATHCrossRefMathSciNetGoogle Scholar
  15. Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73: 13–22zbMATHCrossRefMathSciNetGoogle Scholar
  16. Maurer TB, Ochsner PE, Schwarzer G, Schumacher M (2001) Increased loosening of cemented straight stem prostheses made from titanium alloys. An analysis and comparison with prostheses made of cobalt-chromium-nickel alloy. Int Orthop 25(2): 77–80CrossRefGoogle Scholar
  17. Parr WC (1985) Jackknifing differentiable statistical functionals. J R Stat Soc B 47: 56–66zbMATHMathSciNetGoogle Scholar
  18. R Development Core Team (2006) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, AustriaGoogle Scholar
  19. Satten G, Datta S (2001) The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Am Stat 55(3): 207–210CrossRefMathSciNetGoogle Scholar
  20. Scheike T, Zhang MJ (2007) Direct modelling of regression effects for transition probabilities in multistate models. Scand J Stat 34(1): 17–32CrossRefMathSciNetGoogle Scholar
  21. Scheike TH, Zhang MJ, Gerds TA (2008) Predicting cumulative incidence probability by direct binomial regression. Biometrika 95: 205–220CrossRefGoogle Scholar
  22. Schwarzer G, Schumacher M, Maurer TB, Ochsner PE (2001) Statistical analysis of failure times in total joint replacement. J Clin Epidemiol 54(10): 997–1003CrossRefGoogle Scholar
  23. Vander Vaart A (1998) Asymptotic statistics. Cambridge Univ. Press, CambridgeGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Frederik Graw
    • 1
    • 2
  • Thomas A. Gerds
    • 3
  • Martin Schumacher
    • 2
  1. 1.Institute of Integrative BiologyETH ZurichZurichSwitzerland
  2. 2.Institute of Medical Biometry and Medical InformaticsUniversity Medical Center FreiburgFreiburgGermany
  3. 3.Department of BiostatisticsUniversity of CopenhagenCopenhagen KDenmark

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