Lifetime Data Analysis

, Volume 19, Issue 1, pp 33–58 | Cite as

Applying competing risks regression models: an overview

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Abstract

In many clinical research applications the time to occurrence of one event of interest, that may be obscured by another—so called competing—event, is investigated. Specific interventions can only have an effect on the endpoint they address or research questions might focus on risk factors for a certain outcome. Different approaches for the analysis of time-to-event data in the presence of competing risks were introduced in the last decades including some new methodologies, which are not yet frequently used in the analysis of competing risks data. Cause-specific hazard regression, subdistribution hazard regression, mixture models, vertical modelling and the analysis of time-to-event data based on pseudo-observations are described in this article and are applied to a dataset of a cohort study intended to establish risk stratification for cardiac death after myocardial infarction. Data analysts are encouraged to use the appropriate methods for their specific research questions by comparing different regression approaches in the competing risks setting regarding assumptions, methodology and interpretation of the results. Notes on application of the mentioned methods using the statistical software R are presented and extensions to the presented standard methods proposed in statistical literature are mentioned.

Keywords

Competing risks Cause-specific hazard Subdistribution hazard Mixture model Vertical modelling Pseudo-observation approach 
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References

  1. Allignol A, Schumacher M, Wanner C, Drechsler C, Beyersmann J (2011) Understanding competing risks: a simulation point of view. BMC Med Res Methodol 11:86. doi: 10.1186/1471-2288-11-86
  2. Andersen PK, Keiding N (2012) Interpretability and importance of functionals in competing risks and multistate models. Stat Med 31: 1074–1088. doi: 10.1002/sim.4385 MathSciNetCrossRefGoogle Scholar
  3. Andersen PK, Perme MP (2010) Pseudo-observations in survival analysis. Stat Methods Med Res 19: 71–99. doi: 10.1177/0962280209105020 MathSciNetCrossRefGoogle Scholar
  4. Andersen PK, Klein JP, Rosthøj S (2003) Generalised linear models for correlated pseudo-observations, with applications to multi-state models. Biometrika 90: 15–27. doi: 10.1093/biomet/90.1.15 MathSciNetMATHCrossRefGoogle Scholar
  5. Bakoyannis G, Touloumi G (2011) Practical methods for competing risks data: a review. Stat Methods Med Res. doi: 10.1177/0962280210394479
  6. Barthel P, Schneider R, Bauer A, Ulm K, Schmitt C, Schömig A, Schmidt G (2003) Risk stratification after acute myocardial infarction by heart rate turbulence. Circulation 108: 1221–1226. doi: 10.1161/01.CIR.0000088783.34082.89 CrossRefGoogle Scholar
  7. Bauer A, Kantelhardt JW, Barthel P, Schneider R, Makikallio T, Ulm K, Hnatkova K, Schömig A, Huikuri H, Bunde A, Malik M, Schmidt G (2006) Declaration capacity of heart rate as a predictor of mortality after myocardial infarction: cohort study. Lancet 367: 1674–1681. doi: 10.1016/S0140-6736(06)68735-7 CrossRefGoogle Scholar
  8. Bauer A, Barthel P, Schneider R, Ulm K, Müller A, Joeining A, Stich R, Kiviniemi A, Hnatkova K, Huikuri H, Schömig A, Malik M, Schmidt G (2009) Improved stratification of autonomic regulation for risk prediction in post-infarction patients with preserved left ventricular function (isar-risk). Eur Heart J 30: 576–583. doi: 10.1093/eurheartj/ehn540 CrossRefGoogle Scholar
  9. Belot A, Abrahamowicz M, Remontet L, Giorgi R (2010) Flexible modeling of competing risks in survival analysis. Stat Med 29(23): 2453–2468. doi: 10.1002/sim.4005 MathSciNetGoogle Scholar
  10. Beyersmann J, Schumacher M (2007) Letter to the editor: Misspecified regression model for the subdistribution hazard of a competing risk. Stat Med 26: 1649–1652. doi: 10.1002/sim.2727 MathSciNetCrossRefGoogle Scholar
  11. Beyersmann J, Dettenkofer M, Bertz H, Schumacher M (2007) A competing risks analysis of bloodstream infection after stem-cell transplantation using subdistribution hazards and cause-specific hazards. Stat Med 26: 5360–5369. doi: 10.1002/sim.3006 MathSciNetCrossRefGoogle Scholar
  12. Beyersmann J, Latouche A, Buchholz A, Schumacher M (2009) Simulating competing risks data in survival analysis. Stat Med 28(6): 956–971. doi: 10.1002/sim.3516 MathSciNetCrossRefGoogle Scholar
  13. Beyersmann J, Schumacher M, Allignol A (2012) Competing risks and multistate models with R. Springer, New YorkMATHCrossRefGoogle Scholar
  14. Cox DR (1972) Regression models and life-tables. J R Stat Soc Ser B 34: 187–220. doi: 10.2307/2985181 MATHGoogle Scholar
  15. Cox C, Chu H, Schneider MF, Muñoz A (2007) Tutorial in biostatistics: parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Stat Med 26: 4352–4374. doi: 10.1002/sim.2836 MathSciNetCrossRefGoogle Scholar
  16. Crowder MJ (2001) Classical competing risks. Chapman & Hall/CRC, Boca RatonMATHCrossRefGoogle Scholar
  17. Dignam JJ, Kocherginsky MN (2008) Choice and interpretation of statistical tests used when competing risks are present. J Clin Oncol 26(24): 4027–4034. doi: 10.1200/JCO.2007.12.9866 CrossRefGoogle Scholar
  18. Efron B, Tibshirani RJ (1994) An introduction to the bootstrap. Chapman and Hall/CRC, LondonGoogle Scholar
  19. Escarela G, Bowater RJ (2008) Fitting a semi-parametric mixture model for competing risks in survival data. Commun Stat 37: 277–293. doi: 10.1080/03610920701649134 MathSciNetMATHCrossRefGoogle Scholar
  20. Fahrmeir L, Tutz G (2001) Multivariate statistical modelling based on generalized linear models. Springer, New YorkMATHGoogle Scholar
  21. Fine JP, Gray RJ (1999) A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc 94: 496–509. doi: 10.2307/2670170 MathSciNetMATHCrossRefGoogle Scholar
  22. Friedman M (1982) Piecewise exponential models for survival data with covariates. Ann Stat 10: 101–113MATHCrossRefGoogle Scholar
  23. Grambauer N, Schumacher M, Beyersmann J (2010) Proportional subdistribution hazards modeling offers a summary analysis, even if misspecified. Stat Med 29: 875–884. doi: 10.1002/sim.3786 MathSciNetCrossRefGoogle Scholar
  24. Gray R (1988) A class of k-sample tests for comparing the cumulative incidence function in the presence of a competing risk. Ann Stat 16: 1141–1154. doi: 10.2307/2241622 MATHCrossRefGoogle Scholar
  25. Gray B (2010) cmprsk: Subdistribution analysis of competing risks. URL http://CRAN.R-project.org/package=cmprsk, R package version 2.2-1
  26. Hastie TJ (1997) Generalized additive models. Chapman & Hall/CRC, New YorkGoogle Scholar
  27. Højsgaard S, Halekoh U, Yan J (2005) The R package geepack for generalized estimating equations. J Stat Softw 15:1–11. http://CRAN.R-project.org/package=survival, R package version 2.36-5Google Scholar
  28. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data. Wiley, HobokenMATHCrossRefGoogle Scholar
  29. Kaplan EL, Meier P (1958) Non-parametric estimation from incomplete observations. J Am Stat Assoc 53: 457–481MathSciNetMATHCrossRefGoogle Scholar
  30. Klein JP (2010) Competing risks. WIREs Comput Stat 2: 333–339. doi: 10.1002/wics.83 CrossRefGoogle Scholar
  31. Klein JP, Andersen PK (2005) Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function. Biometrics 61: 223–229. doi: 10.1111/j.0006-341X.2005.031209.x MathSciNetMATHCrossRefGoogle Scholar
  32. Klein JP, Moeschberger ML (2003) Survival analysis—techniques for censored and truncated data. Springer, New YorkMATHGoogle Scholar
  33. Klein J, Gerster M, Andersen P, Tarima S, Perme M (2008) SAS and R functions to compute pseudo- values for censored data regression. Comput Methods Prog Biomed 89(3): 289–300. doi: 10.1016/j.cmpb.2007.11.017 CrossRefGoogle Scholar
  34. Koller MT, Raatz H, Steyerberg EW, Wolbers M (2012) Competing risks and the clinical community: irrelevance or ignorance. Stat Med 31: 1089–1097MathSciNetCrossRefGoogle Scholar
  35. Larson MG, Dinse GE (1985) A mixture model for the regression analysis of competing risks data. J R Stat Soc Ser C 34: 201–211MathSciNetGoogle Scholar
  36. Latouche A, Boisson V, Chevret S, Porcher R (2007) Misspecified regression model for the subdistribution hazard of a competing risk. Stat Med 26(5): 965–974. doi: 10.1002/sim.2600 MathSciNetCrossRefGoogle Scholar
  37. Lau B, Cole SR, Moore SR, Gange SJ (2008) Evaluating competing adverse and beneficial outcomes using a mixture model. Stat Med 27: 4313–4327. doi: 10.1002/sim.3293 MathSciNetCrossRefGoogle Scholar
  38. Lau B, Cole SR, Gange SJ (2009) Competing risk regression models for epidemiologic data. Am J Epidemiol 170: 244–256. doi: 10.1093/aje/kwp107 CrossRefGoogle Scholar
  39. Lau B, Cole S, Gange S (2011) Parametric mixture models to evaluate and summarize hazard ratios in the presence of competing risks with time-dependent hazards and delayed entry. Stat Med 30: 654–665. doi: 10.1002/sim.4123 MathSciNetCrossRefGoogle Scholar
  40. Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73: 13–22. doi: 10.1093/biomet/73.1.13 MathSciNetMATHCrossRefGoogle Scholar
  41. Lunn M, McNeil D (1995) Applying Cox regression to competing risks. Biometrics 51: 524–532. doi: 10.2307/2532940 CrossRefGoogle Scholar
  42. McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New YorkMATHGoogle Scholar
  43. Miller RG (1974) The Jackknife—a review. Biometrika 6(1): 1–15. doi: 0.1093/-biomet/61.1.1 Google Scholar
  44. Ng GK, McLachlan GJ (2003) An em-based semi-parametric mixture model approach to the regression analysis of competing-risks data. Stat Med 22: 1097–1111. doi: 10.1002/sim.1371 CrossRefGoogle Scholar
  45. Nicolaie MA, Houwelingen HC, Putter H (2010) Vertical modeling: a pattern mixture approach for competing risks modeling. Stat Med 29: 1190–1205. doi: 10.1002/sim.3844 MathSciNetGoogle Scholar
  46. Perme MP, Andersen PK (2008) Checking hazard regression models using pseudo-observations. Stat Med 27: 5309–5328. doi: 10.1002/sim.3401 MathSciNetCrossRefGoogle Scholar
  47. Pintilie M (2006) Competing risks: a practical perspective. Wiley, ChichesterMATHGoogle Scholar
  48. Prentice R, Kalbfleisch J, Peterson A, Flournoy N, Farewell V, Breslow N (1978) The analysis of failure times in the presence of competing risks. Biometrics 34: 541–554. doi: 10.2307/2530374 MATHCrossRefGoogle Scholar
  49. Putter H, Fiocco M, Geskus RB (2007) Tutorial in biostatistics: competing risks and multi-state models. Stat Med 26(11): 2389–2430. doi: 10.1002/sim.2712 MathSciNetCrossRefGoogle Scholar
  50. R Development Core Team (2011) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, http://www.R-project.org/, ISBN 3-900051-07-0
  51. Robins J, Rotnitzky A (1992) Recovery of information and adjustment for dependent censoring using surrogate markers. In: Jewell N, Dietz K, Farewell V (eds) AIDS epidemiology—methodological issues. Birkhäuser, Boston, pp 24–33Google Scholar
  52. Scheike TH, Zhang MJJ (2008) Flexible competing risks regression modeling and goodness-of-fit. Lifetime Data Anal 14: 464–483. doi: 10.1007/s10985-008-9094-0 MathSciNetMATHCrossRefGoogle Scholar
  53. Schemper M, Smith T (1996) A note on quantifying follow-up in studies of failure time. Lancet 17: 343–346Google Scholar
  54. Schoenfeld D (1982) Partial residuals for the proportional hazards regression model. Biometrika 69(1): 239–241. doi: 10.1093/biomet/69.1.239 CrossRefGoogle Scholar
  55. Sun Y, Hyun S, Gilbert P (2008) testing and estimation of time-varying cause-specific hazard ratios with covariate adjustment. Biometrics 64: 1070–1079. doi: 10.1111/j.1541-0420.2008.01012.x MathSciNetMATHCrossRefGoogle Scholar
  56. Tai BC, Wee J, Machin D (2011) Analysis and design of randomised clinical trials involving competing risks endpoints. Trials 12: 127. doi: 10.1186/1745-6215-12-127 CrossRefGoogle Scholar
  57. Therneau T (2011) Survival: survival analysis, including penalised likelihood. http://CRAN.R-project.org/package=survival, R package version 2.36-5
  58. Therneau TM, Grambsch PM (2000) Modeling survival data: extending the Cox model (statistics for biology and health). Springer, New YorkMATHGoogle Scholar
  59. Tsiatis A (1975) A nonidentifiability aspect of the problem of competing risks. Proc Natl Acad Sci USA 72(1):20–22Google Scholar
  60. Urbanek S (2011) multicore: parallel processing of R code on machines with multiple cores or CPUs. http://CRAN.R-project.org/package=multicore, R package version 0.1-5

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für Medizinische Statistik und Epidemiologie der Technischen Universität MünchenMunichGermany
  2. 2.1. Medizinische Klinik und Poliklinik der Technischen Universität MünchenMunichGermany

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