Lifetime Data Analysis

, Volume 25, Issue 1, pp 1–25 | Cite as

Vertical modeling: analysis of competing risks data with a cure fraction

  • Mioara Alina NicolaieEmail author
  • Jeremy M. G. Taylor
  • Catherine Legrand


In this paper, we extend the vertical modeling approach for the analysis of survival data with competing risks to incorporate a cure fraction in the population, that is, a proportion of the population for which none of the competing events can occur. The proposed method has three components: the proportion of cure, the risk of failure, irrespective of the cause, and the relative risk of a certain cause of failure, given a failure occurred. Covariates may affect each of these components. An appealing aspect of the method is that it is a natural extension to competing risks of the semi-parametric mixture cure model in ordinary survival analysis; thus, causes of failure are assigned only if a failure occurs. This contrasts with the existing mixture cure model for competing risks of Larson and Dinse, which conditions at the onset on the future status presumably attained. Regression parameter estimates are obtained using an EM-algorithm. The performance of the estimators is evaluated in a simulation study. The method is illustrated using a melanoma cancer data set.


Mixture cure model Competing risks Cumulative incidences 



Funding was provided by IAP grant (Grant No. P7/06).


  1. Andersen PK, Borgan O, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, BerlinCrossRefzbMATHGoogle Scholar
  2. Andersson TML, Dickman PW, Eloranta S, Lambert PC (2011) Estimating and modelling cure in population-based cancer studies within the framework of flexible parametric survival models. BMC Med Res Methodol 11:1–11CrossRefGoogle Scholar
  3. Andrae B, Andersson TML, Lambert PC, Kemetli L, Silverdal L, Strander B, Ryd W, Dillner J, Tolnber S, Sparen P (2012) Screening and cervical cancer cure: population based cohort study. Br Med J 344:e900CrossRefGoogle Scholar
  4. Basu S, Tiwari TC (2010) Breast cancer survival, competing risks and mixture cure model: a Bayesian analysis. J R Stat Soc Ser A 173:307–329MathSciNetCrossRefGoogle Scholar
  5. Canty A, Ripley B (2008) Boot: Bootstrap r (s-plus) functions. R package version 1.2-34Google Scholar
  6. Chao EC (1998) Gibbs sampling for long-term survival data with competing risks. Commun Stat Theory Methods 54:350–366zbMATHGoogle Scholar
  7. Chen CH, Tsay Y, Wu Y, Horng C (2013) Logistic aft location-scale mixture regression models with nonsusceptibility for left-truncated and general interval-censored data. Stat Med 32:4285–4305MathSciNetCrossRefGoogle Scholar
  8. Choi KC, Zhou X (2002) Large sample properties of mixture models with covariates competing risks. J Multivar Anal 82:331–366MathSciNetCrossRefzbMATHGoogle Scholar
  9. Corbiere F, Commenges D, Taylor JMG, Joly P (2009) A penalized likelihood approach for mixture cure models. Stat Med 28:510–524MathSciNetCrossRefGoogle Scholar
  10. Farewell VT (1986) Mixture models in survival analysis: are they worth the risk? Can J Stat 14:257–262MathSciNetCrossRefGoogle Scholar
  11. Jia X, Sima CS, Brennan MF, Panageas KS (2013) Cure models for the analysis of time-to-event data in cancer studies. J Surg Oncol 108:342–347CrossRefGoogle Scholar
  12. Kim S, Zeng D, Li Y, Spiegelman D (2013) Joint modeling of longitudinal and cure survival data. J Stat Theory Pract 7:324–344MathSciNetCrossRefGoogle Scholar
  13. Klebanov L, Yakolev A (2007) A new approach to testing for sufficient follow-up in cure-rate analysis. J Stat Plan Inference 137:3557–3569MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kuk AYC, Chen C (1992) A mixture model combining logistic regression with proportional hazards regression. Biometrika 79:531–541CrossRefzbMATHGoogle Scholar
  15. Larson MG, Dinse GE (1985) A mixture model for the regression analysis of competing risks data. Appl Stat 34:201–211MathSciNetCrossRefGoogle Scholar
  16. Laska EG, Meisner MJ (1992) Nonparametric estimation and testing in a cure model. Biometrics 48:1223–1234CrossRefGoogle Scholar
  17. Li CS, Taylor JMG (2002) A semiparametric accelerated failure time cure model. Stat Med 21:3235–3247CrossRefGoogle Scholar
  18. Maller R, Zhou S (1994) Testing for sufficient follow-up and outliers in survival data. J Am Stat Assoc 89(428):1499–1506MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ng SK, McLachlan GJ (1998) On modifications to the long-term survival mixture model in the presence of competing risks. J Stat Comput Simul 61:77–96MathSciNetCrossRefzbMATHGoogle Scholar
  20. Nicolaie MA, Putter H, van Houwelingen JC (2010) Vertical modeling: a pattern mixture approach for competing risks data. Stat Med 29:1190–1205MathSciNetGoogle Scholar
  21. Nicolaie MA, Putter H, van Houwelingen JC (2015) Vertical modeling: analysis of competing risks data with missing causes of failure. Stat Methods Med Res 24(6):891–908MathSciNetCrossRefGoogle Scholar
  22. Peng Y (2003) Estimating baseline distribution in proportional hazards cure models. Comput Stat Data Anal 42:187–201MathSciNetCrossRefzbMATHGoogle Scholar
  23. Peng Y, Dear KBG (2000) A nonparametric mixture model for cure rate estimation. Biometrics 56:2236–243CrossRefGoogle Scholar
  24. Peng Y, Taylor JMG (2013) Cure models. In: Klein JP, van Houwelingen HC, Ibrahim JG, Scheike TH (eds) Handbook of survival analysis. Chapman and Hall CRC, Boca Raton, pp 113–134Google Scholar
  25. Peng Y, Taylor JMG (2017) Residual-based model diagnosis methods for mixture cure models. Biometrics 73:495–505MathSciNetCrossRefzbMATHGoogle Scholar
  26. R Development Core Team (2010) R: a language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
  27. Scolas S, Ghouch AE, Legrand C, Oulhaj A (2016a) Diagnostic checks in mixture cure models with interval-censoring. Stat Methods Med Res.
  28. Scolas S, Ghouch AE, Legrand C, Oulhaj A (2016b) Variable selection in a flexible parametric mixture cure model with interval-censored data. Stat Med 35:1210–1225MathSciNetCrossRefGoogle Scholar
  29. Sy JP, Taylor JMG (2000) Estimation in a Cox proportional hazards cure model. Biometrics 56:227–236MathSciNetCrossRefzbMATHGoogle Scholar
  30. Taylor JMG (1995) Semi-parametric estimation in failure time mixture models. Biometrics 51:899–907CrossRefzbMATHGoogle Scholar
  31. Taylor JMG, Liu N (2007) Statistical issues involved with extending standard models. In: Nair V (ed) Advances in statistical modeling and inferences: essays in honor of Kjell A. World Scientific Publishing Company, Singapore, Doksum, pp 299–311CrossRefGoogle Scholar
  32. Wei L (1992) The accelerated failure time model: a useful alternative to the cox regression model in survival analysis. Stat Med 11:1871–1879CrossRefGoogle Scholar
  33. Yamaguchi K (1992) Accelerated failure-time regression models with a regression model of surviving fraction: an application to the analysis of permanent employment in Japan. J Am Stat Assoc 87:284–292Google Scholar
  34. Yu M, Law NJ, Taylor JMG, Sandler HM (2004) Joint longitudinal survival-cure models and their application to prostate cancer. Stat Sin 14:835–862MathSciNetzbMATHGoogle Scholar
  35. Yu XQ, De Angelis R, Andersson TML, Lambert PC, Dickman P (2013) Estimating the proportion cured of cancer: some practical advice for users. Cancer Epidemiol 37:836–842CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Mioara Alina Nicolaie
    • 1
    Email author
  • Jeremy M. G. Taylor
    • 2
  • Catherine Legrand
    • 1
  1. 1.Institute of Statistics, Biostatistics and Actuarial SciencesCatholic University of LouvainLouvain-la-NeuveBelgium
  2. 2.School of Public HealthUniversity of MichiganAnn ArborUSA

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