Lifetime Data Analysis

, Volume 20, Issue 3, pp 369–386 | Cite as

A class of semiparametric transformation models for survival data with a cured proportion

  • Sangbum Choi
  • Xuelin Huang
  • Yi-Hau Chen


We propose a new class of semiparametric regression models based on a multiplicative frailty assumption with a discrete frailty, which may account for cured subgroup in population. The cure model framework is then recast as a problem with a transformation model. The proposed models can explain a broad range of nonproportional hazards structures along with a cured proportion. An efficient and simple algorithm based on the martingale process is developed to locate the nonparametric maximum likelihood estimator. Unlike existing expectation-maximization based methods, our approach directly maximizes a nonparametric likelihood function, and the calculation of consistent variance estimates is immediate. The proposed method is useful for resolving identifiability features embedded in semiparametric cure models. Simulation studies are presented to demonstrate the finite sample properties of the proposed method. A case study of stage III soft-tissue sarcoma is given as an illustration.


Counting process Crossing survivals Discrete frailty  Nonparametric likelihood Survival analysis Transformation model 


  1. American Joint Committee on Cancer (2002) Cancer Staging Manual, 6th edn. Springer, New YorkGoogle Scholar
  2. Bickel PJ, Klaassen CA, Ritov Y, Wellner JA (1998) Efficient and adaptive estimation for semiparametric models, 2nd edn. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  3. Broët P, Rycke YD, Tubert-Bitter P, Lellouch J, Asselain B, Moreau T (2001) A semiparametric approach for the two-sample comparison of survival times with long-term survivors. Biometrics 57:844–852CrossRefzbMATHMathSciNetGoogle Scholar
  4. Caroni C, Crowder M, Kimber A (2010) Proportional hazards models with discrete frailty. Lifetime Data Anal 16:374–384CrossRefMathSciNetGoogle Scholar
  5. Chen MN, Ibrahim JG, Sinha D (1999) A new Bayesian model for survival data with a surviving fraction. J Am Stat Assoc 94:909–919CrossRefzbMATHMathSciNetGoogle Scholar
  6. Chen Y-H (2009) Weighted Breslow-type estimator and maximum likelihood estimation in semiparametric transformation models. Biometrika 96:591–600CrossRefzbMATHMathSciNetGoogle Scholar
  7. Coleman TF, Li Y (1994) On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds. Math Prog 67:189–224CrossRefzbMATHMathSciNetGoogle Scholar
  8. Coleman TF, Li Y (1996) An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J Optimiz 6:418–445CrossRefzbMATHMathSciNetGoogle Scholar
  9. Cormier JN, Huang X, Xing Y, Thall PF, Wang X, Benjamin RS, Pollock RE, Antonescu CR, Maki RG, Brennan MF, Pisters PWT (2004) Cohort analysis of patients with localized high-risk extremity soft tissue sarcoma treated at two cancer centers: chemotherapy-associated outcomes. J Clin Oncol 22:4567–4574CrossRefGoogle Scholar
  10. Dabrowska DM, Doksum KA (1988) Estimation and testing in the two-ample generalized odds-rate model. J Am Stat Assoc 83:744–749CrossRefzbMATHMathSciNetGoogle Scholar
  11. Geyer GJ (2009) Trust: Trust Region Optimization. R package 0.1-2Google Scholar
  12. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  13. Lu W, Ying Z (2004) On semiparametric transformation cure models. Biometrika 91:331–343CrossRefzbMATHMathSciNetGoogle Scholar
  14. Mao M, Wang J-L (2010) Semiparametric efficient estimation for a class of generalized proportional odds cure models. J Am Stat Assoc 105:302–311CrossRefMathSciNetGoogle Scholar
  15. Murphy SA (1994) Consistency in a proportional hazards model incorporating a random effect. Ann Stat 22:712–731CrossRefzbMATHGoogle Scholar
  16. Murphy SA (1995) Asymptotic theory for the frailty model. Ann Stat 23:182–198CrossRefzbMATHGoogle Scholar
  17. Parner E (1998) Asymptotic theory for the correlated gamma-frailty models. Ann Stat 26:183–214CrossRefzbMATHMathSciNetGoogle Scholar
  18. Peng Y, Dear KBG (2000) A nonparametric mixture model for cure rate estimation. Biometrics 56:237–243CrossRefzbMATHGoogle Scholar
  19. Sy JP, Taylor JMG (2000) Estimation in a Cox proportional hazards cure model. Biometrics 56:227–236CrossRefzbMATHMathSciNetGoogle Scholar
  20. Tsodikov AD, Ibrahim JG, Yakovlev AY (2003) Estimating cure rates from survival data: an alternative to two-component mixture models. J Am Stat Assoc 98:1063–1079CrossRefMathSciNetGoogle Scholar
  21. Zeng D, Lin DY (2006) Efficient estimation of semiparametric transformation models for counting processes. Biometrika 93:627–640CrossRefzbMATHMathSciNetGoogle Scholar
  22. Zeng D, Yin G, Ibrahim JG (2006) Semiparametric transformation models for survival data with a cure fraction. J Am Stat Assoc 101:670–684CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of BiostatisticsThe University of Texas, MD Anderson Cancer CenterHoustonUSA
  2. 2.Institute of Statistical Science, Academia SinicaTaipeiTaiwan

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