Advertisement

Lifetime Data Analysis

, Volume 25, Issue 1, pp 26–51 | Cite as

A class of semiparametric cure models with current status data

  • Guoqing DiaoEmail author
  • Ao Yuan
Article
  • 164 Downloads

Abstract

Current status data occur in many biomedical studies where we only know whether the event of interest occurs before or after a particular time point. In practice, some subjects may never experience the event of interest, i.e., a certain fraction of the population is cured or is not susceptible to the event of interest. We consider a class of semiparametric transformation cure models for current status data with a survival fraction. This class includes both the proportional hazards and the proportional odds cure models as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the maximum likelihood estimators for the regression coefficients are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in finite samples. For illustration, we provide an application of the models to a study on the calcification of the hydrogel intraocular lenses.

Keywords

Box–Cox transformation Cure fraction Empirical process NPMLE Proportional hazards cure model Proportional odds cure model Semiparametric efficiency 

Notes

Acknowledgements

The authors wish to thank Dr. Donglin Zeng and the referees for their helpful comments and suggestions, which lead to a considerable improvement in the presentation of this manuscript. The authors also would like to thank Drs. A. F. K. Yu, K. F. Lam, and HongQi Xue for providing the Calcification data.

References

  1. Barlow RE, Bartholomew DJ, Bremner JM, Brunk HD (1972) Statistical inference under order restrictions. Wiley, New YorkzbMATHGoogle Scholar
  2. Berkson J, Gage RP (1952) Survival curve for cancer patients following treatment. J Am Stat Assoc 47:501–515CrossRefGoogle Scholar
  3. Betensky RA, Schoenfeld DA (2001) Nonparametric estimation in a cure model with random cure times. Biometrics 57(1):282–286MathSciNetCrossRefzbMATHGoogle Scholar
  4. Box GEP, Cox DR (1982) An analysis of transformation revisited, rebutted. J Am Stat Assoc 77:209–210CrossRefzbMATHGoogle Scholar
  5. Chen MH, Ibrahim JG, Sinha D (1999) A new bayesian model for survival data with a surviving fraction. J Am Stat Assoc 94:909–919MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cook RJ, White BJG, Grace YY, Lee KA, Warkentin TE (2008) Analysis of a nonsusceptible fraction with current status data. Stat Med 27:2715–2730MathSciNetCrossRefGoogle Scholar
  7. Cox DR (1972) Regression model and life-tables (with discussion). J Roy Stat Soc B 34:187–220zbMATHGoogle Scholar
  8. Diao G, Zeng D, Yang S (2013) Efficient semiparametric estimation of short-term and long-term hazard ratios with right-censored data. Biometrics 69(4):840–849MathSciNetCrossRefzbMATHGoogle Scholar
  9. Fang H, Li G, Sun J (2005) Maximum likelihood estimation in a semiparametric logistic/proportional-hazards mixture model. Scand J Stat 32(1):59–75MathSciNetCrossRefzbMATHGoogle Scholar
  10. Farewell VT (1982) The use of mixture models for the analysis of survival data with long-term survivors. Biometrics 38:1041–1046CrossRefGoogle Scholar
  11. Farewell VT (1986) Mixture models in survival analysis: Are they worth the risk? Can J Stat 14:257–262MathSciNetCrossRefGoogle Scholar
  12. Gray RJ, Tsiatis AA (1989) A linear rank test for use when the main interest is in differences in cure rates. Biometrics 45(3):899–904CrossRefzbMATHGoogle Scholar
  13. Groeneboom P (1989) Brownian motion with a parabolic drift and airy functions. Probab Theory Relat Fields 81(1):79–109MathSciNetCrossRefGoogle Scholar
  14. Groeneboom P, Wellner JA (1992) Information bounds and nonparametric maximum likelihood estimation. Birkhauser, BaselCrossRefzbMATHGoogle Scholar
  15. Hoel DG, Walburg HE (1972) Statistical analysis of survival experiments. J Natl Cancer Inst 49:361–372Google Scholar
  16. Huang J (1996) Efficient estimation for the proportional hazards model with interval censoring. Ann Stat 24(2):540–568MathSciNetCrossRefzbMATHGoogle Scholar
  17. Huang J, Rossini AJ (1997) Sieve estimation for the proportional-odds failure-time regression model with interval censoring. J Am Stat Assoc 92(439):960–967MathSciNetCrossRefzbMATHGoogle Scholar
  18. Ibrahim JG, Chen MH, Sinha D (2001) Bayesian survival analysis. Springer, New YorkCrossRefzbMATHGoogle Scholar
  19. Kosorok MR (2008) Bootstrapping the grenander estimator. In: Balakrishnan N, Pena E, Silvapulle MJ (eds) Beyond parametrics in interdisciplinary research: Festschrift in honor of Professor Pranab K. Sen, Institute of Mathematical Statistics, pp 282–292Google Scholar
  20. Kuk AYC, Chen CH (1992) A mixture model combining logistic-regression with proportional hazards regression. Biometrika 79(3):531–541CrossRefzbMATHGoogle Scholar
  21. Lam K, Xue H (2005) A semiparametric regression cure model with current status data. Biometrika 92(3):573–586.  https://doi.org/10.1093/biomet/92.3.573 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lam KF, Fong DYT, Tang O (2005) Estimating the proportion of cured patients in a censored sample. Stat Med 24(12):1865–1879.  https://doi.org/10.1002/sim.2137 MathSciNetCrossRefGoogle Scholar
  23. Li CS, Taylor JMG, Sy JP (2001) Identifiability of cure models. Stat Probab Lett 54:389–395MathSciNetCrossRefzbMATHGoogle Scholar
  24. Lin DY, Oakes D, Ying Z (1998) Additive hazards regression with current status data. Biometrika 85(2):289–298MathSciNetCrossRefzbMATHGoogle Scholar
  25. Liu H, Shen Y (2009) A semiparametric regression cure model for interval-censored data. J Am Stat Assoc 104:1168–1178MathSciNetCrossRefzbMATHGoogle Scholar
  26. Lu W, Ying Z (2004) On semiparametric transformation cure models. Biometrika 91(2):331–343MathSciNetCrossRefzbMATHGoogle Scholar
  27. Ma S (2008) Additive risk model for current status data with a cured subgroup. Ann Instit Stat Math 63:117–134MathSciNetCrossRefzbMATHGoogle Scholar
  28. Ma S (2009) Cure model with current status data. Stat Sin 19(1):233–249MathSciNetzbMATHGoogle Scholar
  29. Ma S, Kosorok MR (2005) Penalized log-likelihood estimation for partly linear transformation models with current status data. Ann Stat 33(5):2256–2290.  https://doi.org/10.1214/009053605000000444 MathSciNetCrossRefzbMATHGoogle Scholar
  30. Maller R, Zhou X (1996) Survival analysis with long-term survivors. Wiley, New YorkzbMATHGoogle Scholar
  31. Peng Y, Dear KBG (2000) A nonparametric mixture model for cure rate estimation. Biometrics 56(1):237–243CrossRefzbMATHGoogle Scholar
  32. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C: the art of scientific computing, 2nd edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
  33. Robertson T, Wright F, Dykstra R (1988) Order restricted statistical inference. Wiley, New YorkzbMATHGoogle Scholar
  34. Rossini AJ, Tsiatis AA (1996) A semiparametric proportional odds regression model for the analysis of current status data. J Am Stat Assoc 91(434):713–721MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sy JP, Taylor JMG (2000) Estimation in a cox proportional hazards cure model. Biometrics 56(1):227–236MathSciNetCrossRefzbMATHGoogle Scholar
  36. Taylor J (1995) Semiparametric estimation in failure time mixture-models. Biometrics 51(3):899–907CrossRefzbMATHGoogle Scholar
  37. Tsodikov A (1998) A proportional hazards model taking account of long-term survivors. Biometrics 54:1508–1516CrossRefzbMATHGoogle Scholar
  38. 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(464):1063–1078.  https://doi.org/10.1198/01622145030000001007 MathSciNetCrossRefGoogle Scholar
  39. van der Laan MJ, Jewell NP (2001) The NPMLE for doubly censored current status data. Scand J Stat 28(3):537–547MathSciNetCrossRefzbMATHGoogle Scholar
  40. van der Vaart A, Wellner J (1996) Weak convergence and empirical processes: with applications to statistics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  41. van der Vaart AW (2002) Semiparametric statistics. In: Lectures on probability theory and statistics (Lecture notes in math), vol 1781, Springer, New York, pp 331–457Google Scholar
  42. Xue H, Lam K, Li G (2004) Sieve maximum likelihood estimator for semiparametric regression models with current status data. J Am Stat Assoc 99(466):346–356.  https://doi.org/10.1198/016214504000000313 MathSciNetCrossRefzbMATHGoogle Scholar
  43. Yakovlev AY, Tsodikov AD (1996) Stochastic models of tumor latency and their biostatistical applications. World Scientific, New JerseyCrossRefzbMATHGoogle Scholar
  44. Yu A, Kwan K, Chan D, Fong D (2001) Clinical features of 46 eyes with calcified hydrogel intraocular lenses. J Cataract Refract Surg 27:1596–1606CrossRefGoogle Scholar
  45. Yuan M, Diao G (2014) Semiparametric odds rate model for modeling short-term and long-term effects with application to a breast cancer genetic study. Int J Biostat 10(2):231–249MathSciNetCrossRefGoogle Scholar
  46. Zeng D, Cai J, Shen Y (2006a) Semiparametric additive risks model for interval-censored data. Stat Sin 16:287–302MathSciNetzbMATHGoogle Scholar
  47. Zeng D, Yin G, Ibrahim J (2006b) Semiparametric transformation models for survival data with a cure fraction. J Am Stat Assoc 101:670–684MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of StatisticsGeorge Mason UniversityFairfaxUSA
  2. 2.Department of Biostatistics, Bioinformatics and BiomathematicsGeorgetown UniversityWashingtonUSA

Personalised recommendations