Skip to main content
SpringerLink
Log in

A general transformation class of semiparametric cure rate frailty models

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

We consider a class of cure rate frailty models for multivariate failure time data with a survival fraction. This class is formulated through a transformation on the unknown population survival function. It incorporates random effects to account for the underlying correlation, and includes the mixture cure model and the proportional hazards cure model as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the nonparametric maximum likelihood estimators for the parameters of these models are consistent and asymptotically normal, and that the limiting variances achieve the semiparametric efficiency bounds. Simulation studies demonstrate that the proposed methods perform well in finite samples. We provide an application of the proposed methods to the data of the age at onset of alcohol dependence, from the Collaborative Study on the Genetics of Alcoholism.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abecasis G.R., Cardon L.R., Cookson W.O.C. (2000) A general test of association for quantitative traits in nuclear families. American Journal of Human Genetics 66: 279–292

    Article  Google Scholar 

  • Bailey-Wilson, J. E., Thomas, D., MacCluer, J. W. (2005). Genetic Analysis Workshop 14: Summarizing analyses comparing microsatellite and SNP marker loci for genome-wide scans. Genetic Epidemiology, 29(Suppl 1), S1–S132.

    Google Scholar 

  • Begleiter H., Reich T., Hesselbrock V., Porjesz B., Li T.K., Schuckit M.A., Edenberg H.J., Rice J.P. (1995) The collaborative study on the genetics of alcoholism. Alcohol Health Res World 19: 228–236

    Google Scholar 

  • Berkson J., Gage R.P. (1952) Survival curve for cancer patients following treatment. Journal of the American Statistical Association 47: 501–515

    Article  Google Scholar 

  • Betensky R.A., Schoenfeld D.A. (2001) Nonparametric estimation in a cure model with random cure times. Biometrics 57: 282–286

    Article  MathSciNet  MATH  Google Scholar 

  • Bickel P.J., Klaassen C.A.J., Ritov Y., Wellner J.A. (1993) Efficient and adaptive estimation for semiparametric models. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Box G.E.P., Cox D.R. (1964) An analysis of transformations (with discussion). Journal of the Royal Statistical Society. Series B 26: 211–252

    MathSciNet  MATH  Google Scholar 

  • Chatterjee N., Shih J. (2001) A bivariate cure-mixture approach for modeling familial association in diseases. Biometrics 57: 779–786

    Article  MathSciNet  MATH  Google Scholar 

  • Chen M.H., Ibrahim J.G., Sinha D. (1999) A new Bayesian model for survival data with a surviving fraction. Journal of the American Statistical Association 94: 909–919

    Article  MathSciNet  MATH  Google Scholar 

  • Chen M.H., Ibrahim J.G., Sinha D. (2002) Bayesian inference for multivariate survival data with a cure fraction. Journal of Multivariate Analysis 80: 101–126

    Article  MathSciNet  MATH  Google Scholar 

  • Clayton D.G. (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65: 141–151

    Article  MathSciNet  MATH  Google Scholar 

  • Cooner F., Banerjee S., Carlin B.P., Sinha D. (2007) Flexible cure rate modeling under latent activation schemes. Journal of the American Statistical Association 102: 560–572

    Article  MathSciNet  MATH  Google Scholar 

  • Cox D.R. (1972) Regression models and life-tables (with discussion). Journal of the Royal Statistical Society, Series B 34: 187–220

    MATH  Google Scholar 

  • Diao G., Lin D.Y. (2006) Semiparametric variance-component models for linkage and association analyses of censored trait data. Genetic Epidemiology 30: 570–581

    Article  Google Scholar 

  • Glidden D.V. (2007) Pairwise dependence diagnostics for clustered failure-time data. Biometrika 94: 371–385

    Article  MathSciNet  MATH  Google Scholar 

  • Gray R.J., Tsiatis A.A. (1989) A linear rank test for use when the main interest is in differences in cure rates. Biometrics 45: 899–904

    Article  MATH  Google Scholar 

  • Kuk A.Y.C., Chen C.H. (1992) A mixture model combining logistic regression with proportional hazards regression. Biometrika 79: 531–541

    Article  MATH  Google Scholar 

  • Li C.S., Taylor J.M.G., Judy P.S. (2001) Identifiability of cure models. Statistics & Probability Letters 54: 389–395

    Article  MathSciNet  MATH  Google Scholar 

  • Locatelli I., Rosina A., Lichtenstein P., Yashin A. (2007) A correlated frailty model with long-term survivors for estimating the heritability of breast cancer. Statistics in Medicine 26: 3722–3734

    Article  MathSciNet  Google Scholar 

  • Maller R., Zhou X. (1996) Survival analysis with long-term survivors. Wiley, New York

    MATH  Google Scholar 

  • Murphy S.A., van der Vaart A.W. (2000) On profile likelihood. Journal of the American Statistical Association 95: 449–465

    Article  MathSciNet  MATH  Google Scholar 

  • Peng Y., Dear K.B.G. (2000) A nonparametric mixture model for cure rate estimation. Biometrics 56: 237–243

    Article  MATH  Google Scholar 

  • Peng Y., Taylor J.M.G., Yu B. (2007) A marginal regression model for multivariate failure time data with a surviving fraction. Lifetime Data Analysis 13: 351–369

    Article  MathSciNet  MATH  Google Scholar 

  • Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P. (1992). Numerical Recipes in C: The art of scientific computing (2nd ed.). Cambridge: Cambridge University Press.

  • Price D.L., Manatunga A.K. (2001) Modelling survival data with a cured fraction using frailty models. Statistics in Medicine 20: 1515–1527

    Article  Google Scholar 

  • Rudin W. (1973) Functional analysis. McGraw-Hill, New York

    MATH  Google Scholar 

  • Sy J.P., Taylor J.M.G. (2000) Estimation in a Cox proportional hazards cure model. Biometrics 56: 227–236

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor J.M.G. (1995) Semi-parametric estimation in failure time mixture models. Biometrics 51: 899–907

    Article  MATH  Google Scholar 

  • Tsodikov A.D. (1998) A proportional hazards model taking account of long-term survivors. Biometrics 54: 1508–1516

    Article  MATH  Google Scholar 

  • Tsodikov A.D., Ibrahim J., Yakovlev A.Y. (2003) Estimating cure rates from survival data: An alternative to two-component mixture models. Journal of the American Statistical Association 98: 1063–1078

    Article  MathSciNet  Google Scholar 

  • van der Vaart A.W., Wellner J.A. (1996) Weak convergence and empirical processes. Springer, New York

    MATH  Google Scholar 

  • Yakovlev, A. Y., Asselain, B., Bardou, V. J., Fourquet, A., Hoang, T., Rochefediere, A., Tsodikov, A. D. (1993). A simple stochastic model of tumour recurrence and its applications to data on premenopausal breast cancer. In B. Asselain, M. Boniface, C. Duby, C. Lopez, J. P. Masson, J. Tranchefort (Ed.), Biometrie et Analyse de Dormees Spatio-Temporelles (Vol. 12, pp. 66–82). France: Société Francaise de Biométrie, ENSA Renned.

  • Yau K.K.W., Ng A.S.K. (2001) Long-term survivor mixture model with random effects: Application to a multicentre clinical trial of carcinoma. Statistics in Medicine 20: 1591–1607

    Article  Google Scholar 

  • Yin G. (2008) Bayesian transformation cure frailty models with multivariate failure time data. Statistics in Medicine 27: 5929–5940

    Article  MathSciNet  Google Scholar 

  • Yin G., Ibrahim J. (2005) Cure rate models: A unified approach. The Canadian Journal of Statistics 33: 559–570

    Article  MathSciNet  MATH  Google Scholar 

  • Yu B., Peng Y. (2008) Mixture cure models for multivariate survival data. Computational Statistics and Data Analysis 52: 1524–1532

    Article  MathSciNet  MATH  Google Scholar 

  • Zeng D., Lin D.Y. (2007) Maximum likelihood estimation in semiparametric regression models with censored data (with discussion). Journal of the Royal Statistical Society: Series B 69: 507–564

    Article  MathSciNet  Google Scholar 

  • Zeng D., Lin D.Y., Lin X. (2008) Semiparametric transformation models with random effects for clustered failure time data. Statistica Sinica 18: 355–377

    MathSciNet  MATH  Google Scholar 

  • Zeng D., Yin G., Ibrahim J. (2006) Semiparametric transformation models for survival data with a cure fraction. Journal of American Statistical Association 101: 670–684

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, George Mason University MS 4A7, 4400 University Drive, Fairfax, VA, 22030, USA

    Guoqing Diao

  2. Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, China

    Guosheng Yin

Authors
  1. Guoqing Diao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guosheng Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guoqing Diao.

About this article

Cite this article

Diao, G., Yin, G. A general transformation class of semiparametric cure rate frailty models. Ann Inst Stat Math 64, 959–989 (2012). https://doi.org/10.1007/s10463-012-0354-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-012-0354-0

Keywords

Search

Navigation