Skip to main content
SpringerLink
Log in

Piecewise Linear Approximations for Cure Rate Models and Associated Inferential Issues

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

Cure rate models offer a convenient way to model time-to-event data by allowing a proportion of individuals in the population to be completely cured so that they never face the event of interest (say, death). The most studied cure rate models can be defined through a competing cause scenario in which the random variables corresponding to the time-to-event for each competing causes are conditionally independent and identically distributed while the actual number of competing causes is a latent discrete random variable. The main interest is then in the estimation of the cured proportion as well as in developing inference about failure times of the susceptibles. The existing literature consists of parametric and non/semi-parametric approaches, while the expectation maximization (EM) algorithm offers an efficient tool for the estimation of the model parameters due to the presence of right censoring in the data. In this paper, we study the cases wherein the number of competing causes is either a binary or Poisson random variable and a piecewise linear function is used for modeling the hazard function of the time-to-event. Exact likelihood inference is then developed based on the EM algorithm and the inverse of the observed information matrix is used for developing asymptotic confidence intervals. The Monte Carlo simulation study demonstrates the accuracy of the proposed non-parametric approach compared to the results attained from the true correct parametric model. The proposed model and the inferential method is finally illustrated with a data set on cutaneous melanoma.

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

  • Arbutiski T (1985) A family of multiplicative survival models incorporating a long-term survivorship parameter c as a function of covariates. Comm Stat Theo Meth 14:1627–1642

    Article  Google Scholar 

  • Balakrishnan N, Pal S (2012) EM algorithm-based likelihood estimation for some cure rate models. J Stat Theo Prac 6:698–724

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Pal S (2013a) Lognormal lifetimes and likelihood-based inference for flexible cure rate models based on COM-Poisson family. Comput Stat Data Anal 67:41–67

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Pal S (2013b) Expectation maximization-based likelihood inference for flexible cure rate models with Weibull lifetimes. Stat Methods Med Res. doi:10.1177/0962280213491641. available online

  • Balakrishnan N, Pal S (2013c) COM-Poisson cure rate models and associated likelihood-based inference with exponential and Weibull lifetimes. In: Balakrishnan N., Pal S (eds) Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference, Chapter 22. Wiley, UK, pp 308–348

    Chapter  Google Scholar 

  • Berkson J, Gage RP (1952) Survival curves for cancer patients following treatment. J Am Stat Assoc 47:501–515

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Boag JM (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc Ser B 11:15–44

    MATH  Google Scholar 

  • Broet P, Rycke Y, 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 57:844–852

    Article  MathSciNet  MATH  Google Scholar 

  • Cantor AB, Shuster JJ (1992) Parametric versus non-parametric methods for estimating cure rates based on censored survival data. Stat Med 11:931–937

    Article  Google Scholar 

  • Castro MD, Cancho VG, Rodrigues J (2009) A Bayesian long-term survival model parametrized in the cured fraction. Biom J 51:443–455

    Article  MathSciNet  Google Scholar 

  • Chen MH, Ibrahim JG (2001) Maximum likelihood methods for cure rate models with missing covariates. Biometrics 57:43–52

    Article  MathSciNet  MATH  Google Scholar 

  • Chen MH, Ibrahim JG, Sinha D (1999) A new Bayesian model for survival data with a surviving fraction. J Am Stat Assoc 94:909–919

    Article  MathSciNet  MATH  Google Scholar 

  • de Freitas LA, Rodrigues J (2013) Standard exponential cure rate model with informative censoring. Comm Stat Sim Comp 42:8–23

    Article  MathSciNet  MATH  Google Scholar 

  • Fang HB, Li G, Sun J (2005) Maximum likelihood estimation in a semiparametric logistic/proportional-hazards mixture model. Scand J Stat 32:59–75

    Article  MathSciNet  MATH  Google Scholar 

  • Farewell VT (1977) A model for a binary variable with time-censored observations. Biometrika 64:43–46

    Article  MATH  Google Scholar 

  • Farewell VT (1982) The use of mixture models for the analysis of survival data with long-term survivors. Biometrics 38:1041–1046

    Article  Google Scholar 

  • Farewell VT (1986) Mixture models in survival analysis: are they worth the risk Can J Stat 14:257–262

    Article  MathSciNet  Google Scholar 

  • Ghitany ME (1993) On the information matrix of exponential mixture models with long-term survivors. Biom J 35:15–27

    Article  Google Scholar 

  • Ghitany ME, Maller RA (1992) Asymptotic results for exponential mixture models with long-term survivors. Statistics 23:321–336

    Article  MathSciNet  MATH  Google Scholar 

  • Goldman AI (1984) Survivorship analysis when cure is a possibility: a Monte Carlo study. Stat Med 3:153–163

    Article  Google Scholar 

  • Hu T, Xiang L (2013) Efficient estimation for semiparametric cure models with interval-censored data. J Multivar Anal 121:139–151

    Article  MathSciNet  MATH  Google Scholar 

  • Ibrahim JG, Chen MH, Sinha D (2001) Bayesian semiparametric models for survival data with a cure fraction. Biometrics 57:383–388

    Article  MathSciNet  MATH  Google Scholar 

  • Ibrahim JG, Chen MH, Sinha D (2005) Bayesian Survival Analysis. Wiley, New York

    Book  MATH  Google Scholar 

  • Kim S, Chen MH, Dey DK, Gamerman D (2007) Bayesian dynamic models for survival data with a cure fraction. Lifetime Data Anal 13:17–35

    Article  MathSciNet  MATH  Google Scholar 

  • Kim S, Xi Y, Chen MH (2009) A new latent cure rate marker model for survival data. Ann Appl Stat 3:1124–1146

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Larson MG, Dinse GE (1985) A mixture model for the regression analysis of competing risks data. Appl Stat 34:201–211

    Article  MathSciNet  Google Scholar 

  • Laska EM, Meisner MJ (1992) Nonparametric estimation and testing in a cure model. Biometrics 48:1223–1234

    Article  Google Scholar 

  • Li CS, Taylor JM, Sy JP (2001) Identifiability of cure models. Stat Prob Lett 54:389–395

    Article  MathSciNet  MATH  Google Scholar 

  • Liu H, Shen Y (2009) A semiparametric regression cure model for interval-censored data. J Am Stat Assoc 104:1168–1178

    Article  MathSciNet  MATH  Google Scholar 

  • Liu M, Lu W, Shao Y (2006) Interval mapping of quantitative trait loci for time-to-event data with the proportional hazards mixture cure model. Biometrics 62:1053–1061

    Article  MathSciNet  MATH  Google Scholar 

  • Lo YC, Taylor JM, McBride WH, Withers HR (1993) The effect of fractionated doses of radiation on mouse spinal cord. Int J Radiat Oncol Biol Phys 27:309–317

    Article  Google Scholar 

  • Lu W (2008) Maximum likelihood estimation in the proportional hazards cure model. Ann Inst Stat Math 60:545–574

    Article  MathSciNet  MATH  Google Scholar 

  • Maller RA, Zhou X (1992) Estimating the proportion of immunes in a censored sample. Biometrika 79:731–739

    Article  MathSciNet  MATH  Google Scholar 

  • Maller RA, Zhou X (1996) Survival Analysis with Long-term Survivors. Wiley, New York

    MATH  Google Scholar 

  • Maller RA, Zhou S (1995) Testing for the presence of immune or cured individuals in censored survival data. Biometrics 1197–1205

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

    Article  MATH  Google Scholar 

  • Peng Y, Dear KB, Denham JW (1998) A generalized F mixture model for cure rate estimation. Stat Med 17:813–830

    Article  Google Scholar 

  • Rodrigues J, de Castro M, Balakrishnan N, Cancho V G (2011) Destructive weighted Poisson cure rate models. Lifetime Data Anal 17:333–346

    Article  MathSciNet  MATH  Google Scholar 

  • Rodrigues J, Cancho V G, de Castro M, Louzada-Neto F (2009) On the unification of long-term survival models. Stat Prob Lett 79:753–759

    Article  MathSciNet  MATH  Google Scholar 

  • Sposto R, Sather HN, Baker SA (1992) A comparison of tests of the difference in the proportion of patients who are cured. Biometrics 48:87–99

    Article  Google Scholar 

  • Struthers CA, Farewell VT (1989) A mixture model for time to AIDS data with left truncation and an uncertain origin. Biometrika 76:814–817

    Article  MATH  Google Scholar 

  • Sy JP, Taylor JM (2000) Estimation in a Cox proportional hazards cure model. Biometrics 56:227–236

    Article  MathSciNet  MATH  Google Scholar 

  • Sy JP, Taylor JM (2001) Standard errors for the Cox proportional hazards cure model. Math Comput Model 33:1237–1251

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Tournoud M, Ecochard R (2008) Promotion time models with time-changing exposure and heterogeneity: application to infectious diseases. Biom J 50:395–407

    Article  MathSciNet  Google Scholar 

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

  • Tsodikov A (1998b) Asymptotic efficiency of a proportional hazards model with cure. Stat Prob Lett 39:237–244

    Article  MathSciNet  MATH  Google Scholar 

  • Tsodikov A (2001) Estimation of survival based on proportional hazards when cure is a possibility. Math Comput Model 33:1227–1236

    Article  MathSciNet  MATH  Google Scholar 

  • Tsodikov AD, Ibrahim JG, Yakovlev AY (2003) Estimating cure rates from survival data. J Am Stat Assoc 98:1063–1078

    Article  MathSciNet  Google Scholar 

  • Yakovlev AY, Cantor AB, Shuster JJ (1994) Parametric versus non-parametric methods for estimating cure rates based on censored survival data. Stat Med 13:983–986

    Article  Google Scholar 

  • Yakovlev AY, Tsodikov AD (1996) Stochastic Models of Tumor Latency and their Biostatistical Applications (Vol 1). World Scientific, Singapore

    Book  MATH  Google Scholar 

  • 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 87:284–292

    Google Scholar 

  • Zhao L, Feng D, Bellile EL, Taylor JM (2014) Bayesian random threshold estimation in a Cox proportional hazards cure model. Stat Med 33:650–661

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

    N. Balakrishnan & F. S. Milienos

  2. Department of Statistics and Insurance Science, University of Piraeus, 80, Karaoli & Dimitriou St., 18534, Piraeus, Greece

    M. V. Koutras

  3. Department of Mathematics, University of Texas at Arlington, Arlington, Texas, 76019, USA

    S. Pal

Authors
  1. N. Balakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. V. Koutras
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. S. Milienos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. S. Milienos.

Additional information

F. S. Milienos research is supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balakrishnan, N., Koutras, M.V., Milienos, F.S. et al. Piecewise Linear Approximations for Cure Rate Models and Associated Inferential Issues. Methodol Comput Appl Probab 18, 937–966 (2016). https://doi.org/10.1007/s11009-015-9477-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-015-9477-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation