Abstract
We propose completely nonparametric methodology to investigate location-scale modeling of two-component mixture cure models that is similar in spirit to accelerated failure time models, where the responses of interest are only indirectly observable due to the presence of censoring and the presence of long-term survivors that are always censored. We use nonparametric estimators of the location-scale model components that depend on a bandwidth sequence to propose an estimator of the error distribution function that has not been considered before in the literature. When this bandwidth belongs to a certain range of undersmoothing bandwidths, the proposed estimator of the error distribution function is root-n consistent. A simulation study investigates the finite sample properties of our approach, and the methodology is illustrated using data obtained to study the behavior of distant metastasis in lymph-node-negative breast cancer patients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aitkin M, Anderson D, Francis B, Hinde J (1989) Statistical modelling in GLIM. Clarendon Press, New York
Beran R (1981) Nonparametric regression with randomly censored survival data. Technical report
Boag J (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc Ser B Stat Methodol 11(1):15–44
Cai C, Zou Y, Peng Y, Zhang J (2012) smcure: an r-package for estimating semiparametric mixture cure models. Comput Methods Programs Biomed 108(3):1255–1260
Collett D (1994) Modelling survival data in medical research. CRC monographs on statistics & applied probability. CRC Press, Boca Raton
Dabrowska D (1987) Non-parametric regression with censored survival time data. Scand J Stat 14(3):181–197
Farewell V (1986) Mixture models in survival analysis: are they worth the risk? Can J Stat 14(3):257–262
González-Manteiga W, Crujeiras R (2013) An updated review of goodness-of-fit tests for regression models. TEST 22(3):361–411
Harris E, Albert A (1990) Survivorship analysis for clinical studies. Statistics: a series of textbooks and monographs. CRC Press, Boca Raton
Haybittle J (1959) The estimation of the proportion of patients cured after treatment for cancer of the breast. Br J Radiol 32(383):725–733
Haybittle J (1965) A two-parameter model for the survival curve of treated cancer patients. J Am Stat Assoc 60(309):16–26
Kuk A, Chen C (1992) A mixture model combining logistic regression with proportional hazards regression. Biometrika 79(3):531–541
Lawless J (1982) Statistical models and methods for lifetime data. Wiley series in probability and mathematical statistics: applied probability and statistics. Wiley, New York
Lázaro E, Armero C, Gómez-Rubio V (2019) Approximate Bayesian inference for mixture cure models. TEST
Li G, Datta S (2001) A bootstrap approach to nonparametric regression for right censored data. Ann Inst Stat Math 53(4):708–729
López-Cheda A, Cao R, Jácome M, Van Keilegom I (2017) Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Comput Stat Data Anal 105:144–165
Lu W (2008) Maximum likelihood estimation in the proportional hazards cure model. Ann Inst Stat Math 60(3):545–574
Lu W (2010) Efficient estimation for an accelerated failure time model with a cure fraction. Stat Sin 20(2):661–674
Patilea V, Van Keilegom I (2019) A general approach for cure models in survival analysis. Ann Stat (to appear)
Portier F, Van Keilegom I, El Ghouch A (2017) On an extension of the promotion time cure model. Ann Stat (under revision)
Sinha D, Chen M, Ibrahim J (2003) Bayesian inference for survival data with a surviving fraction. In: Kolassa JE, Oakes D (eds) Crossing boundaries: statistical essays in Honor of Jack Hall. Lecture notes-monograph series. Institute of Mathematical Statistics, Beachwood, pp 117–138
Stone C (1977) Consistent nonparametric regression. Ann Stat 5(4):595–620
Sy J, Taylor J (2000) Estimation in a Cox proportional hazards cure model. Biometrics 56(1):227–236
Taylor J (1995) Semi-parametric estimation in failure time mixture models. Biometrics 51(3):899–907
Tsodikov A (1998) A proportional hazards model taking account of long-term survivors. Biometrics 54(4):1508–1516
Tsodikov A, Ibrahim J, Yakovlev A (2003) Estimating cure rates from survival data: an alternative to two-component mixture models. J Am Stat Assoc 98(464):1063–1078
Van Keilegom I, Akritas M (1999) Transfer of tail information in censored regression models. Ann Stat 27(5):1745–1784
Wang Y, Klijn J, Zhang Y, Sieuwerts A, Look M, Yang F, Talantov D, Timmermans M, Meijer-van Gelder M, Yu J, Jatkoe T, Berns E, Atkins D, Foekens J (2005) Gene-expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer. Lancet 365(9460):671–679
Xu J, Peng Y (2014) Nonparametric cure rate estimation with covariates. Can J Stat 42(1):1–17
Yakovlev A, Tsodikov A (1996) Stochastic models of tumor latency and their biostatistical applications. Series in mathematical biology and medicine. World Scientific, Singapore
Yakovlev A, Cantor A, Shuster J (1994) Parametric versus non-parametric methods for estimating cure rates based on censored survival data. Stat Med 13(9):983–986
Yin G, Ibrahim J (2005) Cure rate models: a unified approach. Can J Stat 33(4):559–570
Zeng D, Yin G, Ibrahim J (2006) Semiparametric transformation models for survival data with a cure fraction. J Am Stat Assoc 101(474):670–684
Acknowledgements
The authors would like to thank and acknowledge the following sources of financial support. This research has been supported by the European Research Council (2016–2021, Horizon 2020/ERC Grant Agreement No. 694409), the IAP research network Grant No. P7/06 of the Belgian government (Belgian science policy), the Collaborative Research Center “Statistical modelling of nonlinear dynamic processes” (SFB 823, Project C1) of the German Research Foundation, and the Bundesministerium für Bildung und Forschung (project “MED4D: Dynamic medical imaging: Modeling and analysis of medical data for improved diagnosis, supervision, and drug development”).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Chown, J., Heuchenne, C. & Van Keilegom, I. The nonparametric location-scale mixture cure model. TEST 29, 1008–1028 (2020). https://doi.org/10.1007/s11749-019-00698-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-019-00698-8