Abstract
Cure rate models have been used in a number of fields. These models are applied to analyze survival data when the population has a proportion of subjects insusceptible to the event of interest. In this paper, we propose a new cure rate survival model formulated under a competing risks setup. The number of competing causes follows the negative binomial distribution, while for the latent times we posit the power piecewise exponential distribution. Samples from the posterior distribution are drawn through MCMC methods. Some properties of the estimators are assessed in a simulation study. A dataset on cutaneous melanoma is analyzed using the proposed model as well as some existing models for the sake of comparison.
Similar content being viewed by others
References
Aitkin M, Laird N, Francis B (1983) A reanalysis of the stanford heart transplant data. J Amer Stat Assoc 78:264–292
Balakrishnan N, Koutras MV, Milienos FS, Pal S (2016) Piecewise linear approximations for cure rate models and associated inferential issues. Methodol Comput Appl Probab 18:937–966
Berkson J, Gage RP (1952) Survival curve for cancer patients following treatment. J Am Stat Assoc 47:501–515
Boag JW (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc B 11:15–53
Breslow NE (1974) Covariance analysis of censored survival data. Biometrics 30:89–99
Cancho VG, Rodrigues J, de Castro M (2011) A flexible model for survival data with a cure rate: a Bayesian approach. J Appl Stat 38:57–70
Cancho VG, Louzada F, Ortega EM (2013) The power series cure rate model: An application to a cutaneous melanoma data. Commun Stat Simul Comput 42:586–602
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
Clark DE, Ryan LM (2002) Concurrent prediction of hospital mortality and length of stay from risk factors on admission. Health Serv Res 37:631–645
Conlon ASC, Taylor JMG, Sargent DJ (2014) Multi-state models for colon cancer recurrence and death with a cured fraction. Stat Med 33:1750–1766
de Castro M, Cancho VG, Rodrigues J (2009) A Bayesian long-term survival model parametrized in the cured fraction. Biom J 51:443–455
Demarqui FN, Loschi RH, Dey DK, Colosimo EA (2012) A class of dynamic piecewise exponential models with random time grid. J Stat Plann Inference 42:728–742
Gallardo DI, Gómez HW, Bolfarine H (2017a) A new cure rate model based on the Yule-Simon distribution with application to a melanoma data set. J Appl Stat 44:1153–1164
Gallardo DI, Gómez YM, Arnold BC, Gómez HW (2017b) The Pareto IV power series cure rate model with applications. SORT - Stat Oper Res Trans 41:297–318
Gallardo DI, Gómez YM, de Castro M (2018) A flexible cure rate model based on the polylogarithm distribution. J Stat Comput Simul 88:2137–2149
Gamernan D (1991) Dynamic Bayesian models for survival data. Appl Stat 40:63–79
Geisser S, Eddy WF (1979) A predictive approach to model selection. J Am Stat Assoc 74:153–160
Gelfand AE, Dey DK (1994) Bayesian model choice: asymptotics and exact calculations. J R Stat Soc B 56:501–514
Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7:457–511
Gómez YM, Gallardo DI, Arnold BC (2018) The power piecewise exponential model. J Stat Comput Simul 88:825–840
Gupta RD, Kundu D (2001) Exponentiated exponential family: An alternative to gamma and Weibull distributions. Biom J 43:117–130
Hardy GH, Littlewood JE, Pólya G (1952) Inequalities, 2 edn. Cambridge
Hashimoto EM, Ortega EMM, Cordeiro GM, Cancho VG (2014) The Poisson Birnbaum-Saunders model with long-term survivors. Statistics 48:1394–1413
Ibrahim JG, Chen MH, Sinha D (2001) Bayesian survival analysis. Springer, New York
Kim S, Chen MH, Dey DK, Gamerman D (2006) Bayesian dynamic models for survival data with a cure fraction. Lifetime Data Anal 13:17–35
Klein JP, van Houwelingen HC, Ibrahim JG, Scheike TH (eds) (2014) Handbook of survival analysis. Chapman & Hall/CRC, Boca Raton
Madi MT, Raqab MZ (2009) Bayesian inference for the generalized exponential distribution based on progressively censored data. Commun Stat Theory Methods 38:2016–2029
Maller RA, Zhou X (1996) Survival analysis with Long-Term survivors. Wiley, New York
McKeague IW, Tighiouart M (2000) Bayesian estimators for conditional hazard functions. Biometrics 56:1007–1015
Peng Y, Taylor JMG (2014) Cure models. In: Klein JP, van Houwelingen HC, Ibrahim JG, Scheike TH (eds) Handbook of survival analysis. Chapman & Hall/CRC, Boca Raton, pp 113–134
Piegorsch WW (1990) Maximum likelihood estimation for the negative binomial dispersion parameter. Biometrics 46:863–867
Plummer M (2017) JAGS Version 4.3.0 user manual. http://mcmc-jags.sourceforge.net
R Core Team (2018) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, http://www.R-project.org
Robert CP, Casella G (2004) Monte carlo statistical methods, 2nd edn. Springer, New York
Rodrigues J, Cancho VG, de Castro M, Louzada-Neto F (2009) On the unification of long-term survival models. Stat Probab Lett 79:753–759
Ross GJS, Preece DA (1985) The negative binomial distribution. Stat 34:323–336
Saha K, Paul S (2005) Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter. Biometrics 61:179–185
Sahu SK, Dey DK, Aslanidu H, Sinha D (1997) A Weibull regression model with gamma frailties for multivariate survival data. Lifetime Data Anal 3:123–137
Sinha D, Chen MH, Gosh SK (1999) Bayesian analysis and model selection for interval-censored survival data. Biometrics 55:585–590
Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc B 64:583–639
Su Y, Yajima M (2015) R2jags. R package version 0.5-7. Vienna, Austria: The Comprehensive R Archive Network. https://cran.r-project.org/package=R2jags
Tsodikov AD, Ibrahim JG, Yakovlev AY (2003) Estimating cure rates from survival data: an alternative to two-component mixture models. J Am Stat Assoc 98:1063–1078
Yakovlev AY, Tsodikov AD (1996) Stochastic models of tumor latency and their biostatistical applications. World Scientific, Singapore
Yin G, Ibrahim JG (2005) Cure rate models: a unified approach. Can J Stat 33:559–570
Zhao X, Zhou X (2008) Discrete0time survival models with long-term survivors. Stat Med 27:1261–1281
Acknowledgements
We would like to thank two reviewers for their comments, which helped us to improve our paper. The work of the first author is partially supported by CNPq, Brazil. Research carried out using the computational resources of the Center for Mathematical Sciences Applied to Industry (CeMEAI) funded by FAPESP, Brazil (grant 2013/07375-0).
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.
Rights and permissions
About this article
Cite this article
de Castro, M., Gómez, Y.M. A Bayesian Cure Rate Model Based on the Power Piecewise Exponential Distribution. Methodol Comput Appl Probab 22, 677–692 (2020). https://doi.org/10.1007/s11009-019-09728-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-019-09728-2