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On the unification of long-term survival models

Author

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  • Rodrigues, Josemar
  • Cancho, Vicente G.
  • de Castro, Mrio
  • Louzada-Neto, Francisco

Abstract

In this paper we extend the long-term survival model proposed by Chen etal. [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] via the generating function of a real sequence introduced by Feller [Feller, W., 1968. An Introduction to Probability Theory and its Applications, third ed., vol. I, Wiley, New York]. A direct consequence of this new formulation is the unification of the long-term survival models proposed by Berkson and Gage [Berkson, J., Gage, R.P., 1952. Survival cure for cancer patients following treatment. Journal of the American Statistical Association 47, 501-515] and Chen etal. (see citation above). Also, we show that the long-term survival function formulated in this paper satisfies the proportional hazards property if, and only if, the number of competing causes related to the occurrence of an event of interest follows a Poisson distribution. Furthermore, a more flexible model than the one proposed by Yin and Ibrahim [Yin, G., Ibrahim, J.G., 2005. Cure rate models: A unified approach. The Canadian Journal of Statistics 33, 559-570] is introduced and, motivated by Feller's results, a very useful competing index is defined.

Suggested Citation

  • Rodrigues, Josemar & Cancho, Vicente G. & de Castro, Mrio & Louzada-Neto, Francisco, 2009. "On the unification of long-term survival models," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 753-759, March.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:6:p:753-759
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    Cited by:

    1. Borges, Patrick & Rodrigues, Josemar & Balakrishnan, Narayanaswamy, 2012. "Correlated destructive generalized power series cure rate models and associated inference with an application to a cutaneous melanoma data," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1703-1713.
    2. Amanda D’Andrea & Ricardo Rocha & Vera Tomazella & Francisco Louzada, 2018. "Negative Binomial Kumaraswamy-G Cure Rate Regression Model," JRFM, MDPI, vol. 11(1), pages 1-14, January.
    3. Carvalho Lopes, Celia Mendes & Bolfarine, Heleno, 2012. "Random effects in promotion time cure rate models," Computational Statistics & Data Analysis, Elsevier, vol. 56(1), pages 75-87, January.
    4. Adriano Suzuki & Vicente Cancho & Francisco Louzada, 2016. "The Poisson–Inverse-Gaussian regression model with cure rate: a Bayesian approach and its case influence diagnostics," Statistical Papers, Springer, vol. 57(1), pages 133-159, March.
    5. Francisco Louzada & M�rio de Castro & Vera Tomazella & Jhon F.B. Gonzales, 2014. "Modeling categorical covariates for lifetime data in the presence of cure fraction by Bayesian partition structures," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(3), pages 622-634, March.
    6. Mauro Ribeiro de Oliveira & Fernando Moreira & Francisco Louzada, 2017. "The zero-inflated promotion cure rate model applied to financial data on time-to-default," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1395950-139, January.
    7. Mário Castro & Yolanda M. Gómez, 2020. "A Bayesian Cure Rate Model Based on the Power Piecewise Exponential Distribution," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 677-692, June.
    8. Bao Yiqi & Cibele Maria Russo & Vicente G. Cancho & Francisco Louzada, 2016. "Influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(6), pages 1027-1060, May.
    9. Bader Aljawadi & Mohd Rizam A. Bakar & Noor Akma Ibrahim & Habshah Midi, 2011. "Parametric Estimation of the Cure Fraction Based on BCH Model Using Left-Censored Data with Covariates," Modern Applied Science, Canadian Center of Science and Education, vol. 5(3), pages 103-103, June.
    10. N. Balakrishnan & M. V. Koutras & F. S. Milienos & S. Pal, 2016. "Piecewise Linear Approximations for Cure Rate Models and Associated Inferential Issues," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 937-966, December.
    11. Janette Larney & James Samuel Allison & Gerrit Lodewicus Grobler & Marius Smuts, 2023. "Modelling the Time to Write-Off of Non-Performing Loans Using a Promotion Time Cure Model with Parametric Frailty," Mathematics, MDPI, vol. 11(10), pages 1-17, May.
    12. Vicente G. Cancho & Dipak K. Dey & Francisco Louzada, 2016. "Unified multivariate survival model with a surviving fraction: an application to a Brazilian customer churn data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(3), pages 572-584, March.
    13. Rocha, Ricardo & Nadarajah, Saralees & Tomazella, Vera & Louzada, Francisco, 2017. "A new class of defective models based on the Marshall–Olkin family of distributions for cure rate modeling," Computational Statistics & Data Analysis, Elsevier, vol. 107(C), pages 48-63.
    14. Diego I. Gallardo & Heleno Bolfarine & Atonio Carlos Pedroso-de-Lima, 2017. "A clustering cure rate model with application to a sealant study," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(16), pages 2949-2962, December.
    15. Elizabeth Hashimoto & Gauss Cordeiro & Edwin Ortega, 2013. "The new Neyman type A beta Weibull model with long-term survivors," Computational Statistics, Springer, vol. 28(3), pages 933-954, June.
    16. Alex Mota & Eder A. Milani & Jeremias Leão & Pedro L. Ramos & Paulo H. Ferreira & Oilson G. Junior & Vera L. D. Tomazella & Francisco Louzada, 2023. "A new cure rate frailty regression model based on a weighted Lindley distribution applied to stomach cancer data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 32(3), pages 883-909, September.
    17. Diego I. Gallardo & Mário de Castro & Héctor W. Gómez, 2021. "An Alternative Promotion Time Cure Model with Overdispersed Number of Competing Causes: An Application to Melanoma Data," Mathematics, MDPI, vol. 9(15), pages 1-14, July.
    18. Balakrishnan, N. & Pal, Suvra, 2013. "Lognormal lifetimes and likelihood-based inference for flexible cure rate models based on COM-Poisson family," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 41-67.
    19. Vicente G. Cancho & Márcia A. C. Macera & Adriano K. Suzuki & Francisco Louzada & Katherine E. C. Zavaleta, 2020. "A new long-term survival model with dispersion induced by discrete frailty," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(2), pages 221-244, April.

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