The Geometric Generalized Birnbaum–Saunders Model with long-Term Survivors
Abstract
Introduction: A cure rate survival model was developed based on the assumption that the number of competing reasons for the event of interest has the Geometric distribution and the time allocated to the event of interest follows the Generalized Birnbaum-Saunders distribution. Methods: The Geometric GB-S distribution was defined and two useful representations were represented for its density function which contributes to the creation of some mathematical properties. Furthermore, the parameters of the model with cure rate were estimated by using the maximum likelihood method. Results: Several simulations were performed and a real data set was analyzed from the medical area for different sample sizes and censoring percentages. Conclusion: By considering the advantages of the GGB-S model, the model can be implemented as an appropriate alternative to explain or predict the survival time for long-term individuals.
References
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1. Birnbaum ZW, Saunders SC. A new family of life distributions. Journal of Applied Probability. 1969;6(2):319-27.
2. Pescim RR, Cordeiro GM, Nararajah S, Demétrio CG, Ortega EM. The Kummer beta Birnbaum-Saunders: An alternative fatigue life distribution. Hacettepe Journal of Mathematics and Statistics. 2014;43:473-510.
3. Johnson NL, Kotz S, Balakrishnan N. Continuous univariate distributions, vol. 2 of wiley series in probability and mathematical statistics: applied probability and statistics. Wiley, New York; 1995.
4. Dıaz-Garcıa JA, Leiva V. A new family of life distributions based on Birnbaum-Saunders distribution. Technical report I-02-17 (PE/CIMAT), Mexico. www. cimat. mx/biblioteca/RepTec; 2002.
5. Dı́az-Garcı́a JA, Leiva-Sánchez Vc. A new family of life distributions based on the elliptically contoured distributions. Journal of Statistical Planning and Inference. 2005;128(2):445-57.
6. Owen W. Another look at the Birnbaum-Saunders distribution. Available from: http://www.stat.lanl.gov/MMR2004/Extended%20Abstract/WOwnn.pdf; 2004.
7. Owen WJ. A new three-parameter extension to the Birnbaum-Saunders distribution. IEEE Transactions on Reliability. 2006;55(3):475-9.
8. Leiva V. Chapter 1 - Genesis of the Birnbaum–Saunders Distribution. The Birnbaum-Saunders Distribution: Academic Press; 2016. p. 1-15.
9. Elsayed E, editor Accelerated Life Testing Model for a Generalized Birnbaum-Saunders Distribution. QUALITA2013; 2013.
10. Barros M, Paula GA, Leiva V. A new class of survival regression models with heavy-tailed errors: robustness and diagnostics. Lifetime Data Analysis. 2008;14(3):316-32.
11. Barros M, Paula GA, Leiva V. An R implementation for generalized Birnbaum–Saunders distributions. Computational Statistics & Data Analysis. 2009;53(4):1511-28.
12. Boag JW. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society Series B (Methodological). 1949;11(1):15-53.
13. Berkson J, Gage RP. Survival curve for cancer patients following treatment. Journal of the American Statistical Association. 1952;47(259):501-15.
14. Farewell VT. The use of mixture models for the analysis of survival data with long-term survivors. Biometrics. 1982:1041-6.
15. Kuk AY, Chen C-H. A mixture model combining logistic regression with proportional hazards regression. Biometrika. 1992;79(3):531-41.
16. Peng Y, Dear KB, Denham J. A generalized F mixture model for cure rate estimation. Statistics in Medicine. 1998;17(8):813-30.
17. Yakovlev AY, Tsodikov AD. Stochastic models of tumor latency and their biostatistical applications: World Scientific; 1996.
18. Chen M-H, Ibrahim JG, Sinha D. A new Bayesian model for survival data with a surviving fraction. Journal of the American Statistical Association. 1999;94(447):909-19.
19. Yin G, Ibrahim JG. A general class of Bayesian survival models with zero and nonzero cure fractions. Biometrics. 2005;61(2):403-12.
20. Cooner F, Banerjee S, Carlin BP, Sinha D. Flexible cure rate modeling under latent activation schemes. Journal of the American Statistical Association. 2007;102(478):560-72.
21. Rodrigues J, de Castro M, Cancho VG, Balakrishnan N. COM–Poisson cure rate survival models and an application to a cutaneous melanoma data. Journal of Statistical Planning and Inference. 2009;139(10):3605-11.
22. Castro Md, Cancho VG, Rodrigues J. A Bayesian Long‐term Survival Model Parametrized in the Cured Fraction. Biometrical Journal. 2009;51(3):443-55.
23. Borges P, Rodrigues J, Balakrishnan N. Correlated destructive generalized power series cure rate models and associated inference with an application to a cutaneous melanoma data. Computational Statistics & Data Analysis. 2012;56(6):1703-13.
24. Cancho VG, Louzada F, Barriga GD. The Geometric Birnbaum–Saunders regression model with cure rate. Journal of Statistical Planning and Inference. 2012;142(4):993-1000.
25. Hashimoto EM, Ortega EM, Cordeiro GM, Cancho VG. The Poisson Birnbaum–Saunders model with long-term survivors. Statistics. 2014;48(6):1394-413.
26. Cordeiro GM, Cancho VG, Ortega EM, Barriga GD. A model with long-term survivors: negative binomial Birnbaum-Saunders. Communications in Statistics-Theory and Methods. 2016;45(5):1370-87.
27. Meshkat M, Baghestani A, Zayeri F. The Poisson Generalized Birnbaum-Saunders Cure Model and Application in Breast Cancer Data. J Biom Biostat. 2018;9(1):389.
28. Taketomi N, Yamamoto K, Chesneau C, Emura T. Parametric distributions for survival and reliability analyses, a review and historical sketch. Mathematics. 2022;10(20):3907.
29. Casella G, Berger RL. Statistical inference: Duxbury Pacific Grove, CA; 2002.
30. Klein JP, Moeschberger ML. Survival analysis: techniques for censored and truncated data: Springer Science & Business Media; 2005.
31. Betts JT. Practical methods for optimal control and estimation using nonlinear programming: SIAM; 2010.
32. Coles S, Bawa J, Trenner L, Dorazio P. An introduction to statistical modeling of extreme values: Springer; 2001.
33. Dunn PK, Smyth GK. Randomized quantile residuals. Journal of Computational and Graphical Statistics. 1996;5(3):236-44.
34. Rigby RA, Stasinopoulos DM. Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society: Series C (Applied Statistics). 2005;54(3):507-54.
| Files | ||
| Issue | Vol 9 No 1 (2023) | |
| Section | Original Article(s) | |
| DOI | https://doi.org/10.18502/jbe.v9i1.13976 | |
| Keywords | ||
| Cure fraction models Generalized Birnbaum-Saunders distribution Geometric distribution Lifetime data. | ||
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How to Cite
1.
Baghestani AR, Zayeri F, Meshkat M. The Geometric Generalized Birnbaum–Saunders Model with long-Term Survivors. JBE. 2023;9(1):51-67.
