DOI QR코드

DOI QR Code

A Bayesian cure rate model with dispersion induced by discrete frailty

  • Cancho, Vicente G. (Department of Applied Mathematics and Statistics, University of Sao Paulo) ;
  • Zavaleta, Katherine E.C. (Department of Statistics, Federal University of Sao Carlos) ;
  • Macera, Marcia A.C. (Department of Applied Mathematics and Statistics, University of Sao Paulo) ;
  • Suzuki, Adriano K. (Department of Applied Mathematics and Statistics, University of Sao Paulo) ;
  • Louzada, Francisco (Department of Applied Mathematics and Statistics, University of Sao Paulo)
  • Received : 2018.02.20
  • Accepted : 2018.08.21
  • Published : 2018.09.30

Abstract

In this paper, we propose extending proportional hazards frailty models to allow a discrete distribution for the frailty variable. Having zero frailty can be interpreted as being immune or cured. Thus, we develop a new survival model induced by discrete frailty with zero-inflated power series distribution, which can account for overdispersion. This proposal also allows for a realistic description of non-risk individuals, since individuals cured due to intrinsic factors (immunes) are modeled by a deterministic fraction of zero-risk while those cured due to an intervention are modeled by a random fraction. We put the proposed model in a Bayesian framework and use a Markov chain Monte Carlo algorithm for the computation of posterior distribution. A simulation study is conducted to assess the proposed model and the computation algorithm. We also discuss model selection based on pseudo-Bayes factors as well as developing case influence diagnostics for the joint posterior distribution through ${\psi}-divergence$ measures. The motivating cutaneous melanoma data is analyzed for illustration purposes.

Keywords

References

  1. Adamidis K and Loukas S (1998). A lifetime distribution with decreasing failure rate, Statistics & Probability Letters, 39, 35-42. https://doi.org/10.1016/S0167-7152(98)00012-1
  2. Ata N and Ozel G (2013). Survival functions for the frailty models based on the discrete compound Poisson process, Journal of Statistical Computation and Simulation, 83, 2105-2116. https://doi.org/10.1080/00949655.2012.679943
  3. Barral AM (2001). Immunological studies in malignant melanoma: importance of TNF and the thioredoxin system (Doctorate Thesis), Linkoping University, Linkoping.
  4. Barreto-Souza W, De Morais AL, and Cordeiro GM (2011). The Weibull-geometric distribution, Journal of Statistical Computation and Simulation, 81, 645-657. https://doi.org/10.1080/00949650903436554
  5. Barriga GDC and Louzada F (2014). The zero-inflated Conway-Maxwell-Poisson distribution: Bayesian inference, regression modeling and influence diagnostic, Statistical Methodology, 21, 23-34. https://doi.org/10.1016/j.stamet.2013.11.003
  6. Berkson J and Gage RP (1952). Survival curve for cancer patients following treatment, Journal of the American Statistical Association, 47, 501-515. https://doi.org/10.1080/01621459.1952.10501187
  7. Boag JW (1949). Maximum likelihood estimates of the proportion of patients cured by cancer therapy, Journal of the Royal Statistical Society Series B, 11, 15-53.
  8. Brooks SP (2002). Discussion on the paper by Spiegelhalter, Best, Carlin, and van der Linde (2002). Journal of the Royal Statistical Society Series B, 64, 616-618.
  9. Cancho VG, Louzada F, and Ortega EM (2013). The power series cure rate model: an application to a cutaneous melanoma data, Communications in Statistics-Simulation and Computation, 42, 586-602. https://doi.org/10.1080/03610918.2011.639971
  10. Caroni C, Crowder M, and Kimber A (2010). Proportional hazards models with discrete frailty, Lifetime Data Analysis, 16, 374-384. https://doi.org/10.1007/s10985-010-9151-3
  11. Casella G and George EI (1992). Explaining the Gibbs sampler, The American Statistician, 46, 167-174.
  12. Chahkandi M and Ganjali M (2009). On some lifetime distributions with decreasing failure rate, Computational Statistics & Data Analysis, 53, 4433-4440. https://doi.org/10.1016/j.csda.2009.06.016
  13. Chen MH, Ibrahim JG, and Sinha D (1999). A new Bayesian model for survival data with a surviving fraction, Journal of the American Statistical Association, 94, 909-919. https://doi.org/10.1080/01621459.1999.10474196
  14. Chib S and Greenberg E (1995). Understanding the Metropolis-Hastings algorithm, The American Statistician, 49, 327-335.
  15. Cho H, Ibrahim JG, Sinha D, and Zhu H (2009). Bayesian case influence diagnostics for survival models, Biometrics, 65, 116-124. https://doi.org/10.1111/j.1541-0420.2008.01037.x
  16. Choo-Wosoba H, Levy SM, and Datta S (2015). Marginal regression models for clustered count data based on zero-inflated Conway-Maxwell-Poisson distribution with applications, Biometrics, 2, 606-618.
  17. Consul PC and Jain GC (1973). A generalization of the Poisson distribution, Technometrics, 15, 791-799. https://doi.org/10.1080/00401706.1973.10489112
  18. Cook RD and Weisberg S (1982). Residuals and Influence in Regression, Chapman and Hall, New York.
  19. Cordeiro GM, Cancho VG, Ortega EMM, and Barriga GDC (2016). A model with long-term survivors: negative binomial Birnbaum-Saunders, Communications in Statistics-Theory and Methods, 45, 1370-1387. https://doi.org/10.1080/03610926.2013.863929
  20. Coskun K (2007). A new lifetime distribution, Computational Statistics & Data Analysis, 51, 4497-4509. https://doi.org/10.1016/j.csda.2006.07.017
  21. Cowles MK and Carlin BP (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review, Journal of the American Statistical Association, 91, 883-904. https://doi.org/10.1080/01621459.1996.10476956
  22. del Castillo J and Perez-Casany M (2005). Overdispersed and underdispersed Poisson generalizations, Journal of Statistical Planning and Inference, 134, 486-500. https://doi.org/10.1016/j.jspi.2004.04.019
  23. Dey DK and Birmiwal LR (1994). Robust Bayesian analysis using divergence measures. Statistics & Probability Letters, 20, 287-294. https://doi.org/10.1016/0167-7152(94)90016-7
  24. Eudes AM, Tomazella VLD, and Calsavara VF (2013). Modelagem de sobrevivencia com fracao de cura para dados de tempo de vida Weibull modificada, Revista Brasileira de Biometria, 30, 326-342.
  25. Geisser S and Eddy WF (1979). A predictive approach to model selection, Journal of the American Statistical Association, 74, 153-160. https://doi.org/10.1080/01621459.1979.10481632
  26. Gelfand AE, Dey DK, and Chang H (1992). Model determination using predictive distributions with implementation via sampling-based methods. In Bayesian Statistics: Proceedings of the Fourth Valencia International Meeting, April 15-20, 1991, volume 4, pages 147-167. Oxford University Press, USA.
  27. Gupta PL, Gupta RC, and Tripathi RC (1995). Inflated modified power series distributions with applications, Communications in Statistics-Theory and Methods, 24, 2355-2374. https://doi.org/10.1080/03610929508831621
  28. Hougaard P (1986). A class of multivariate failure time distributions, Biometrika, 73, 671-678.
  29. Ibrahim JG, Chen MH, and Sinha D (2005). Bayesian Survival Analysis, Springer, New York.
  30. Kirkwood JM, Ibrahim JG, Sondak VK, et al. (2000). High- and low-dose interferon alfa-2b in high-risk melanoma: first analysis of intergroup trial E1690/S9111/C9190, Journal of Clinical Oncology, 18, 2444-2458. https://doi.org/10.1200/JCO.2000.18.12.2444
  31. Li CS, Taylor JMG, and Sy JP (2001). Identifiability of cure models, Statistics & Probability Letters, 54, 389-395. https://doi.org/10.1016/S0167-7152(01)00105-5
  32. Milani EA, Tomazella VLD, Dias TCM, and Louzada F (2015). The generalized time-dependent logistic frailty model: an application to a population-based prospective study of incident cases of lung cancer diagnosed in Northern Ireland, Brazilian Journal of Probability and Statistics, 29, 132-144. https://doi.org/10.1214/13-BJPS232
  33. Moger TA, Aalen OO, Halvorsen TO, Storm HH, and Tretli S (2004). Frailty modelling of testicular cancer incidence using Scandinavian data, Biostatistics, 5, 1-14. https://doi.org/10.1093/biostatistics/5.1.1
  34. Morel JG and Neerchal NK (2012). Overdispersion models in SAS. SAS Institute Inc., Cary, NC.
  35. Morita LHM, Tomazella VL, and Louzada-Neto F (2016). Accelerated lifetime modelling with frailty in a non-homogeneous Poisson process for analysis of recurrent events data, Quality Technology & Quantitative Management, 1-21.
  36. Ortega EMM, Cordeiro GM, Campelo AK, Kattan MW, and Cancho VG (2015). A power series beta Weibull regression model for predicting breast carcinoma, Statistics in Medicine, 34, 1366-1388. https://doi.org/10.1002/sim.6416
  37. Peng F and Dey DK (1995). Bayesian analysis of outlier problems using divergence measures, Canadian Journal of Statistics, 23, 199-213. https://doi.org/10.2307/3315445
  38. R Development Core Team (2010). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
  39. Rodrigues J, Cancho VG, de Castro M, and Louzada-Neto F (2009a). On the unification of long-term survival models, Statistics & Probability Letters, 79, 753-759. https://doi.org/10.1016/j.spl.2008.10.029
  40. Rodrigues J, de Castro M, Cancho VG, and Balakrishnan N (2009b). COM-Poisson cure rate survival models and an application to a cutaneous melanoma data, Journal of Statistical Planning and Inference, 139, 3605-3611. https://doi.org/10.1016/j.jspi.2009.04.014
  41. Samani EB, Amirian Y, and Ganjali M (2012). Likelihood estimation for longitudinal zero-inflated power series regression models, Journal of Applied Statistics, 39, 1965-1974. https://doi.org/10.1080/02664763.2012.699951
  42. Spiegelhalter DJ, Best NG, Carlin BP, and van der Linde A (2002). Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society Series B, 64, 583-639. https://doi.org/10.1111/1467-9868.00353
  43. Sun FB and Kececloglu DB (1999). A new method for obtaining the TTT-plot for a censored sample. In Proceedings of the Annual Reliability and Maintainability Symposium, 112-116.
  44. Tahmasbi R and Rezaei S (2008). A two-parameter lifetime distribution with decreasing failure rate, Computational Statistics & Data Analysis, 52, 3889-3901. https://doi.org/10.1016/j.csda.2007.12.002
  45. Tsodikov AD, Ibrahim JG, and Yakovlev AY (2003). Estimating cure rates from survival data: an alternative to two-component mixture models, Journal of the American Statistical Association, 98, 1063-1078. https://doi.org/10.1198/01622145030000001007
  46. Van den Broek J (1995). A score test for zero inflation in a Poisson distribution, Biometrics, 51, 738-743. https://doi.org/10.2307/2532959
  47. Vaupel JW, Manton KG, and Stallard E (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality, Demography, 16, 439-454. https://doi.org/10.2307/2061224
  48. Weiss R (1996). An approach to Bayesian sensitivity analysis, Journal of the Royal Statistical Society Series B, 58, 739-750.
  49. Wienke A (2010). Frailty Models in Survival Analysis, Chapman and Hall/CRC, New York.
  50. Yakovlev AY and Tsodikov AD (1996). Stochastic Models of Tumor Latency and Their Biostatistical Applications, World Scientific, New Jersey.
  51. Yang Z, Hardin JW, Addy CL, and Vuong QH (2007). Testing approaches for overdispersion in Poisson regression versus the generalized Poisson model, Biometrical Journal, 49, 565-584. https://doi.org/10.1002/bimj.200610340