Mixture cure rate models are commonly used to analyze lifetime data with long-term survivors. On the other hand, frailty models also lead to accurate estimation of coefficients by controlling the heterogeneity in survival data. Gamma frailty models are the most common models of frailty. Usually, the gamma distribution is used in the frailty random variable models. However, for survival data which are suitable for populations with a cure rate, it may be better to use a discrete distribution for the frailty random variable than a continuous distribution. Therefore, we proposed two models in this study. In the first model, continuous gamma as the distribution is used, and in the second model, discrete hyper-Poisson distribution is applied for the frailty random variable. Also, Bayesian inference with Weibull distribution and generalized modified Weibull distribution as the baseline distribution were used in the two proposed models, respectively. In this study, we used data of patients with gastric cancer to show the application of these models in real data analysis. The parameters and regression coefficients were estimated using the Metropolis with Gibbs sampling algorithm, so that this algorithm is one of the crucial techniques in Markov chain Monte Carlo simulation. A simulation study was also used to evaluate the performance of the Bayesian estimates to confirm the proposed models. Based on the results of the Bayesian inference, it was found that the model with generalized modified Weibull and hyper-Poisson distributions is a suitable model in practical study and also this model fits better than the model with Weibull and Gamma distributions.
BradburnMJClarkTGLoveSB, et al.
Survival analysis part II: multivariate data analysis–an introduction to concepts and methods.Br J Cancer2003;
89: 431–436.
2.
CoxDR.Regression models and life-tables. J R Stat Soc Ser B1972;
34: 187–202.
3.
BradburnMJClarkTGLoveSB, et al.
Survival analysis Part III: multivariate data analysis–choosing a model and assessing its adequacy and fit.Br J Cancer2003;
89: 605–611.
4.
MartinezEZAchcarJAJácomeAAA, et al.
Mixture and non-mixture cure fraction models based on the generalized modified Weibull distribution with an application to gastric cancer data.Comput Meth Programs Biomed2013;
112: 343–355.
5.
RafatiSBaneshiMRBahrampourA.Factors affecting long-survival of patients with breast cancer by non-mixture and mixture cure models using the Weibull, log-logistic and Dagum distributions: a Bayesian approach.Asian Pacific J Cancer Prev2020;
21: 485–490.
6.
CanchoVGRodriguesJde CastroM.A flexible model for survival data with a cure rate: a Bayesian approach. J Appl Stat2011;
38: 57–70.
7.
PetoRLeePNPaigeWS.Statistical analysis of the bioassay of continuous carcinogens.Br J Cancer1972;
26: 258–261.
8.
HoseiniMBahrampourAMirzaeeM.Comparison of weibull and lognormal cure models with cox in the survival analysis of breast cancer patients in Rafsanjan. J Res Health Sci2017;
17: 1–6.
9.
CarrascoJMFOrtegaEMMCordeiroGM.A generalized modified Weibull distribution for lifetime modeling. Comput Stat Data Anal2008;
53: 450–462.
10.
CorbiereFCommengesDTaylorJMG, et al.
A penalized likelihood approach for mixture cure models.Stat Med2009;
28: 510–524.
11.
De SouzaDCanchoVGRodriguesJ, et al.
Bayesian cure rate models induced by frailty in survival analysis.Stat Meth Med Res2017;
26: 2011–2028.
12.
DuchateauLJanssenP.The frailty model.
Berlin:
Springer Science & Business Media, 2007.
13.
VaupelJWMantonKGStallardE.The impact of heterogeneity in individual frailty on the dynamics of mortality.Demography1979;
16: 439–454.
14.
LancasterT.Econometric methods for the duration of unemployment. Econom J Econom Soc1979;
47: 939–956.
15.
NikaeenRKhalilianABahrampourA.Determining the effective factors on gastric cancer using frailty model in South-East and North of Iran. Iran J Heal Sci2017;
5: 35–48.
16.
HougaardP.Life table methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika1984;
71: 75–83.
17.
PriceDLManatungaAK.Modelling survival data with a cured fraction using frailty models.Stat Med2001;
20: 1515–1527.
18.
CaroniCCrowderMKimberA.Proportional hazards models with discrete frailty.Lifetime Data Anal2010;
16: 374–384.
19.
Sáez-CastilloAJConde-SánchezA.A hyper-Poisson regression model for overdispersed and underdispersed count data. Comput Stat Data Anal2013;
61: 148–157.
20.
AtaNÖzelG.Survival functions for the frailty models based on the discrete compound Poisson process. J Stat Comput Simul2013;
83: 2105–2116.
21.
DokhiMOhtakiMHiyamaE.A cure Weibull gamma-frailty survival model and its application to exploring the prognosis factors of neuroblastoma. Hiroshima J Med Sci2009;
58: 25–35.
22.
PengYTaylorJMG.Mixture cure model with random effects for the analysis of a multi-center tonsil cancer study.Stat Med2011;
30: 211–223.
23.
WienkeARipattiSPalmgrenJ, et al.
A bivariate survival model with compound Poisson frailty. Stat Med2010;
29: 275–283.
24.
PengYZhangJ.Estimation method of the semiparametric mixture cure gamma frailty model.Stat Med2008;
27: 5177–5194.
25.
ChoiSHuangX.A general class of semiparametric transformation frailty models for nonproportional hazards survival data.Biometrics2012;
68: 1126–1135.
26.
ScudilioJCalsavaraVFRochaR, et al.
Defective models induced by gamma frailty term for survival data with cured fraction. J Appl Stat2019;
46: 484–507.
27.
LeaoJLeivaVSauloH, et al.
Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data.Stat Med2018;
37: 4421–4440.
28.
TsodikovADIbrahimJGYakovlevAY.Estimating cure rates from survival data: an alternative to two-component mixture models.J Am Stat Assoc2003;
98: 1063–1078.
29.
JackmanS.Estimation and inference via Bayesian simulation: an introduction to Markov chain Monte Carlo. Am J Pol Sci2000;
44: 375–404.
30.
ChibSGreenbergE.Understanding the metropolis-hastings algorithm. Am Stat1995;
49: 327–335.
31.
BalkaJDesmondAFMcNicholasPD.Bayesian and likelihood inference for cure rates based on defective inverse Gaussian regression models. J Appl Stat2011;
38: 127–144.
32.
GamermanDLopesHF.Markov chain Monte Carlo: stochastic simulation for Bayesian inference.
Boka Raton, FL:
Chapman & Hall, 2006.
33.
HosseinnatajARezaBaneshiMBahrampourA.Mortality risk factors in patients with gastric cancer using Bayesian and ordinary Lasso logistic models: a study in the Southeast of Iran.Gastroenterol Hepatol Bed Bench2020;
13: 31–36.
34.
BardwellGECrowEL.A two-parameter family of hyper-Poisson distributions. J Am Stat Assoc1964;
59: 133–141.
35.
CoxDROakesD.Analysis of survival data.
London:
Taylor & Francis, 1984.
36.
BuchholzH.The confluent hypergeometric function: with special emphasis on its applications.
Berlin:
Springer Science & Business Media, 2013.
37.
MathaiAMSaxenaRK.Extensions of an Euler’s integral through statistical techniques. Math Nachrichten1971;
51: 1–10.
38.
DavisPJ.Leonhard Euler’s integral: a historical profile of the gamma function: In memoriam: Milton Abramowitz. Am Math Mon1959;
66: 849–869.
39.
BaryshnikovYGhristR.Euler integration over definable functions.Proc Natl Acad Sci2010;
107: 9525–9530.
40.
MallerRAZhouX.Survival analysis with long-term survivors.
New York:
Wiley, 1996.
41.
BoagJW.Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc Ser B1949;
11: 15–53.
42.
CarlinBPLouisTA.Bayes and empirical Bayes methods for data analysis.
Boca Raton, FL:
Chapman & Hall/CRC, 2000.
43.
YinGIbrahimJG.Cure rate models: a unified approach. Can J Stat2005;
33: 559–570.
44.
KimSChenMHDeyDK, et al.
Bayesian dynamic models for survival data with a cure fraction.Lifetime Data Anal2007;
13: 17–35.
45.
ChenM-HIbrahimJGSinhaD.A New Bayesian Model for Survival Data with a Surviving Fraction. J Am Stat Assoc1999;
94: 909–919.
46.
CastroM deCanchoVGRodriguesJ.A Bayesian long-term survival model parametrized in the cured fraction. Biometrical J2009;
51: 443–455.
47.
IbrahimJGChenMSinhaD.Bayesian survival analysis.
New York, NY:
Springer, 2001.
48.
SpiegelhalterDJBestNGCarlinBP, et al.
Bayesian measures of model complexity and fit. J R Stat Soc Ser B (Stat Methodol)2002;
64: 583–639.
49.
BrooksS.Discussion on the paper by Spiegelhalter, Best, Carlin and van der Linde. J R Stat Soc Ser B (Stat Methodol)2002;
64: 616–639.
