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A Bayesian cure rate model with dispersion induced by discrete frailty
Commun. Stat. Appl. Methods, CSAM 2018;25:471-488
Published online September 30, 2018
© 2018 Korean Statistical Society.

Vicente G. Canchoa, Katherine E. C. Zavaletab, Márcia A. C. Maceraa, Adriano K. Suzuki1, a, Francisco Louzadaa

aDepartment of Applied Mathematics and Statistics, University of São Paulo, Brazil;
bDepartment of Statistics, Federal University of São Carlos, Brazil
Correspondence to: Department of Applied Mathematics and Statistics, University of São Paulo, Avenida Trabalhador São-carlense, 400 - Centro CEP: 13566-590, São Carlos-SP, Brazil. E-mail: suzuki@icmc.usp.br
Received February 20, 2018; Revised August 12, 2018; Accepted August 21, 2018.
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 Ψ-divergence measures. The motivating cutaneous melanoma data is analyzed for illustration purposes.
Keywords : Bayes factor, Bayesian inference, cure rate models, frailty models, Kullback-Leibler, zero-inflated power series distribution