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Relaxed Poisson cure rate models. (English) Zbl 1381.62281
Summary: The purpose of this article is to make the standard promotion cure rate model [A. Yu. Yakovlev and A. D. Tsodikov, Stochastic models of tumor latency and their biostatistical applications. Singapore: World Scientific Publishing (1996; Zbl 0919.92024)] more flexible by assuming that the number of lesions or altered cells after a treatment follows a fractional Poisson distribution [N. Laskin, Commun. Nonlinear Sci. Numer. Simul. 8, No. 3–4, 201–213 (2003; Zbl 1025.35029)]. It is proved that the well-known Mittag-Leffler relaxation function [M. N. Berberan-Santos, J. Math. Chem. 38, No. 4, 629–635 (2005; Zbl 1101.33015)] is a simple way to obtain a new cure rate model that is a compromise between the promotion and geometric cure rate models allowing for superdispersion. So, the relaxed cure rate model developed here can be considered as a natural and less restrictive extension of the popular Poisson cure rate model at the cost of an additional parameter, but a competitor to negative-binomial cure rate models [the first author et al., J. Stat. Plann. Inference 139, No. 10, 3605–3611 (2009; Zbl 1173.62074)]. Some mathematical properties of a proper relaxed Poisson density are explored. A simulation study and an illustration of the proposed cure rate model from the Bayesian point of view are finally presented.

62P10 Applications of statistics to biology and medical sciences; meta analysis
62F15 Bayesian inference
Full Text: DOI
[1] Berberan-Santos, Properties of the Mittag-Leffler relaxation function, Journal of Mathematical Chemistry 38 pp 629– (2005) · Zbl 1101.33015
[2] Cahoy, Parameter estimation for fractional Poisson processes, Journal of Statistical Planning and Inference 140 pp 106– (2010) · Zbl 1205.62118
[3] Cancho, A flexible model for survival data with a cure rate: a Bayesian approach, Journal of Applied Statistics 38 pp 57– (2011)
[4] Cooner, Flexible cure rate modeling under latent activation schemes, Journal of the American Statistical Association 102 pp 560– (2007) · Zbl 1172.62331
[5] Cowles, Markov chain Monte Carlo convergence diagnostics: a comparative review, Journal of the American Statistical Association 91 pp 883– (1996) · Zbl 0869.62066
[6] Gamerman, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference (2006)
[7] Geweke, Evaluating the Accuracy of Sampling-based Approaches to Calculating Posterior Moments (1992)
[8] Gorenflo, Computation of the Mittag-Leffler function E{\(\alpha\)},{\(\beta\)}(z) and its derivative, Fractional Calculus and Applied Analysis 5 pp 491– (2002) · Zbl 1027.33016
[9] Ibrahim, Bayesian Survival Analysis (2001) · Zbl 0978.62091
[10] Jayakumar, The first-order autoregressive Mittag-Leffler process, Journal of Applied Probability 30 pp 462– (1993) · Zbl 0777.60063
[11] Jose, A count model based on Mittag-Leffler interarrival times, Statistica 4 pp 501– (2011)
[12] Kirkwood, High- and low-dose interferon alfa-2b in high-risk melanoma: First analysis of intergroup trial E1690/S9111/C9190, Journal of Clinical Oncology 18 pp 2444– (2000)
[13] Kokonendji, Connections of the Poisson weight function to overdispersion and underdispersion, Journal of Statistical Planning and Inference 138 pp 1287– (2008) · Zbl 1133.62007
[14] Laskin, Fractional Poisson process, Communication in Non Linear Science and Numerical Simulation 8 pp 201– (2003) · Zbl 1025.35029
[15] Mauro, Full characterization of the fractional Poisson process, Europhysics Letters 96 pp 20004– (2011)
[16] Mittag-Leffler, Une generalisation de lâintegrale de Laplace-Abel, Comptes Rendus de lAcadémie des Sciences, Series IIC 137 pp 537– (1903)
[17] Mudholkar, The exponentiated Weibull family: a reanalysis of the bus-motor failure data, Technometrics 37 pp 436– (1995) · Zbl 0900.62531
[18] Rodrigues, COM-Poisson cure rate survival models and an application to a cutaneous melanoma data, Journal of Statistical Planning and Inference 139 pp 3605– (2009a) · Zbl 1173.62074
[19] Rodrigues, On the unification of the long-term survival models, Statistics and Probability Letters 79 pp 753– (2009b) · Zbl 1349.62485
[20] Tucker, How well is the probability of tumor cure after fractionated irradiation described by Poisson statistics, Radiation Research 24 pp 273– (1990)
[21] Yakovlev, Stochastic Models of Tumor Latency and Their Biostatistical Applications (1996) · Zbl 0919.92024
[22] Yin, Cure rate models: a unified approach, Canadian Journal of Statistics 33 pp 559– (2005) · Zbl 1098.62127
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