Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data

Abstract

Cure rate models have been widely studied to analyze time-to-event data with a cured fraction of patients. Our proposal consists of incorporating frailty into a cure rate model, as an alternative to the existing models to describe this type of data, based on the Birnbaum-Saunders distribution. Such a distribution has theoretical arguments to model medical data and has shown empirically to be a good option for their analysis. An advantage of the proposed model is the possibility to jointly consider the heterogeneity among patients by their frailties and the presence of a cured fraction of them. In addition, the number of competing causes is described by the negative binomial distribution, which absorbs several particular cases. We consider likelihood-based methods to estimate the model parameters and to derive influence diagnostics for this model. We assess local influence on the parameter estimates under different perturbation schemes. Deriving diagnostic tools is needed in all statistical modeling, which is another novel aspect of our proposal. Numerical evaluation of the considered model is performed by Monte Carlo simulations and by an illustration with melanoma data, both of which show its good performance and its potential applications. Particularly, the illustration confirms the importance of statistical diagnostics in the modeling.

Birnbaum‐Saunders distribution: A review of models, analysis, and applications

“…Several different models relating to univariate and multivariate BS distributions can be found in the works of Athayde et al, Balakrishnan and Saulo, Barros et al, Bhatti, Cancho et al, Cordeiro et al, Cordeiro and Lemonte, Desmond et al, Díaz‐García and Domínguez‐Molina, Díaz‐García and Leiva, Ferreira et al, Fierro et al, Fonseca and Cribari‐Neto, Genç, Gomes et al, Guiraud et al, Gupta and Akman, Hashimoto et al, Jamalizadeh and Kundu, Khosravi et al, Kundu, Leiva et al, Lemonte and Cordeiro, Lemonte and Ferrari, Martínez‐Flórez et al, Marchant et al, Olmos et al, Onar and Padgett, Ortega et al, Owen, Owen and Padgett, Park and Padgett, Patriota, Pescim et al, Pourmousa et al, Raaijmakers, Romeiro et al, Reina et al, Reyes et al, Sanhueza et al, Vilca and Leiva‐Sánchez, Volodin and Dzhungurova, and Ziane et al Recently, some survival analytic methods have been developed based on BS and related models. For example, one may refer to the works of Leão, Leão et al, and Balakrishnan and Liu . This is one direction in which there is a lot more research work that could be carried out.…”

A new cure rate model with flexible competing causes with applications to melanoma and transplantation data

“…References 23,24 discuss some strategies to accommodate zero frailty, for example, if W = 0 is used to describe long-term survivors, then units will never fail. Other examples of this approach are given in References 12,[25][26][27][28] A third and recent way to introduce cure rate models is known in the literature as the improper models introduced by Balka et al 29 In these models, the parameter space of the distribution is extended producing an improper model in which the cumulative distribution function no longer approaches 1, but to p ∈ (0, 1). More details in improper models extensions can be found in References 30-34.…”

“…References 23,24 discuss some strategies to accommodate zero frailty, for example, if W = 0 is used to describe long‐term survivors, then units will never fail. Other examples of this approach are given in References 12,25‐28.…”

Multivariate Birnbaum-Saunders Distributions: Modelling and Applications

“…The third period (2011 to the present) is characterized by a new inventiveness, breaking the link with lifetime data analysis and hence extended application in new areas such as: biology, crop yield assessment, econometrics, energy production, forestry, industry, informatics, insurance, inventory management, medicine, psychology, neurology, pollution monitoring, quality control, sociology and seismology; see, for example, Bhatti (2010); Kotz et al (2010); Balakrishnan et al (2011); Leiva et al (2010Leiva et al ( , 2011Leiva et al ( , 2012; Vilca et al (2010); Villegas et al (2011); Azevedo et al (2012); Ferreira et al (2012); Paula et al (2012); Santos-Neto et al (2012; Marchant et al (2013; Saulo et al (2013Saulo et al ( , 2018; Barros et al (2014); Rojas et al (2015); Wanke and Leiva (2015); Bourguignon et al (2017); Garcia-Papani et al (2017); ; Lillo et al (2018) and the references therein. In addition, risk and hazard analysis applications, in engineering and medicine, using the BS distribution were performed by Bebbington et al (2008); Kundu et al (2008); Azevedo et al (2012); Athayde (2017); Leão et al (2017Leão et al ( , 2018aLeão et al ( , 2018b; Athayde et al (2018) and Desousa et al (2018). Furthermore, the issue of robust parameter estimation has been considered, for example, by Wang et al (2013Wang et al ( , 2015 and Lemonte (2016).…”