Defective models induced by gamma frailty term for survival data with cured fraction

ABSTRACT

In this paper, we propose a defective model induced by a frailty term for modeling the proportion of cured. Unlike most of the cure rate models, defective models have advantage of modeling the cure rate without adding any extra parameter in model. The introduction of an unobserved heterogeneity among individuals has bring advantages for the estimated model. The influence of unobserved covariates is incorporated using a proportional hazard model. The frailty term assumed to follow a gamma distribution is introduced on the hazard rate to control the unobservable heterogeneity of the patients. We assume that the baseline distribution follows a Gompertz and inverse Gaussian defective distributions. Thus we propose and discuss two defective distributions: the defective gamma-Gompertz and gamma-inverse Gaussian regression models. Simulation studies are performed to verify the asymptotic properties of the maximum likelihood estimator. Lastly, in order to illustrate the proposed model, we present three applications in real data sets, in which one of them we are using for the first time, related to a study about breast cancer in the A.C.Camargo Cancer Center, São Paulo, Brazil.

KEYWORDS:

• Breast cancer data
• defective distributions
• frailty models
• Gompertz distribution
• inverse Gaussian distribution
• long-term survivors
• survival analysis

Acknowledgments

The authors thank CAPES and FAPESP for the financial support received during the course of this project and A.C.Camargo Cancer Center for the breast cancer data set.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Juliana Scudilio http://orcid.org/0000-0003-3274-2407

Vinicius F. Calsavara http://orcid.org/0000-0003-2332-5863

Ricardo Rocha http://orcid.org/0000-0003-3057-9482

Francisco Louzada http://orcid.org/0000-0001-7815-9554

Vera Tomazella http://orcid.org/0000-0002-6780-2089

Agatha S. Rodrigues http://orcid.org/0000-0003-1112-8264

Additional information

Funding

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior and Fundação de Amparo à Pesquisa do Estado de São Paulo.