Analysis of Copula Frailty defective models in presence of Cure Fraction

Introduction: Analyzing long term survivors such as diabetic patients can't be done using the usual survival models. One approach to analyze it is using defective distribution that doesn't force a pre-assumption of cure fraction to the model. To study more than one random variable interacting together, multivariate distributions may be used. However, most of multivariate distributions have complicated forms, which make the computations difficult. Besides, it may be hard to find a multivariate distribution that fits the data properly, especially in health care field. To get over this problem, one can use copula approach. In literature, to the best of our knowledge, only one paper handled copula defective models and didn't consider the effect of covariates. In this paper, we take into consideration not only existed covariates but also unobserved ones by including frailty term. Methods: Two new models are introduced. The first model, used Gumbel copula to take the dependence into consideration together with the observed covariates. The second one take into consideration not only the dependence but also the unobserved covariates by integrating frailty term in to the model. Results: A diabetic retinopathy data is analyzed. The two models indicated the existence of long-term survivals through negative parameters without the need of pre-assuming the existence of it. Including frailty term to the model helped in capturing more dependence between the variables. We compared the results using goodness of fit methods, and the results suggested that the model with frailty term is the best to be used. Conclusion: The two introduced models correctly detected the existence of cure fraction with less estimated parameters than that in mixture cure fraction models. Also, it has the advantage of not pre-assuming the existence of cure fraction to the model. comparing both models, the model with frailty term fitted the data better.