Copula based dependent censoring in cure models

In this paper we consider a time-to-event variable T that is subject to random right censoring, and we assume that the censoring time C is stochastically dependent on T and that there is a positive probability of not observing the event. There are various situations in practice in which this happens, and appropriate models and methods need to be considered to avoid biased estimators of the survival function or incorrect conclusions in clinical trials. In this work we propose a fully parametric mixture cure model for the bivariate distribution of (T, C), which deals with all these features. The model depends on a parametric copula and on parametric marginal distributions for T and C. A major advantage of our approach in comparison to existing approaches in the literature is that the copula which models the dependence between T and C is not assumed to be known, nor is the association parameter. Furthermore, our model allows for the identification and estimation of the cure fraction and the association between T and C, despite the fact that only the smallest of these variables is observable. Sufficient conditions are developed under which the model is identified, and an estimation procedure is proposed. The asymptotic behaviour of the estimated parameters is studied, and their finite sample performance is illustrated by means of a thorough simulation study and an analysis of breast cancer data.