The single-index/Cox mixture cure model

In survival analysis, it often happens that a certain fraction of the subjects under study never experience the event of interest, that is, they are considered "cured." In the presence of covariates, a common model for this type of data is the mixture cure model, which assumes that the population consists of two subpopulations, namely the cured and the non-cured ones, and it writes the survival function of the whole population given a set of covariates as a mixture of the survival function of the cured subjects (which equals one), and the survival function of the non-cured ones. In the literature, one usually assumes that the mixing probabilities follow a logistic model. This is, however, a strong modeling assumption, which might not be met in practice. Therefore, in order to have a flexible model which at the same time does not suffer from curse-of-dimensionality problems, we propose in this paper a single-index model for the mixing probabilities. For the survival function of the non-cured subjects we assume a Cox proportional hazards model. We estimate this model using a maximum likelihood approach. We also carry out a simulation study, in which we compare the estimators under the single-index model and under the logistic model for various model settings, and we apply the new model and estimation method on a breast cancer data set.