The Single-Index/Cox Mixture Cure Model

Abstract

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.

A simulation-extrapolation approach for the mixture cure model with mismeasured covariates

“…where V is the covariance matrix of the error in (2). Distributions different from Gaussian can be used too but here we focus on normal errors.…”

“…where E (1) denotes the covariates in X that are not present in Z, E (2) denotes the common components of X and Z, E (3) denotes the covariates in Z that are not present in X and q 1 is the number of covariates in E (3) . In other words, we are removing the repeated covariates from the vector (X T , Z T ) T without loosing any information.…”

“…Choosing a parametric model for the incidence seems quite standard in the literature of mixture cure models ( [29,8,32]) because of its simplicity and ease of interpretability (particularly for multiple covariates). Recently there have been also a few nonparametric proposals ( [39,2,21]) but we choose to focus on the previous models since they are more widely used in practice ( [18,40,38,34]). In addition, nonparametric estimation of π 0 (•) would require repeating the simex procedure for multiple points of the support, which is very computationally intensive.…”

Cure models to estimate time until hospitalization due to COVID-19

“…Although parametric methods show some important advantages, such as their ease of interpretation or the simplicity of the parameter estimation, some authors have proposed semiparametric approaches to estimate the cure rate. Some of these semiparametric approaches are based on splines [ 29 ] or on single-index structures [ 30 ]. The flexibility can be further improved if completely nonparametric forms are introduced for the latency, being independent of any external covariate [ 31 ] or even taking these external factors into account [ 15 ], while the incidence is still modeled using parametric formulations.…”

Bivariate distributions based on copulas functions: developments and applications in medical studies

“…Other approaches to model data with cure rate can be found in Peng et al (1998) Rodrigues et al (2009, Rodrigues et al (2011), Gallardo et al (2018, Pescim et al (2019), Amico et al (2019) and Leão et al (2020).…”