Latency function estimation under the mixture cure model when the cure status is available.

This paper addresses the problem of estimating the conditional survival function of the lifetime of the subjects experiencing the event (latency) in the mixture cure model when the cure status information is partially available. The approach of past work relies on the assumption that long-term survivors are unidentifiable because of right censoring. However, in some cases this assumption is invalid since some subjects are known to be cured, e.g., when a medical test ascertains that a disease has entirely disappeared after treatment. We propose a latency estimator that extends the nonparametric estimator studied in López-Cheda et al. (TEST 26(2):353-376, 2017b) to the case when the cure status is partially available. We establish the asymptotic normality distribution of the estimator, and illustrate its performance in a simulation study. Finally, the estimator is applied to a medical dataset to study the length of hospital stay of COVID-19 patients requiring intensive care.

Nonparametric kernel estimation of the probability of cure in a mixture cure model when the cure status is partially observed.

Cure models are a class of time-to-event models where a proportion of individuals will never experience the event of interest. The lifetimes of these so-called cured individuals are always censored. It is usually assumed that one never knows which censored observation is cured and which is uncured, so the cure status is unknown for censored times. In this paper, we develop a method to estimate the probability of cure in the mixture cure model when some censored individuals are known to be cured. A cure probability estimator that incorporates the cure status information is introduced. This estimator is shown to be strongly consistent and asymptotically normally distributed. Two alternative estimators are also presented. The first one considers a competing risks approach with two types of competing events, the event of interest and the cure. The second alternative estimator is based on the fact that the probability of cure can be written as the conditional mean of the cure status. Hence, nonparametric regression methods can be applied to estimate this conditional mean. However, the cure status remains unknown for some censored individuals. Consequently, the application of regression methods in this context requires handling missing data in the response variable (cure status). Simulations are performed to evaluate the finite sample performance of the estimators, and we apply them to the analysis of two datasets related to survival of breast cancer patients and length of hospital stay of COVID-19 patients requiring intensive care.

A product-limit estimator of the conditional survival function when cure status is partially known.

We introduce a nonparametric estimator of the conditional survival function in the mixture cure model for right-censored data when cure status is partially known. The estimator is developed for the setting of a single continuous covariate but it can be extended to multiple covariates. It extends the estimator of Beran, which ignores cure status information. We obtain an almost sure representation, from which the strong consistency and asymptotic normality of the estimator are derived. Asymptotic expressions of the bias and variance demonstrate a reduction in the variance with respect to Beran's estimator. A simulation study shows that, if the bandwidth parameter is suitably chosen, our estimator performs better than others for an ample range of covariate values. A bootstrap bandwidth selector is proposed. Finally, the proposed estimator is applied to a real dataset studying survival of sarcoma patients.