Conference Agenda

In the field of survival data analysis, datasets that include both a cure fraction (i.e., individuals who will never experience the event of interest) and dependent censoring (loss to follow-up for a reason related to the event of interest before the occurrence of that event) are not scarce. It is therefore essential to consider appropriate models and methods in order to avoid biased estimators of the survival function or incorrect conclusions in clinical trials. Delhelle and Van Keilegom (2024) proposed a fully parametric mixture cure model for the bivariate distribution of survival and censoring times (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 significant advantage of this 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, the 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.

This talk presents an improvement of this model. Administrative censoring is considered separately from dependent censoring, and covariates are included in the model.

References
[1] Delhelle, M. and Van Keilegom, I. (2024), Copula based dependent censoring in cure models. TEST, (under revision), arXiv preprint arXiv:2403.07963.

In survival analysis, certain scenarios involve cases where not all individuals are at risk of experiencing the event of interest. Advanced methods have been developed to address such data, commonly referred to as cure model analysis. A cure model allows for direct modeling of the cure proportion and the effect of covariates on it. Based on how the cure proportion is introduced, cure models can be broadly categorized into two types: mixture cure models and non-mixture cure models. Mixture cure models enable the estimation of both the probability of being cured and the survival function for uncured individuals. A key objective in these models is to determine whether covariates influence the cure rate.

To develop a new nonparametric test, the focus shifts to a novel measure: distance correlation. Distance correlation has several important properties, including its applicability to vectors X and Y in arbitrary dimensions, and the fact that a distance correlation of zero characterizes independence. One notable extension of distance correlation is martingale difference correlation, which evaluates deviations from conditional mean independence between a scalar response variable Y and a vector predictor variable X. Moreover, martingale difference correlation and its empirical counterpart retain many advantageous properties of distance correlation and sample distance correlation.

This study proposes a new test for assessing the significance of multiple covariates, leveraging martingale difference correlation. The effectiveness of the proposed test is evaluated through a Monte Carlo simulation study under various scenarios, and the method is applied to a rheumatoid arthritis dataset.

In survival analysis, there are situations in which not all subjects are susceptible to the final event. For example, if the event is a cancer therapy-related adverse effect, there will be a fraction of patients (considered as cured) that will never experience it. Mixture cure models address this by estimating both the probability of cure and the survival function for the uncured subjects. In the literature, nonparametric estimation of these functions focuses on continuous univariate covariates. However, in clinical practice, it is common to collect several patient characteristics and even medical images. The R package sicure provides a set of functions related to the implementation of single-index mixture cure models, that can handle a vector covariate under the assumption that the survival function depends on it through an unknown linear combination that can be estimated by maximum likelihood. This approach can be easily extended to functional covariates. The implementation of a nonparametric estimator for the density function of the uncured individuals is also included. Although the use of this package is illustrated with a medical dataset, it may be useful in any other field that involves a time variable, an uncensoring indicator, more than one covariate and the presence of a cure fraction.

Survival data in the presence of a cure fraction has recently attracted growing interest from both methodological and application perspectives. To estimate the fraction of 'immune' or 'cured' subjects who will never experience the event of interest, it is necessary to have a sufficiently long follow-up period. A few statistical tests have been introduced to test the assumption of sufficient follow-up, indicating that the right extreme of the censoring distribution exceeds that of the survival time of the uncured subjects. A relaxed notion of 'practically' sufficient follow-up has been proposed recently, suggesting that the follow-up would be considered sufficiently long if the probability for the event happening after the end of the study is very small. However, all these tests do not consider covariate information, which might affect the cure rate and the survival times.
We develop a novel statistical test for 'practically’ sufficient follow-up that accounts for covariates. Our approach relies on the assumption that the density in the tail of the conditional distribution of uncured survival time is a non-increasing function of time given covariate. A kernel smoothed Grenander-type estimator for the non-increasing conditional density is used to construct the test statistics. We study the asymptotic normality of the test statistics and a bootstrap procedure is used to approximate the critical value of the test. The performance of the test is investigated through a simulation study, and we illustrate the practical use of the proposed method on a breast cancer dataset.