Abstract

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 by Yuen and Musta (2024), 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.