Testing for sufficient follow-up in survival data with a cure fraction

In order to estimate the proportion of `immune' or `cured' subjects who will never experience failure, a sufficiently long follow-up period is required. Several statistical tests have been proposed in the literature for assessing the assumption of sufficient follow-up, meaning that the study duration is longer than the support of the survival times for the uncured subjects. These tests do not perform satisfactorily, especially in terms of Type I error. In addition, they are constructed based on the assumption that the survival time for the uncured subjects has a compact support, i.e. the existence of a `cure time'. However, for practical purposes, the assumption of `cure time' is not realistic and the follow-up would be considered sufficiently long if the probability for the event to happen after the end of the study is very small. Based on this observation, we formulate a more relaxed notion of `practically' sufficient follow-up characterized by the quantiles of the distribution and develop a novel nonparametric statistical test. The proposed method relies mainly on the assumption of a non-increasing density function in the tail of the distribution. The test is then based on a shape constrained density estimator such as the Grenander or the kernel smoothed Grenander estimator and a bootstrap procedure is used for computation of the critical values. The performance of the test is investigated through an extensive simulation study, and the method is illustrated on breast cancer data.