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

TL;DR

This work tackles the challenge of assessing whether follow-up is sufficiently long to identify a cure fraction in time-to-event data with censoring. It introduces a practically meaningful notion of sufficient follow-up based on a small tail probability $\epsilon$, and develops a nonparametric test that leverages the non-increasing tail density $f_u$ of the uncured survival distribution. The core methodology uses Grenander-type density estimators, both the classic (G) and a boundary-corrected kernel-smoothed variant (SG), with asymptotic theory and a smoothed bootstrap for critical values. Through extensive simulations and real data from two breast cancer studies, the proposed smoothed Grenander test generally achieves better level control and competitive power compared with existing extreme-value based tests, providing a robust, nonparametric tool for practitioners to judge follow-up sufficiency and to obtain more reliable cure fraction estimates. The approach offers practical guidance on choosing the tail threshold $\tau$ and the tolerance $\epsilon$, and shows promise for extension to covariates and more complex censoring structures in survival analysis.

Abstract

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.

Figures (20)

• Figure 1: Rejection rate of the null hypothesis of insufficient follow-up for different methods (solid: $\hat{f}_{nh}^{SG}$, dashed: $\hat{f}_{n}^{G}$, dotted: $Q_{n}$) in Setting \ref{['enum:sim_exp_unif']} with $n=500$, $\Delta G(\tau_{G})=0.02$ and $p=0.2$ (left), $p=0.6$ (center), $p=0.8$ (right).
• Figure 2: Rejection rate of the null hypothesis of insufficient follow-up for different methods (solid: $\hat{f}_{nh}^{SG}$, dashed: $\hat{f}_{n}^{G}$, dotted: $Q_{n}$) in Setting \ref{['enum:sim_exp_unif']} with $n=500$, $p=0.6$ and $\Delta G(\tau_{G})=0$ (left), 0.02 (center), and 0.2 (right).
• Figure 3: Rejection rate of the null hypothesis of insufficient follow-up for different methods (solid: $\hat{f}_{nh}^{SG}$, dashed: $\hat{f}_{n}^{G}$, dotted: $Q_{n}$) in Setting \ref{['enum:sim_exp_unif']} with $p=0.6$, $\Delta G(\tau_{G})=0.02$ and a sample size of 200 (left), 500 (center), and 1000 (right).
• Figure 4: Rejection rate of the null hypothesis of insufficient follow-up for different methods (solid: $\hat{f}_{nh}^{SG}$ with $\tau_{1}\approx4.5569$, dashed: $\hat{f}_{nh}^{SG}$ with $\tau_{2}=1.25\tau_{1}$, dash-dotted: $\hat{f}_{nh}^{SG}$ with $\tau_{3}=1.5\tau_{1}$, dotted: $Q_{n}$) in Setting \ref{['enum:sim_texp_unif']} with $n=500$, $p=0.6$ and $\Delta G(\tau_{G})=0$ (left), $\Delta G(\tau_{G})=0.02$ (center), and $\Delta G(\tau_{G})=0.2$ (right).
• Figure 5: Kaplan--Meier estimate (solid) and its least concave majorant (dotted) for the breast cancer observational study data.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Proposition 1
• Proposition 2
• Lemma 1
• proof : Proof of Theorem \ref{['thm:sg_normality']}
• proof : Proof of Proposition \ref{['prop:level_grenander']}
• proof : Proof of Proposition \ref{['prop:level_SG']}