Testing for sufficient follow-up in cure models with categorical covariates

TL;DR

The paper extends the notion of practically sufficient follow-up from unconditional to conditional settings with categorical covariates in mixture cure models. It introduces two testing procedures: a conservative intersection-union method (M1) and a more powerful max-type approach (M2) that selects a covariate value via bootstrap to test the conditional tail-density criterion using a smoothed Grenander estimator. Under regularity, both tests achieve asymptotic level $\alpha$, with M1 prioritizing size control and M2 offering higher power, albeit with potential size inflation in borderline cases; extensive simulations and a SEER melanoma application illustrate their finite-sample behavior and practical utility. The methods provide a principled way to assess follow-up sufficiency across covariate strata, which is crucial for reliable conditional cure-rate estimation in survival analysis.

Abstract

In survival analysis, estimating the fraction of 'immune' or 'cured' subjects who will never experience the event of interest, requires a sufficiently long follow-up period. A few statistical tests have been proposed to test the assumption of sufficient follow-up, i.e. whether the right extreme of the censoring distribution exceeds that of the survival time of the uncured subjects. However, in practice the problem remains challenging. To address this, a relaxed notion of 'practically' sufficient follow-up has been introduced recently, suggesting that the follow-up would be considered sufficiently long if the probability for the event occurring after the end of the study is very small. All these existing tests do not incorporate covariate information, which might affect the cure rate and the survival times. We extend the test for 'practically' sufficient follow-up to settings with categorical covariates. While a straightforward intersection-union type test could reject the null hypothesis of insufficient follow-up only if such hypothesis is rejected for all covariate values, in practice this approach is overly conservative and lacks power. To improve upon this, we propose a novel test procedure that relies on the test decision for one properly chosen covariate value. Our approach relies on the assumption that the conditional density of the uncured survival time is a non-increasing function of time in the tail region. We show that both methods yield tests of asymptotically level $α$ and investigate their finite sample performance through simulations. The practical application of the methods is illustrated using a skin melanoma dataset.

Figures (1)

• Figure 1: Kaplan--Meier curves (solid) and their least concave majorants (dashed) for each subsample of the melanoma data: Distant and Female (Black), Regional and Female (Blue), Distant and Male (Green), Regional and Male (Red).

Theorems & Definitions (20)

• Remark 1
• Theorem 1
• Theorem 2
• Lemma 1: Lemma 1 of YM2024
• proof : Proof of Theorem \ref{['thm:sg_discrete_normality']}
• Lemma 2
• proof : Proof of Lemma \ref{['lemma:max_surv_time_discrete_conv']}
• proof : Proof of Theorem \ref{['theorem:test_discrete_level']}
• Proposition 1
• proof : Proof of Proposition \ref{['prop:DKW_ineq_disc_x']}