A Non-Mixture Cure Model with a Change Point in a Co-Variate for Right Censored Data

Abstract

We study a non-mixture cure model with a covariate change-point for right-censored survival data and develop maximum-likelihood estimation under a smoothed likelihood to handle the non-differentiability induced by the threshold. Assuming exponential latency for susceptibles, we derive closed-form scores through a stable reparameterization and jointly estimate the change-point, cure fraction, and rates. In simulations spanning multiple sample sizes, censoring levels, and covariate distributions, the estimator exhibits small bias and competitive RMSE, with accurate change-point recovery. To assess robustness, we further conduct sensitivity analyses under latency misspecification, in which susceptible failure times follow a Weibull distribution while the model is fitted assuming exponential latency; the results show that estimation of the cure fractions and change-point remains stable, whereas the latency rate parameters converge to pseudo-true values as expected under misspecification. We illustrate the method on two biomedical datasets: (i) the colon cancer dataset using the number of positive lymph nodes as the threshold covariate, and (ii) a melanoma cohort using Breslow thickness. In both applications, the fitted model provides clinically interpretable strata, with subgroup-specific cure fractions that are negligible to modest and a threshold estimate consistent with established prognostic cut-offs. These results demonstrate that the smoothed change-point non-mixture cure model is practical, interpretable, and reliable for detecting threshold effects in the presence of long-term survivors.