Bayesian Cure Rate Model: Exploring Laplacian-P-Splines and Metropolis- Langevin-within-GIBBS Sampling Approaches

Abstract

Introduction: In the context of survival analysis, the mixture cure model provides a powerful framework for analyzing time-to-event data where a subset of the population may never experience the event of interest. This model assumes that the study population is composed of two latent subgroups: cured individuals, who will never encounter the event regardless of the follow-up duration, and susceptible individuals, who remain at risk and may eventually experience the event. Traditional survival models like Cox’s proportional hazards model are not designed to handle such data, thereby necessitating models that can estimate not only time-to-event but also the probability of cure. Aim & Objective: This paper aims to develop and evaluate a sampling-free, computationally efficient Bayesian inference approach for the mixture cure model by utilizing penalized B-splines and Laplace approximations. The objective is to achieve rapid and accurate posterior inference for both incidence and latency components of the model, without relying on computationally intensive Markov Chain Monte Carlo (MCMC) sampling. Methodology: We propose a novel approach called the Laplacian P-splines for Mixed Cure (LPSMC) model, which integrates Laplace approximations with P-spline-based hazard modeling. This method allows smooth estimation of survival curves and cure probabilities with credible intervals, while significantly reducing computation time. To benchmark the proposed approach, a fully stochastic algorithm, Metropolis-Langevin-within-Gibbs (MLWG) sampling, is implemented as a comparison baseline. Simulation studies are conducted using two sample sizes (200 and 300), each evaluated under two MCMC chain lengths (5000 and 7000), allowing for a thorough investigation of accuracy, uncertainty quantification, and convergence behavior across scenarios. Results: The simulation results demonstrate that the LPSMC method consistently provides accurate estimates of regression coefficients, cure fractions, and survival curves, with reduced bias, empirical standard error (ESE), and root mean square error (RMSE) as the sample size increases. Additionally, credible intervals obtained through Laplace approximations show coverage probabilities close to nominal levels. Compared to the MLWG sampler, the LPSMC approach achieves comparable statistical performance with a substantial gain in computational efficiency, making it a practical alternative for routine Bayesian inference in mixed cure models. Conclusion: The proposed LPSMC method effectively leverages the flexibility of penalized splines and the efficiency of Laplace’s method to provide a robust, scalable alternative to MCMC for posterior inference in cure models. This framework offers a valuable tool for survival analysis in settings where a cured proportion is present, especially in clinical research where both model interpretability and computational speed are essential.