A Bayesian Log‐Cauchy Mixture Cure Fraction Model for Heavy‐Tailed Survival Data: Implementation via OpenBUGS - Khalid Salah (Statistics in Medicine, 2026)
ABSTRACT Bayesian estimation implemented via OpenBUGS is used to establish the novelty and practical relevance of the log‐Cauchy distribution within a mixture cure fraction modeling framework. Mixture cure fraction models are widely used in survival analysis to accommodate the presence of long‐term survivors; however, standard parametric assumptions often fail when survival data exhibit heavy‐tailed behavior and substantial right censoring. In such settings, conventional distributions such as the exponential, Weibull, gamma, and lognormal may lead to biased estimation and inadequate uncertainty quantification. In this paper, we propose a Bayesian mixture cure fraction model based on the log‐Cauchy distribution to flexibly model survival times of uncured subjects in the presence of extreme observations. Bayesian inference is conducted using Markov chain Monte Carlo (MCMC) methods implemented in OpenBUGS, allowing full posterior inference for model parameters through explicit specification of the log‐Cauchy likelihood. A comprehensive simulation study is performed to assess the finite‐sample performance of the proposed model under varying cure fractions and censoring levels. The results demonstrate accurate parameter estimation, stable MCMC convergence, and reliable posterior coverage, even in challenging scenarios characterized by high censoring and long‐term survival. Comparative analyses further show that the proposed model consistently outperforms competing parametric cure models in terms of estimation accuracy and model fit, particularly under heavy‐tailed survival settings. The practical utility of the model is further illustrated through application to a real breast cancer dataset, where the proposed approach provides improved model fit and interpretable inference compared with conventional parametric alternatives. Overall, the results confirm that the Bayesian log‐Cauchy mixture cure model, implemented via OpenBUGS, provides a robust, flexible, and computationally efficient framework for analyzing survival data with cure fractions and heavy‐tailed characteristics, with clear relevance for medical and epidemiological research.