A Penalized Bayesian Semiparametric Cumulative Hazard Transformation Model for Long-Term Survival

In this paper, we propose a unified Bayesian semiparametric transformation cure model that simultaneously relaxes proportionality assumptions and accommodates long-term survivors within a coherent inferential framework. The proposed approach integrates three components: (i) a cumulative hazard transformation model that generalizes proportional hazards, proportional odds, and related structures; (ii) a mixture cure formulation that separates incidence (probability of cure) and latency (susceptible survival) mechanisms; and (iii) a nonparametric baseline cumulative hazard specified through positive increments at observed event times, regularized via a second-order random-walk prior on the log-scale. To enhance structural adaptability, the transformation function is modeled using a data-adaptive monotone spline representation, ensuring strict monotonicity while allowing nonlinear warping of cumulative risk. Posterior inference is conducted through a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm that jointly updates regression parameters, cure coefficients, transformation components, and baseline hazard increments.  As an illustration of proposed methodology, we analyze real survival data from patients diagnosed with pelvic sarcoma, a rare and aggressive malignancy characterized by substantial heterogeneity in long-term outcomes. The proposed model captures deviations from proportional hazards and provides clinically interpretable estimates of cure probability and covariate effects on latency.