Survival analysis — also called time-to-event analysis — is central to medical research. Whether studying overall survival in cancer, time to relapse in addiction medicine, or graft survival after organ transplantation, the statistical challenge is the same: some patients have not yet experienced the event when the study ends (right censoring), the hazard may change over time, and multiple risk factors interact. Bayesian methods bring three key advantages: natural incorporation of prior information from earlier trials, exact inference for complex models with competing risks and frailties, and posterior predictive distributions that directly answer clinical questions.
Bayesian Cox Proportional Hazards Model
The Cox model relates the hazard function to covariates through a log-linear model, leaving the baseline hazard unspecified. In the Bayesian version, regression coefficients receive prior distributions, and the baseline hazard is modeled nonparametrically (using a gamma process prior or piecewise exponential model) or semiparametrically. The posterior distribution over hazard ratios provides credible intervals that are directly interpretable as probability statements — unlike frequentist confidence intervals.
β ~ N(μ₀, Σ₀) (prior on regression coefficients)
h₀(t) ~ Gamma Process(c · H₀(t), c) (prior on cumulative baseline hazard)
Parametric and Flexible Models
When stronger assumptions are warranted, parametric Bayesian survival models — Weibull, log-normal, log-logistic, generalized gamma — provide smooth hazard functions and enable extrapolation beyond the observed follow-up period. This extrapolation capability is critical for health technology assessment, where decision-makers need lifetime cost-effectiveness projections. Bayesian model averaging across parametric families quantifies structural uncertainty about the hazard shape, producing more honest long-term predictions.
Patients face multiple potential events: a cancer patient may die from the tumor, from treatment toxicity, or from unrelated causes. Bayesian competing risks models simultaneously estimate cause-specific hazards, properly accounting for the dependence structure. Multi-state models extend this to complex disease pathways — healthy to diagnosed to treated to remission to relapse — with Bayesian inference providing transition probability estimates with full uncertainty.
Frailty and Hierarchical Models
Unobserved heterogeneity among patients — some are inherently more robust, others more frail — is captured by frailty models, which add random effects to the hazard function. Bayesian hierarchical frailty models naturally accommodate the multilevel structure of clinical data: patients nested within hospitals, hospitals within regions. The partial pooling of Bayesian hierarchical models shrinks extreme hospital-level estimates toward the overall mean, producing more stable hospital performance comparisons.
Dynamic Prediction and Joint Models
Joint models of longitudinal biomarkers and survival outcomes are a frontier where Bayesian methods excel. By linking a longitudinal submodel (tracking tumor markers, viral load, or functional status over time) with a survival submodel, clinicians can make dynamic predictions that update as new biomarker measurements arrive. The JMbayes package implements this in a fully Bayesian framework, producing individualized survival curves that evolve with each clinic visit.
"Bayesian survival analysis doesn't just estimate how long patients live — it tells us how uncertain we should be about that estimate, which is precisely the information clinicians need for honest conversations with patients." — Ibrahim, Chen, and Sinha, authors of Bayesian Survival Analysis
Current Frontiers
Bayesian cure-rate models handle the increasingly common situation where a fraction of patients are effectively cured. Bayesian landmark analysis and pseudo-observation methods provide dynamic treatment effect estimation. And the integration of survival models with causal inference frameworks — particularly Bayesian g-computation and marginal structural models — enables estimation of causal survival effects from observational data.