Bayesian analyses of multiple random change points in survival models with applications to clinical trials

Single and multiple random change points (RCPs) in survival analysis have arisen naturally in oncology trials, where the time to hazard rate change differs from one subject to another. Recently, Xu formulated and discovered important properties of these survival models using a frequentist approach, allowing us to estimate the hazard rates, rate parameters of the exponential distributions for the RCPs, expected survival and hazard functions. However, these methods did not provide an estimation of the uncertainty or the confidence intervals for the parameters and their differences or ratios. Therefore, statistical inferences were not able to be drawn on the parameters and their comparisons. To solve this issue, this article implements a Gibbs sampler method to estimate the above parameters and the differences or ratios alongside the 100(1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>-</mml:mo></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math>)% highest posterior density (HPD) intervals calculated from Chen-Shao's algorithm. The estimated rate parameters from the methods in Xu serve as empirical values in the Gibbs sampler method. Thus, formal statistical inferences can now be readily drawn. Simulation studies demonstrate that the proposed methods yield robust estimates, with the samples from the marginal posterior distributions converging rapidly and exhibiting favorable behavior. The 95% HPD intervals also demonstrate excellent coverage probabilities. This proposed method has a multitude of applications in clinical trials such as efficient clinical trial design and sample size adjustment based on the estimated parameter values at interim analyses.