Bayesian Regression Methods for Crossing Survival Curves

Abstract

In survival data analysis, the proportional hazards (PH), accelerated failure time (AFT), and proportional odds (PO) models are commonly used semiparametric models for the comparison of survivability in subjects. These models assume that the survival curves do not cross. However, in some clinical applications, the survival curves pertaining to the two groups of subjects under the study may cross each other, especially for long-duration studies. Hence, these three models stated above may no longer be suitable for making inference Yang and Prentice (2005) proposed a model which separately models the short-term and long-term hazard ratios nesting both PH and PO. This feature allows for the survival functions to cross. First, we study the estimation procedure in the Yang-Prentice model with regards to the two-sample case. We propose two different approaches: (1) Bayesian bootstrap and (2) smoothing methods. The first approach involves Bayesian bootstrap with likelihoods corresponding to binomial and Poisson forms while the second approach involves kernel smoothing methods as well as smoothing spline methods. A simulation is conducted to compare various methods under the two-sample case. Next, we extend the Yang-Prentice model to a regression version involving predictors and examine three likelihood approaches including Poisson form, pseudo-likelihood, and Bayesian smoothing. The effects of model misspecification on asymptotic relative efficiency are also studied empirically. The results from simulation studies indicate that the PH, AFT, and PO models are not robust to model misspecifications when the survival functions are allowed to cross.

Finally, we calculate the marginal density via variational methods to determine the Bayes factor. Either a full Bayesian or Bayesian approach is implemented to perform model selection. Both approaches accurately identify the correct model, even under slight misspecification, and are computationally more efficient than MCMC techniques.