Bayesian Variable Selection in Semiparametric Proportional Hazards Model for High Dimensional Survival Data

Abstract

Variable selection for high dimensional data has recently received a great deal of attention. However, due to the complex structure of the likelihood, only limited developments have been made for time-to-event data where censoring is present. In this paper, we propose a Bayesian variable selection scheme for a Bayesian semiparametric survival model for right censored survival data sets. A special shrinkage prior on the coefficients corresponding to the predictor variables is used to handle cases when the explanatory variables are of very high-dimension. The shrinkage prior is obtained through a scale mixture representation of Normal and Gamma distributions. Our proposed variable selection prior corresponds to the well known lasso penalty. The likelihood function is based on the Cox proportional hazards model framework, where the cumulative baseline hazard function is modeled a priori by a gamma process. We assign a prior on the tuning parameter of the shrinkage prior and adaptively control the sparsity of our model. The primary use of the proposed model is to identify the important covariates relating to the survival curves. To implement our methodology, we have developed a fast Markov chain Monte Carlo algorithm with an adaptive jumping rule. We have successfully applied our method on simulated data sets under two different settings and real microarray data sets which contain right censored survival time. The performance of our Bayesian variable selection model compared with other competing methods is also provided to demonstrate the superiority of our method. A short description of the biological relevance of the selected genes in the real data sets is provided, further strengthening our claims.