Comparison of objective Bayes factors for variable selection in parametric regression models for survival analysis

Abstract

This paper considers the problem of selecting a set of regressors when the response variable is distributed according to a specified parametric model and observations are censored. Under a Bayesian perspective, the most widely used tools are Bayes factors (BFs), which are undefined when improper priors are used. In order to overcome this issue, fractional (FBF) and intrinsic (IBF) BFs have become common tools for model selection. Both depend on the size, Nt , of a minimal training sample (MTS), while the IBF also depends on the specific MTS used. In the case of regression with censored data, the definition of an MTS is problematic because only uncensored data allow to turn the improper prior into a proper posterior and also because full exploration of the space of the MTSs, which includes also censored observations, is needed to avoid bias in model selection. To address this concern, a sequential MTS was proposed, but it has the drawback of an increase of the number of possible MTSs as Nt becomes random. For this reason, we explore the behaviour of the FBF, contextualizing its definition to censored data. We show that these are consistent, providing also the corresponding fractional prior. Finally, a large simulation study and an application to real data are used to compare IBF, FBF and the well-known Bayesian information criterion.

Keywords: improper priors; intrinsic prior; model selection; survival analysis.