Additive Bayesian variable selection under censoring and misspecification
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We study the effect and interplay of two important issues on Bayesian model selection (BMS): the presence of censoring and model misspecification. Misspecification refers to assuming the wrong model or functional effect on the response, or not recording truly relevant covariates. We focus on additive accelerated failure time (AAFT) models, as these are fairly flexible to capture covariate effects and facilitate interpreting our theoretical analysis. Also, as an important byproduct we exploit a direct link between AAFT and additive probit models to extend our results and algorithms to binary responses. We study BMS under different priors, including local priors and a prior structure that combines local and non-local priors as a means of enforcing sparsity. We show that, as usual, BMS asymptotically chooses the model of smallest dimension minimizing Kullback-Leibler divergence to the data-generating truth. Under mild conditions, such model contains any covariate that has predictive power for either the outcome or censoring times, and discards the remaining covariates. We characterize asymptotic Bayes factor rates and help interpret the role of censoring and misspecification: both have an asymptotically negligible effect on false positives, but their impact on power is exponential. We help understand the practical relevance of the latter via simple descriptions of early/late censoring and the drop in the predictive accuracy provided by covariates. From a methods point of view, we develop algorithms to capitalize on the AAFT tractability, a simple augmented Gibbs sampler to hierarchically explore the linear and non-linear effects of each covariate, and an implementation in the R package mombf. We conduct an extensive simulation study to illustrate the performance of the proposed methods and others based on the Cox model and likelihood penalties under misspecification and censoring.
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