Bayesian Variable Selection With l 1 $$ {l}_1 $$ -Ball for Spatially Partly Interval-Censored Data

The objective of this study is to perform variable selection and parameter estimation for analyzing partly interval-censored data based on a proportional hazards model that incorporates spatial effects. To broaden the model's applicability across diverse scenarios, we consider two types of spatial structures: adjacency and distance information. Leveraging the differentiable properties of the $l_1$ -ball prior developed through projection-based methods, we have devised an efficient Bayesian algorithm by introducing latent variables and applying stochastic gradient Langevin dynamics principles. This algorithm can rapidly deliver results without resorting to complex sampling steps. Through simulations encompassing various scenarios, we have validated the performance of this method in both variable selection and parameter estimation. In our real data application, the proposed approach selects important variables associated with the emergence time of permanent teeth. Additionally, it identifies the spatial structure that best fits these data characteristics. This selection and identification are based on two Bayesian model selection criteria: the log pseudo-marginal likelihood and the deviance information criterion.