Multi-Level Variable Selection Using a BART-Enhanced Mixed-Effects Framework

Selecting important individual- and cluster-level predictors has become increasingly critical in healthcare research, where data often exhibit hierarchical structures due to collection from multiple clusters. Mixed-effects models, which account for within-cluster correlation and between-cluster heterogeneity, are a natural approach for multilevel variable selection. However, currently available variable selection methods for multilevel data are predominantly based on mixed-effects models that impose restrictive parametric assumptions, potentially limiting their utility when the underlying relationships are nonlinear or involve interactions. While nonparametric methods have shown promise for variable selection in non-clustered data, they have been much less studied in the multilevel setting. Moreover, nonparametric methods that explicitly account for multilevel structure have largely been designed for prediction, rather than for simultaneous selection of relevant covariates at both the individual and cluster levels. To address these limitations, we propose a flexible, fully Bayesian unified framework for simultaneous variable selection of both fixed and random effects. Our framework integrates the nonparametric flexibility of Bayesian Additive Regression Trees (BART) for fixed-effect predictor selection with a hierarchical Bayesian component that identifies random-effect predictors via covariance decomposition and permutation strategies. To address scenarios common in multilevel data, where cluster-level covariates are constant within clusters and can induce near-collinearity and instability in selection, we further propose a computationally efficient two-step procedure. This method disentangles the contributions of individual- and cluster-level predictors, thereby mitigating collinearity and improving stability in variable selection. Comprehensive simulation studies demonstrate the effectiveness and robustness of our proposed methods across diverse scenarios. We further illustrate the practical utility of these approaches by applying them to a multilevel Alzheimer's disease dataset.