Subset selection for linear mixed models

Linear mixed models (LMMs) are instrumental for regression analysis with structured dependence, such as grouped, clustered, or multilevel data. However, selection among the covariates--while accounting for this structured dependence--remains a challenge. We introduce a Bayesian decision analysis for subset selection with LMMs. Using a Mahalanobis loss function that incorporates the structured dependence, we derive optimal linear actions for any subset of covariates and under any Bayesian LMM. Crucially, these actions inherit shrinkage or regularization and uncertainty quantification from the underlying Bayesian LMM. Rather than selecting a single "best" subset, which is often unstable and limited in its information content, we collect the acceptable family of subsets that nearly match the predictive ability of the "best" subset. The acceptable family is summarized by its smallest member and key variable importance metrics. Customized subset search and out-of-sample approximation algorithms are provided for more scalable computing. These tools are applied to simulated data and a longitudinal physical activity dataset, and in both cases demonstrate excellent prediction, estimation, and selection ability.