  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 coefficients for (i) any
given subset of variables and (ii) all subsets of variables that satisfy a
cardinality constraint. Crucially, these estimates inherit shrinkage or
regularization and uncertainty quantification from the underlying Bayesian
model, and apply for any well-specified Bayesian LMM. More broadly, our
decision analysis strategy deemphasizes the role of a single "best" subset,
which is often unstable and limited in its information content, and instead
favors a collection of near-optimal subsets. This collection is summarized by
key member subsets and variable-specific 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 demonstrate excellent prediction,
estimation, and selection ability.
