  Linear Mixed-Effects (LME) models are a fundamental tool for modeling
clustered data, including cohort studies, longitudinal data analysis, and
meta-analysis. The design and analysis of variable selection methods for LMEs
is considerably more difficult than for linear regression because LME models
are nonlinear. The approach considered here is motivated by a recent method for
sparse relaxed regularized regression (SR3) for variable selection in the
context of linear regression. The theoretical underpinnings for the proposed
extension to LMEs are developed, including consistency results, variational
properties, implementability of optimization methods, and convergence results.
In particular we provide convergence analyses for a basic implementation of SR3
for LME (called MSR3) and an accelerated hybrid algorithm (called MSR3-fast).
Numerical results show the utility and speed of these algorithms on realistic
simulated datasets. The numerical implementations are available in an open
source python package pysr3.
