Minimum Description Length Principle for Linear Mixed Effects Models

The minimum description length (MDL) principle originated from data compression literature and has been considered for deriving statistical model selection procedures. Most of the existing methods that use the MDL principle focus on models with independent data, particularly in the context of linear regression. This paper considers data with repeated measurements and studies the selection of fixed effect covariates for linear mixed effect models. We propose a class of MDL procedures which incorporate the dependence structure within individual or cluster and uses data-adaptive penalties that suit both finite and infinite dimensional data generating mechanisms. Theoretical justifications are provided from both data compression and statistical perspectives, where the covariance of random effects is treated as known or estimated by maximum likelihood. Numerical experiments are conducted to demonstrate the usefulness of the proposed MDL procedure and the influence of the estimated covariance, and an application to U.S. EPA data for air quality control is provided.