A scalable and efficient covariate selection criterion for mixed effects regression models with unknown random effects structure

A new model selection criterion for mixed effects regression models is introduced. The criterion is computable even when the model is fitted with a two-step method or when the structure and the distribution of the random effects are unknown. The criterion is especially useful in the early stage of the model building process when one needs to decide which covariates should be included in a mixed effects regression model, but has no knowledge of the random effect structure. This is particularly relevant in substantive fields where variable selection is guided by information criteria rather than regularization. The calculation of the criterion requires only the evaluation of cluster-level log-likelihoods and does not rely on heavy numerical integration. Theoretical and numerical arguments are used to justify the method and its usefulness is illustrated by analysing data from a youth behaviour study.