  We propose a new model selection criterion for mixed effects regression
models that is computable when the model is fitted with a two-step method, even
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. We provide theoretical and
numerical arguments to justify the method and we illustrate its usefulness by
analyzing data on a socio-economic study of young American Indians.
