Marginal and Conditional Akaike Information Criteria in Linear Mixed Models

In linear mixed models, the Akaike information criterion (AIC) is often used to decide on the inclusion of a random effect. An important special case is the choice between linear and nonparametric regression models estimated using mixed model penalized splines. We investigate the behavior of two commonly used versions of the AIC, derived either from the implied marginal model or the conditional model formulation. We find that the marginal AIC is not asymptotically unbiased for twice the expected relative Kullback-Leibler distance, and favors smaller models without random effects. For the conditional AIC, it is computationally costly for large sample sizes to correct for estimation uncertainty. However, ignoring it, as is common practice, induces a bias that yields the following behavior: Whenever the random effects variance estimate is positive (even if small), the more complex model is preferred. We illustrate our results in a simulation study, and investigate their impact in modeling childhood malnutrition in Zambia.