On the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models

Sonja Greven, Johns Hopkins Bloomberg School of Public Health, Department of Biostatistics
Thomas Kneib, Ludwig-Maximilians-Universität Munich

A reworked and updated version of this technical report, including new reslts can be found at http://www.bepress.com/jhubiostat/paper202


The Akaike information criterion (AIC) is often used in regression models to determine the model specifcation most suitable for describing a specifc data generating mechanism. In particular, in linear mixed models, the AIC is frequently employed to differentiate between models including and excluding a specific random effect. An important special case is the model choice between linear and more general smooth terms, when nonparametric regression models using penalized splines are estimated as certain linear mixed models. Two versions of the AIC have been proposed for the linear mixed model. First, the marginal AIC derived from the implied marginal model, and second, the conditional AIC built upon the conditional model formulation. We investigate theoretical properties of both, and shed light on their differences. We find that the marginal AIC is no longer an asymptotically unbiased estimator for twice the expected relative Kullback-Leibler distance, and in fact favors smaller models without random effects. This behavior is related to recent findings on the non-standard asymptotics of likelihood ratio tests for variance components, which are on the boundary of the parameter space under the null hypothesis. For the conditional AIC, we show that while it is computationally costly for large sample sizes to correct for estimation uncertainty in the effective degrees of freedom, this uncertainty cannot be neglected even asymptotically. Ignoring it, as is common practice, induces a bias that yields the following behavior: Whenever the random effects variance is estimated to be positive, the more complex model is preferred, regardless of the value of the estimated variance. All theoretical results are illustrated in simulation studies, and their impact on practical data analyses is investigated in an application on childhood malnutrition in a developing country.