CIRJE-F-575. Srivastava, Muni S. and Tatsuya Kubokawa, "Conditional Information Criteria for Selecting Variables in Linear Mixed Models", July 2008.

In this paper, we consider the problem of selecting the variables of the fixed effects in the linear mixed models where the random effects are present and the observation vectors have been obtained from many clusters. As the variable selection procedure, we here use the Akaike Information Criterion, AIC. In the context of the mixed linear models, two kinds of AIC have been proposed: the marginal AIC and the conditional AIC. In this paper, we show that the conditional AIC derived by Vaida and Blanchard is invalid. Thus, we derive three correct versions of the conditional AIC depending upon different estimators of the regression coefficients and the random effects. Through the simulation studies, it is shown that the conditional AIC's are superior to the marginal AIC in the sense of selecting the true model. Finally, the results are extended to the case when the random effects in all the clusters are of the same dimension but have a common unknown covariance matrix.