A cautionary note on the use of information fit indexes in covariance structure modeling with means

Information fit indexes such as Akaike Information Criterion, Consistent Akaike Information Criterion, Bayesian Information Criterion, and the expected cross validation index can be valuable in assessing the relative fit of structural equation models that differ regarding restrictiveness. In cases in which models without mean restrictions (i.e., saturated mean structure) are compared to models with restricted (i.e., modeled) means, one should take account of the presence of means, even if the model is saturated with respect to the means. The failure to do this can result in an incorrect rank order of models in terms of the information fit indexes. We demonstrate this point by an analysis of measurement invariance in a multigroup confirmatory factor model.