A criterion-based model comparison statistic for structural equation models with heterogeneous data

Heterogeneous data are common in social, educational, medical and behavioral sciences. Recently, finite mixture structural equation models (SEMs) and two-level SEMs have been respectively proposed to analyze different kinds of heterogeneous data. Due to the complexity of these two kinds of SEMs, model comparison is difficult. For instance, the computational burden in evaluating the Bayes factor is heavy, and the Deviance Information Criterion may not be appropriate for mixture SEMs. In this paper, a Bayesian criterion-based method called the L"v measure, which involves a component related to the variability of the prediction and a component related to the discrepancy between the data and the prediction, is proposed. Moreover, the calibration distribution is introduced for formal comparison of competing models. Two simulation studies, and two applications based on real data sets are presented to illustrate the satisfactory performance of the L"v measure in model comparison.