Mixture multigroup structural equation modelling for ordinal data

Social scientists often compare groups in terms of relations between latent variables (LV) (i.e. structural relations) using Structural Equation Modelling (SEM). LVs are measured indirectly by questionnaires; thus, measurement invariance must be evaluated before comparisons can be made. To efficiently compare many groups, the recently proposed Mixture Multigroup SEM (MMG-SEM) clusters groups based on their structural relations while accounting for measurement (non-)invariance. However, the current MMG-SEM relies on standard SEM implementations of maximum likelihood (ML) estimation in R, which assume continuous indicators. This can introduce bias when dealing with ordinal data. In this paper, we extend MMG-SEM to accommodate ordinal data, relying on the stepwise Structural-After-Measurement estimation approach. In the first step, we implement a multigroup categorical confirmatory factor analysis (MG-CCFA) with diagonally weighted least squares (DWLS) to estimate the measurement model. The second step uses ML to perform the clustering and estimate cluster-specific structural relations. Two simulation studies compare (1) the performance of this approach to that of ML-based MMG-SEM and (2) model selection performance. The results show better recovery of measurement model parameters with DWLS, particularly with fewer response categories, whereas both approaches perform similarly in recovering the clusters and structural relations as well as in model selection.