Social scientists often compare groups in terms of relations between latent variables (LV) (often called structural relations) using Structural Equation Modeling (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 maximum likelihood (ML) estimation, which assumes continuous indicators. This can introduce bias when dealing with ordinal data, especially for variables with fewer item categories. 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 (MM). The second step uses ML to perform the clustering and estimate cluster-specific structural relations. A simulation study compares the performance of this approach to that of ML-based MMG-SEM under various conditions. The results show a better recovery of MM parameters with DWLS, particularly with fewer response categories, whereas both approaches perform similarly regarding the recovery of the clusters and structural relations.
Alonso et al. (Thu,) studied this question.