Mixture Multilevel SEM versus Multilevel SEM for comparing structural relations across groups in presence of measurement non-invariance

Structural equation modeling (SEM) is commonly used to explore relationships between latent variables, such as beliefs and attitudes. However, comparing structural relations across a large number of groups, such as countries, can be challenging. Existing SEM approaches may fall short, especially when measurement non-invariance is present. In this project, we propose Mixture Multilevel SEM (MixML-SEM), a novel approach to comparing relationships between latent variables across many groups. MixML-SEM gathers groups with the same structural relations in a cluster, while accounting for measurement non-invariance in a parsimonious way by means of random effects. Specifically, MixML-SEM captures measurement non-invariance using multilevel CFA and, then, it estimates the structural relations and mixture clustering of the groups by means of the structural-after-measurement (SAM) approach. In this way, MixML-SEM ensures that the clustering is focused on structural relations and unaffected by differences in measurement. In contrast, multilevel SEM estimates measurement and structural models simultaneously, and both with random effects. If desired, a post-hoc clustering can be applied to the group-specific regression estimates derived from the random effects. In comparison to ML-SEM, MixML-SEM proves particularly advantageous when small groups are combined with large groups. This is because combining information from multiple groups within a cluster leads to more accurate estimates of the structural relations, whereas, in case of ML-SEM, these estimates are affected by shrinking. We demonstrate the advantages of MixML-SEM through simulations and an empirical example on how social pressure to be happy relates to life satisfaction.