Mixture multilevel SEM vs. multilevel SEM for comparing structural relations across groups in presence of measurement non-invariance

Structural equation modeling (SEM) is commonly used to explore relations between latent variables, such as beliefs and attitudes. However, comparing structural relations across a large number of groups, such as countries or classrooms, can be challenging. Existing SEM approaches may fall short, especially when measurement non-invariance is present. In this paper, 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 confirmatory factor analysis and, then, it estimates the structural relations and mixture clustering of the groups by means of the structural-after-measurement 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 (ML-SEM) estimates measurement and structural models simultaneously, and both with random effects. In comparison to ML-SEM, MixML-SEM provides better estimates of the structural relations, especially when (some of) the groups are large. 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 shrinkage bias. We demonstrate the advantages of MixML-SEM through simulations and an empirical example on how social pressure to be happy relates to life satisfaction across 40 countries.