# Comparing Bayesian estimation and structural-after-measurement approaches for structural equation models with latent interactions and complex data structures

## TL;DR

This study compares Bayesian and structural-after-measurement approaches for estimating structural equation models with latent interactions and complex data structures.

## Contribution

The paper systematically compares Bayesian and SAM approaches in multilevel and partially nested SEMs with latent interactions.

## Key findings

- SAM approaches performed well across various latent interactions in multilevel and partially nested SEMs.
- Bayesian approaches struggled with cross-level latent interactions and were not easily extended to partially nested SEMs.
- SAM approaches are suggested as a versatile alternative or complement to conventional estimators.

## Abstract

Bayesian and structural-after-measurement (SAM) approaches have been developed, in part, to address limitations of conventional estimators in the context of structural equation models (SEMs) with latent interactions. Although both approaches have shown promise in a variety of contexts including small-sample studies, there is very little literature systematically comparing the relative benefits, limitations, and trade-offs among these approaches. In this study, we (a) compared the performance of estimators under each approach in multilevel SEMs with a within-, between-, or cross-level latent interaction and (b) demonstrated the flexibility of SAM approaches by extending and investigating them in partially nested SEMs with latent moderated mediation. The results suggest substantial differences between estimator performance as a function of the type of latent interaction. SAM approaches performed well with a variety of latent interactions in multilevel and partially nested SEMs, while Bayesian approaches, including those with informative priors, struggled with models that included a cross-level latent interaction and were not easily extended to partially nested SEMs. Overall, the results suggest that SAM approaches are a versatile and highly adaptable alternative or complement to conventional full-information estimators. To conclude, we outline estimator considerations based on the SEM type, latent interaction, and data structure.

The online version contains supplementary material available at 10.3758/s13428-025-02840-x.