# Testing measurement and structural invariance in latent mediation models – A comparison of IPCR and Bayesian MNLFA

## TL;DR

This paper compares two methods for testing measurement and structural invariance in latent mediation models with continuous moderators using real and simulated data.

## Contribution

The paper introduces a Bayesian framework for MNLFA and compares it with IPCR for testing invariance in moderated mediation models.

## Key findings

- Both IPCR and Bayesian MNLFA can test invariance across continuous moderators.
- Simulation results show differences in parameter bias between the two methods.
- Bayesian MNLFA benefits from posterior predictive checks and cross-validation for model selection.

## Abstract

Moderated mediation models are frequently used in psychological research to examine direct, indirect, and total effects across an external moderating variable. When these models involve latent variables, measurement invariance should be tested first to ensure that measures function equivalently across subpopulations. If measurement invariance is violated, conclusions drawn about the moderation effects can be biased. However, measurement invariance is seldom tested across the moderator variable itself, especially if it is continuous. In this paper, we present two approaches that allow testing measurement and structural invariance simultaneously and across continuous covariates. They are termed individual parameter contribution regression (IPCR; Arnold et al., Structural Equation Modeling: A Multidisciplinary Journal, 27, 613–628, 2019) and moderated nonlinear latent factor analysis (MNLFA; Bauer & Hussong, Psychological Methods, 14(2), 101–125, 2009). We showcase both approaches with empirical data of \documentclass[12pt]{minimal}
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\begin{document}$$N = 399$$\end{document}N=399 couples in the German Family Panel (Brüderl et al., 2022). We show how MNLFA can be estimated in a Bayesian framework and explain Bayesian model selection with posterior predictive model checks and leave-one-out cross-validation (Vehtari et al., Statistics and Computing, 27(5), 1413–1432, 2017). Afterwards, we present the results of a simulation study comparing IPCR and Bayesian MNLFA with regard to parameter bias. We close with a comparison of both approaches regarding the empirical analysis and the simulation study and provide recommendations for applied researchers working with latent moderated mediation models.

The online version contains supplementary material available at 10.3758/s13428-025-02781-5.