Facing off with Scylla and Charybdis: a comparison of scalar, partial, and the novel possibility of approximate measurement invariance

TLDR

Measurement invariance is essential for comparing latent scores across groups, and recent work shows that Bayesian SEM can replace strict zero constraints with approximate zeros, enabling well‑fitting models even when exact invariance fails. This paper introduces approximate measurement invariance based on BSEM and offers a step‑by‑step guide for selecting the appropriate MI type. The authors employ BSEM in Mplus and use simulated data to determine when approximate MI can be applied and yields unbiased results. Empirical and simulation results demonstrate that approximate MI outperforms full or partial MI in detecting true latent mean differences with small intercept and loading variations, providing a viable alternative when strict or partial MI leads to poor fit.

Abstract

Measurement invariance (MI) is a prerequisite for comparing latent variable scores across groups. The current paper introduces the concept of approximate measurement invariance building on the work of Muthén and Asparouhov and their application of Bayesian Structural Equation Modeling (BSEM) in the software Mplus. They showed that with BSEM exact zeros constraints can be replaced with approximate zeros to allow for minimal steps away from strict MI, still yielding a well-fitting model. This new opportunity enables researchers to make explicit trade-offs between the degree of MI on the one hand, and the degree of model fit on the other. Throughout the paper we discuss the topic of approximate MI, followed by an empirical illustration where the test for MI fails, but where allowing for approximate MI results in a well-fitting model. Using simulated data, we investigate in which situations approximate MI can be applied and when it leads to unbiased results. Both our empirical illustration and the simulation study show approximate MI outperforms full or partial MI In detecting/recovering the true latent mean difference when there are (many) small differences in the intercepts and factor loadings across groups. In the discussion we provide a step-by-step guide in which situation what type of MI is preferred. Our paper provides a first step in the new research area of (partial) approximate MI and shows that it can be a good alternative when strict MI leads to a badly fitting model and when partial MI cannot be applied.