title: Facing off with Scylla and Charybdis: a comparison of scalar, partial, and the novel possibility of approximate measurement invariance.
abstract: Measurement invariance (MI) is a pre-requisite for comparing latent variable scores across groups. The current paper introduces the concept of approximate MI 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.
authors.affiliation: Department of Methods and Statistics, Faculty of Social Sciences, Utrecht University Utrecht, Netherlands ; Optentia Research Program, Faculty of Humanities, North-West University Vanderbijlpark, South Africa.
authors.forename: Rens
authors.initials: R
authors.lastname: van de Schoot
authors.forename: Anouck
authors.initials: A
authors.lastname: Kluytmans
authors.forename: Lars
authors.initials: L
authors.lastname: Tummers
authors.forename: Peter
authors.initials: P
authors.lastname: Lugtig
authors.forename: Joop
authors.initials: J
authors.lastname: Hox
authors.forename: Bengt
authors.initials: B
authors.lastname: Muthén
keywords: Bayesian structural equation modeling; Mplus; informative/subjective prior; measurement invariance; prior variance