Choosing Priors in Bayesian Measurement Invariance Modeling: A Monte Carlo Simulation Study

ABSTRACT

Multi-group Bayesian structural equation modeling (MG-BSEM) gained considerable attention among substantive researchers investigating cross-group differences and methodologists exploring challenges in measurement invariance testing. MG-BSEM allows for greater flexibility by applying elastic rather than strict equality constraints on item parameters across groups. This, however, requires a specification of user-defined prior variances for cross-group differences in item parameters. Although prior selection in general Bayesian settings is well-studied, guidelines with respect to tuning the normal prior variances in MG-BSEM approximate measurement invariance (AMI) analysis are still largely missing. In a Monte Carlo simulation study we find that correctly specifying prior variances results in more precise credibility intervals (CI) and posterior standard deviations, while prior misspecification has little influence on point estimates. We compared the BIC, DIC, and PPP fit measures and found in our simulation scenarios that the DIC measure was most effective, when a proper threshold for model selection was applied.

KEYWORDS:

• Bayesian structural equation modeling (BSEM)
• crossgroup comparisons
• measurement invariance
• Monte Carlo simulation study

Acknowledgments

Eldad Davidov would like to thank the University of Zurich Research Priority Program Social Networks. The authors would like to thank Lisa Trierweiler for the English proof of the manuscript.

Notes

¹ There is a large body of literature on prior selection (see, e.g. Berger & Sun, 2008). In this paper we only refer to prior selection in AMI. We avoid the term “optimal prior choice”. Instead, we prefer the terms “tuned prior variance” of the group differences in MG-CFA models or simply “the correct or appropriate prior choice” for several reasons. First, our study covers a selected set of conditions, but there are many other possible conditions and scenarios not examined in our study. So, our guidelines may not apply to other conditions. Second, our guidelines, as will be shown later, are different for different conditions. Third, our study is based on Monte Carlo simulations, and thus does not provide any general theoretical proof about the “optimality” of the chosen prior.

² This guidance is obviously limited given the finite and small number of conditions included in our study.

³ In contrast to prior variances, item parameters in different groups are assumed to be independent of each other (with a zero covariance between them), therefore, $V\left(a-b\right)=V\left(a\right)+V\left(b\right),$ where v is the variance and a and b are the parameters.

⁴ Although Cronbach’s alpha has been criticized in the literature (see, e.g. Brown, 2015; Sijtsma, 2009), we use it here because it is still much in use by applied researchers.

⁵ For the classification, RMSE is defined as $\sqrt{1/R\sum _{r=1}^R{\left(Prio{r}_r-AMI\right)}^2}$ where $Prio{r}_r$ is selected by the procedure prior value, and $AMI$ is the level of simulated invariance, that is, the prior. R is the number of replications.

Additional information

Funding

This work has been prepared under the project Scales Comparability in Large Scale Cross-Country Surveys, which is funded by the Polish National Science Centre, as part of the grant competition Sonata 8 [UMO-2014/15/D/HS6/04934].