Psychometric Society Webinar Series

Bayesian covariance structure modelling for measurement invariance testing


Presenter: Jean-Paul Fox, University of Twente

Bayesian covariance structure modeling (BCSM) is a new approach for modeling clustered data. In this multivariate modeling approach a dependence structure is directly modeled through a structured covariance matrix. The BCSM can be used to model measurement (in)variance, which shows to have many advantages over traditional methods. The (cluster) effects of group-specific item parameters (Verhagen and Fox, 2013) are described by modeling their dependencies on (clustered) item response observations through a structured covariance matrix. The covariance parameters of the structured covariance matrix can be tested with the Bayes factor to test for (uniform and/or non-uniform) measurement invariance.

The BCSM for measurement invariance testing is defined for mixed response types, where the additional cluster correlation is tested with the Bayes factor (Fox et al., 2020). It is shown that measurement invariance can be tested simultaneously across items and thresholds for multiple groups. This avoids the risk of capitalization on chance that occurs in multiple-step procedures and avoids cumbersome procedures where items are examined sequentially. The proposed measurement invariance procedure is applied to PISA data of the 2015 cycle, where the advantages of the method are illustrated. The selected PISA items were developed to assess students’ mathematic achievements and consisted of dichotomously and polytomously scored items. The data is used to illustrate the BCSM method for a two-group and multi-group situation, and also to compare results with other DIF methods.

The BCSM can identify dependence structures describing a negative correlation among clustered observations (Nielsen et al, 2021) – this represents within-cluster measurement variance – and therefore the encompassing prior approach (savage-dickey density ratio) can be used for Bayes factor testing of measurement invariance. This approach avoids testing sequentially many measurement invariance assumptions, testing parameters on the boundary of the parameter space, and comparing non-nested hierarchical models through information criteria with complex penalty terms.

Finally, it is shown that the BCSM procedure for measurement invariance testing has great potential to test for instance simultaneously for uniform and non-uniform invariance, and simultaneously for multiple effects of invariance (i.e., cross-classified and nested clusters).

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Fox, J.-P., Koops, J., Feskens, R., Beinhauer, L. (2020). Bayesian covariance structure modelling for measurement invariance testing. Behaviormetrika 47, 385–410. DOI 10.1007/s41237-020-00119-3.

Nielsen, N.M., Smink, W.A.C. & Fox, J.-P. (2021). Small and negative correlations among clustered observations: limitations of the linear mixed effects model. Behaviourmetrika, 48, 51-77. DOI: 10.1007/s41237-020-00130-8.

Verhagen, J. and Fox, J.-P. (2013). Longitudinal measurement in health-related surveys. A Bayesian joint growth model for multivariate ordinal responses. Statistics in Medicine, 32 Issue 17, p. 2988-3005.

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