Bayesian covariance structure modelling for measurement invariance testing

In a Bayesian Covariance Structure Model (BCSM) the dependence structure implied by random item parameters is modelled directly through the covariance structure. The corresponding measurement invariance assumption for an item is represented by an additional correlation in the item responses in a group. The BCSM for measurement invariance testing is defined for mixed response types, where the additional correlation is tested with the Bayes factor. 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, where the advantages of the method are illustrated.