Understanding non-linear modeling of measurement invariance in heterogeneous populations

This study examined how a non-linear modeling of ordered categorical variables within multiple-group confirmatory factor analysis supported measurement invariance. A four-item classroom disciplinary climate scale used in cross-cultural framework was empirically investigated. In the first part of the analysis, a separated categorical confirmatory factor analysis was initially applied to account for the complex structure of the relationships between the observed measures in each country. The categorical multiple-group confirmatory factor analysis (MGCFA) was then used to conduct a cross-country examination of full measurement invariance namely the configural, metric, and scalar levels of invariance in the classroom discipline climate measures. The categorical MGCFA modeling supported configural and metric invariances as well as scalar invariance for the latent factor structure of classroom disciplinary climate. This finding implying meaningful cross-country comparisons on the scale means, on the associations of classroom disciplinary climate scale with other scales and on the item-factor latent structure. Application of the categorical modeling appeared to correctly specify the factor structure of the scale, thereby promising the appropriateness of reporting comparisons such as rankings of many groups, and illustrating league tables of different heterogeneous groups. Limitations of the modeling in this study and future suggestions for measurement invariance testing in studies with large numbers of groups are discussed.