In social sciences, the study of group differences concerning latent constructs is ubiquitous. These constructs are generally measured by means of scales composed of ordinal items. In order to compare these constructs across groups, one crucial requirement is that they are measured equivalently or, in technical jargon, that measurement invariance holds across the groups. This study compared the performance of multiple group categorical confirmatory factor analysis (MG-CCFA) and multiple group item response theory (MG-IRT) in testing measurement invariance with ordinal data. A simulation study was conducted to compare the true positive rate (TPR) and false positive rate (FPR) both at the scale and at the item level for these two approaches under an invariance and a non-invariance scenario. The results of the simulation studies showed that the performance, in terms of the TPR, of MG-CCFA- and MG-IRT-based approaches mostly depends on the scale length. In fact, for long scales, the likelihood ratio test (LRT) approach, for MG-IRT, outperformed the other approaches, while, for short scales, MG-CCFA seemed to be generally preferable. In addition, the performance of MG-CCFA's fit measures, such as RMSEA and CFI, seemed to depend largely on the length of the scale, especially when MI was tested at the item level. General caution is recommended when using these measures, especially when MI is tested for each item individually. A decision flowchart, based on the results of the simulation studies, is provided to help summarizing the results and providing indications on which approach performed best and in which setting.
Item level diagnostics and model - data fit in item response theory (IRT) using BILOG - MG v3.0 and IRTPRO v3.0 programmes
