We have examined measurement invariance across gender in a multi-group model with categorical indicators (binary data) using the procedures as described in the User Guide (p. 485) in which thresholds and factor loadings are constrained in tandem.
We are now examining measurement invariance across parents (reporting on the same child, so dependent data) using a one-group model following procedures described by Chiorri et al. (2015, see: http://asm.sagepub.com/content/early/2015/01/19/1073191114568301.abstract) as we could not find such a model in the Mplus User Guide. In this article a procedure for examining MI with categorical indicators is described in which 3 models are tested. Model 1, the least restrictive model (configural invariance): item thresholds and factor loadings are free across groups, residual variances are fixed at 1 in all groups and factor means are fixed at 0 in all groups. The second model (weak invariance): factor loadings are constrained to be equal. In model 3 (strong invariance) the thresholds are constrained to be equal and the residual variances and latent means were freed in one group.
Is this a more accurate procedure given that this is a one-group model? Or should we use the procedure described for multi-group models with categorical indicators meaning that we should constrain factor loadings and thresholds in tandem? Can you explain what scale factors are?