In a recent paper – Edossa, Schroeders, Weinert, & Artelt, 2018 – we came across the issue of longitudinal measurement invariance testing with categorical data. There are quite good primers and textbooks on longitudinal measurement invariance testing with continuous data (e.g., Geiser, 2013). However, at the time of writing the manuscript there wasn’t an application of measurement invariance testing in the longitudinal run with categorical data. In case your are interest in using such an invariance testing procedure, we uploaded the R syntax for all measurement invariance steps.
Basically, the testing procedure has the same parameter restrictions as the cross-sectional multi-group confirmatory factor analysis (for more information, see Schroeders & Wilhelm, 2011), with the exception of additional residual correlations across time points. In contrast, to measurement invariance testing with continuous data, one of the main difference is that with categorical data thresholds and factor loadings have to be varied in tandem (see Table 1), which is not always acknowledged. Consequently, the step of metric measurement invariance testing is dropped.
|Configural invariance||*||*||*||Fixed at 0|
|Weak invariance||Fixed||*||*||Fixed at 0|
|Strong invariance||Fixed||Fixed||*||Fixed at 0/*|
|Strict invariance||Fixed||Fixed||Fixed||Fixed at 0/*|
|Configural invariance||(*||*)||Fixed at 1||Fixed at 0|
|Strong invariance||(Fixed||Fixed)||Fixed at 1/*||Fixed at 0/*|
|Strict invariance||(Fixed||Fixed)||Fixed at 1||Fixed at 0/*|
Note. The asterisk (*) indicates that the parameter is freely estimated. Fixed = the parameter is fixed to equity over time points; Fixed at 1 = the residual variances are fixed to 1 at all time points; Fixed at 0 = factor means are fixed at 0 at all time points. Fixed at 0/* = factor means are fixed at 0 at the first time point and freely estimated at the other time points. Fixed at 1/* = the residual variances are fixed to 1 at the first time point and freely estimated at the other time points. Parameters in parentheses need to be varied in tandem.
During the revision of the manuscript, Liu et al. (2016) published another approach for longitudinal measurement invariance testing with ordered-categorical data in Psychological Methods, which actually yields the same results and df.