# Comparing a Fuzzy Hybrid Approach with Invariant MGCFA to Study National Identity
## 4 Methods
### Multi-Group Confirmatory Factor Analysis  
Artificial significant mean differences among a set of subsamples (population segments) may be the consequence of various methodological issues such as organization of the survey, item translations that change the meaning across groups or even respondents' response styles [25; 39; 40]. In such cases, the results could be biased, and the conclusions could mislead users and practitioners. [PERSON] et al. [40] advised researchers to analyse measurement equivalence as a necessary step before comparing or pooling data from distinct groups.  
Measurement equivalence is generally based on a group of tests that goes from the least constrained model (configural base scenario) to the most constrained model (full scalar invariance model). [PERSON] [41] advised first to analyse via an Exploratory Factor Analysis that the structure (number of factors and patterns of salient and non-salient factor loadings) is similar across groups. Unfortunately, the similarity concept is left to researchers' best judgement, and the results regarding the number of factors and the set of indicators included in each of them has varied substantially. Indeed, [7] recommended researchers to discard those items that could distort configural equivalence as these will likely provoke poor adjustment fits in MGCFA. This practice has rendered that the results could not be easily compared as the studies are neither usually based on the same constructs nor the same items are included in their respective reflective or formative specification. This feature is a very restrictive condition that does not exist in FHA.  
The rest of the steps for measurement invariance can be directly tested by MGCFA. Therefore, for example, metric equivalence requires item loadings to be equal. This restriction is scarcely met in real applications, especially if many groups are included in the analysis. For this reason, partial metric equivalence was defined when at least the factor loadings across groups of two items per latent construct are equal [41], and many empirical applications are based on partial metric equivalence rather than on full metric equivalence. In any case, the metric equivalence is insufficient when, as in the present case, researcherswant to compare mean differences. The comparison of latent construct means relies on a more restrictive equivalence--the scalar equivalence, in which the item intercepts are added to the item factor loadings across groups in order to be equal. Scalar equivalence means that there is a pattern of similar high and low scores on the set of indicators, and as this is not normal for some indicators which present a high variability among groups, scalar equivalence is not frequently reached with a large number of indicators [40; 42]. Similarly, as before, partial scalar equivalence with two indicators per construct is usually addressed, and researchers tend to frequently rely on partial scalar equivalent scores. Nevertheless, although numerous studies rely on partial scalar equivalence models, there exists a certain controversy regarding whether it is necessary or not to have a full scalar invariance model in order to compare meaningfully the latent mean scores. [PERSON] et al. [19] adopted a pragmatic approach and compared the latent civic and ethnic latent construct means using two models: a partial scalar equivalent model and a full scalar equivalent model. The authors finally came up with a recommendation that latent means comparison can be based on a partial scalar equivalent model if the rank order of the means between the full and partial scalar equivalent models does not fluctuate.