# Comparing a Fuzzy Hybrid Approach with Invariant MGCFA to Study National Identity
## 5 Results  
### MGCFA  
As discussed above, EFAs (principal component factor analyses with a Promax rotation) were performed with R to analyse if similar factor structures were present in the twenty respective population segments obtained from ten countries and two different years. The EFA revealed that the number and content of the dimensions in each region did not have a similar structure across groups, neither in the number of factors, nor in the contents of each factor, or the patterns of salient and non-salient factor loadings. The number of factors ranged from 2 to 4, and the split of the indicators in the factors was not stable, but, in most of the cases, the ethnic and the civic dimensions obtained by previous studies were also obtained. For the pool of the observations, three factors were obtained: (1) the first factor contained the first three indicators to have been born in the country, to have the country citizenship, and to have lived in the country for most of one's life; (2) the second factor contained the indicators 5 and 8 (to practice the majoritarian religion and to have ancestry from the country; and (3) the third factor contained the indicators 4, 6 and 7, which were more related to the civic dimension (to speak the language, to respect the country's political institutions and laws, and to feel country nationality). Besides the dark prerogatives given by [41], we decided to move on with the analysis of measurement equivalence across the territories under analysis by means of MGCFA using R software. We analysed two different scenarios. First, all the items were assigned to one factor as EFA did not provide any initial similar allocation across countries, and second, we decided to use a second order latent model using the factors obtained for the pool of the data according to the structure provided by the EFA with the pool of observations. The steps discussed above were followed with the idea of being able to compare factor meansacross territories and political orientation in case of full or partial scalar equivalent models if those could be achieved. The adequacy of the model fit was based whenever it was possible following the standard practice in the discipline, that is when: (1) both the comparative fit index (CFI) and Tucker-Lewis index (TLI) were higher than 0.95; and (2) both the standardized root mean square residual (SRMR) and root mean square error of approximation (RMSEA) were lower than 0.05. We depart from these standard values using relatively close values to the reference points mentioned above, as some cases, CFI greater than 0.9 and RMSEA lower than 0.08, were considered acceptable [41]. Meanwhile, the loss and gain in successive measurement invariance steps was based on the differences of the fit indices [39; 63]. Thus, equivalence was accepted whenever both the decrease in CFI was equal to or smaller than 0.01 and the increase in RMSEA was equal to or smaller than 0.015 [63; 64]. Both studies showed that small changes (between the steps) in CFI and RMSEA reliably indicate that a further step in equivalence is reached. If in any step, the difference of fit indices was outside the recommended values, the constraints on some parameters were relaxed according to the values of the modification indices (MIs) to get a partial measurement invariance equivalent model. Table 1 shows the initial models that tested configural (pool), configural (group), metric and scalar equivalent models. Unsurprisingly, all the uni-dimensional models presented a very poor adjustment to the data. On the other hand, the initial models for configural (pool), configural (group), metric and scalar models for the 3-factor national identity model presented very acceptable indices for both group variables (country-year and political orientation-year). There were some important differences observed for both group variables.