# Comparing a Fuzzy Hybrid Approach with Invariant MGCFA to Study National Identity
## 1 Introduction  
Modern states, nations and countries vary in the origin and the composition of national population majorities, such as linguistic, ethnic, and religious groups. For this reason, national identities, and the degree of attachment to the nation have been developed very differently in the historical course of each country in which migration and globalization have been affecting the national identity formation [1, 2]. [PERSON] [3] contended that some nationals could even develop such a strong nationhood identity that it makes them restrictive regarding the foreigners' naturalization process. Based on data from two waves of the International Social Survey Program (ISSP) in 2003 and 2013 for ten countries (Denmark, France, Germany, Ireland, Norway, Portugal, Russia, Spain, United Kingdom, and the United States), the current study analyses the national identity construct with the use of two well-known methodologies: Multi-group Confirmatory Factor Analysis (MGCFA) and a Fuzzy Hybrid Analysis (FHA). The first method is based on latent constructs, which are unobserved variables that are usually measured indirectly using several items or indicators in a questionnaire. It is an econometric method that needs to be estimated and several assumptions are implicitly made about the error distributions. It also needs to test measurement equivalence for theanalysis of the main differences observed between groups in order to obtain reliable results that are not a consequence of measurement biases [4; 5; 6; 7]. On the other hand, the essence of the second method is analytical, and it is mainly based on the properties of the Fuzzy Set Theory (FST). The method is well-known and has been applied to measure latent constructs in many fields such as the hotel industry [8], education [9], supplier selection [10], and Green energy [11]. [PERSON] et al. [12] argued that FST is a good analytical tool to analyse construct formation as researchers could not find a unique objective function to measure latent concepts that are common in social science. There is already extensive academic literature that has studied nationhood, grounding the concept under different social theories, such as the distinction made between civic and ethnic dimensions that are aligned with the ideas of state and cultural nations. However, [PERSON] [13] argued that \"the distinction is conceptually unclear, too simple to capture the complex nature of perceptions of nationhood and of little relevance for contemporary attitudes to migration (p. 970)\". The measurement of national identity at the transnational level suffers from methodological and comparability problems, which have caused great controversy in the past. The literature widely adopts methodologies that limit the results comparison, such as the MGCFA [14]. For this reason, this study introduced a new methodological approach in the study of National Identity, the Fuzzy-Hybrid Approach, which is more flexible and permits the comparison of the results at national level. Moreover, the paper presented here contributes more to the debate on the study of nationhood from an empirical perspective providing a new methodological approach that has not been commonly used in the field. Consequently, the main goal of the study was to analyse whether the Fuzzy Hybrid Approach is applicable in the field of the social sciences, using the case of nationhood analysis. Additionally, the study aimed to investigate to what extent FHA results are near other robust invariant MGCFA measurements. The main contribution of the current paper is to show that alternative methods based on fuzzy logic approaches can be used in the field of social science beyond the more traditional methods used in the literature, such as cluster analysis and MGCFA. We apply an FHA approach providing empirical evidence that the well-known FHA can be applied in this field without producing any distortion on the basic results obtained by other classical methods.