# Comparing a Fuzzy Hybrid Approach with Invariant MGCFA to Study National Identity
## 4 Methods
### Fuzzy Hybrid Analysis
#### 4.2.2 The Last Four Steps  
This section will cover the part of the hybrid analysis which is based on TOPSIS. As explained above, after the defuzzification step, a crisp information matrix V with dimension (8, 143) contains the defuzzified value for each indicator and population group. Consequently, it is now possible to determine the positive and negative-ideal solutions that are observed after the aggregation process. During the data curation, it was explained that all the criteria to measure the national identity were chosen in a way in which higher values were associated with stricter national identity formation. Therefore, in TOPSIS parlance, it can be said that all the criteria can be considered as benefit values [62]. Consequently, the positive ideal solution is obtained by the maximum figures observed in the whole set of groups, and the negative ideal solution is characterized by the minimum figures.  
Once the positive and negative ideal solutions are obtained, the fifth step consists of obtaining the Euclidean distances between each observation and the ideal solutions, as a way to compare the relative distance which will be used to calculate the relative national identity synthetic indicator (RNISI) per each observation. The Euclidean distances, \(S_{j}^{+}\) and \(S_{j}^{-}\), measure the distances of the aggregated value of each population segment with respect to the positive and negative ideal solutions respectively. Then, RNISI is calculated as a relative index between the distance, with respect to the negative ideal solution, and the sum of the obtained distances with respect to both ideal solutions, that is \(\boldsymbol{RNISI_{j}}=\frac{S_{j}^{-}}{S_{j}^{+}+S_{j}}\rightarrow\left[ \boldsymbol{0},\boldsymbol{1}\right]\).  
In this way, a particular observation (population-segment) perceives national identity in a stricter way if the relative index is closer to one, and the index can be used to rank all the observations according to the descending order to find which segment is more or less strict with respect to the national identity concept. The logic of the index is clear because the indicator is higher for those segments which are closer to the positive ideal solution and farther from the negative ideal solution.  
Finally, the seventh step is used to calculate the elasticities of the index for each population segment with respect to each of the eight criteria included in the national identity formation. The values measure the sensibility of the indicator with respect to changes in the crisp values of each criterion. This information allows us to understand which of the criteria included in the index formation produces more or less changes in the respective indicator. The information also permits to analyse whether some population segments are more or less elastic to some of the criteria. Mathematically, elasticities are calculated as \(\eta_{ij}=\frac{\Delta^{\infty}\partial\boldsymbol{\Pi}S_{i}I_{j}}{\Delta^{ \infty}\boldsymbol{C}_{i}}=\frac{d(\partial\boldsymbol{\Pi}S_{i})}{d(\boldsymbol {C}_{i})}\frac{C_{i}}{\partial\boldsymbol{\Pi}S_{i}}\), where \(i\) denotes the criterion (1... 8) and \(j\) the population segment (1... 143). In the next section, the results of the study will be presented and compared with those obtained by MGCFA.