## 4 Survey Exploratory Analysis  
Factor analysis (FA) and principal component analysis (PCA) are multivariate statistical methods that are used in educational research to analyse measurement scales. Although both methods involve reducing the number of dimensions, they differ from a conceptual and mathematical point of view [52]. FA assumes that the correlations observed between variables result from unobserved latent factors influencing the observed variables. The main goal of this type of analysis is to identify, understand and model these underlying structures. In an FA model, each variable \(X_{i},\ i=1,2,\ldots,n\), is written as a linear combination of the common factors (\(F_{j},\ j=1,2,\ldots,m\)) plus a specific factor \(u_{i}\), that is, \(X_{i}=a_{i1}F_{1}+a_{i2}F_{2}+\ldots+a_{im}F_{m}+u_{i}\), \(i=1,2,\ldots,n\), where the scalars of these linear combinations \(a_{ij},\ i=1,2,\ldots,n\), \(j=1,2,\ldots,m\), are called factor loadings. It is assumed that the common factors are standardised (E\((F_{j})=0\) and Var\((F_{j})=1,\ j=1,2,\ldots,m\)), and that the specific factors of the original variables are uncorrelated (cov\((u_{i},u_{j})=0\), if \(i\
eq j\)) and have a mean equal to zero (E\((u_{i})=0,\ i=1,2,\ldots,n\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Cronbach’s Alpha**} & **Confidence Interval** & **Cronbach’s Alpha** & \multirow{2}{*}{**Number of items**} \\  & **(95\%)** & & **Bosed on Standardised** & \\  & & & **Items** & \\ \hline
0.894 & [0.864, 0.920] & 0.896 & 18 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Reliability statistics. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & **Scale Mean if** & \begin{tabular}{c} **Scale Variance** \\ **if Item** \\ **Deleted** \\ \end{tabular}  & \begin{tabular}{c} **Corrected** \\ **if Item** \\ **Deleted** \\ \end{tabular}  & \begin{tabular}{c} **Squared** \\ **Item-Total** \\ **Correlation** \\ \end{tabular}  & \ \begin{tabular}{c} **Cronbach’s** \\ **Multiple** \\ **Correlation** \\ \end{tabular}  & \begin{tabular}{c} **Cronbach’s** \\ **Alpha if Item** \\ **Deleted** \\ \end{tabular}  \\ \hline
**E1** & 67.3621 & 83.816 & 0.516 & 0.561 & 0.889 \\
**E2** & 67.1466 & 83.587 & 0.486 & 0.563 & 0.890 \\
**E3** & 67.7155 & 80.744 & 0.537 & 0.444 & 0.888 \\
**E4** & 67.3793 & 82.742 & 0.543 & 0.420 & 0.888 \\
**E5** &There are two types of FA: exploratory FA (EFA) and confirmatory FA (CFA). EFA is a multivariate statistical technique used for dimension reduction that seeks to discover the latent factors in a set of quantitative variables. In CFA, the factors are known a priori (and are usually previously described theoretically), and the objective is to verify whether this existing theoretical structure fits the data. On the other hand, the goal of PCA is to reduce the dimensionality of a dataset while retaining the maximum possible variance from the original data. The aim is to represent the data as a new set of uncorrelated variables or principal components (\(C_{i},\ i=1,2,\ldots,m\), with \(\operatorname{cov}(C_{i},C_{j})=0\), if \(i\
eq j\)) that are linear combinations of the original variables, that is, \(C_{i}=a_{1}X_{1}+a_{2}X_{2}+\ldots+a_{in}X_{m},\ i=1,2,\ldots,m\), such that they efficiently capture the variability in the data. Both FA and PCA have some limitations: firstly, they assume linear relationships between variables, and secondly, their interpretation is only reasonable if the variables are scaled at the numeric level (interval or ratio level of measurement) [53]. These assumptions are not always justified, which can render these reduction methods less suitable for certain contexts. Nonetheless, in specific fields such as social and education sciences, ordinal variables are sometimes treated as continuous variables, and the same principles and assumptions that apply to this type of data apply to ordinal data [56]. In addition, to overcome the limitations of these methods when using ordinal variables, we carried out a categorical PCA (CATPCA) [53].