Using multiple correspondence analysis to determine recommended professional profiles for Smart Cities projects

Multiple correspondence analysis for the variables related to functions in technicians

In this section, the results of the ${\chi }^2$ and Fisher’s Exact tests for variables associated with technicians’ functions are presented. Prior to conducting the MCA, it was necessary to confirm the presence of statistically significant dependencies among pairs of categorical variables. For each pair, the ${\chi }^2$ statistic ( $X_0$) and its corresponding p-value were computed, with Bonferroni correction applied to control for multiple comparisons. In addition, the effect size ( ${\eta }^2$) was calculated to evaluate the strength of associations, classified as weak ( ${\eta }^2<0.06$), moderate ( $0.06\le {\eta }^2<0.14$), or strong ( ${\eta }^2\ge 0.14$). The results are presented in Table 5, where adjusted p-values ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}$) below 0.05 indicate statistically significant dependencies. Bold entries denote the strongest relationships according to the ${\eta }^2$ classification.

Based on the results obtained, several significant associations were identified among the functional variables of technicians. The variable Cloud shows strong dependence with IoT and Security, as well as moderate associations with Data and PM-Civil, highlighting its central influence in shaping the functional competence structure of this professional group. The variable Security is also dependent on Data and IoT, both displaying moderate ${\eta }^2$ values. In contrast, PM-Civil appears largely independent from most other variables except Cloud, suggesting that it represents a distinct and more specialised professional dimension. Finally, the weak association between Data and IoT indicates that these dimensions contribute independently within the functional profile of technicians.

Next, the eigenvalues explained variance, and the cumulative percentage of explained variance are analysed to determine the number of retained dimensions. The results obtained are shown in Table 6. Additionally, Fig. 4 represents the scree plot, which is a graphical representation of the eigenvalues, with the eigenvalues plotted against the dimension number. The point at which the eigenvalues level off or drop below a certain threshold can be used to determine the optimal number of dimensions to retain. One of the distinctive characteristics of MCA is that the dimensions created do not necessarily explain a large percentage of the total variance (Agresti, 2013).

Based on the results in Table 6, two dimensions are retained because they account for 63.62% of the explained variance. Additionally, as seen in the scree plot in Fig. 4, the contribution of the subsequent dimensions to the explained variance becomes too small to justify their inclusion in the model.

To interpret the dimensions retained in the MCA, it is first necessary to determine which variables exert the greatest influence on their definition. This is assessed using the ${\eta }^2$ coefficient, which measures the proportion of inertia of each dimension explained by a variable. Following the guidelines of Lebart, Morineau & Piron (1995) and Husson, Lê & Pagès (2011), variables with ${\eta }^2\ge 0.2$ are considered to make a substantial contribution to the inertia of a dimension and are therefore essential for its interpretation.

As shown in Table 7 and Fig. 5, all variables present ${\eta }^2$ values above the threshold in at least one of the two dimensions. The first dimension (Dimension 1), which explains 44.1% of the total inertia, is primarily structured by the variables Cloud ( ${\eta }^2=0.6805$), Security ( ${\eta }^2=0.5095$), and IoT ( ${\eta }^2=0.5188$). The variable Data ( ${\eta }^2=0.3797$) shows a moderate contribution, while PM ( ${\eta }^2=0.7967$) stands out in Dimension 2 (which explains 19.6% of inertia). These results indicate that Dimension 1 captures variability linked to technological functions, whereas Dimension 2 reflects differences associated with project management roles.

To ensure that only well-represented points are used for interpretation, the squared cosine ( $co{s}^2$) values were examined. According to Lebart, Morineau & Piron (1995), Abdi & Williams (2010), and Greenacre (2017), categories with $co{s}^2\ge 0.4$ are considered well projected on the factorial map.

From Table 8, it can be observed that, for Dimension 1, the categories Cloud (Essential) and Cloud (Relevant + Useful) ( $co{s}^2=0.6805$), Security (Essential) and Security (Relevant + Useful) ( $co{s}^2=0.5095$), and IoT (Essential) ( $co{s}^2=0.5188$) exceed the threshold and are therefore retained in Fig. 6 as graphical representation. The categories of the variable Data, however, present $co{s}^2$ values below 0.4 and are thus excluded from the biplot due to their limited representational quality. For Dimension 2, both categories of PM-Civil (Essential and Relevant + Useful) show very high values ( $co{s}^2=0.7967$), confirming their relevance for interpreting this axis.

Based on these criteria ( ${\eta }^2\ge 0.2$ for variables and $co{s}^2\ge 0.4$ for categories), the biplot in Fig. 7 includes only those categories that meet both thresholds, ensuring a clear and interpretable representation focused on the most informative variables and categories. Additionally, the percentage contribution of each category to each retained dimension is presented in Table 9.

In the biplot, Dimension 1 differentiates between two distinct groups of categories. On the positive side, we find Essential categories for Cloud, Security, and IoT, whereas on the negative side, the Relevant + Useful categories of these same variables are grouped together. The categories of Data are not displayed in the plot because they are under the $co{s}^2\ge 0.4$ threshold. This pattern suggests that Dimension 1 reflects a contrast between essential and additional relevance perceptions of technological factors, indicating that technicians distinguish strongly between technologies considered indispensable vs. those perceived as supplementary or supportive.

Dimension 2 is primarily defined by the variable PM-Civil, whose two categories (Essential and Relevant + Useful) are projected at opposite extremes along this axis. This implies that Dimension 2 captures variability specifically associated with project management functions, differentiating individuals according to their valuation of PM-related aspects.

Regarding individuals, their distribution across the four quadrants indicates heterogeneity in response profiles. The clustering of individuals near particular category points reflects groups sharing similar evaluations. For instance, individuals located toward the upper-right quadrant tend to associate technological dimensions (Cloud, IoT, Security) with essential importance, while those situated toward the lower-left quadrant are closer to categories rated as Relevant + Useful, suggesting a more moderate perception of importance.

In summary, Dimension 1 represents the technological valuation axis, contrasting essential vs. complementary technological factors (Cloud, Security, IoT, Data), while Dimension 2 corresponds to a project management valuation axis, dominated by PM-Civil. Both categories of the variable Data, despite its moderate contribution to inertia, were excluded from the graphical display due to its low representational quality ( $co{s}^2<0.4$). This methodological refinement enhances the explanatory clarity of the MCA results and allows a more focused understanding of how technicians differentiate between essential and supportive functions in their professional activities.

Finally, a bootstrap resampling procedure ( $R=1000$) was conducted to assess the robustness of the factorial configuration obtained for the set of functional categories corresponding to technicians. This approach made it possible to estimate the variability of category coordinates across replicated datasets, offering an empirical measure of the stability of their factorial positioning. The Mean Standard Deviation (Mean SD) across the first two factorial dimensions was calculated as a summary indicator of coordinate dispersion, where lower values denote higher positional stability. The results of this analysis are presented in Table 10.

The results indicate that most functional categories display satisfactory coordinate stability, with Mean SD values generally falling within a moderate range (0.17–0.46). Categories associated with Cloud show the lowest variability, confirming their stable positioning across bootstrap replications. Although Data (Essential) and PM-Civil (Essential) exhibit slightly greater dispersion, these deviations remain within acceptable limits and are restricted to specific instances. Overall, the factorial configuration obtained for technicians in relation to functions demonstrates substantial robustness, as most categories maintain consistent positions throughout the resampling process. Some categories are excluded in the bootstrap output because the resampling was performed at the individual level, and categories with very low frequencies may not be represented in all bootstrap iterations.

Multiple correspondence analysis for the variables related to functions in engineers

Similarly, the dependence analysis was extended to the set of functional variables for engineers. The ${\chi }^2$ statistic ( $X_0$) and its associated p-values were computed for all possible pairs, applying Bonferroni correction to account for multiple comparisons. The adjusted p-values ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}$) and effect sizes ( ${\eta }^2$) are summarized in Table 11, highlighting the most significant dependencies according to the ${\eta }^2$ classification. Note that bold values indicate statistical significance at $\alpha =0.05$.

The results indicate a more intricate pattern of dependencies among the functional variables of engineers compared with those of technicians. The variable Security shows a strong dependency on Data ( ${\eta }^2=0.18$), while maintaining moderate associations with PM and CivilEng, which confirms its integrative role across technical and managerial dimensions. The variable PM exhibits moderate associations with Data, IoT, Security, and CivilEng, suggesting that project management interacts with several technological functions within the competence structure of engineers. Moreover, Cloud displays moderate dependencies with Security, IoT, and Data, reinforcing its transversal role across digital functions. In contrast, CivilEng shows only weak relationships with most variables, whereas Data and IoT remain significantly associated but behave independently with respect to CivilEng, indicating a clear distinction between general engineering activities and domain-specific roles.

Next, the eigenvalues explained variance, and the cumulative percentage of explained variance are analysed to determine the number of retained dimensions. The results obtained are shown in Table 12 and Fig. 8.

According to the results in Table 12, three dimensions are retained, as they account for 70.82% of the cumulative explained variance. Reducing to one dimension less only retains 56.86%, which is not adequate. Furthermore, as illustrated in the scree plot in Fig. 8, starting from the second dimension, the contribution to the explained variance is too small to justify including them in the model.

To interpret the dimensions retained in the MCA, it is first necessary to determine which variables exert the greatest influence on each axis. This influence is assessed using the ${\eta }^2$ coefficient. As previously, variables with ${\eta }^2\ge 0.2$ are considered to make a substantial contribution to the inertia of a dimension and are therefore essential for its interpretation.

As shown in Table 13 and Figs. 9, 10, and 11, all variables display non-negligible ${\eta }^2$ values in at least one of the three retained dimensions. Specifically, Dimension 1, which explains 39.5% of the total inertia, is primarily structured by the variables PM ( ${\eta }^2=0.4818$), Security ( ${\eta }^2=0.4963$), and Data ( ${\eta }^2=0.5183$). IoT ( ${\eta }^2=0.3858$) and Cloud ( ${\eta }^2=0.3255$) also make moderate contributions. Conversely, CivilEng ( ${\eta }^2=0.1606$) exerts little influence on this first dimension.

In contrast, Dimension 2, which accounts for 17.4% of the total inertia, is dominated by the variable CivilEng ( ${\eta }^2=0.7111$), followed by Data ( ${\eta }^2=0.1477$). The remaining variables show low to negligible associations with this dimension, indicating that it captures variation mainly linked to engineering functions.

Finally, Dimension 3, which explains 14.9% of the inertia, is primarily associated with the variables Cloud ( ${\eta }^2=0.4026$) and PM ( ${\eta }^2=0.1910$), while IoT ( ${\eta }^2=0.1370$) and Security ( ${\eta }^2=0.0782$) have a more moderate effect. These results suggest that the first dimension represents a general technological valuation axis, the second dimension reflects engineering specialisation, and the third dimension captures complementary variability associated with cloud-based and project management functions.

Once the most influential variables are identified, it is necessary to assess the quality of the representation of their categories in the factorial space. This is evaluated using the squared cosine ( $co{s}^2$), which measures the proportion of inertia of each category explained by the retained dimensions. As previously, categories with $co{s}^2\ge 0.4$ are considered well represented on the factorial map and thus retained for graphical interpretation.

From Table 14 and Fig. 12, it can be observed that several categories reach high $co{s}^2$ values in at least one dimension. For Dimension 1, the categories of PM ( $co{s}^2=0.4818$), Security ( $co{s}^2=0.4963$), and Data ( $co{s}^2=0.5184$) are particularly well represented, confirming their strong association with this axis. The categories of Cloud ( $co{s}^2=0.3255$) and IoT ( $co{s}^2=0.3858$) show moderate representation, while those of CivilEng ( $co{s}^2=0.1606$) fall below the threshold, indicating limited projection quality.

Consequently, the two categories of CivilEng and Cloud were excluded from the biplot of Dimensions 1 and 2 (Fig. 13), as their $co{s}^2$ values do not meet the 0.4 criterion. Similarly, the categories of CivilEng and IoT were removed from the biplot of Dimensions 1 and 3 (Fig. 14), and those of PM, Security, Data, and IoT were excluded from the biplot of Dimensions 2 and 3 (Fig. 15) due to inadequate representation quality.

Based on these criteria ( ${\eta }^2\ge 0.2$ for variables and $co{s}^2\ge 0.4$ for categories), the biplots presented in Figs. 13, 14, and 15 include only categories that meet both thresholds, ensuring a clear and interpretable representation focused on the most informative elements.

Additionally, the percentage contribution of each category to each retained dimension is detailed in Table 15. In Dimension 1, the categories PM (Essential and Relevant + Useful), Security (Essential and Relevant + Useful), and Data (Essential and Relevant + Useful) stand out for their strong contributions, confirming their importance in defining the main technological axis. For Dimension 2, the two categories of CivilEng make the largest contribution, consistent with their high ${\eta }^2$ value and confirming their role as key descriptors of engineering specialisation. In Dimension 3, the categories of Cloud and PM are the most influential, shaping a secondary axis associated with cloud and project management aspects.

In the biplot of Dimensions 1 and 2 (Fig. 13), Dimension 1 separates categories perceived as Essential (positive side) from those considered Relevant + Useful (negative side) for PM, Security, and Data, illustrating a contrast between indispensable and supportive functions. Dimension 2, in turn, is driven by the CivilEng variable, with its Essential category projected on the negative side and its Relevant + Useful category on the positive side, reflecting diverging evaluations of engineering roles.

In the biplot of Dimensions 1 and 3 (Fig. 14), the Essential categories of PM and Cloud are positioned toward the positive end of Dimension 3, while their Relevant + Useful counterparts appear on the opposite side, indicating a contrast between core and complementary functions within project–cloud interactions.

In the biplot of Dimensions 2 and 3 (Fig. 15), the factorial space is primarily shaped by CivilEng on Dimension 2 and Cloud on Dimension 3, showing how the combination of both axes highlights relationships between engineering specialisation and technological infrastructure.

Regarding individuals, their distribution across the factorial planes indicates substantial variability in response profiles. Despite the dispersion across quadrants, clusters of individuals can be observed around particular category points, suggesting the existence of groups with similar perceptions of function relevance.

In summary, Dimension 1 represents a general technological axis dominated by PM, Security, and Data; Dimension 2 captures engineering specialisation led by CivilEng; and Dimension 3 reflects project–cloud interactions defined by Cloud and PM. Categories that did not meet the representational quality criterion ( $co{s}^2<0.4$) were excluded from the biplots to ensure analytical rigour and visual clarity. This methodological refinement enhances the explanatory power of the MCA results and provides a focused interpretation of how engineers differentiate between essential and supportive roles across functions.

Finally, a bootstrap resampling procedure ( $R=1\text{,}000$) was applied to assess the robustness of the factorial configuration obtained for the function categories of engineers. This approach provides an empirical foundation for evaluating the variability of category coordinates and estimating the consistency of their positions across resampled datasets. The results are presented in Table 16, which reports the standard deviations of category coordinates along the first two factorial dimensions, together with their corresponding Mean SD values.

The results indicate a satisfactory degree of coordinate stability, with Mean SD values predominantly within a moderate range (0.24–0.56). The category CivilEng (Essential) exhibits notably high positional consistency, whereas PM (Relevant + Useful) and Cloud (Relevant + Useful) display slightly greater variability, reflecting sensitivity to sample composition in certain cases. Nevertheless, these deviations remain within acceptable limits and do not compromise the factorial configuration. Overall, the findings confirm that the factorial solution for engineers is both stable and reliable, supporting the robustness and interpretability of the structure across resampled datasets. Similar to the previous analysis, some categories are not displayed in the bootstrap results because resampling was conducted at the individual level, and low-frequency categories were not consistently represented in all replicated samples.

Multiple correspondence analysis for the variables related to skills in technicians

This section presents the results of the dependency analysis for variables related to skills among technicians. Pairwise ${\chi }^2$ and Fisher’s Exact tests were conducted to identify significant associations between skill categories. To control for the inflation of Type I error due to multiple comparisons, the Bonferroni correction was applied. When applicable, the effect size ( ${\eta }^2$) was calculated to quantify the strength of the associations. The adjusted p-values ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}$) and corresponding effect classifications are summarised in Table 17. The Fisher’s Exact test was employed when the expected frequencies in any cell were below 5, in which case ${\eta }^2$ could not be computed and is reported as NA. In Table 17, bolded $p_{\mathrm{a}\mathrm{d}\mathrm{j}}$ values indicate statistically significant dependencies after Bonferroni correction ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}<0.05$), while NA entries for ${\eta }^2$ correspond to Fisher’s tests where effect sizes could not be estimated.

As shown in Table 17, only one significant dependency was detected among the skill variables of technicians. The pair “SS”–“GS” exhibits a statistically significant association after Bonferroni correction ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}=0.003$), indicating that social and green skills tend to co-occur within this professional group. However, since this relationship was assessed using Fisher’s Exact test, the effect size ( ${\eta }^2$) could not be estimated and is therefore unavailable. All other pairs, including “ET”–“B/M”, “ET”–“SS”, and “ET”–“GS”, show non-significant results with weak or negligible ${\eta }^2$ values. Overall, these findings suggest that, for technicians, transversal skill domains remain largely independent, with minimal overlap between digital, managerial, and environmental competences.

Next, the eigenvalues explained variance, and the cumulative percentage of explained variance are analysed to determine the number of dimensions to consider. The results obtained are shown in Table 18 and Fig. 16.

According to the results presented in Table 18, two dimensions will be considered because they explain 64.31% of the variance. Furthermore, as illustrated in the scree plot shown in Fig. 16, starting from the second dimension, the added percentage of explained variance is not significant enough to warrant its inclusion in the model.

To interpret the dimensions retained in the MCA, it is first necessary to determine which variables exert the greatest influence on each axis. This influence is evaluated using the ${\eta }^2$ coefficient, which measures the proportion of inertia of each dimension explained by each variable. Following the same criterion used in previous sections, variables with ${\eta }^2\ge 0.2$ are considered to make a substantial contribution to the inertia of a dimension and are therefore essential for its interpretation.

As shown in Table 19 and Fig. 17, all variables display ${\eta }^2$ values above or near the 0.2 threshold in at least one of the two retained dimensions. Dimension 1, which explains 37.4% of the total inertia, is primarily structured by the variables SS ( ${\eta }^2=0.6283$) and GS ( ${\eta }^2=0.6357$), both of which exert a strong influence on this axis. The variable B/M ( ${\eta }^2=0.1814$) has a moderate effect, while ET ( ${\eta }^2=0.0490$) shows little association with Dimension 1.

In contrast, Dimension 2, which explains 27.0% of the total inertia, is mainly defined by the variable ET ( ${\eta }^2=0.6681$), followed by B/M ( ${\eta }^2=0.3779$). The remaining variables, SS and GS, have minimal or no effect on this dimension. These results suggest that Dimension 1 represents a general skills evaluation axis dominated by soft and general skills, while Dimension 2 captures variability associated with technical and management skills.

Once the most influential variables are identified, it is necessary to evaluate the quality of the representation of their categories in the factorial space. This is measured using the squared cosine ( $co{s}^2$), which quantifies the proportion of inertia of each category explained by the retained dimensions. As previously stated, categories with $co{s}^2\ge 0.4$ are considered well projected and thus retained for interpretation.

From Table 20 and Fig. 18, it can be observed that several categories reach high $co{s}^2$ values in at least one dimension. In Dimension 1, the categories of SS ( $co{s}^2=0.6283$) and GS ( $co{s}^2=0.6357$) are particularly well represented, confirming their strong influence on this axis. In Dimension 2, the two categories of ET ( $co{s}^2=0.6681$) and those of B/M ( $co{s}^2=0.3779$) show moderate to high representation, while the categories of SS and GS display very low $co{s}^2$ values.

As a result, the categories of the variable B/M were excluded from the biplot of Dimensions 1 and 2 (Fig. 19), since their $co{s}^2$ values do not meet the 0.4 criterion. This ensures that the graphical representation focuses only on categories with satisfactory projection quality, improving interpretability.

Based on these criteria ( ${\eta }^2\ge 0.2$ for variables and $co{s}^2\ge 0.4$ for categories), the biplot presented in Fig. 19 includes only those categories that meet both thresholds, providing a clear and reliable visual representation centred on the most informative categories.

The percentage contribution of each category to each retained dimension is shown in Table 21. For Dimension 1, the Essential categories of SS (30.10%) and GS (34.32%) are the main contributors, followed by their Relevant + Useful counterparts, which also make notable contributions. This pattern indicates that Dimension 1 reflects a contrast between essential and supportive soft and general skills.

In Dimension 2, the strongest contributions correspond to ET (Essential) (19.01%) and ET (Relevant + Useful) (42.96%), followed by B/M (Essential) (29.88%), confirming that this dimension is mainly associated with technical and management skills. The remaining categories contribute marginally to this axis.

In the biplot (Fig. 19), Dimension 1 differentiates between categories positioned on the positive side—corresponding to Essential perceptions of SS and GS (and those located on the negative side) associated with Relevant + Useful categories. This indicates a contrast between highly valued core skills and those considered complementary. Dimension 2, in turn, separates the categories of ET and B/M according to their perceived relevance: Essential categories are projected on the positive side, while Relevant + Useful appear on the negative side. This reflects differing assessments of technical vs. management-oriented skills.

Regarding individuals, although they are dispersed across all quadrants, clusters can be observed around the category points. Specifically, a concentration appears near the Relevant + Useful categories of GS, SS, and B/M, and near the Essential category of ET, suggesting shared evaluation patterns among these groups.

In summary, Dimension 1 represents the axis of soft and general skills, contrasting essential vs. supportive perceptions, while Dimension 2 captures the valuation of technical and management skills. The categories of the variable B/M were excluded from the biplot due to their insufficient projection quality ( $co{s}^2<0.4$), ensuring analytical consistency and visual clarity. This methodological refinement strengthens the interpretability of the MCA results.

Finally, a bootstrap resampling procedure ( $R=1\text{,}000$) was applied to evaluate the robustness of the factorial configuration obtained for the skill categories of technicians. This approach estimated the variability of category coordinates across replicated samples. The Mean Standard Deviation (Mean SD) computed over the first two factorial dimensions served as an indicator of coordinate stability, with lower values denoting higher positional consistency. The results are presented in Table 22.

The bootstrap results indicate that most skill categories display satisfactory coordinate stability. In particular, the group labelled ‘Relevant + Useful” shows low Mean SD values (below 0.15), confirming that their positions within the factorial space remain consistent across resampling. Slightly higher variability is observed among the “Essential” categories—especially for “B/M (Essential)”—although these deviations are confined to specific cases and do not compromise the overall factorial configuration. Overall, the findings suggest that the factorial solution obtained from the technician skill data is robust, with a stable underlying structure supported by the majority of categories. As in previous analyses, certain categories do not appear in the bootstrap results because the resampling procedure was performed at the individual level, and low-frequency categories may not be represented in all replicated samples.

Multiple correspondence analysis for the variables related to skills in engineers

The dependency analysis for the skill-related variables in engineers was performed using pairwise ${\chi }^2$ tests ( $X_0$), with Bonferroni correction applied to account for multiple comparisons. Effect sizes ( ${\eta }^2$) were computed from the ${\chi }^2$ values to evaluate the strength of the detected associations. The results are presented in Table 23. Adjusted p-values ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}$) below 0.05 (bold entries) indicate statistically significant dependencies, whereas the ${\eta }^2$ classification provides an additional measure of the magnitude of each relationship.

As shown in Table 23, two significant dependencies were identified among the skill-related variables for engineers. The strongest association was observed between “SS” and “GS” ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}=0.025$, ${\eta }^2=0.086$), suggesting that social and green skills frequently co-occur within engineer competence profiles. A moderate relationship was also detected between “B/M” and “SS” ( $p_{\mathrm{a}\mathrm{d}\mathrm{j}}=0.056$, ${\eta }^2=0.071$), indicating a partial overlap between business management and social skills. In contrast, pairs such as “B/M–GS” and those involving “ET” displayed weak ${\eta }^2$ values and non-significant adjusted p-values, implying that entrepreneurial thinking (ET) functions largely as an independent dimension. Overall, the results indicate that while social and green skills show a degree of integration, the main skill domains remain relatively distinct within the competence structure of engineers.

Next, the eigenvalues explained variance, and the cumulative percentage of explained variance are analysed to determine the number of dimensions to consider. The results are shown in Table 24 and Fig. 20.

Based on the results obtained in Table 24, the number of retained dimensions is two, as it accounts for 66.00% of the explained variance. Additionally, as shown in the scree plot represented in Fig. 20, from the second dimension onwards, the contribution to the model’s explained variance is insufficient to justify including further dimensions.

To interpret the dimensions retained in the MCA, it is first necessary to determine which variables exert the greatest influence on each axis. This is assessed using the ${\eta }^2$ coefficient, which measures the proportion of inertia of each dimension explained by a variable. As in previous sections, variables with ${\eta }^2\ge 0.2$ are considered to make a substantial contribution to the inertia of a dimension and are therefore key for its interpretation.

As shown in Table 25 and Fig. 21, all variables display non-negligible ${\eta }^2$ values in at least one of the two retained dimensions. Specifically, Dimension 1, which explains 38.9% of the total inertia, is mainly structured by the variables SS ( ${\eta }^2=0.5538$), GS ( ${\eta }^2=0.5275$), and B/M ( ${\eta }^2=0.4719$), all of which make strong contributions to this axis. Conversely, the variable ET ( ${\eta }^2=0.0019$) has little influence on Dimension 1.

In contrast, Dimension 2, which accounts for 27.1% of the total inertia, is primarily defined by the variable ET ( ${\eta }^2=0.8610$), followed by B/M ( ${\eta }^2=0.1371$). The remaining variables (SS and GS) show weak associations with this dimension. These findings indicate that Dimension 1 represents an axis of general and soft skills evaluation, while Dimension 2 captures variation related to technical and management-oriented skills.

Once the most influential variables have been identified, the next step is to evaluate the quality of representation of their categories in the factorial space. This is done using the squared cosine ( $co{s}^2$), which quantifies the proportion of inertia of each category explained by the retained dimensions. In accordance with established criteria, categories with $co{s}^2\ge 0.4$ are considered well represented and suitable for graphical interpretation.

From Table 26 and Fig. 22, it can be observed that several categories exhibit high $co{s}^2$ values in at least one dimension. In Dimension 1, the categories of SS ( $co{s}^2=0.5538$), GS ( $co{s}^2=0.5275$), and B/M ( $co{s}^2=0.4719$) are particularly well represented, confirming their strong association with this axis. In Dimension 2, the two categories of ET ( $co{s}^2=0.8610$) show very high representation quality, while those of B/M ( $co{s}^2=0.1371$) fall below the threshold, indicating limited projection quality.

Consequently, the categories of the variable B/M were excluded from the biplot of Dimensions 1 and 2 (Fig. 23), since their $co{s}^2$ values did not meet the 0.4 criterion. This ensures that the graphical display focuses on the most informative categories with satisfactory projection quality.

Based on these criteria ( ${\eta }^2\ge 0.2$ for variables and $co{s}^2\ge 0.4$ for categories), the biplot in Fig. 23 includes only categories meeting both thresholds, providing a clear and interpretable visualisation centred on the most relevant elements.

The percentage contribution of each category to each retained dimension is shown in Table 27. In Dimension 1, the Essential categories of SS (22.26%) and GS (22.74%) stand out as the main contributors, followed by their Relevant + Useful counterparts (13.36% and 11.18%, respectively), confirming their strong role in defining this axis. Additionally, the Essential category of B/M (20.69%) also contributes notably. This pattern highlights that Dimension 1 distinguishes between core and complementary perceptions of general and soft skills.

For Dimension 2, the most influential categories correspond to ET (Essential) (19.84%) and ET (Relevant + Useful) (59.52%), reaffirming that this dimension represents technical skill valuation. Other categories show minor contributions to this dimension.

In the biplot (Fig. 23), Dimension 1 differentiates between categories located on the positive side, corresponding to Essential perceptions of B/M, SS, and GS, and those positioned on the negative side, linked to Relevant + Useful categories of these same variables. This reveals a contrast between highly valued core skills and complementary ones. Meanwhile, Dimension 2 is structured by the ET variable, with its Essential category projected on the positive side and its Relevant + Useful category on the negative side, reflecting divergent evaluations of technical skills.

Regarding individuals, although they are distributed across all quadrants, notable concentrations can be observed around the points corresponding to the Relevant + Useful categories of GS, SS, and B/M, as well as near the Essential category of ET. This suggests the existence of response groups with similar evaluations of skill importance.

In summary, Dimension 1 represents an axis contrasting essential vs. complementary general and soft skills, whereas Dimension 2 captures the valuation of technical skills. The exclusion of B/M categories due to low $co{s}^2$ values enhances the clarity and analytical rigour of the MCA results, ensuring that the interpretation focuses on the most representative and well-projected elements.

Finally, a bootstrap resampling procedure ( $R=1\text{,}000$) was conducted to evaluate the robustness of the factorial configuration obtained for the skill categories associated with engineers. This procedure assessed the consistency of category positions across resampled datasets, offering an empirical measure of the model stability. The results are presented in Table 28.

The results indicate that most skill categories exhibit satisfactory coordinate stability, with Mean SD values predominantly within a low-to-moderate range. Categories such as “B/M (Relevant + Useful)” and “SS (Relevant + Useful)” maintain particularly stable positions across bootstrap replications, confirming their consistent placement within the factorial space. Although “ET (Relevant + Useful)” displays slightly greater dispersion, this variability remains limited and does not compromise the interpretability of the overall structure. Overall, the factorial configuration obtained for the skill categories of engineers demonstrates a high level of stability, supporting the robustness of the MCA results. As in the previous analyses, some categories do not appear in the bootstrap outcomes because resampling is performed at the individual level, and low-frequency categories may not be represented in all replicated samples.