## 5 Results and Analysis Using Learning Analytics Techniques
### Regression Analysis and Correlations  
Although the clustering analysis may provide insight on dimensions that have a greater impact on academic performance, the conclusions emerging from this analysis are not conclusive yet. Regression analysis provides a generalization of the results on a probabilistic basis, which considers a sample of the data to conduct hypothesis testing to provide conclusions about the entire population. To verify the statistical rigor of the relationships between academic performance and students' MI and SRLAS profiles, regressions were performed between students' final grades and the dimensions of the following: (a) MI, (b) SRLAS, and (c) a combination of both. _Minitab_ was used to select the best set of explanatory variables using a stepwise method, and the best regressions obtained by least-squares were determined. The coefficients in the equations below represent the relative weight of the dimensions that appear in the regression equation, and the figures in parentheses indicate their statistical significance, given by their corresponding \(p\)-values. Positive coefficient values indicate dimensions that have a favorable impact on the final grades, while negative values have the opposite effect. 1. Regression 1 (Final grade vs. MI dimensions) \[ \begin{split} FinalGrade&=73.10+6.66 LogMath-2.15 Mus-2.08 Lin-2.06 Nat\...\\ &(0.0000)\quad(0.0000)\quad(0.0010)\quad(0.0342)\quad(0.0260)\\ N=618,R^{2}=16.01\%\\ \end{split} \] (2)
2. Regression 2 (Final grade vs. SRLAS dimensions) \[ \begin{split} FinalGrade&=80.50-2.76 Anx+3.88 SelfReg-1.60 ExtMot-1.345 ocInt\...\\ &(0.0000)\quad(0.0001)\quad(0.0010)\quad(0.0164)\quad(0.0532)\\ N=618,R^{2}=16.50\%\\ \end{split} \] (3) \[ \begin{split} N=618,R^{2}=16.50\%\\ \end{split} \] (4)
3. Regression 3 (Final grade vs. MI \(\cup\) SRLAS dimensions) \[ \begin{split} FinalGrade&=74.06-2.94 Anx+4.81 LogMath-2.4 1 Lin+2.58 SelfReg-2.17 Mus\...\\ &(0.0000)\quad(0.0000)\quad(0.0000)\quad(0.0127)\quad(0.0020)\quad(0.0 008)\\ N=618,R^{2}=19.05\%\\ \end{split} \] (5) \[ \begin{split} N=618,R^{2}=19.05\%\\ \end{split} \] (6)  
Based on the regression equations, the following conclusions can be derived: (a) the logical-mathematical intelligence exerts an important positive impact on students' final grades, (b) the self-regulation dimension also has a positive influence on the final grade, whereas the anxiety and external motivation dimensions have a negative relationship, and (c) considering both MI and SRLAS constructs together, the most important dimensions that present a positive relationship with academic performance are the logical-mathematical and self-regulation dimensions, whereas anxiety most negatively influences academic performance. In multiple regression, the correlated explanatory variables interact by competing to reach a place in the regression in such a manner that the significance of the present variables is modified when a new variable is introduced. Consequently, a correlation analysis is essential to perceive the dynamics of the stepwise algorithm to obtain an appropriate regression analysis. Tables 7-9 present correlation matrices, which enable a better understanding of the three regressions mentioned above. The entries illustrate the correlation coefficient \(r\) between the corresponding dimensions, as well as the correlations of the final grade with each of the MI and/or SRLAS dimensions. In Table 7, it is observed that all intelligences are positively correlated, where the highest correlation observed is between _LogMath_ and _Intra_ (\(r\) = 0.74). This means that, on average, the higher a student's _LogMath_ dimension is, the higher their _Intra_ dimension, and vice versa. Moreover, it is found that the _LogMath_ dimension is the most correlated with the remaining ones. The average of the correlations among _LogMath_ and the remaining intelligences is \(r\) = 0.57. On the contrary, the lowest correlation is found for _Mus_ and _Inter_ (\(r\) = 0.15).