# A Scoring Algorithm for the Early Prediction of Academic Risk in STEM Courses
## 5 Results  
The Youden index is used to determine the threshold value \(c\) when converting a numerical variable into a dichotomous one. The value of \(c\), for which the Youden index is the highest, represents the optimal threshold for distinguishing between negative and positive outcomes. For example, if the variable _Colloquium 1_, which records the percentage of points obtained on the first midterm exam (ranging from 0% to 100%), is to be converted into a dichotomous variable that takes values 0 or 1, the optimal threshold (cut-off point) can be determined using the Youden index. All percentage values lower than the optimal threshold \(c\) will be assigned a value of 0, while all values equal to or greater than \(c\) will be assigned a value of 1. For different threshold values \(c\in[0\%,100\%]\) of the percentage of points obtained on the first midterm exam, the numerical variable is transformed into a dichotomous variable. Then, for this independent dichotomous variable and the known values of the dependent dichotomous variable _Success_, the Youden index is calculated. The value of \(c\) for which the Youden index is the highest is the optimal threshold value for converting the numerical variable into a dichotomous one. Table 7 presents several different threshold values for the variable _Colloquium 1_ along with the corresponding Youden index values. The maximum Youden index \begin{table}
\begin{tabular}{l c c} \hline \hline
**Variable** & **Mean (Std Dev)** & **Median (Min–Max)** \\ \hline To know & 4.74 (1.37) & 4.75 (1.75–7.00) \\ Toward accomplishment & 4.47 (1.74) & 4.50 (1.00–7.00) \\ To experience stimulation & 3.88 (1.57) & 3.75 (1.25–7.00) \\ External regulation & 5.62 (1.34) & 6.00 (1.75–7.00) \\ Introjected & 5.50 (1.33) & 6.00 (1.75–7.00) \\ Identified & 3.76 (1.84) & 3.50 (1.00–7.00) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Descriptive statistical indicators for the answers to the questions on study motivation. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Variable** & **Mean (Std Dev)** & **Median (Min–Max)** \\ \hline School grade & 4.00 (0.55) & 4.00 (2.30–5.00) \\ Matriculation grade & 3.22 (0.70) & 3.00 (2.00–5.00) \\ Matriculation percentage & 0.64 (0.13) & 0.65 (0.27–0.85) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Descriptive statistical indicators for the answers about the previous learning level. value is achieved at a threshold of 0.82, making it the optimal threshold value for the variable _Colloquium 1_. All values of the variable _Colloquium 1_ that are less than 0.82 are assigned the value 0 \((n=42)\), while all other values are assigned the value 1 \((n=34)\), thus converting _Colloquium 1_ into a dichotomous variable. Using this method, all numerical variables for each of the 15 weeks of teaching were converted into dichotomous variables and then used for further analysis. To identify the most relevant predictors of student success, Pearson's chi-square test was applied to all dichotomous variables. A variable was considered significant if its associated \(p\)-value was below the threshold of 0.05. For example, the value of the Pearson chi-square statistic for the variable _Colloquium 1_ is \(\chi^{2}=26.71\) (according to expression (1)). The corresponding \(p\)-value is very small (\(p<0.001\)), indicating a significant association between the variables _Colloquium 1_ and _Success_. This suggests that _Colloquium 1_ is a relevant variable for predicting the variable _Success_. Table 8 presents the list of variables that met this criterion, organized by week, highlighting which aspects of student engagement and performance exhibit a statistically significant association with course success at different stages of the semester. Of the 20 variables analyzed, 16 were identified as relevant and included in the subsequent analysis to develop the scoring system. The number of relevant variables fluctuated throughout the semester, with the fewest (\(n=7\)) observed in the second week and the highest (\(n=15\)) in the final week.