# Identifying At-Risk Students for Early Intervention--A Probabilistic Machine Learning Approach
## 4 Results  
This section starts by responding to the first objective, which was to determine features that can serve as predictors of at-risk students. Therefore, to accomplish the first objective, the study used [PERSON]'s correlation and an extra trees classifier to select features that were useful for predicting at-risk students in each stage. The [PERSON]'s correlation and ETC results are shown in Tables 2 and 3, respectively. The variables with asterisks (*) are assessments predicted at each stage--assessment 1 (in stages 1 and 2), test 1 (in stage 3), test 2 (in stage 4), and continuous assessment 1 (in stage 5). Thus, correlation and feature importance were observed between variables with and without an asterisk. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Variables** & **Stage 1** & **Stage 2** & **Stage 3** & **Stage 4** & **Stage 5** \\ \hline TITLE & 0.1429 & 0.1631 & 0.3422 & 0.3412 & 0.2272 \\ QUAL CODE & 0.0581 & 0.0276 & \(-\)0.0597 & \(-\)0.0543 & 0.0403 \\ CLASS GROUP & \(-\)0.0612 & \(-\)0.0597 & 0.1509 & 0.0252 & \(-\)0.0039 \\ \# OF COURSES & 0.0581 & 0.0616 & 0.1488 & 0.0160 & 0.1588 \\ TIME ON SITE & 0.3003 & 0.3839 & 0.7380 & 0.6277 & 0.5053 \\ TIME ON COURSES & 0.2927 & 0.3775 & 0.7497 & 0.6107 & 0.5167 \\ TIME ON ACTIVITIES & 0.2426 & 0.2798 & 0.7427 & 0.4853 & 0.4856 \\ AS 1 (S1 \& S2) * & & 0.8195 & 0.6532 & 0.5707 \\ TM 1 (S3) * & & & & 0.6820 & 0.4848 \\ TA 2 (S4) * & & & & 0.5590 \\ CA 1 (S5) * & & & & & \\ \hline \hline \end{tabular}  
* The variables with asterisks (*) are assessments predicted at each stage.  
\end{table} 
Table 2: [PERSON]’s correlation between variables with and without an asterisk. In stage 1, the student engagement variables and the TITLE variable showed stronger positive correlations than QUAL CODE, CLASS GROUP, and # OF COURSES. Higher correlation values for these variables were also seen in stage 2. From stages 3 to 5, strong and moderately positive correlations were observed, including for assessment variables. However, correlation values were at their highest in stage 3, week 6. Weak positive and negative relationships and lower feature importance values were observed among QUAL CODE, CLASS GROUP, # OF COURSES, and the asterisk variables throughout the five stages. Thus, these variables were not considered for predicting at-risk students. A strong positive correlation was observed among the student engagement variables, and thus, including them all would have resulted in a multicollinearity problem. Since TIME ON COURSES had a stronger correlation and a higher feature importance value for most stages than the other student engagement variables, it was considered for predicting at-risk students, and the other variables were omitted. While student engagement variables had higher feature importance values for stage 1 and stage 2, a significant drop in feature importance was noted after including performance variables (assessment 1, test 1, and test 2) in stages 3 to 5. After identifying critical variables for predicting at-risk students, a five-stage PML model was constructed by incorporating the ongoing assessments. The study's second objective was to build a PML model to predict at-risk students and to evaluate its performance. The performance of the PML model was assessed and compared with that of a standard logistic regression model, as presented in Table 4. While both models demonstrated similar performance levels in all five stages, the PML model offers the additional advantage of enabling the quantification of uncertainty pertaining to both model parameters and predictions. In this context, the uncertainty quantification is useful because it provides a measure of reliability in the model's predictions. This is important when identifying at-risk students, where decisions will often be made based on a model's predictions. By quantifying the uncertainty, decision makers can better understand the limitations and potential risks associated with a particular decision or action.