# Re-Defining, Analyzing and Predicting Persistence Using Student Events in Online Learning +
## 5 Results
### Analysis of Prevalence of Local Persistence  
The first question is about how local persistence is distributed among the students in the SPOCs (i.e., prevalence). In order to evaluate this, the model to determine local persistence, presented in Section 4 has been used. Figure 1 provides the histogram of local persistence. This histogram reflects that most of the students had either a fair/moderate local persistence (between 0.3 and 0.7) or the maximum local persistence (local persistence above 0.9). Among the 3062 pairs of course-student where local persistence was considered after filtering exercises, there were 432 cases where the student always got the answer right and thus there was no information about local persistence. From the 2630 remaining cases, 1216 (33%) cases represented students who were occasionally persistent at the local level, i.e., their local persistence was between 0.3 and 0.7. Among those learners, most of them were between 0.4 and 0.5. Moreover, there were 988 (32%) with very high local persistence (above 0.9) while there were only 155 (5%) students with low local persistence (below 0.3). The mean of local persistence was 0.70, and the median was 0.67, which means that many students did not give up when they faced difficulties with an exercise and kept on trying the exercise. The high prevalence of local persistence might be due to the considered context since most exercises did not have many possible options so these types of exercises might have engaged students to make different attempts until they succeed since they knew they could get the correct answer with a relative low effort. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ID** & **Sequence** & **Attempts** & **Grade** & **Local Persistence (Accumulated)** \\ \hline
1 & 0 & 1 & 0 & \(0/4=0\) \\
2 & 0 1 & 2 & 1 & \((0+1)/(4+1)=1/5\ =\ 0.2\) \\
3 & 0 0 0 0 0 & & & \\
3 & 1 & & & \((0+1+5)/(4+1+5)\ =\ 6/10\ =\ 0.6\) \\
4 & 0 0 0 0 & 4 & 0 & \((0+1+5+3)/(4+1+5+7)\ =\ 9/17\ =\ 0.53\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Example of computation of local persistence. In order to delve into the prevalence of local persistence, several profiles of students were identified. These profiles were identified using the clustering algorithm K-Means and a merged dataset including variables related to interactions with exercises (per_attempt and avg_grade) and videos (per_videos). As not all students interacted with videos and exercises, the merged dataset had 2544 pairs or course-student (after removing the 432 cases with undefined local persistence). Particularly, five profiles (related to five clusters) were identified. The choice of five clusters was made empirically so as to obtain the most visible and clearly distinguishable student profiles. Table 4 provides the average value of the variables in each cluster, and the range between percentile 5 and 95 of local persistence (lp_5_95). The range of values and percentiles of local persistence for each profile can be observed in Figure 2. What follows is a description of each profile:  
* Profile 1 (\(n=307\)). They are the students who most complete videos and attempt exercises on average, and the grade of the exercises they attempt is 0.84 on average, which is good. However, their local persistence is 0.53 on average. This means that they have their exercises right at first attempt many often (67% of times) so their average grade is good, but they are not always persistent at the local level. In fact, from the 33% of exercises they have it wrong at first attempt, they eventually solve correctly 16%, which means that they are persistent at the local level about half of the times. * Profile 2 (\(n=594\)). They are students with very high local persistence and average grade in the exercises they try. This means that they want to complete what they start. However, they engage with few exercises and videos on average. Instructors should motivate these students to engage more with the SPOC. * Profile 3 (\(n=578\)): They are good students.