personalized multi-regression models for predicting
TRANSCRIPT
Personalized Multi-Regression Models for PredictingStudents’ Performance in Course Activities
Asmaa ElbadrawyComputer Science
University of MinnesotaTwin Cities, MN
R. Scott StudhamUniversity of Minnesota
Twin Cities, [email protected]
George KarypisComputer Science
University of MinnesotaTwin Cities, MN
ABSTRACTMethods that accurately predict the grade of a student at agiven activity and/or course can identify students that areat risk in failing a course and allow their educational insti-tution to take corrective actions. Though a number of ap-proaches have been developed for building such performanceprediction models, they either estimate a single model for allstudents based on their past course performance and interac-tions with learning management systems (LMS), or estimatestudent-specific models that do not take into account LMSinteractions; thus, failing to exploit fine-grain informationrelated to a student’s engagement and effort in a course. Inthis work we present a class of linear multi-regression modelsthat are designed to produce models that are personalizedto each student and also take into account a large number offeatures that relate to a student’s past performance, coursecharacteristics, and student’s engagement and effort. Thesemodels estimate a small number of regression models thatare shared across the different students along with student-specific linear combination functions to facilitate personal-ization. Our experimental evaluation on a large set of stu-dents, courses, and activities shows that these models arecapable of improving the performance prediction accuracyby over 20%. In addition, we show that by analyzing theestimated models along with the student-specific combina-tion functions we can gain insights on the effectiveness of theeducational material that is made available at the courses ofdifferent departments.
KeywordsRegression with multi-regression models, Analyzing studentbehavior, Predicting student performance
1. INTRODUCTIONMotivated by the ongoing desire to improve the quality ofeducation and address the ever increasing cost of higher edu-cation by ensuring that students graduate within four years,data mining techniques have been increasingly deployed to
analyze the vast amounts of historical data being collectedat various Colleges and Universities that pertain to students’academic performance. One of the problems that these tech-niques are trying to solve is to identify the students that areat risk of failing a course and thus allow the institution totake corrective actions by providing additional services andresources to the students and/or instructors.
Two general classes of approaches have been developed forsolving this problem, both of which rely on supervised learn-ing. The first uses a set of features related to the student’spast course performance and interactions with online learn-ing management systems (LMS) to estimate a single regres-sion model [3]. This model is estimated to predict the stu-dent’s course grade as a function of these features. The sec-ond uses factorization models, initially developed in the con-text of recommender systems, to predict students’ grades fordifferent course activities. Specifically, multi-relational mod-els were used to predict students’ performance by learninglatent factors that satisfy student-task and task-skill rela-tions [5] while approaches based on tensor factorization wereused to take temporal aspects into consideration by model-ing the fact that, over time, students acquire knowledge anddevelop expertise [7]. Unlike the models based on a singleregression, factorization models can achieve better predic-tion accuracy as their prediction models are personalized toeach student. However, these approaches entirely ignore thevarious features associated with how the students interactwith the material/information provided in the LMS, whichcan potentially be used to improve the overall predictionaccuracy.
In this work we investigate the effectiveness of a class oflinear multi-regression models for predicting the students’performance at various course activities (e.g., quizzes and as-signments). These models, that were inspired by previouslydeveloped approaches in the area of recommender systems[8, 2, 6], estimate a small number of linear regression mod-els along with a student-specific linear function to combinethem. The advantage of this approach is that the regressionmodels are estimated by taking into account the historicalinformation of all students that allows for cross-student in-formation sharing and thus overcome issues related to datasparsity while providing accurate modeling of each student’sunique characteristics via the user-specific linear combina-tion function. The regression models utilize a wide-range offeatures that include the students’ past course performance,
their interactions with the LMS, and information related tothe type of course and activity. We experimentally eval-uated the performance of these models on a large datasetextracted from the University of Minnesota’s Moodle instal-lation [1] that contains 832 courses, 11,556 students, and189,641 graded activities. The multi-regression models wereable to achieve an RMSE of 0.147 whereas the RMSE of thecorresponding single regression model was 0.177.
An advantage of the multi-regression model is that by clus-tering the students based on their combination weights wecan segment them into groups whose prediction models arequite similar. By analyzing these groups we can gain insightson the factors that determine the students’ performance, dis-cover systematic differences across the groups, and identifyareas for further analysis. Towards this end, we analyzedthe combination weights for the multi-regression model con-sisting of just two regression models and identified threegroups of students. The underlying regression models fortwo of these groups were different from each other in howmuch they rely on the LMS interaction features. In addition,some of the Departments in which the courses that these stu-dent took showed a high specificity to one of these groups.These results may suggest that the type of information thatis provided in the LMS for certain departments may not bebeneficial in improving the grades of the students and assuch accessing it does not lead to better understanding andthus grades.
The rest of the paper is organized as follows. Section 2 de-scribes the multi-regression model that we used. Section 3describes the dataset that we used along with the variousfeatures that we extracted. Section 4 provides the experi-mental evaluation and analysis of the results. Finally, Sec-tion 5 provides some concluding remarks.
2. MODEL DESCRIPTIONWe wish to learn a model that predicts the student gradeswithin the different course activities, like assignments andquizzes, given some input features. To achieve this, we de-veloped a linear multi-regression model inspired by [8] and[2]. In this model, the grade gs,a for student s in activity ais estimated as
gs,a = bs + bc + ptsWfsa
= bs + bc +
l∑d=1
(ps,d
nF∑k=1
fsa,kwd,k
),
(1)
where bs and bc are student and course bias terms, respec-tively, fsa is a vector of length nF that holds the inputfeatures (the predictors), l is the number of linear regres-sion models, W is a matrix of dimensions l× nF that holdsthe coefficients of the l linear regression models, and ps isa vector of length l that holds the memberships of students within the l different regression models. The term wd,k
represents the weight of feature k under the dth regressionmodel, whereas the term ps,d represents the membership ofstudent s in the dth regression model; that is, how much thedth regression model contributes to the grade estimation forstudent s. Throughout the rest of the paper, we will refer tothe model parameters as the bias terms, the regression mod-els’ feature weights (referred to by the vectors w1, . . . ,wl)and the students’ memberships within the different regres-
sion models (referred to as ps for each student s).
This multi-regression model has an advantage over learn-ing a single linear regression model for all students sincethe multi-regression model is more personalized. Personal-ization is achieved through the student-specific membershipweights which determine how much each linear model con-tributes to the grade estimation for the target student. Itis also achieved through learning student-specific bias termsthat reflect upon individual student differences. The coursebias terms capture the grade patterns within the differentcourses. This multi-regression model also has an advantageover learning a different model for each student as it learns asmaller number of models that capture the performance pat-terns of the different student groups. Accordingly, it makesuse of the similarities among the students (with respect toperformance) and can better handle the data sparsity issue.
The parameters of the multi-regression model are estimatedby solving a minimization process of the form
minimize(W,P,B)
L(W,P,B) + λ(||P ||2F + ||W ||2F ), (2)
where the loss function L(.) is the Root Mean Squared Er-ror (RMSE) and W , P and B are the feature weights, stu-dents memberships and bias terms, respectively. The termλ(||P ||2F + ||W ||2F ) controls the magnitude of the featureweights and the student memberships and thus preventsover-fitting. The scalar λ is fine-tuned in the process ofestimating the model parameters.
In addition to accurately predicting students performance,the multi-regression model can be used to analyze how thedifferent features contribute to the predicted grades and thusgain some insights about the students behavior. For properanalysis of the estimated model parameters, it is more con-venient that all the returned feature weights, student mem-berships, and bias terms have non-negative values which willmake all the model’s components to contribute additively tothe predicted grades. Accordingly, the optimization problemfor learning the model parameters takes the form
minimize(W,P,B)
L(W,P,B) + λ(||P ||2F + ||W ||2F ), s.t.
wd,c ≥ 0, 1 ≤ d ≤ l, 1 ≤ c ≤ nF ,
ps,d ≥ 0, 1 ≤ d ≤ l,∀s,bc ≥ 0, ∀c,bs ≥ 0, ∀s.
(3)
The two minimization problems are solved using stochasticcoordinate descent.
3. DATASET AND EVALUATIONWe used a dataset extracted from the University of Min-nesota’s Moodle installation; which is one of the largestMoodle installations world wide. The dataset spans twosemesters and it contains 832 course instances and 11,556students. The courses belong to 157 different departments,each student has registered in at least 4 courses, and thetotal of number of assignment and quiz submissions are114,498 and 75,143, respectively. The dataset also containeda total of 251,348 forum posts. We will refer to the assign-ments and quizzes as activities. The activity grades are nor-malized to be in the range [0, 1].
3.1 Feature DescriptionFor each student-activity pair (s, a), we constructed a fea-ture vector fsa whose features fsa fall into three categories:student-specific features, activity-specific features and Moodle-interaction features. Each of these features are describednext.
3.1.1 Student-specific featuresThese are features related to the student and they mainlydescribe the student’s previous grade history. We use twostudent-specific features:
- cumGPA: The GPA accumulated over the courses pre-viously taken by the student.
- cumGrade: The average grade achieved over all of thepervious activities in the course. For the first activityin the course, cumGrade is set to the cumGPA.
3.1.2 Activity-specific featuresThese are features that relate to the activity and the coursethat this activity belongs to. We use three activity-specificfeatures:
- activity type: This can either be quiz or assignment.The activity type is handled by having two indicatorvalues, one for quiz and one for assignment.
- course level: The course level takes an integer value of1, 2, 3 or 4 and describes the course’s level of difficultywith 4 being the most difficult. These levels are derivedfrom the numeric designation of the courses.
- department: The department to which the course be-longs to. Departments are handled by having one indi-cator feature per department. The feature correspond-ing to the department of the training instance is set to1 and the rest are set to 0.
3.1.3 Moodle interaction featuresThese features describe the student’s interaction with Moo-dle prior to the due date of the activity. These features wereextracted from Moodle’s log files and are the following:
- n-init-disc: The number of discussions initiated by thestudent.
- n-engaged-disc: The number of times that the studentposted to an open discussion.
- n-read-posts: The number of forum discussions thatare read by the student.
- n-viewed-mater : The number of times the student viewedsome course material.
- n-add-contrib: The number of times the student con-tributed to the course by adding something to thecourse page (e.g., a wiki-page).
- n-other-accesses: The number of times the studentmade any other kind of access to the course pages.This feature is concerned with the student’s interac-tion with the other Moodle modules (e.g., surveys).
For each of the above Moodle interaction features, we cre-ated five different features that measured the specified inter-action at various time intervals prior to the due date. Fourof them measure the interaction at [0, 1), [1, 2), [2, 4), and[4, 7) days prior to the due date, whereas the fifth measuresthe interaction up to the due date of the previous assign-ment. These features will be denoted by appending “-x” tothe feature name, where x is the intervals’ upper bound (e.g.,n-init-disc-1, n-init-disc-2, n-init-disc-4, and n-init-disc-7 ),and the fifth will be denoted without the “-x” suffix. Notethat for the forum interaction features, the collected num-bers were normalized with respect to the total number ofavailable forum discussions/posts.
3.2 EvaluationThe dataset was randomly split into training and test sub-sets containing 80% and 20% of the student-activity pairs,respectively. The model was trained on the training setand evaluated on the test set. This process was repeated4 times and the obtained results on the test set were aver-aged and reported. The model is evaluated in terms of theroot mean squared error (RMSE) between the actual andpredicted grades on the test set.
3.3 Baseline ApproachWe compare the performance of a multi-regression modelagainst the performance of a linear regression model. Thelinear regression model estimates the student grades as
gsa = w0 +
nF∑k=1
wkfk, (4)
where fk is the value of feature k and the wk’s are the re-gression coefficients of the linear regression model. Notethat this linear regression model is different from a multi-regression model with one linear model since the latter esti-mates the student grade as
gsa = bs + bc +ms,1
nF∑k=1
w1,kfk,
where bs and ms,1 are the student-specific bias and mem-bership terms and bc is the course-specific bias term.
4. RESULTS AND ANALYSISThe results and analysis are presented in three parts. Inthe first part, we explore how the multi-regression modelperforms given different features and how it performs againsta single linear regression model. In the second part we showhow the different bias terms affect the performance of themulti-regression model. In the third part we analyze theestimated model parameters for the cases in which we learnone, two and three linear models in order to gain insightsabout the different student populations.
4.1 Multi-Regression Models Prediction Accu-racy
In order to see the importance of the Moodle interactionfeatures and how much they influence the accuracy of thepredicted grades, we trained the multi-regression models andthe baseline model twice; once using the student and activ-ity features and once using the student, activity and theMoodle-interaction features.
Figure 1 shows the performances of the linear and the multi-regression models with and without using Moodle-interactionfeatures1. The figure also shows the change in RMSE ob-tained by the multi-regression model as the number of linearmodels increases.
These results show that using a multi-regression model withone linear model gives an RMSE of 0.168 whereas using thelinear regression model described by Equation 4 gives anRMSE of 0.223. This is due to the student-specific member-ship and bias terms which enable the multi-regression modelto better capture individual student performances. More-over, the course-specific bias terms can capture the gradedistribution within the different courses.
Figure 1 also shows that the RMSE obtained by the multi-regression model decreases with increasing number of linearmodels. A larger number of linear models with student-specific memberships allow for more personalization. Usingten regression models, the obtained RMSE falls to 0.145.The incremental gains in RMSE saturate with increasingnumber of linear models.
Comparing the performance of the two multi-regression mod-els in Figure 1, we can see that the model that uses theMoodle features performs better than the one that does notuse them. A multi-regression model with ten linear mod-els gives and an RMSE of 0.168 without using the Moodlefeatures and gives an RMSE of 0.145 using the Moodle fea-tures. It can also be seen from the RMSE curves of the twomulti-regression models that as the number of linear modelsincreases, the incremental gains achieved by the model thatdoes not use the Moodle features saturate faster than theincremental gains achieved by the other model. The use ofMoodle features lead to more drop in RMSE with increas-ing number of regression models. We believe this is becausethe model that uses the Moodle features have more student-Moodle interaction information to learn from as the numberof regression models increase.
4.2 Effect of the Bias Terms on the PredictionAccuracy of the Multi-Regression Models
In order to understand how the different bias terms con-tribute to the prediction accuracy, we trained the multi-regression model using each of the student and course biasterms separetely. Figure 2 shows the performance of multi-regression models that use different bias terms and differentnumber of linear models. The plots show that the coursebias contributes to the model accuracy more than the stu-dent bias.
4.3 Analyzing Feature WeightsFor analyzing how the different features contribute to thepredicted grades, we learn the multi-regression model usingEquation 3 that has the non-negativity constrains. As men-tioned in Section 2, and according to Equation 3, all theparameters returned by the model are non-negative. This
1These results were generated by learning the model with-out the non-negativity constrains according to Equation 2.Recall that these constrains are only used when we wish toanalyze the model parameters and not when we predict thestudent grades.
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.21
0.22
0.23
1 2 3 4 5 6 7 8 9 10
RMSE
Regression Models
Single Regression-No Moodle Features
Single Regression-Moodle Features
Multi Regression-No Moodle Features
Multi Regression-Moodle Features
Figure 1: Change in RMSE as the number of regres-sion models increases.
0.145
0.15
0.155
0.16
0.165
0.17
0.175
1 2 3 4 5 6 7 8 9 10
RMSE
Regression Models
Course+Student Biases
Course Bias
Student Bias
Figure 2: Effect of the different bias terms on themodel performance.
way it is easy to compare the importance of the differentfeatures among the different estimated models. Note thatthe non-negativity constrains did not degrade the RMSE ina significant way. For the case of two linear models, theRMSE obtained with and without the non-negativity con-strains are 0.165 and 0.160 respectively.
4.3.1 Determining Importance of Model ParametersFor each feature weight wd,k, we would like to estimate howmuch it contributed to all the estimated grades. Given agrade gs,a, and according to Equation 2, the weight wd,k
contributes to gs,a by (ms,lwd,kfsa,k). Accordingly, the im-portance id,k of the feature weight wd,k is accumulated usingall the estimated grades as
id,k =
∑gs,a∈G(ms,dwd,kfsa,k)/gs,a
|G| ,
where G is the set of all grades in the test set and |G| is thesize of G.
Similarly, we estimate the importance of the student andcourse biases by how much they contribute to all the pre-
0 0.1 0.2
Feature Weight
M1 M2 M3
0 0.1 0.2
Feature Weight
M1 M2
0 0.1 0.2
assignment
quiz
num-attempts
course-level
cumGPA
cumGrade
n-viewed-mater
n-viewed-mater-1
n-viewed-mater-2
n-viewed-mater-4
n-viewed-mater-7
n-other-accesses
student-bias
course-bias
Feature Weight
M1
Figure 3: Feature weights for learning a multi-regression model with one linear model (left), two linear models(center) and three linear models (right).
dicted grades. The importance of a student bias term isestimate as
iS =
∑gs,a∈G bs/gs,a
|G| ,
and the importance of a course bias term is estimate as
iC =
∑gs,a∈G bc/gs,a
|G| .
4.3.2 ResultsWe analyze the estimated feature weights for learning amulti-regression model with one, two and three linear mod-els. Figure 3 shows the estimated feature weights for one(left), two (center) and three regression models (right). Thebinary features representing the departments were omittedas well as the features with zero or very low importance val-ues. The features related to the forum activities had verylow importance values and thus were omitted from the fig-ure. These features have very low importances as they onlyappear in a small fraction of the training data (between 10%and 25% of the training instances), whereas the features re-lated to viewing the course material appeared in almost allthe training instances.
In all three cases shown in Figure 3, the student bias, course
bias and the features related to viewing the course materialcontribute the most to the predicted grades. In the caseof two regression models, which we will refer to as M1 andM2, the quiz, number of attempts and course level are im-portant under M1 and not M2. Another interesting pointis that the features related to viewing the course materialhave higher importance under M2. In the case of three re-gression models, which we will refer to as M1, M2 and M3,the quiz, number of attempts and course level are importantunder M1 but not under M2 or M3, whereas the assignmentand most of the features related to viewing the course ma-terial have higher importance under M2 and M3. The factthat we have some models concerned with assignments andothers concerned with quizzes and their number of attemptsreflects that the type and properties of the activity have animpact on predicting student performance.
4.4 Analyzing Student MembershipsAnalyzing student memberships can give insights about thedifferent student populations. We focus on the case in whichwe learn a multi-regression model with two linear modelssince this case is easy to visualize, and moreover, we haveseen from Figure 3 that the features related to viewing thecourse material is more important under one of the two mod-
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mod
el 2
-Con
tribu
tion
Model 1-Contribution
Figure 4: Student memberships for a multi-regression model with two linear models. Everypoint represents a student.
els. The latter point indicates that viewing the course ma-terial does not have the same impact on all students.
4.4.1 Determining Models Contributions to StudentGrades
Given a multi-regression model with two linear models M1and M2, we would like to estimate for each student s howmuch each of the two models contributes to the grades pre-dicted for s. Given a grade gs,a, and according to Equa-tion 2, model d, where d ∈ {1, 2}, contributes to gs,a by(ms,d
∑nFk=1 wd,kfsa,k). Accordingly, the contribution of model
d to the grades of student s is estimated as
js,d =
∑gs,a∈Gs
ms,d
∑nFk=1 wd,kfsa,k/gs,a
|Gs|,
where Gs is the set of all grades of student s, and |Gs| isthe size of Gs. The value js,d estimates how much modeld, where d ∈ {1, 2}, contributes to the grades predicted forstudent s, taking into account the membership of s in d,ms,d. The value js,d lies in the range [0, 1], where js,d =0 means model d does not contribute at all to the gradespredicted for s, and js,d = 1 means that the grades of sare only estimated via model d. Accordingly, in our case wehave 0 ≤ (js,1 + js,2) ≤ 1.
We have plotted js,1 against js,2 for each student s as shownin Figure 4. Each point corresponds to a student, and thex- and y-axis represent js,1 and js,2, respectively. Some stu-dents have almost the same contributions by the two models,whereas other students have a higher contribution by one ofthe two models. Since the two models differ in how muchviewing course material influences the predicted grades, thiscan indicate that students with high contribution by onemodel can be different from students with high contributionby the other model.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mod
el 2
-Con
tribu
tion
Model 1-Contribution
Group 1Group 2Group 3
Figure 5: Student memberships for a multi-regression model with two linear models. Everypoint represents a student. See Section 4.4 for adescription of the point coloring.
In order to explore for student differences, we divided thestudents into three different groups based on their (js,1, js,2)values using the following procedure: First, we normalizedthe model contributions js,1 and js,2 for each student tohave a one-norm of 1. Second, we estimated the meanand standard deviation of js,1 for all students, and we got(µ, σ) = (0.453, 0.142). Finally, we clustered the studentsinto three different groups as follows:
- Student (Group 1) contains all students withjs,1 > µ+ σ.
- Student (Group 2) contains all students withjs,1 < µ− σ.
- Student (Group 3) contains all students withµ− σ ≤ js,1 ≤ µ+ σ.
Student Groups 1, 2 and 3 are shown in Figure 5 in blue,green and red colors, respectively. We have omitted all stu-dents with (js,1 + js,2) < 0.2, that is, students with bothmodels contributing to their grades by less than 0.2. Groups1, 2 and 3 contain 847, 797 and 7824 students, respectively.
4.4.2 Analysis of Student GroupsThe distribution of the GPA for the three student groups isshown in Figure 6. Group 1 has a GPA average that is lowerthan Groups 2 and 3, and a higher GPA standard deviationthan the other two groups.
We have finally investigated the relative appearance of thedifferent departments within the three student groups. Foreach (department, group) pair (d, g), we have computed theappearance as
qd,g =nd,g/ng∑3
h=1 nd,h/nh
,
0
1W
RIT
SRM
NU
RS
MAT
HM
KTG
AGEC
CH
EMC
OM
MIN
ETAN
THSL
HS
BBE
CA
HLT
HSP
ED BMSP
ANM
GM
TC
LSP
ACC
TIN
SC
SCI
OLP
DED
UC
NAT
RH
SMPH
ILG
EOG
PHAR
STAT
ECO
NKI
N EEFM
ISAH
SAG
RO
SOC
ABU
SID
SCSM
GT
FSO
SEC
HED
HD
BIO
LFS
CN
RM
JOU
RPU
BHC
MG
TPS
TLPS
YH
OR
TES
PM ESL
Dep
t Dist
ribut
ion
Group 1 Group 2 Group 3
Figure 7: Relative appearance for the different departments within the three student groups.
1.5 2 2.5 3 3.5 40
1000
2000
1.5 2 2.5 3 3.5 40
50100150
1.5 2 2.5 3 3.5 40
50
100
Group 3: mean=3.18, std=0.53
Group 2: mean=3.0, std=0.52
Group 1: mean=2.84, std=0.67
Figure 6: GPA distribution for the three studentgroups.
where nd,g is the number of students belonging to group gand are enrolled in courses belonging to department d, andng is the number of students belonging to gtoup g. Theqd,g metric can be interpreted as the probability of groupg given department d. Figure 7 shows how the differentdepartments tend to appear within each group. Each de-partment is represented by a vertical line. The verticalline has a red, a blue and a green part corresponding toqd,G1, qd,G2 and qd,G3, that is the department’s appearancewithin student Groups 1, 2 and 3, respectively. An inter-esting finding is that some departments primarily appearunder Group 1 and almost never appear under Group 2 (thedepartments towards the left of the figure like Writing, Nurs-ing and Math). These departments tend to have studentswhose access to course material is less influencing of theirpredicted grades. This finding may suggest that the typeof material provided in the LMS for the departments thatintensively appear with Group 1 may not be addressing theright student needs and therefore are not beneficial in im-proving the grades of the students. Accordingly, students’access to such non-beneficial material does not lead to betterunderstanding and thus does not lead to better grades.
5. CONCLUSIONSIn this work, we have used a multi-regression model to pre-dict student performance in course activities and analyze theresulting student populations. We have shown that a multi-
regression model performs better than single linear regres-sion as it captures personal student differences through thestudent-specific membership weights. We have also shownthat the RMSE tends to decrease with increasing the num-ber of linear regression models and thus allowing room formore personalized predictions. We have also shown thatusing the Moodle interaction features lead to an improvedprediction accuracy.
Analyzing the estimated parameters of the multi-regressionmodel showed that the student bias, course bias and fea-tures related to viewing the course material are the factorsthat mostly contribute to the predicted grades. The analysisalso showed that the activity-specific features had differentcontributions within the different linear models. Moreover,the analysis of the different student populations showed thatthe features relating to viewing of course material contributeto the predictions of a certain student subpopulation higherthan other students. It also appeared that some depart-ments tend to have students whose viewing of course ma-terial contribute to their predicted grades less than otherstudents. This might indicate that the material providedin the LMS for these departments may not be addressingthe right student needs and thus are not helping studentsachieving better grades.
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