=Paper=
{{Paper
|id=Vol-2704/paper3
|storemode=property
|title=Modeling Trajectories to Understand the Delayed Completion of Sequential Curricula Undergraduate Programs
|pdfUrl=https://ceur-ws.org/Vol-2704/paper3.pdf
|volume=Vol-2704
|authors=Renato Boegeholz,Julio Guerra,Eliana Scheihing
|dblpUrl=https://dblp.org/rec/conf/ectel/BoegeholzGS20
}}
==Modeling Trajectories to Understand the Delayed Completion of Sequential Curricula Undergraduate Programs==
https://ceur-ws.org/Vol-2704/paper3.pdf
Modeling Trajectories to Understand the
Delayed Completion of Sequential Curricula
Undergraduate Programs
Renato Boegeholz[0000−0002−6826−1210] , Julio Guerra[0000−0002−8296−9848] , and
Eliana Scheihing[0000−0003−1801−9167]
Instituto de Informatica, Universidad Austral de Chile, Chile
renato.boegeholz@uach.cl,{jguerra,escheihi}@inf.uach.cl
Abstract. Taking more time than expected to complete university de-
gree programs is a global and known problem, and in Chile, has relevance
because pressure exists to complete degrees on time. In this work, we
explore academic delay in higher education programs, in particular an
engineering program, and its relation with academic information sum-
marizing the trajectory of students along with the academic program.
Academic information is represented by semester-by-semester features
that reflect different aspects such as performance, workload, and dif-
ficulty. Exploratory analyses of these variables reveal two orthogonal
groups: performance and workload; then used to build models predicting
the relative delay of a student at her 8th term at the program relative to
the expected completion at 8th term. To further explore the trajectory
of delay and analyze how the delay relates to other academic aspects
such as term by term performance or workload, a sequential model was
built. Results show different patterns of behaviors across different lev-
els of delay in the 8th term. The methods and results of this research
can be used by educational institutions or the government to support its
decisions about the use of resources and attrition rates reduction.
Keywords: academic analytics · curricular analytics · learning analyt-
ics, · educational data mining · academic trajectories · time to degree.
1 Introduction
Taking more time than expected to complete university degree programs is a
global and known problem. The average time to degree1 is 1.36 [1] in Latin
America, and 1.31 in Chile, a number that has not experienced a relevant vari-
ation in the last 10 years. This statistic not only means that obtaining a degree
1
The ratio between the average time it takes for students to graduate; and the theo-
retical duration of the study program
Copyright c 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 R. Boegeholz, J. Guerra, E. Scheihing
takes on average about 31% more time than expected but also reveals that career
delays are a serious problem with economic considerations: this situation causes
in Chile an additional expense for families and the government estimated at US$
500 million [2,3]. Delay in completing a degree program may have multiple causes
such as the failure and repetition of subjects, temporary suspension of studies,
or assuming less course load than expected according to the curricular plan [4].
Special importance is given to academic reasons behind academic delay because
pressure exists to complete degrees on time: high education in Chile is financed
by the student (or her family) with the help of scholarships or other funding
benefits that require good performance and usually do not tolerate delays [5].
In this work, we explore academic delay in higher education programs, in
particular an engineering program, and its relation with academic information
summarizing the trajectory of students along with the academic program. While
similar work focused in performance variables such as grades [6, 17, 19], we in-
clude other relevant aspects of the academic information such as the course load
taken by the student each term, the difficulty associated with the courses taken,
and repetition of courses which academic situation may determine the risk of
punitive actions (for example, elimination of the study program by failing a sub-
ject more than twice, or failing, in the same semester, more than two subjects),
among others. The first research question of this work is:
1) What is the relation between delay in obtaining the degree and academic
information along the trajectory of the student?
We build prediction models on academic delay considering different features
that can describe academic trajectories. As mentioned before, we seek to repre-
sent such trajectories in terms of different academic information spanning per-
formance, course workload, and difficulty. These academic factors are relevant
not only because they could predict academic delay, but because they could
characterize the trajectories in an actionable manner, that is, further analysis
of such trajectories could bring insights that could support counseling practices
and curricular re-design actions. Thus, we state a second research question:
2) Can the academic trajectories relate to delay be characterized in a manner
that provides information about student behavior?
2 Related Work
Researchers have used different approaches to analyze academic information and
typically centered around performance measures. Based on grading data and dis-
missing the background characteristics of the students, [9] identified curriculum
subjects that can serve as effective indicators of academic performance. Using
X-means [7] to group students yearly, [9] discovered typical progress patterns
and evaluated the predictive capacity of the explanatory subjects. Considering
the results in an entrance self-assessment test and the academic performance of
Modeling Trajectories of Delayed Degree Completion 3
the first year, [11] used K-means [10] to group students and follow their perfor-
mance trajectories in the following 2nd and 3rd year, measuring the influence
of the first year behavior in the progression of the curriculum [11]. [6] explored
multiple personal and social factors that can affect the academic performance of
university students and, using the Grade Point Average (GPA) as an explained
variable through decision trees [14] proposed a qualitative model to classify and
predict it [6].
Dropout has also been investigated. Aiming to relieve dropout, [16] used rec-
ommendation systems techniques to predict the grades that students will obtain
in future subjects. The predictions were made using personalized multiple linear
regressions (PLMR) [15] on student participation data in both traditional classes
and Massive Open Online Courses (MOOC) [16]. [13] examined demographic
variables as family characteristics; pre-university and university academic per-
formance factors; and the participation or not in recovery courses, to predict
the persistence of the students in the study programs. Using as explanatory
variables the scores of the ACT standardized test for college admission, the av-
erage of grades in high school, the average grades of the first semester of the
university; and using analysis of variance (ANOVA), Pearson product-moment
correlations, and multiple regression analysis [12] showed that students who were
academically prepared to take college-level courses were more likely to persist
than students assigned to mandatory recovery courses [13].
Researchers have also focused on performing analyses of students’trajectories
to inform the design of curricula. [17] used student performance data from a spe-
cific program to perform an analysis of the curriculum design of the program.
In particular, they modeled the difficulty of each subject as its contribution
(negative or positive) to the students’ GPA and then contrasting this measure
to a survey of student perception. Using the same performance data they also
performed a dropout and enrollment path analysis [17]. Important to consider is
that most of the previous work has been carried related to a flexible course–credit
systems where the degrees are obtained as the sum of core and optional approved
subjects [8, 11, 13, 19]. This context differs from our sequential-non-flexible cur-
ricula, where the flow of subjects to take is pre-defined for all the terms of the
program.
The representation of the academic trajectories of the students is not triv-
ial, and it is necessary to consider the temporal dependence of the variables
under study. Following this idea, [19] used frequent pattern mining [18] to re-
veal academic trajectories to understand the sequences of subjects to take that
could improve student performance. In [20], a sequential data model is proposed
that explicitly captures the temporal dependencies of the academic performance
characteristics to form fingerprints or signatures that are constructed allowing
different analytical interpretations and the development of predictive models for
the risk of academic delay.
As presented, several of the proposed models -all of the regression type- con-
sidered the performance of the students as explanatory variables, without taking
into account the dependence that exists in the performance of a student in var-
4 R. Boegeholz, J. Guerra, E. Scheihing
ious subjects within the same semester, as well as in successive semesters. Our
work distinguished from previous work in several aspects: in its multivariate na-
ture considers in addition to academic performance aspects such as the academic
workload and the difficulty of the subjects in each semester; incorporating the
temporal dimension in the modeling of the students’ trajectories; and explor-
ing such trajectories in the context of a curriculum with a sequential structure,
where about 90% of the courses to be completed are mandatory.
3 Methods
3.1 Academic Features and Data Descriptions
Academic information at the level of the degree program includes data of courses
taken, passed, failed (with their grades), and dropped in each term of the aca-
demic life of the student. We combine these records with historic data and the
expected curricular progress 2 to generate a series of academic features in each
term of the student academic life. These features represent different dimensions
related to academic performance, academic workload (courseload), the relative
difficulty of the courses, and the consistency between the courses taken and their
theoretical order in the curriculum. Nine features for each term of each student
were computed and are defined in Table 1. More details about the definition of
the variables can be found in Appendix A.
We understand by academic trajectory to all this information organized in a
term by term sequence. To allow comparisons between trajectories, we considered
only the activity of the first 8 semesters (8 terms) for each student, counted
from their first enrollment. These 8 semesters also represent a milestone of the
study program because contain the required subjects to obtain the Licenciatura
degree3 . Considering this, the explained variable will be the delay in the 8th
term (DELAY8), which measures how far is the student of having completed all
courses of the first 8 terms of the plan in her first 8 terms of academic life. A
student who passed all planned courses of the first 8 semesters of the program
in her first 8 semesters of academic life, has delay zero.
To reduce inconsistencies in the comparisons of the trajectories (because of
the dependency between the features and the structure of the program curric-
ula), it was decided to analyze only one study program: Engineering in Computer
Science. For the period of available data, this program implements three curricu-
lum versions (2008, 2010, and 2015) each with 11 semesters of duration and an
average of 6 subjects per semester. To study the performance of the students
during their first 8 semesters, only whose admitted between 2008 and 2015 were
2
In Chile most of higher education programs have a semi-flexible curricular plan,
where the study sequence in pre-defined term by term.
3
In Chile, “Licenciatura” is similar to a bachelor’s degree. To obtain this grade is nec-
essary to complete 8 semesters of subjects that are part of an academic major. This
degree allows you to continue an academic career. To be qualified for professional
practice there are necessary between 2 and 4 semesters of additional subjects.
Modeling Trajectories of Delayed Degree Completion 5
Table 1: Description and possible values of the features built for this study. Definition
of the variables can be found in Appendix A.
Feature Description Possible val-
ues
GP A Not cumulative weighted grade average for the semester. [1.0, 7.0] with 4.0
the passing grade
P ASSRAT E Passing rate of the semester (the ratio of passed subjects [0, 1]
to passed plus failed subjects).
F IRST IM E The proportion of subjects enrolled for the very first time [0, 1]
in the semester.
P ROGRE Contribution of subjects passed in the semester to the [0, 1]
total of subjects required to obtain the degree.
W KLOAD Academic workload rate for the semester measured in [0, LENP rog ]
CST to the average semester CST of the program.
DIF F IC A Difficulty of the semester, as an additive measure (“Al- [0, max(HF Rj )SU Bi ]
pha difficulty”).
DIF F IC B Difficulty of the semester, as a geometric measure (“Beta [0, 1]
difficulty”).
DISP AR The disparity of subjects enrolled in the semester: the [0, 1]
difference, in semesters, between the highest level subject
and the lowest level subject.
DELAY Measurement of the delay between the theoretical and [0, 1]
actual (average) semester of the student given their date
of admission.
considered. The resulting data set was composed of 14,199 records of academic
activity of 365 different students.
3.2 Data Modeling
We are interested in modeling the behavior of the 8th-semester delay (DELAY8)
as a function of the other eight features defined for each semester. We limit the
scope of the independent variables (the features) to the first 4 semesters because
of two reasons. First, the idea of predicting delay (and also predicting dropout)
gain relevance if a prediction can be done early, thus it seems reasonable to
predict a delay in the 8th semester with information from the four first terms.
Second, the four initial semesters correspond to the “Bachillerato” milestone
6 R. Boegeholz, J. Guerra, E. Scheihing
in the engineering programs of the university, where the foundational courses
of math and physics are concentrated and which have the higher failure rates,
thus are strongly related with academic failure or success. The analyses will be
performed in three steps.
The first step is to perform an exploratory data analysis (EDA) for all the
variables. We summarize the main characteristics of each variable (mean, me-
dian, quantiles, and range) together with their box plots, following with correla-
tion matrix and principal component analysis (PCA) to gain an understanding
of the structure of the set of variables and identify the most significant variables
which can explain the academic delay.
The second step includes building predictive models using two supervised
algorithms, linear regression (LR) and support vector machine (SVM) on the
delay at 8th semester with other features of the first 4 semesters, such as difficulty
and workload as predictors. These analyses target research question 1.
The third step is to characterize academic trajectories in terms of the relation
of the term features and delay. An adaptation of the sequential model proposed
by [20] is implemented. In our case, this model is built as follows. Each student
trajectory is represented as a sequence of 4 nodes (semesters 1 to 4), where each
node is a single value representing a delay score. To compute the i semester’s
delay score, first all students are clustered using k-means on their i semester’s
features. Then the score of the i semesters is the average delay of all students
in the cluster (cluster members).
4 Results and Discussion
4.1 Exploratory Data Analysis
We explored the behavior of all the features in their semester-by-semester pro-
gression. Box plots and summary statistics for the variables can be found in
Appendix B. Figure 2 presented here as a sample shows box plots of the fea-
tures GPA (not cumulative weighted grade average) and DIFFIC A (an additive
measure of difficulty) for the first 8 semesters.
From the analysis can be observed that the DELAY variable shows distribu-
tions with increasing medians and variability through the terms, as students on
average accumulate more delay as they stay more terms. It can be observed that
the distributions of the average grade (GPA) have medians that increase slightly
but consistently through the terms. PASSRATE shows the lowest median in the
2nd semester with a value of 0.5. In the following semesters, the values of the
medians increase progressively meanwhile the median passing rate for the 1st
semester is much higher than the rest, with a value of 0.7. Dispersion is similar
in all semesters.
In the case of the academic workload (WKLOAD), the median distribution
in semesters 2nd to 4th is below the academic workload defined by the cur-
riculum and it approaches that value in semester 5. The dispersion of these
distributions increases between the 6th and 8th semesters. The relative diffi-
culty, both DIFFIC A, and DIFFIC B, show medians that descend as students
Modeling Trajectories of Delayed Degree Completion 7
Fig. 2: Variability in the samples across the 8 semesters for (a) The not cumulative
weighted grade average (GPA) and (b) The alpha difficulty (DIFFIC A).
progress in their semesters, with quite similar dispersion. The disparity (DIS-
PAR) shows only two median values: 0.143 for semesters 2 through 5; and 0.286,
for semesters 6 through 8. Dispersion appears biased (positively or negatively)
for all semesters, except for semester 8 in which symmetry is appreciated.
Correlation matrices and principal component analysis (PCA) were obtained
to explore options for reducing the set of variables. The results of the analyzes for
semester 3 are described in Figure 4. Biplots and correlograms for all the features
can be found in Appendix B. In all semesters it is observed that the delay is
negatively correlated with the variables of performance, curricular progress, and
academic load, namely: GPA, PASSRATE, PROGRE, AVG, and WKLOAD.
On the other hand, it has a very weak positive correlation with the measures
of difficulty of the subjects (DIFFIC A and DIFFIC B) and weak negative with
the measure of disparity (DIS). Two main components of the PCA shows two
Fig. 4: (a) Biplot and (b) correlogram of the 3rd-semester variables.
groups of variables: i) those that are more related to the individual performance
8 R. Boegeholz, J. Guerra, E. Scheihing
of the students: AVG, PASSRATE, PROGRE and GPA, and ii) those which are
related to the characteristics of the study program: WKLOAD, DIFFIC A, and
DIFFIC B. It is interesting the distinct components represented by performance
and workload. To represent each of the groups in the subsequent analyzes of this
work, we selected GPA and DIFFIC A respectively based on a better degree of
interpretation than they may have compared to the other features.
Table 2: Summary measures of the predictive capacity of the models. K indicates the
number of predictors in the model.
Regression SVM
Model K AICc ∆AICc RMSE RSq MAE
SGD1 2 -123.430 98.913 0.154 0.588 0.127
SGD2 4 -146.596 75.746 0.142 0.645 0.114
SGD3 6 -161.323 61.020 0.139 0.679 0.116
SGD4 8 -177.462 44.881 0.137 0.688 0.115
SA1 8 -118.020 104.322 0.153 0.562 0.127
SA2 16 -153.047 69.296 0.140 0.657 0.117
SA3 24 -180.135 42.208 0.122 0.738 0.102
SA4 32 -222.343 0 0.106 0.805 0.088
4.2 Prediction Models
To analyze the predictive capacity of the explanatory variables, two families of
models where built. The SGDi models use predictors GP Ai and DIF F IC Ai
adding them incrementally from first to fourth semester. In this way,
SGD1 implements: DELAY8 ∼ GP A1 + DIF F IC A1
SGD2 implements: DELAY8 ∼ GP A1 + DIF F IC A1 + GP A2 + DIF F IC A2
and so on.
The SAi models use as predictors all the features in each semester in the same
incremental way. Thus,
SA1 implements: DELAY8 ∼ GP A1 +P ASSRAT E1 +W KLOAD1 +DIF F IC A1 +
DIF F IC B1 + DISP AR1 + F IRST IM E1 + P ROGRE1
SA2 implements: DELAY8 ∼ GP A1 +P ASSRAT E1 +W KLOAD1 +DIF F IC A1 +
DIF F IC B1 +DISP AR1 +F IRST IM E1 +P ROGRE1 +GP A2 +P ASSRAT E2 +
W KLOAD2 + DIF F IC A2 + DIF F IC B2 + DISP AR2 + F IRST IM E2 +
P ROGRE2
and so forth.
Table 2 summarizes the results obtained. Regressions were built using a Gen-
eralized Linear Model (GLM) with Gaussian errors and an identity link func-
tion. The corrected Akaike Information Criterion (AICc) of every model and the
Modeling Trajectories of Delayed Degree Completion 9
difference with the lower value was computed. We can observe that this indica-
tor declines, as expected, with the inclusion of the variables of the consecutive
semesters. An important finding is that the predictive power in the first two
semesters is quite similar between the model with two variables and the model
with all variables. In contrast, by including semesters 3 and 4, the predictive
power of the complete model is much greater. In the case of SVM, the RSME,
RSq, and MAE indicators are calculated and ratify the observed with AICc. In
particular, with the complete model until the fourth semester, an RSq of 0.805
is obtained which is high.
4.3 Characterization of the Delay Trajectories
Trajectories were built for each student as a sequence of 4 values, one per each
of the 4 first terms, each representing the average delay of all students who have
similar academic features in a semester. To do this, we performed a clustering at
each semester with all its features. Each term, the student falls into one cluster,
from which the average delay marks his/her delay trajectory. The number of
clusters obtained varied from 3 to 4.
Figure 5a shows the trajectories for all students who completed 8 semesters.
To understand the delay behavior along time, the trajectories were presented
into 3 groups: those students who reached their 8th semester with 2 semesters
of delay or less (DELAY 8 ≤ 0.29), students who reached the 8th semester with
a delay between 2 and 4 semesters (0.29 < DELAY 8 ≤ 0.57), and students who
have a delay of more than 4 semesters (DELAY 8 > 0.57).
Figures 5b to 5d shows the trajectories for every one of those groups. It can
be observed -as in Figure 5a- that for the first semester, the prediction of delay
for all students is 2 or 4 semesters.
Most of the students who complete their 8th semester with a delay lower or
equal than 2 semesters (Figure 5b), were projected with 2 semesters or less of
delay throughout the entire program.
On the other hand, students who finished the 8th semester with a delay
greater than 4 semesters (Figure 5d), maintained similar forecasts during the 4
semesters under study.
It can be seen that in both groups, the less and more delayed, the proportion
of “good” and “bad” results for their 1st semester is quite similar, which would
make the 1st semester a not reliable indicator of the final delay result.
5 Conclusions
In this research paper, we applied statistical and data mining methods to under-
stand the different behaviors in the progression of students across a sequential
curriculum program related to delay in obtaining a degree. First, features that
summarize different aspects of the academic records information such as perfor-
mance, workload, and course difficulty were built for each term a student stays
10 R. Boegeholz, J. Guerra, E. Scheihing
Fig. 6: Delay trajectories for (a) All the students. (b) Students with an 8th-semester
delay less or equal than 2 semesters. (c) Students with an 8th-semester delay between
2 and 4 semesters. (d) Students with an 8th-semester delay greater than 4 semesters.
in the academic program. A measure of the delay was custom-made in rela-
tion to the expected progress at term 8th. Second, we performed an exploratory
data analysis of the features which revealed two different sets of variables that
appeared orthogonal within the two principal components of a PCA: a group
with all performance variables, and a group with workload and difficulty. The
weighted average grade (GPA) and one measure of difficulty (DIFFIC A) were
selected to represent these groups. Third, the predictive capacity of the fea-
tures was explored, revealing that the selected two variables can predict the
8th-semester delay close enough as a model using all predictors. Fourth, delay
sequences were modeled and represented as trajectories showing distinguishable
groups of students’behavior. The evidence provided shows that the delay is not
strictly determined by the students’performance during their first semester. Low
initial performances can follow a path of progressive improvement and reduce
their potential delay while ends the program. In conclusion, this work provides
valuable insight into a better understanding of the dynamics of the progress in
Modeling Trajectories of Delayed Degree Completion 11
a sequential curricular program, potentially contributing to the decision-making
of institutions, directors, and students.
6 Acknowledgments
Work funded by Universidad Austral de Chile and the LALA project (grant no.
586120-EPP-1-2017-1-ES-EPPKA2-CBHE-JP). This project has been funded
with support from the European Commission. This publication reflects only the
views of the authors, and the Commission cannot be held responsible for any
use which may be made of the information contained therein.
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Modeling Trajectories of Delayed Degree Completion A1
Appendix A Definition of Explanatory Variables
The following equations describe how the features were built for each semester
i of the student’s stay and every subject j enrolled in that semester:
PSU Bi
j=1 (GRAj · CT Sj )
GP Ai = PSU Bi ∈ [1.0, 7.0] (1)
j=1 CT Sj
P ASSi
P ASSRAT Ei = ∈ [0, 1] (2)
P ASSi + F AILi
SU B1Ti
F IRST IM Ei = ∈ [0, 1] (3)
SU Bi
P ASSi
P ROGREi = ∈ [0, 1] (4)
SU BP rog
PSU Bi
j=1 CT Sj
W KLOADi = ∈ [0, LENP rog ] (5)
CT SAvg
SU
X Bi
DIF F IC Ai = HF Rj ∈ [0, SU Bi · max(HF Rj )] (6)
j=1
SU
Y Bi
DIF F IC Bi = 1 − (1 − HF Rj ) ∈ [0, 1] (7)
j=1
max(SEMj ) − min(SEMj )
DISP ARi = ∈ [0, 1] where j = 1..SU Bi (8)
LENP rog − 1
i ≤ AV Gi
(
0
DELAYi = i−AV Gi ∈ [0, 1] (9)
LENP rog −1 i > AV Gi
Where:
LENP rog = The total number of semesters of the study program.
SU BP rog = The total number of subjects to be completed in the program to obtain the degree.
CT SP rog = The total number of CTS credits of the study program.
SU Bi = The number of subjects enrolled in semester i.
SU B1Ti = The number of subjects enrolled in the semester i for the very first time.
P ASSi = The number of passed subjects in semester i.
F AILi = The number of failed subjects in semester i.
GRAj = The final grade obtained by the student in the subject j.
SEMj = Semester in which the subject j is located within the study program.
PSU Bi
j=1 SEMj
AV Gi = Average semester in which the student is, with AV Gi = SU Bi ∈ [1, LENP rog ]
A2 R. Boegeholz, J. Guerra, E. Scheihing
CT Sj = The number of CTS credits of the subject j.
CT SAvg = The average number of CTS credits per semester of the study program.
HF Rj = The historical failure rate of the subject j.
Modeling Trajectories of Delayed Degree Completion B1
Appendix B Exploratory Data Visualization
Fig. B.1: Variability in the samples for the weighted semiannual average (GPA),
throughout the 8 semesters.
Table of Fig. B.1: Not cumulative weighted grade average (GPA)
Statistic N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 4.087 1.099 1.000 3.440 4.120 4.870 6.630
2 324 3.718 1.106 1.000 3.173 3.870 4.442 6.350
3 277 3.818 0.997 1.000 3.250 3.910 4.560 6.070
4 234 3.896 1.041 1.000 3.482 4.060 4.545 6.360
5 200 4.110 0.981 1.300 3.470 4.120 4.680 6.400
6 184 4.143 1.158 1.000 3.500 4.225 4.942 6.510
7 163 4.305 1.028 1.000 3.630 4.260 4.955 6.490
8 154 4.531 1.030 1.000 3.883 4.570 5.255 6.410
B2 R. Boegeholz, J. Guerra, E. Scheihing
Fig. B.2: Variability in samples for the passing rate (PASSRATE), over the 8 semesters.
Table of Fig. B.2: Passing rate (PASSRATE)
Semester N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 0.672 0.295 0 0.4 0.7 1 1
2 337 0.526 0.321 0.000 0.250 0.500 0.750 1.000
3 280 0.554 0.338 0.000 0.333 0.585 0.833 1.000
4 241 0.558 0.331 0.000 0.250 0.600 0.800 1.000
5 204 0.618 0.336 0.000 0.333 0.600 1.000 1.000
6 191 0.593 0.354 0.000 0.333 0.600 1.000 1.000
7 164 0.661 0.338 0.000 0.333 0.750 1.000 1.000
8 158 0.693 0.314 0.000 0.500 0.775 1.000 1.000
Table of Fig. B.3: First-time-enrolled subjects rate (FIRSTIME)
Semester N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 0.672 0.295 0 0.4 0.7 1 1
2 337 0.526 0.321 0.000 0.250 0.500 0.750 1.000
3 280 0.554 0.338 0.000 0.333 0.585 0.833 1.000
4 241 0.558 0.331 0.000 0.250 0.600 0.800 1.000
5 204 0.618 0.336 0.000 0.333 0.600 1.000 1.000
6 191 0.593 0.354 0.000 0.333 0.600 1.000 1.000
7 164 0.661 0.338 0.000 0.333 0.750 1.000 1.000
8 158 0.693 0.314 0.000 0.500 0.775 1.000 1.000
Modeling Trajectories of Delayed Degree Completion B3
Fig. B.3: Variability in samples for the first-time-enrolled subjects rate (FIRSTIME),
over the 8 semesters.
Fig. B.4: Variability in the samples for the academic workload (WKLOAD), throughout
the 8 semesters.
B4 R. Boegeholz, J. Guerra, E. Scheihing
Table of Fig. B.4: Academic workload (WKLOAD)
Semester N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 1.036 0.108 0.131 1.067 1.067 1.067 1.067
2 337 0.770 0.226 0.000 0.667 0.800 0.984 1.133
3 280 0.754 0.196 0.000 0.656 0.767 0.875 1.167
4 241 0.713 0.251 0.000 0.600 0.733 0.885 1.148
5 204 0.797 0.253 0.000 0.667 0.867 0.942 1.333
6 191 0.832 0.316 0.000 0.633 0.918 1.049 1.333
7 164 0.827 0.203 0.000 0.700 0.900 0.938 1.167
8 158 0.926 0.289 0.000 0.738 0.967 1.167 1.500
Fig. B.5: Variability in the samples for the alpha difficulty (DIFFIC A), throughout
the 8 semesters.
Table of Fig. B.5: Difficulty (alpha) (DIFFIC A)
Semester N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 2.009 0.331 0.053 1.822 2.248 2.248 2.248
2 324 1.914 0.386 0.465 1.737 2.005 2.209 2.615
3 277 1.741 0.340 0.742 1.488 1.830 2.030 2.610
4 235 1.552 0.490 0.155 1.357 1.627 1.876 2.484
5 201 1.490 0.472 0.096 1.151 1.552 1.846 2.484
6 184 1.336 0.487 0.107 1.018 1.285 1.747 2.450
7 163 1.307 0.509 0.450 0.805 1.297 1.772 2.415
8 154 1.068 0.436 0.400 0.747 1.014 1.299 2.443
Modeling Trajectories of Delayed Degree Completion B5
Fig. B.6: Variability in samples for beta difficulty (DIFFIC B), over the 8 semesters.
Table of Fig. B.6: Difficulty (beta) (DIFFIC B)
Semester N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 0.922 0.089 0.053 0.909 0.956 0.956 0.956
2 324 0.913 0.084 0.402 0.908 0.938 0.952 0.973
3 277 0.897 0.058 0.615 0.862 0.921 0.939 0.976
4 235 0.882 0.067 0.470 0.854 0.900 0.926 0.969
5 201 0.853 0.096 0.452 0.814 0.890 0.923 0.962
6 184 0.812 0.121 0.107 0.727 0.841 0.914 0.962
7 163 0.773 0.156 0.397 0.611 0.841 0.911 0.958
8 154 0.709 0.175 0.347 0.567 0.742 0.865 0.959
Table of Fig. B.7: Disparity in the semester (DISPAR)
Semester N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 0.000 0.000 0 0 0 0 0
2 337 0.110 0.112 0.000 0.000 0.143 0.143 0.429
3 280 0.132 0.106 0.000 0.000 0.143 0.143 0.429
4 241 0.190 0.136 0.000 0.143 0.143 0.286 0.571
5 204 0.196 0.124 0.000 0.143 0.143 0.286 0.571
6 191 0.237 0.161 0.000 0.143 0.286 0.286 0.714
7 164 0.257 0.150 0.000 0.143 0.286 0.286 0.857
8 158 0.302 0.189 0.000 0.143 0.286 0.429 0.714
B6 R. Boegeholz, J. Guerra, E. Scheihing
Fig. B.7: Variability in the samples for the disparity in the semester (DISPAR),
throughout the 8 semesters.
Fig. B.8: Variability in the samples for the delay related to the curriculum (DELAY),
throughout the 8 semesters.
Modeling Trajectories of Delayed Degree Completion B7
Table of Fig. B.8: Delay related to curriculum (DELAY)
Statistic N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 0.000 0.000 0 0 0 0 0
2 337 0.052 0.055 −0.048 0.000 0.048 0.107 0.143
3 280 0.131 0.085 −0.020 0.057 0.143 0.190 0.286
4 241 0.173 0.105 −0.036 0.086 0.179 0.257 0.429
5 204 0.232 0.134 0.000 0.114 0.286 0.333 0.476
6 191 0.275 0.165 0.000 0.120 0.314 0.387 0.643
7 164 0.316 0.214 −0.048 0.114 0.371 0.486 0.714
8 158 0.362 0.234 0.000 0.107 0.371 0.571 0.857
Fig. B.9: Variability in the samples for the curricular progress (PROGRE), throughout
the 8 semesters.
Table of Fig. B.9: Curricular progress (PROGRE)
Semester N Mean St. Dev. Min Pctl(25) Median Pctl(75) Max
1 365 0.094 0.041 0.000 0.061 0.100 0.140 0.143
2 337 0.055 0.037 0.000 0.020 0.061 0.082 0.143
3 280 0.053 0.037 0.000 0.020 0.041 0.082 0.143
4 241 0.053 0.037 0.000 0.020 0.061 0.082 0.122
5 204 0.062 0.039 0.000 0.020 0.061 0.102 0.143
6 191 0.065 0.047 0.000 0.020 0.061 0.102 0.163
7 164 0.074 0.046 0.000 0.041 0.080 0.122 0.143
8 158 0.082 0.046 0.000 0.041 0.082 0.122 0.163
B8 R. Boegeholz, J. Guerra, E. Scheihing
Fig. B.11: Biplot and correlogram for semester 1.
Fig. B.13: Biplot and correlogram for semester 2.
Fig. B.15: Biplot and correlogram for semester 3.
Modeling Trajectories of Delayed Degree Completion B9
Fig. B.17: Biplot and correlogram for semester 4.
Fig. B.19: Biplot and correlogram for semester 5.
Fig. B.21: Biplot and correlogram for semester 6.
B10 R. Boegeholz, J. Guerra, E. Scheihing
Fig. B.23: Biplot and correlogram for semester 7.
Fig. B.25: Biplot and correlogram for semester 8.