Self-consistent Peer Ranking for Assessing Student Work
Dealing with Large Populations
Kees van Overveld
1
and Tom Verhoeff
2
1
Dept. of Industrial Engineering & Innovation Sciences, Eindhoven University of Technology, Eindhoven, Netherlands
2
Dept. of Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands
Keywords:
Large Scale Assessment, Peer Reviewing, Ranking Algorithm.
Abstract:
Assessing large populations of students puts a serious burden on teaching staff capacity. For open-format
assignments, automation of the reviewing process can offer only limited support. Peer ranking is a partial
solution to the problem, with the added benefit that students’ critical reading skills are developed. We see two
remaining problems, however: (1) for students, it is a major challenge to assign marks on an absolute scale, and
(2) students’ competence in reviewing may vary significantly—so not all peer reviews should have a similar
weight in the process. To remedy these shortcomings, we suggest an approach to peer ranking, inspired by
Jon Kleinberg’s HITS-algorithm, where both the students’ assignment results and the quality of their double
anonymous peer reviews are algorithmically ranked. Based on preliminary model calculations, we estimate
that this strategy may reduce the required effort for reviewing open-format assignments approximately by a
factor of ten. A first large-scale pilot with this method will take place in undergraduate courses at Eindhoven
University of Technology, spring 2013. Since this involves about 900 students, automated support is a must.
We describe the peer reviewing facilities that were introduced in our web-based education support system
named peach
3
.
1 MOTIVATION AND PROBLEM
DEFINITION
Assessing large populations of students puts a seri-
ous burden on teaching staff capacity. This is even
more so if strict deadlines need to be observed with re-
spect to providing feedback to students. In a practical
scenario, set at Eindhoven University of Technology
in early 2013, some 900 students will be submitting
elaborations of homework assignments, each corre-
sponding to about two A4 pages of text, in a weekly
rhythm, where marks need to be provided no later
than two weeks after submission, and no more than
two staff members are available for reviewing.
If reviewing a single work is estimated to take 20
minutes, completing the entire correction takes 300
person hours, or 150 hours per individual teacher. Al-
though one week contains 24 × 7 = 168 hours, it is
obvious that straightforward reviewing is no option.
Peer reviewing, i.e., students reviewing each
other’s work using a protocol that ensures anonymity,
seems a plausible first option (Sadler and Good, 2006;
Lu and Bol, 2007). A naive scheme, however, where
students give marks to their peers, suffers from two
obvious drawbacks:
1. Unless the assignments admit only a single correct
answer, there is subjectivity involved in marking.
In the current casus, the assignments are deliber-
ately open ended. They contain questions of the
form ‘give an example for X’, ‘give some argu-
ments in favor of, and some arguments against Y’,
or ‘what is your substantiated opinion regarding
Z’. Although a student can be expected to form a
global opinion (‘this is quite good’), we ask too
much if this opinion should be made quantitative,
say, on a 10-point scale.
2. More importantly,not all students can be expected
to be equally competent reviewers. This prob-
lem could be mitigated by having every work re-
viewed by sufficiently many students, so that non-
systematic errors can be expected to average out.
This will not work in practice, however, since it is
unrealistic to have students review more than, say,
ve works each.
Problem 1 is partially solved by having students
merely rank works, that is, to put the (say) five works
they review in order of quality, rather than to give ab-
399
van Overveld K. and Verhoeff T..
Self-consistent Peer Ranking for Assessing Student Work - Dealing with Large Populations.
DOI: 10.5220/0004352903990404
In Proceedings of the 5th International Conference on Computer Supported Education (CSEDU-2013), pages 399-404
ISBN: 978-989-8565-53-2
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
solute marks. From the methodology of social sci-
ences (Mellenbergh, 2011), it is known that compara-
tive ranking is generally easier than absolute ranking.
We use the term “peer ranking” (following (Allain
et al., 2006)) for comparative ranking in the context
of peer review.
Peer ranking, however, does not completely solve
Problem 1: as part of the assessment process, our stu-
dents need an absolute marking.
The research question of this paper, combining
Problems 1 and 2, is now stated as:
‘How can peer ranking be used, taking differ-
ences in students’ reviewing competences into
account, in order to obtain absolute marks in
assessments?’
For peer ranking that accounts for differences in re-
viewing competences among peers, we coin the term
‘self-consistent peer ranking’.
In Section 2, we formally define self-consistent
peer ranking and an approach to it, loosely based on
Jon Kleinberg’s HITS algorithm (Kleinberg, 1999).
Some implementation details are described in Sec-
tion 3. Prior to the actual implementation in a real-life
setting, we want to gain some feeling for the merits
of the approach. Therefore, we performed a model
study; this is discussed in Section 4. Section 6 lists
a number of possible variations of the method, Sec-
tion 5 discusses the conditions for application of the
algorithm in an educational context, and Section 7
discusses the web-based support facility peach
3
. Fi-
nally, in Section 8 we summarize our conclusions and
indicate directions of future work.
2 PROPOSED APPROACH
The problem of ranking the quality of submitted
works, based on judgments by reviewers with un-
known and varying reviewing competence, somewhat
resembles the problem that Google is solving by
means of page ranking:
a web page is good if many web pages link to it;
not every link should contribute equally to the
‘goodness’ of a webpage;
a link from a good webpage should contribute
more;
this gives a cyclic definition of what constitutes
‘good for web pages.
In the case of peer reviewing, the reasoning goes:
a student’s work is good if peers have a high es-
teem of it;
not every peer’s opinion should contribute equally
to the ‘goodness’ of a work;
the opinion of a competent peer should contribute
more;
this gives a cyclic definition of what constitutes
‘good’ (for works) and ‘competent’ (for peers).
The definitions for the goodness of a work and the
competence of a peer can now be given formally.
Students have a reviewing competence, called c
i
for student number i, i = 1...N. Competences are
initially unknown.
Works have a quality (‘goodness’), called q
j
for
work number j, j = 1.. .N. Note that a q
j
is not nec-
essarily a final grade; that is, once we have an esti-
mate for q
j
, we still have the problem of converting
it into a grade. Qualities are initially unknown. Re-
view competence and quality of work are assumed to
be independent variables.
An assessment where student i reviews work j
produces an indicator, called a
ij
. A larger a
ij
value
means that student i rates work j as better. The in-
dicator a
ij
gives information both about student i and
work j. Again, this is not necessarily a grade. When a
collection of a
ij
is known, the challenge is to recover
the c
i
and the q
j
.
For a first, na¨ıve approach, we treat c
i
and q
j
sym-
metrically; we scale them between 1 and 1; we as-
sume a full set of a
ij
(that is, every student has re-
viewed every work), and we prepare the a
ij
so that
they are also scaled between 1 and 1. The values c
i
,
q
j
, and a
ij
are called self-consistent, when (a) the q
j
are the weighted averages of the a
ij
, where the c
i
are
the weight factors, i.e. q
j
=
i
a
ij
c
i
, and (ii) similarly
with the roles of c
i
and q
j
reversed, i.e., c
i
=
j
a
ij
q
j
.
The following algorithm, if it converges, produces a
set of c
i
and q
j
that are self-consistent for given a
ij
.
1. Initialize all c
i
to random values between 1
and +1.
2. Calculate first estimate
j
: q
0
j
=
i
a
ij
c
0
i
.
3. Update
i
: c
n+1
i
=
j
a
ij
q
n
j
.
4. Update
j
: q
n+1
j
=
i
a
ij
c
n+1
i
.
5. Renormalize c
i
and q
j
to keep them between 1
and +1.
6. Repeat steps 3 through 5 until convergence, that
is, c
n
i
j
a
ij
q
n
j
; and q
n
j
i
a
ij
c
n
i
.
This algorithm is in fact a so-called power iteration
(Golub and Van Loan, 1996). Power iteration con-
verges under weak conditions. Indeed, in case of
convergence, q = AA
T
q holds, where q is a vector
of q
j
, and matrix A holds all a
ij
. We see that q is
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400
an eigenvector of the positive-definite AA
T
; the re-
peated scaling ensures that the largest eigenvalue is 1,
and power iteration is a well-known stable route to
find the eigensystem with the largest eigenvalue for
positive-definite matrices.
The above algorithm has the same structure as Jon
Kleinberg’s HITS algorithm (Kleinberg, 1999), used
for self-consistent ranking of scientific citations. In
the next section, we examine the modifications and
additions needed to make the algorithm work for self-
consistent peer ranking.
3 IMPLEMENTATION DETAILS
To apply the algorithm from the previous section to
reviewing students’ works, we have to resolve three
issues.
i. If students’ reviewing comprises ranking instead
of marking their peers’ works, we have to con-
struct a numeric value for a
ij
for every pair (stu-
dent i, work j) from all orderings on the collection
of works as found by all students;
ii. Since students will review and rank no more than,
say, five works each, the majority of a
ij
is un-
known. If an unknown a
ij
is represented by 0
(encoding a neutral judgment for work j by stu-
dent i), the matrix A is sparse. We must cope with
the sparseness of A;
iii. We demand that, eventually, students receive
marks for their works on some given scale, say
0 through 10. The q
j
only carry information in
their ordering; hence, we have to convert ranks to
absolute marks.
The resolution of these three issues is closely related.
We start with item iii, then i, and finally item ii.
3.1 From Ordered q
i
to Marks
After completion of the algorithm, we re-order the q
j
so that they are monotonically increasing in j. Now
that we have obtained the vector q, we know the order
of the quality of the works. This means that the even-
tual marks should be such that the work j = 1 should
receive the lowest mark, and the work with j = N re-
ceives the highest mark. The marks of the other works
could be obtained, for instance, by linear interpolation
between these two. The mark m
k
for work k then is
given by
m
k
= m
1
+ (m
N
m
1
)(k 1)/(N 1) (1)
So, with merely correcting two works, we can assign
marks to all works.
To obtain a more reliable set of marks, however,
we may prefer to have a few more works corrected
and marked by teaching staff. In case more works
are marked by hand, the interpolation could be more
advanced: with four hand-corrected works, we might
choose the numbers 1, N/3, 2N/3, N and use a piece-
wise linear function or a spline in k for the interpola-
tion instead of (1).
3.2 From Ranking Results to a
ij
Values
Students each rank a small collection of works. The
result of ranking by student i is equivalent to a set
of relations, a
ij
1
< a
ij
2
for j
1
and j
2
in the set of in-
dices of works, reviewed by this student. We may op-
tionally allow ex aequo ranking, that is a
ij
1
= a
ij
2
for
some maximum number of pairs (j
1
, j
2
). Ranking in-
formation can be encoded in an anti-symmetric N×N
matrix, say S
i
, where +1 occurs in entry ( j
1
, j
2
)
when, according to student i, a
ij
1
< a
ij
2
;
, and 1 oc-
curs in entry ( j
2
, j
1
). All other entries are 0.
For example, if student i ranked a
ij
3
< a
ij
1
< a
ij
2
,
then we will have
S
i
=
j
1
j
2
j
3
.. .
j
1
0 +1 1 0
j
2
1 0 1 0
j
3
+1 +1 0 0
(2)
Next, all matrices S
i
need to be aggregated to obtain
the matrix A for the algorithm.
This aggregation is not trivial. For instance, the S
i
need not all be mutually consistent. That is, an en-
try ( j
1
, j
2
) may contain +1 in one of the S
i
, whereas
it is 1 in another S
i
. Now, prior to running the al-
gorithm, the weights c
i
are unknown. Still, it seems
that the c
i
are necessary to resolve conflicts due to in-
consistencies. Therefore, for full self-consistency, the
construction of A should take place simultaneous with
obtaining c and q.
Although we plan to derive a fully self-consistent
aggregation algorithm to obtain A from the matrices S
i
in the future, we intend to run first trials with a sim-
ple approximation to this scheme. This approxima-
tion amounts to setting a
ij
1
= 1 and a
ij
2
= +1 for
respectively the lowest and highest ranking works j
1
and j
2
, according to student i, and to give the other
works a
ij
values that linearly interpolate these values.
So, for five reviewed works per student, the a
ij
are set
to the sequence 1,0.5,0,0.5,1, irrespective of any
rank assignments by other students to these works.
1
Constructing the a
ij
from the initial ranking inputs in
this way is obviously ad-hoc, and we will use it for a
first trial only to see if the approach is promising.
1
When we admit ex aequo ranking, one or more of the
values may be left out of the sequence 1,0.5,0,0.5, 1.
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3.3 Sparse A
Convergence of the algorithm can be proven for full
rank matrix A. Due to sparseness, however, A is
highly rank deficient. Fortunately, power iteration is
relatively robust. This means that, as long as a min-
imum percentage of a
ij
is known, the algorithm still
can approximately recover c, and, more importantly,
q from A. There are two considerations, however, that
we need to take into account.
Obviously, the fraction of non-empty entries in A
cannot be arbitrarily low. Therefore, given that
each student reviews five works, the total popu-
lation of students (to be called ‘cluster’) in one
peer-ranking trial cannot be too high. In order to
estimate the size of the largest allowable cluster,
we perform a model study, described in the next
section.
With increasing cluster size, the convergence of
our algorithm becomes increasingly problematic.
‘Problematic convergence implies the following.
We need more iterations (perhaps infinitely
many) until convergence. This is no fundamen-
tal issue: it is easy to detect convergence; by
admitting a maximal number of iterations, we
can conclude if convergence fails.
With full rank, the solution of the power iter-
ation algorithm is unique. This can no longer
be proven for rank deficient A. This again is no
fundamental issue, however: when we run the
iteration several times with different starting
conditions, we can easily verify if converged
solutions are sufficiently close.
2
If A gets increasingly rank deficient, the ob-
tained vector q will contain increasingly more
noise. This means that the accuracy of the al-
gorithm decreases, where the accuracy is de-
fined as the extent to which the found order of
the works matches with the order as it would
be found with hand-correction. The match be-
tween the hand-corrected order and the order
found by the algorithm can be empirically as-
sessed by doing a hand correction of the en-
tire cluster. Small mismatches—that is, mis-
matches where the rank position of any q
j
does
not differ too much from a rank position as
2
There is one curious subtlety. If q is a solution to q = AA
T
q,
then so is q. Since the elements of q are scaled between 1 and 1,
we cannot distinguish q and q beforehand. If the teacher reviews
both extreme works (that is, after renumbering, the works with j =
1 and j = N), however, it should be immediately clear which of the
two is the best and which is the worst. This unambiguously fixes
the sign of q.
would be found with hand correction—can be
partially compensated for by doing a larger
fraction of hand corrections—to the extreme
where all works are corrected by hand, and
there is no added value of peer ranking. We
plan to find the optimal cluster size, such that
the accuracy of the algorithm is sufficient, by
means of empirical assessment prior to full-
scale implementation of the algorithm.
4 MODEL STUDY
To get a first, global, idea of attainable maximal clus-
ter size, and hence the maximal efficiency improve-
ment that can be attained by self-consistent peer rank-
ing, we perform a model study. In this model, we
postulated a relation between the c
i
(student’s review-
ing competence) and the a
ij
(the scores, attributed to
works j by student i) as follows.
A student with higher c
i
contributes values for a
ij
that are closer to the true q
j
. By the ‘true q
j
we
mean the q
j
that would result if a teacher would
have reviewed work j.
A student with lower c
i
inputs values for a
ij
that are closer to a uniform random number be-
tween 1 and +1. That is, failing competence is
modeled as an unbiased noise term.
Next, to test the algorithm, we set up a collection of
size N of works, every work with a known quality q
j
,
and a collection of size N of students, every student
with a known reviewing competence c
i
. Cluster size N
will be varied to see what cluster sizes give acceptable
accuracy, where the number of reviewed works per
student is kept fixed to five. Known qualities and re-
viewingcompetencesare taken randomlybetween 1
and +1. The known c and q are called c
known
, q
known
,
respectively.
With c
known
and q
known
, the matrix a
ij
is computed
as follows. For every i, five random js are selected
such that every work j is ‘reviewed by exactly five
different students i. The scores a
ij
are calculated as
a
ij
=
q
known j
(1+ c
known i
) + R (1 c
knowni
)
2
, (3)
where R = rand(1,1) is a uniformly distributed ran-
dom number between 1 and +1. All other a
ij
are set
to 0.
With matrix A set up in this way, the algorithm is
run, and, if convergent, the resulting q and c are plot-
ted against q
known
and c
known
. For ideal reconstruc-
tion, the graph should be monotonically increasing.
The precise shape is determined by the normalization
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used. In our case, the normalization is a Euclidean
distance norm, resulting in a roughly sigmoid shape
for the graph.
Experiments reproduce the predicted behavior,
where increasing sparseness in A causes increasing
deviations from purely monotonic. If we find 5% de-
viations acceptable (that is, 5% of the works receive
an out-of-order q
j
), the matrix A can be as sparse
as 10%. In other words, for a cluster size as large
as 50, with five reviews per work, the algorithm is
capable to find an approximation to the correct order
with no more than 5% errors.
It turns out, however, that the outcome of these tri-
als is sensitive to the assumptions with respect to the
precise form of (3). If we assume students are slightly
more competent in reviewing, the performance of the
algorithm is drastically better; if students are less
competent, the performance is considerably worse—
which is not too unexpected. Therefore, although
these model trials suggest a cluster size of 50 with
ve reviewed works per student, we may want to be
a bit more conservative when we actually implement
the scenario for the first time.
5 DISCUSSION
An algorithm for calculating students’ reviewing
competence and the quality of students’ work may
be a necessary ingredient of peer reviewing, but it is
definitely not sufficient. In (van Zundert, 2012), ed-
ucational considerations regarding peer reviewing are
studied. In this section, we list a number of assump-
tions that should hold for an algorithm like the present
one to be trustworthy.
Peer groups should be unbiased and uncorrelated
so that every assessment can be seen as an in-
dependent measurement of each student’s perfor-
mance. Careful randomization helps to remove
correlations; bias is more subtle to deal with,
though. For instance, in case of misconceptions,
shared by a majority of the students (‘homework
is boring’), correct answers (such as ‘homework
is exciting’) may score systematically low, and the
algorithm has no means to detect this error. It will
manifest itself in that the order, calculated by the
algorithm, consistently differs from the order ob-
tained by staff. In preparing assignments, there-
fore, questions with likely answers that are objec-
tively wrong’, but that could result from collec-
tively shared misconceptions, should be avoided.
Rather, assignments should be such that students
can base their scores on how much detail is pro-
vided, how elaborate an answer is, how clearly the
answer has been written, how convincing the an-
swer is, et cetera.
Peer ranking should be applied to a series of as-
signments rather than a single assignment, so that
statistical evidence can be used to assess the reli-
ability of the final outcome. Statistical evidence
could be, e.g., the standard deviation σ of the
marks over a series of assignments in one term.
If σ decreases as one over the square root of the
number assignments, N, it may be the case that the
outcome indeed measures students’ performance
during that term. In case σ does not decrease with
increasing N when averaging over the series of as-
signments, the per-assignment scores apparently
do not measure the actual performance level of
a student, and the peer review gives no informa-
tion about this level. There could be various rea-
sons for such inconclusive outcomes: perhaps the
assignments do not accurately measure students’
performance levels, or students performance lev-
els vary wildly over the term. From a method-
ological point, it would be good to include the σs
in the final marks.
6 POSSIBLE VARIATIONS
We briefly present three possible variations.
1. The algorithm calculates both c and q from
scratch, using the matrix A as only input. We
may expect, however, that the students’ reviewing
competence will not vary much over time. This
suggests to bootstrap the algorithm with the re-
sults in the first week, and use the found c as a
first estimate in the next week. We may even con-
sider to use the running average of the cs over
subsequent weeks, representing the intuition that
we get increasingly more accurate estimates of the
individual students’ reviewing competence.
2. Teachers may consider to have one or more ‘ex-
ample elaborations’ to be, unknowingly, reviewed
by the students. Since works are reviewed anony-
mously, students will not know that they review a
teacher’s work instead of one of their peers. As-
suming that teachers works have insurmountable
quality, the associated q
j
must keep a constant
value of 1 during the iterations. Therefore, they
serve to further stabilize the algorithm.
3. Despite the efficiency improvementoffered by the
algorithm, reviewing still requires works to be as-
sessed by teachers, which takes time. To reduce
waiting time for students, feedback can be given
in three tiers. The first tier is immediately after the
Self-consistentPeerRankingforAssessingStudentWork-DealingwithLargePopulations
403
raw ranking: a student then can be informed about
the ‘five relative rankings among four other works
that this student’s work received. Although this
carries no absolute information, the difference be-
tween ve times number one’ or ‘five times num-
ber five’ is probably significant. The second tier is
immediately after running the algorithm: students
then can get a percentile score (‘85% of your clus-
ter has lower scores than you’). Only the third
tier feedback, where a student receives an abso-
lute mark, needs to wait until teachers correct the
few representative works per cluster.
7 WEB-BASED SUPPORT: peach
3
At Eindhoven University of Technology, we use a
web-based education support system peach
3
, since
2001 (Scheffers and Verhoeff, 2012). Students sub-
mit their work for deadlined assignments to peach
3
through a web browser. peach
3
monitors the dead-
lines, stores submitted work, performs configurable
automatic checks on the content, disseminates it to
those involved in the course, and allows entering of
manual feedbackand grades. Recently, we added sup-
port for peer reviews, including peer ranking.
To carry out a peer review of an assignment Z, a
new assignment is created that is designated as a peer
review of Z. Students who submitted work for Z are
allocated random works by other students within their
cluster, in such a way that each work is reviewed by
a configurable number of students (in this paper, we
have used five as bundle size). They read anonymized
versions of work under review in a browser, and pro-
vide review reports, grades, and/or a ranking with re-
spect to each other, through a web GUI. All review
results can then be exported, processed, and imported
back into the system as grade. Afterwards, if so de-
sired, students can see anonymized review reports,
grades, and rankings of their work.
8 CONCLUSIONS,
FUTURE WORK
We propose a strategy for reducing the amount of re-
viewing, to be done by teachers, for open-ended as-
signments. An algorithm, called self-consistent peer
ranking, requires students in a cluster of peers to
anonymously rank, say, five peer works. The differ-
ences between students’ ranking competence (the c
i
in the algorithm) are estimated, and used to compute
a weighted final rank score (the order of the q
j
in
the algorithm). Next, teachers review the highest and
lowest ranking work (and perhaps few more for in-
creased reliability) in a cluster, to establish the abso-
lute marks; marks of works not reviewed by teachers
are found by interpolation.
A preliminary model study suggests that clusters
can contain some 40 to 50 students, which would in-
dicate a factor of 8 to 10 reduction of manual correc-
tion work, if students rank five works each, while the
amount of out-of-order errors of the algorithm is no
more than 5%. A group of 1000 students would then
be split into 20–25 clusters.
A first field trial will take place early 2013 at Eind-
hoven University of Technology, involving about one
thousand students. This will involve our web-based
education support system peach
3
, that provides sup-
port for peer reviews and peer ranking. If the results
are promising, we will fine tune the cluster size and
other parameters in the algorithm to get the optimally
achievable efficiency improvement. Also, we will de-
velop the algorithm further so that the matrix A can be
obtained from the individual ranking inputs without
having to resort to the ad-hoc assignment of a range
of numerical values to the a
ij
for given i.
ACKNOWLEDGEMENTS
We would like to acknowledge the helpful feedback
of colleagues on a draft version of this paper.
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