1 Introduction
Peer grading refers to a process where students grade the work of their costudents based on a scoring rubric provided by the instructor. While it has been experimented with in the past (see [8]), it has become increasingly popular in the context of MOOCs, where thousands of students hand in homework that has to be graded (e.g., [3]). Similar to other crowdsourcing scenarios, the hope is that even though students may not be perfect graders, it might be possible to come up with a fair or accurate final grade for each submitted homework by aggregating many such imperfect grades. The challenge of finding good aggregation algorithms has been taken up by the machine learning community and a number of suggestions have been made in the literature (e.g., [5, 10, 6, 7, 2, 11, 9]). The bottom line of these papers is that statistical models or machine learning algorithms are successful at solving this task.
Intrigued by its idea and potential, we used peer grading in an undergraduate course in computer science at the University of Hamburg: the course on algorithms and data structures (AD). Throughout one semester, students solved 6 exercise sheets with 25 exercises each. 6 Teaching assistants (TAs) graded all submissions in the traditional way, but at the same time we applied peer grading on the same set of submissions. The data generated during the whole semester has been made publicly available, the link can be found on our homepages. We refer to it as AD data below. At the end of the semester, we applied various algorithms to aggregate the peer grades and to analyze our data. Contrary to other researchers we found that none of these methods show a satisfactory improvement over a simple baseline. We will discuss a number of possible explanations for these results and analyze sources of problems that prevent more sophisticated methods to succeed.
2 Our setup
For our peer grading experiment we selected the course on algorithms and data structures at the Department of Computer Science, University of Hamburg in the winter term 2014/15. The course is compulsory for all bachelor students in computer science and had 219 active participants in total. We had 6 TAs grade all homework in the traditional way. Roughly once every two weeks, students received an exercise sheet as homework, resulting in 6 sheets with 19 individual exercises in total. The exercises consisted of puzzles on algorithms or proofs about algorithmic properties. For example, the students were asked to apply a given algorithm by hand on a small input or they were asked to design an algorithm to solve a particular problem. In each exercise, students could collect a certain number of points. Getting at least 50% of all homework points in the semester was a necessary requirement to pass the course. Exercises were solved and submitted by groups of up to three students. This semester we had a total of 79 groups. For each individual exercise, every group handed in their submission via an online system (we used a modification of the moodle open source platform^{1}^{1}1https://moodle.org/). After the submission deadline, a sample solution together with assessment criteria was posted online by the course instructor. We tried to describe all possible correct solutions to the exercises, point out pitfalls or common errors and discuss details about the grading procedure. Grading took place simultaneously in the following three different ways.
(i) Self grading. The students were asked to read and understand the sample solution and then grade their own submission first. While they solved exercises in groups of three, the self grading took place individually, resulting in 3 self grades per submission.
(ii) Peer grading. Each student graded 2 randomly selected submissions of other groups, resulting in about 6 peer grades per submission. Grading was doubleblind.
(iii) TA grading. All submissions were furthermore randomly assigned to and graded by one of the 6 TAs.
In two exercise sheets, we experimented with slightly altered settings. In one case, we increased the number of peer grades by each reviewer from 2 to 5, which results in about 15 peer grades for each submission. On another exercise sheet, we applied ordinal peer grading: Instead of assigning every submission an absolute grade, we gave each student a total of 5 peer submissions and asked them to only rank these submissions from worst to best.
In order to ensure that the students take their peer grading duties seriously, we made a reasonable peer grading performance a mandatory requirement to pass the course: students were required to participate in the peer grading of at least 5 exercise sheets, and we announced that sloppy grading behavior would not be tolerated. Moreover, the peer grades contributed to the final grades for the submissions with a weight of 20% (with the other 80% being the respective TA grade). An anonymous questionnaire at the end of the semester revealed that about half the students liked the concept of peer grading, had the impression that it helped them deepen their understanding, and would like to repeat it in another course. The other half of the students did not like it, mainly because it had taken too much of their time.
errors of the estimated grades with respect to the actual underlying true grades. The box plots show the distribution over the errors of 100 independently generated datasets while the number of peer grades per submission increases along the xaxis. Left: Dataset generated according to UMT’s model assumptions with reasonably realistic hyperparameters (true scores generated with
, bias and reliability). Right: Skewed true score distribution according to
and additionally, 20 graders always draw scores uniformly at random while bias and reliability for the remaining 80 graders is modeled as before.3 Algorithms for estimating true grades
From a statistical or machine learning point of view, the most interesting questions in the context of peer grading are the following.
(i) Unsupervised setting. In the complete absence of TA or instructor grades, is it possible to take several “imperfect” peer grades and to aggregate them into a “fair” or “accurate” final grade for each submission? MOOCs traditionally have this setting.
(ii) Supervised setting In a setting where partial grading is available by an instructor, is it possible to predict or recover the instructor’s grades based on aggregated peer grades? If instead of a single instructor, partial grading is performed by several TAs, is it possible to predict missing grades in the quality of TA grades or even better? This is a realistic scenario in large university courses.
(iii) Adversarial setting Is it possible to come up with a grading scheme that is robust against adversarial behavior of individuals (such as students deliberately giving everyone very high grades or trying to downgrade others) or adversarial attacks of groups of students (such as a cartel of students that act together in order to fool the grading procedure)?
Our work focuses on the first two questions. Let us introduce some notation. Over the course of the semester, students solved and graded several exercises. Fix one exercise and suppose that submissions to this exercise have been handed in by the students. Consider a set of graders . By we denote the score given to submission by grader . In the unsupervised scenario, we do not have any data on what the “true” score of each submission should be. Instead, the standard approach is to simply define the true score as the population average over the scores of all potential graders. As we only observe the scores of few graders in practice, the goal of an unsupervised algorithm is to correct for inaccuracies that are introduced by the actual grading procedure. In the supervised setting the true score may be given by the instructor or TAs. The goal of supervised peer grading algorithms is to give an estimate of the true scores for all submissions based on the set of the given peer grades.
Depending on whether the focus is on the absolute values of the scores or just on the ranking of the submissions, the model performance is evaluated using different error functions. We use the error to compare the absolute score values and the Kendall error to compare the rankings induced by the scores. The Kendall error between two rankings counts the number of pairs of items for which the ranking order is inversed. For each pair of items an error of 0 is given for agreement, 1 for inversion, and an error of 0.5 if exactly one of the rankings gives both items the same rank. The final Kendall error is the mean over the errors for all possible pairs.
3.1 Unsupervised models
Mean and Median. We simply estimate the true score of a submission by taking the mean (resp. median) of all peer grades for this submission. These algorithms serve as baseline.
Unsupervisedsingletask (UST). Following [5]
, we make some modelbased assumptions. Fix a single exercise. We assume that the true scores of all submissions to this exercise are normally distributed as
. Each grader has an inherent and a certain. The intuition is that the bias models the tendency of a grader to generally give high scores or to be very strict, whereas the reliability accounts for the variance in the grading performance. Formally, if a submission
has a true , then the grade reported by is normally distributed asAs the actual true score is not known, it is replaced by the current estimate in practice. The bias of all graders is distributed as
and the reliability is Gammadistributed as
The hyperparameters , , , , can be used to control the weight of bias and reliability estimation as well as the strength of regularization. The goal is to fit the model to the observed data and to learn the parameters , and . For this purpose we use the EM algorithm, which [5] found to give results similar to those obtained by more elaborate Gibbs sampling procedures.Unsupervisedmultipletasks (UMT). In our course, each student graded a total of about 57 different submissions for 19 exercises over the whole semester. While the UST model just learns from one exercise at a time, the UMT model learns the parameters jointly over all exercises. UMT’s basic assumption that the bias and reliability of a grader are inherent attributes and do not vary over different exercises could lead to more accurate estimates as there is much more data to work with.
Evaluating UST and UMT on artificial data confirms the finding in [5] that such models do a remarkable job at recovering the true underlying score, see Figure 1 for an illustration. As more grades per submission are added, UST performs almost as well as UMT and they both beat simple algorithms such as mean and median by far. For example, UMT only needs 4 grades per submission to reach the error rates of the mean algorithm with 9 peer grades as seen in the left figures. In particular, the models even work reasonably well on artificial data with a model mismatch, i.e. data that has not been generated according to the model assumptions.
Ordinal models. It has been suggested that instead of collecting numeric grades for each submission (cardinal peer grading), it might be more reliable to ask each student to only rank a set of submissions. Using those partial rankings of each reviewer, the purpose of the ordinal algorithms is to generate an overall ranking of all submissions and to assign them cardinal rankings following a certain score distribution.
A very simple algorithm for this purpose is Borda Count (BC). Given a ranked set of submissions by a reviewer (where is the number of reported grades by each reviewer), the algorithm gives submission the score that is the number of submissions that were ranked lower than . The sums of these scores for each submission determine the overall BC score. A number of more elaborate models have been described for this purpose in [6]: Mallows (MAL) and scored Mallows (MALS), BradleyTerry (BT), Thurstone (THUR) and PlackettLuce (PL).
We will focus here on the BT model, which assumes that the likelihood of switching the ranking of a pair of submissions depends on the distance of their true scores and on the reliability of the grader:
The scores and reliabilities are then estimated using an alternating stochastic gradient descent algorithm. The performance of the ordinal algorithms in comparison to the Mean baseline on artificial datasets is shown in Figure
2. Note that the Mean baseline is computed on the actual grades as opposed to the ordinal models that only have access to the rankings. Nonetheless, all ordinal models but MAL beat the Mean algorithm. The reliability estimation only shows improvements in the right figure, which is unsurprising, as ordinal grading implicitly applies a threshold for inaccuracies in the reported grades: while it is likely that a reviewer misses the true cardinal grade, getting just the order right is much easier, so the effects of the reliability estimation are only visible here when a large portion of the reviewers is reporting random grades. Increasing the magnitude of the bias values for the reviewers decreases the performance of the Mean algorithm while that only has a negligible effect on the ordinal models (not shown in the figure).3.2 Supervised models
In the supervised models, the goal is to learn to use the peer grades to predict the true grades as given by an instructor. In our case, we take the grades provided by the TAs as ground truth. We consider the following two approaches.
Supervisednaive (SN). As baseline we use the following naive algorithm. For all exercises, the submissions are split into a training set and a test set. We use the TA grades as ground truth for the training sets to estimate the overall student grader biases. With those bias estimates, we compute the biascorrected mean scores. Denote the submissions that student graded over the whole course in the training set by . Taking the TA grades as true scores, we calculate the overall bias for the student as
On the test set, let be all students who graded submission . The biascorrected mean score is then given as
Supervisedmultipletasks (SMT). Another approach is to incorporate the TA grades directly into the UMT model. For some subset of the submissions, the respective TA grades are put into the model like any other peer grade, but the TA reliabilities are set to a high constant. This automatically corrects the student reviewer biases towards the TA grades and also gives higher reliabilities to students that grade similar to the TAs. To see how much this improves the overall accuracy, the error is only computed on submissions whose TA grades are unknown to the model.
3.3 Linking the data
While all introduced models estimate the scores, bias and reliability in a very direct way, more accurate results might be obtained by linking up the data in other ways. For example, it stands to reason that students with higher grades on their own homework submissions are likely to be more reliable at grading than students with lower grades. One may even take it a step further by making the homework scores dependent on their grading reliability in order to motivate them to put more effort into the peer grading. Another example is that better performing students might have higher standards when grading other submissions, effectively resulting in a negative bias.
Assuming the data shows strong correlations in this regard, there are several ways to incorporate these additional ties into the model. A very simple approach is fitting a linear function between the own homework scores to the bias/reliability and to use that in the computation of the final grades. Using UMT, one can also build a hybrid model by scaling the estimated reliability with the output of the linear function that takes the own score as input. Such approaches have been shown to improve the model accuracy, see e.g. [5, 4]. We will study these correlations in our dataset and analyze whether they can be used to improve the score estimates.
4 Our dataset and its analysis
4.1 AD data: first observations
We will begin with an overview over the raw data. To be able to compare performances across different exercises, we rescale the scores of each exercise to lie in the interval
. A different standardization of shifting and rescaling the scores to have mean 0 and variance 1 (zscores) leads to very similar results. To get a first impression of our data, consider the plots on the left in Figure
3. In each panel, the figure shows histograms of scores for a particular exercise. These already show a number of interesting points. Not very surprisingly, we see that self grades are often higher than peer grades, which in turn tend to be higher than the grades given by the TAs. For some of the exercises, it looks like the peer grade histograms are “shifted versions” of the TA histograms but this is not always the case. To the contrary, sometimes the overall characteristics and shapes of the histograms are quite different. In general the scores do not seem to be normally distributed. The first obvious reason is that the scores are bounded to a fixed interval which can lead to several artifacts. The score distributions are often skewed, for example when the exercise was easy and many students got full marks for their submission. In many cases the score distribution is clearly bi or multimodal: this can arise if a large number of students solved only part of the exercise whereas others solved it completely.A second aspect illustrated in Figure 3 (right) visualizes the relationship between TA grades and peer grades for individual submissions. While the tendency of peer grades generally being higher than the TA grades is again evident here, we can also see quite a number of submissions that received a score of 0 by TAs but moderate to large scores by peers. Looking into the corresponding submissions reveals that a typical reason is a wrong solution to the exercise where many reviewers miss the error and give full marks instead. We can also see in the figure that there is a large number of submissions with full TA grades but slightly lower peer grades. This happens because it is simply unlikely for all 6 peer grades to be full scores and a single lower grade will drag down the mean value.
(top right), but with white noise added for the evaluation step to simulate the effect of using several TAs: errors were computed against noisy true grades (independent Gaussian noise
).On a more abstract level, the figure reveals that the “sources of error” for a grader are not only a bias due to different taste or different levels of strictness as suggested in the probabilistic models, but also a serious lack of understanding or information. This is problematic for statistical algorithms: if a submission gets high scores by most peer graders, there is no way to know whether this is really justified because the given solution is correct or whether the reason is that all peer graders have overlooked a crucial mistake in the submission. On the other hand, it is hard to detect cases of otherwise reliable graders getting a score completely wrong, in particular in the realistic scenario where only few grades are given for each submission.
4.2 Fitting the models to the data
We analyze our data using the introduced models. In the unsupervised scenario, we simply take all grades, fit the models, and estimate a “true score”. Note that in the unsupervised setting, we cannot optimize the model to fit any ground truth (such as the TA grades). If the histogram of peer grades is shifted with respect to the TA grades^{2}^{2}2See e.g. in the third panel of Figure 3 (left).
, there is no way for an unsupervised model to correct for this. Hence, comparing the estimated true scores to TA scores by any loss function that compares the scores directly, such as the
error, might be dominated by the overall bias shift. To cover for this, we use the Kendall rank correlation as a second error measure.The hyperparameters of the models are chosen as follows. The parameters , and for UST and UMT only control the strength of regularization and were found to have little to no impact on the overall accuracy. The reliability parameters and control whether the model gives all students a similar reliability or fits them to a large variety of reliability values. In our evaluations, we use the sample mean and variance of the given peer grades for each exercise to set and and fix the remaining hyperparameters at , and . For BT, we again use the sample mean and variance as priors and choose , for the reliability. Note that in the exercise sheet where we applied the pure ordinal setting, students only report rankings of submissions, so sample mean and variance are unknown there. In that case, the parameters can be chosen arbitrarily to control the distribution of the resulting scores.
4.3 Analysis of unsupervised models
We will begin with a comparison of the ordinal algorithms (For this section, the implementation provided by [6] was used.) A plot of the algorithm performance on each individual exercise is shown in Figure 4. In our case study we usually collected numeric grades – only their induced rankings are fed into the ordinal algorithms. For that reason, we again use the Mean algorithm as baseline. Exercises 51 and 52 are an exception to that, as we collected the grades in an ordinal fashion there, so we use the simple BC algorithm as baseline for those two exercises. With the exception of Mallow’s model, all ordinal algorithms perform very similarly. In particular, using only the peer grades they all perform worse than the mean algorithm by a significant margin. The difference is smaller when the peer and self grades are combined, as there are significantly more pairwise comparisons in this dataset which helps the ordinal models. Adding the reliability estimation to the models curiously increased the Kendall errors in most exercises by 0.010.03 (not shown in the figure). On the exercises 61 and 62, we asked each student to grade 5 other submissions instead of 2. The effect is visible on the peer grades, as the performance of the ordinal algorithms is substantially better on those tasks, strengthening the point that ordinal models inherently need more grades to match the performance of cardinal algorithms.
As a further experiment, we collected the grades in an ordinal fashion in exercises 51 and 52. Much to our surprise, the embarrassingly simple approach of the BC algorithm yields no worse results than the rankings of the more complex ordinal algorithms on exercise 51 and even beats them on exercise 52. This could imply that more complicated models are not needed for ordinal peer grading. Additionally, we can see that the performance of the ordinal algorithms on exercises 51 and 52 is rather bad in general. One reason for this might be the grading behavior of the students in ordinal tasks. Many students reported that they made less of an effort for ordinal peer grading than for cardinal peer grading because they considered it easier to quickly come up with a ranking than to report absolute grades. We believe that this might be a serious drawback of collecting peer grades in an ordinal fashion. If students do not look at the details of a submission, it is unlikely that the overall grading performance improves over cardinal grading.
Figure 5 shows the results of the cardinal models on our AD dataset along with BT representing the ordinal models. Note that BT cannot be run exclusively on the self assessment grades, as each student only graded one submission which does not imply any ranking. As opposed to the results on the artificial data in Figure 1 where the modelbased approaches clearly outperform the baseline, we can now see that UST and UMT provide no improvement over the simple mean. This finding is disappointing and contradictory to the results in the literature. We will discuss possible explanations for this behavior in the following section.
4.4 Why do the models provide no improvements over mean?
Amount of data. We have about 6 peer grades and 3 self grades per submission, collected over 19 exercises. Each student submitted around 57 grades in total. The results on artificial datasets as well as other studies on peer grading suggest that this should be enough to get a reasonably reliable estimate. A lack of data is not the problem here.
Model assumptions mismatch. As seen above, our data usually does not satisfy the model assumptions (normal distributions, etc). However, experiments with artificial data whose distributions do not agree with the model assumptions show that the model typically still works reasonably well. We do not believe that the model mismatch is the major source of the problem.
Model fitting. We set the hyperparameters as described above, though in our experiments, we found that the models are not sensitive to the choice of the hyperparameters. The actual model fitting was done with the EM algorithm. [5] reported that the results of the EM algorithm are almost equal to the ones of Gibbs sampling. Furthermore, simulations with a number of different artificial datasets resulted in consistent estimates for the score, bias and reliability values.
TAs as baseline. We evaluate our errors against the TA grades, that is we consider the scores or orderings given by the TA grades as ground truth. However, these grades have been given by 6 different TAs, so the variance within these grades might make them unsuitable to serve as ground truth (in the extreme case, if we compared against random grades, then none of the models would outperform the others). We first study this effect with artificial experiments. Consider the setting in Figure 1
, but we now add noise to the true grades before computing the errors (we use Gaussian noise with standard deviation
). The results can be seen in Figure 6 (right). They still look similar to the original results in Figure 1, just the overall performance worsened slightly due to the noise. In particular, the UST and UMT models still considerably outperform the mean estimate. This is even the case if we use an unrealistically large standard deviation for the noise, say .As a next step, we look at our actual data to evaluate the consistency among the TA grades in comparison to the consistency among the peer grades. Due to constraints during data collection, we did not have the possibility to conduct an extensive experiment to compare the grading performance of the TAs with each other. Instead, we consider the following evaluations. We first compare the average reported grade of the TAs to the average grades given by the student reviewers. As can be seen in Figure 7 (left), the TAs have a very low bias amongst each other, in particular it is much lower than the biases amongst the students. Next, we look at the overall histograms of all given grades by each TA, see Figure 6 (left). There is little variance amongst the TA histograms, in particular compared to the variance in the peer histograms. All in all it looks like the TAs grade reasonably consistently, so we believe that the use of different TA grades as ground truth cannot be the major reason for the lack of improvement of the probabilistic models over the simple baseline.
Bias vs. reliability. As seen in Figure 7 (mid, right), the overall biases are not very large, only few exceed an absolute value of —0.1—. On the other hand, the variance in reporting grades is quite high. More than half the students reported grades that deviate from the mean peer grades with a standard deviation of at least 0.14 (a variance of roughly 0.02). To gain some intuition on the values, note that a student who theoretically always reported the same score 0.72 for all submissions (the overall mean peer grade on all exercises together) would end up with a variance of 0.05.
It was reported by [5] that more than 90% of UST’s improvements are due to the fact that the modelbased approaches correct for the bias, an effect that we also confirmed on artificial data. However, in our AD data, the errors induced by low reliabilities dominate the errors due to bias. This may be part of the reason why the models do not perform well here.
Sources of grading error. As discussed above, reasons for differences in grading behavior are not only that people have “different tastes” or are “differently strict”. Rather, it is often the case that graders make serious errors due to lack of information or lack of understanding in the topic. This problem is bound to come up when using peer grading in a course such as algorithms and data structures and might be much less of an issue when grading is used to evaluate project reports [6] or design questions [3].
To check whether the modelbased algorithms improve if we just use “easytograde” exercises, we selected a number of exercises where the errors were low, indicating that the students had no difficulties in grading the submissions. We found that even in this scenario, the modelbased algorithms do not perform better than the mean. This may be due to the fact that in easytograde exercises, the mean algorithm does a good enough job at eliminating the different biases or reliabilities, so there is not much room left for improvements. Similarly, if we just train on “difficulttograde” exercises, we do not find an improvement of modelbased algorithms compared to the mean. The reason is that in this scenario, the errors are dominated by the submissions that are graded totally wrong by a large number of students, so none of the algorithms can do a good job.
Unmotivated students. One might suspect that some proportion of students tried to get away with minimal effort and produced grades that are pretty much random. However, looking closely into the data reveals that we only had a very small number of these students, so the models should succeed at giving them low reliabilities and lead to a better performance.
4.5 Analysis of supervised models
Considering the results for the supervised learning models in Figure
5, we see a similarly disappointing picture: the supervised models do not improve over the simple mean estimator. While the Kendall errors do not change much compared to the unsupervised models, the variance in errors over different tasks gets smaller. This is an effect of calculating the student reviewer bias values against the TA grades which improves the overall error in tasks that have a very high error and overfits in tasks that were graded well by most students in the first place. All in all, supervision does not seem to have a significantly positive effect on the performance of our algorithms.4.6 Analysis of further correlations in the data
To see whether we can meaningfully improve the performance of the models, we first check the correlation between the mean homework performance and the overall bias or mean deviation of the students’ reported grades compared to the TA grades. For bias, we use the same formula as in the SN model. The mean deviation for a student is likewise calculated on all reported grades over the whole course in relation to the TA grades as
The homework performance and the bias of the students are only weakly correlated with an value^{3}^{3}3
Pearson productmoment correlation coefficient.
of , i.e. students with higher grades on their own submissions tend to report lower grades. However, this is not due to better students being stricter or having higher standards but rather because better students are more likely to find flaws in the submissions and therefore report lower grades on average. While the former would necessitate a correction, this would punish students for doing a good job at finding large mistakes and results in a worse overall performance concerning grade accuracy.The correlation between the mean deviation and the homework grades is even weaker at an value of only , indicating a marginally higher agreement between well performing students and the TAs. A possible explanation for the low correlations is the fact that students handed in submissions in groups which means that the scores of each student do not necessarily reflect their knowledge on the topic. A more personal measure is given by the exam that was written at the end of the course. First, we note that the exam grades and the mean homework grade for the students are correlated with an value of , a rather low value. Taking the exam grades instead of the mean homework grade, the correlation with the bias is now slightly higher at while the mean deviation is correlated with the exam grades at which is much stronger than the correlation with the homework scores.
We incorporate the exam grades as a measure of reliability into the model using both mentioned approaches. To use the exam grades directly as the (constant) reliability for each student in the UMT model, we simply scale them to lie in the range , resulting in a distribution similar to the Gammadistribution for the estimated reliability in UMT. For the second approach, we estimate the reliability as usual in UMT but then multiply these values with the normalized exam grades (exam grade divided by mean exam grade). Both approaches yield similar results that are only marginally better than UMT. It seems that using the exam grade for the reliability compensates for some of the error that UMT introduces but still fails to significantly improve over a simple mean.
5 Conclusions
We have mixed feelings towards peer grading after evaluating all the data collected in our algorithms and data structures course. The positive point of view is that even though peer grades tend to be more optimistic than TA grades, the size of this effect is not very large. The peer grades give a reasonably informed picture of the true grades. For this reason, using simple estimates such as the mean grade is competitive to more elaborate modelbased algorithms. From an application point of view this finding is helpful: an easy to understand mechanism such as a simple mean is more acceptable to students than a complicated model when it comes to generating their final grades.
From a statistical or machine learning point of view, our results are somewhat disappointing. None of the models we tested outperforms the simple mean estimator on our data. Our general feeling is that it will be very difficult to come up with algorithms that do a much better job in this particular setting. The reasons for this difficulty might be the heterogeneity of the score distributions of different exercises, the high variance among graders and the different and rather unpredictable sources of grading errors (“lack of understanding” rather than a “slightly different taste”). It seems very hard to model all these aspects unless one has a much larger amount of grades per submission. However, the peer grading setup does not allow us to simply scale up the number of grades per submission, because students are not willing to invest even more time into this process. Finally, let us mention that our negative findings may be special to a course such as algorithms and data structures. Peer grading could be more successful for tasks where grading is a matter of taste rather than of understanding.
As opposed to other papers in the literature, we could not confirm the hypothesis that collecting grades in an ordinal rather than in a cardinal fashion leads to improved estimates. To the contrary, the students themselves reported that they tend to be more sloppy when being asked to provide ordinal grades, so we are somewhat pessimistic about ordinal grading in general.
The data set generated by our class as well as a moodle plugin that supports peer grading in a groupbased scenario as used in the AD course have been been made publicly available on our homepages.
Acknowledgments
This work has been supported by the German Federal Ministry of Education and Research (01PL12033, Universitätskolleg/TP 16 Lehrlabor at Universität Hamburg), the German Research Foundation (LU1718/1) and the Institutional Strategy of the University of Tübingen (Deutsche Forschungsgemeinschaft, ZUK 63). The authors alone are responsible for the content of this publication.
References
 [1]
 [2] J. Dıez, O. Luaces, A. AlonsoBetanzos, A. Troncoso, and A. Bahamonde. 2013. Peer assessment in MOOCs using preference learning via matrix factorization. In NIPS Workshop on Data Driven Education.
 [3] C. Kulkarni, K. P. Wei, H. Le, D. Chia, K. Papadopoulos, J. Cheng, D. Koller, and S. R. Klemmer. 2015. Peer and self assessment in massive online classes. In Design Thinking Research. 131–168.

[4]
F. Mi and D.Y. Yeung. 2015.
Probabilistic graphical models for boosting
cardinal and ordinal peer grading in MOOCs. In
29th AAAI Conference on Artificial Intelligence
.  [5] C. Piech, J. Huang, Z. Chen, C. Do, A. Ng, and D. Koller. 2013. Tuned models of peer assessment in MOOCs. In International Conference on Educational Data Mining.
 [6] K. Raman and T. Joachims. 2014. Methods for ordinal peer grading. In International Conference on Knowledge Discovery and Data Mining (SIGKDD). 1037–1046.
 [7] N. B. Shah, J. K. Bradley, A. Parekh, M. Wainwright, and K. Ramchandran. 2013. A case for ordinal peerevaluation in MOOCs. In NIPS Workshop on Data Driven Education.
 [8] K. Topping. 1998. Peer assessment between students in colleges and universities. In Review of Educational Research, Vol. 68. 249–276.
 [9] M. Venanzi, J. Guiver, G. Kazai, P. Kohli, and M. Shokouhi. 2014. Communitybased bayesian aggregation models for crowdsourcing. In Proceedings of the 23rd International Conference on World Wide Web. 155–164.
 [10] A. Vozniuk, A. Holzer, and D. Gillet. 2014. Peer assessment based on ratings in a social media course. In International Conference on Learning Analytics And Knowledge. 133–137.
 [11] T. Walsh. 2014. The PeerRank method for peer assessment. In European Conference on Artificial Intelligence.