2 The Problem: Having Multiple Graders and Achieving Fair Assessment

We consider grading bias stemming from having multiple graders as a violation of a specific form of fairness, fairness-through-symmetry (Blackburn Reference Blackburn2003). In general, a process is considered fair when it can be expected to produce symmetrical outcomes for identical inputs. In our case, a grading process is “fair” when the grader assigned to a specific student does not, in expectation, affect the grade that this student ultimately receives.

When assessments are carried out in a multi-grader environment, bias can result from a difference in severity across graders or grader error. In this paper, we focus on limiting the first source of bias (severity) and assume that all best practices (i.e., blinding to reduce student-specific error and assessment training to reduce grader-specific error) are being followed to allay the second. We illustrate how difference in grader severity can lead to unfair student assessments in Figure 1.

Figure 1 Illustration of the bias that can be introduced by having multiple graders.

3 Proposed Solution: “Bridging” between Graders

An obvious solution to this type of individual grader bias is to let the same grader review all assessments for a class. Unfortunately, this is an unrealistic solution for large lectures given common restrictions on grader time.Footnote ³

Instead, we put forth the best solution given grader time and resource constraints: bridging across graded groups using a minimal number of bridging observations. By this, we mean that multiple graders assess the same assignment, creating a “bridge” between graders that a model can use to adjust for grader differences in the remaining unbridged observations. In a bridging scenario, some—but not all—of the students in a class will receive grades from each grader. Figure 2 displays a simplified illustration of this solution. In the example, we use a single shared (or “bridged”) student to observe that one grader is stricter than another. Thus, fairness demands that we adjust the students in both sections to reflect relative performance accurately.

Figure 2 Illustration of how bridging can minimize bias stemming from having multiple graders.

Bridging assures that multiple graders can assess the same course and assignments while at the same time minimizing the potential bias that comes from having multiple graders.

Bridging is likely to be familiar to a wide variety of political science faculty, although they have not used it in this specific context. For example, comparativists use bridging to ensure that expert evaluators place political parties from different countries on the same left–right scale even if few experts can evaluate multiple countries’ parties (Bakker et al. Reference Bakker, Jolly, Polk and Poole2014).Footnote ⁴ Bridging has also been used to ensure that evaluators with different scales for “democracy” can reliably place countries on a common scale (King et al. Reference King, Murray, Salomon and Tanon2004; Marquardt and Pemstein Reference Marquardt and Pemstein2018; Pemstein, Tzelgov, and Wang Reference Pemstein, Tzelgov and Wang2015). Americanists use bridging to place politicians operating in different environments in the same ideological space (Poole Reference Poole2007). This familiarity with the concept means that bridging is comfortable to use and, at the same time, relatively easy to explain to curious students.

Using bridging to adjust students’ grades in this way requires that we maintain some standard assumptions of student assessment. These assumptions include the following:

• • Each grader will be internally consistent in their assessments.

• • Graders have to be somewhat consistent, that is, have the same sense of what differentiates a high-quality answer and a low-quality answer.

• • Graders reward better quality answers roughly “linearly.” Performance is treated similarly for high- and low-quality assignments.

These assumptions are not likely to be overly restrictive and must only weakly hold to ensure the success of the process. In the next section, we explain the specific bridging approach we use and test it on real assessment data to show its promise for reducing bias.

4 Data and Analysis

We assert that the grading bias described in this paper is a form of “differential item functioning” (DIF), or interpersonal incomparability. Grades given by one grader are not directly comparable to grades given by another, making it challenging to judge students’ relative mastery of the material when they are assigned different graders. Aldrich and McKelvey suggested that one could overcome this issue by treating the rankings given to particular stimuli (here, grades given to specific exams) as somewhat distorted perceptions of the true, underlying latent value of the stimuli—here, student’s mastery of the material (Aldrich and McKelvey Reference Aldrich and McKelvey1977). This type of modeling has been frequently used in political science research to adjust for survey responses by individuals who might perceive survey questions differently, even when the questions asked and scales used are technically identical (see, e.g., Hollibaugh, Rothenberg, and Rulison (Reference Hollibaugh, Rothenberg and Rulison2013) and Lo, Proksch, and Gschwend (Reference Lo, Proksch and Gschwend2014), and particularly Hare et al. (Reference Hare, Armstrong, Bakker, Carroll and Poole2015), whom we follow in casting the process inside a Bayesian framework).

In this specific case, we adopt a simple model of assessment, where an individual student’s grade on a particular assignment is a function of the student’s underlying skill, attributes of the grader assessing the assignment, and some randomness. We consider two different attributes of the grader. First, we expect that graders might have different baselines for their perceptions of the underlying skill. That is, an “average” performance on a particular assignment may receive a lower or higher score for each grader, depending on this personal attribute. Second, we also expect that graders may be more or less willing to use all parts of the scoring range. That is, even if the graders start an average student at the same score, they may be more or less rewarding to improvements on that score.Footnote ⁵

Thus, we model for each grader $_{i}$ and student $_{j}$ as follows: $\textit{Grade}_{ij}=\alpha _{i} + \beta _{i} \gamma _{j}+\mu _{ij},$ where $\gamma _{j}$ is the underlying true skill of the student, $\alpha _{i}$ is the intercept or “shift term” assigned to each grader, $\beta _{i}$ is the weight term assigned to each grader, and $\mu _{ij}$ is the stochastic error term. The two grader-specific terms reflect what we discussed above: the intercept term ( $\alpha _{i}$ ) discerns whether the graders have different baseline levels for their grades, whereas the weight term ( $\beta _{i}$ ) measures the “stretch,” or how tightly or loosely an improvement in underlying skill is rewarded with an increase in grade.

In this context, bridged exams serve as common stimuli that allow the grader to approximate a student’s ability, adjusting for distorted grader perceptions. By adding bridges, we are adding information for the algorithm that allows it not only to better estimate the underlying latent skill of the student, but also to compare how graders filter the same input through different attributes and estimate those attributes. By then extending those attributes to students who did not serve as bridges, we can judge the relative performance of all students.

We estimate this model in a Bayesian framework (as opposed to using a maximum likelihood approach) for two reasons. First, the Bayesian approach better handles inherently missing data, which is important here because all students who do not serve as bridges have grades “missing.” The Bayesian approach also allows us to understand the uncertainty around our estimated grade distribution better, as we draw from a posterior distribution of all estimated parameters.

In order to estimate the model using a Bayesian approach, we require priors on our estimated parameters—here $\alpha $ , $ \beta $ , $\gamma $ , and $\mu $ . We employ weak priors on each of the grader-specific parameters ( $\alpha _i \sim \mathcal {N}(0,30)$ and $\beta _i \sim N(0,30)$ ) and provide a standard normal prior ( $\gamma _j \sim \mathcal {N}(0,1)$ ) on the underlying skill of a student, so that one can easily rank students, as well as perceive large jumps in the distribution via differences in deviations from the mean.Footnote ⁶ In each estimation, we utilize five chains, each running 30,000 iterations, with the first 2,000 iterations serving as burn-in and thinning the remaining iterations in intervals of 20. Although more complex approaches exist,Footnote ⁷ we believe that simplicity can better ensure that the method is explicable to students with minimal statistics knowledge. Interested practitioners can consult our R package that implements the method.

To evaluate the gains from bridging, we use a simulation exercise on real-world student grading data.Footnote ⁸ We collected grades during a Fall 2018 semester Introduction to Comparative Politics course at a large private research university located in the Northeast,Footnote ⁹ a relatively large course involving 140 students, six review sections, and three teaching assistants. Each teaching assistant graded all students’ performance on a midterm examination, a short 5—7-page paper covering material from the course, and a final exam. Free-response and essay-style answers accounted for the vast majority of the available points in all assignments, suggesting that differences in grader perceptions were likely, and likely to be impactful.

We use the averaged grades of all graders on each assessment as the baseline of fairness, as it removes the possibility that grader assignment affected the student’s grade. Despite implementing best-practice grading protocols to reduce the potential for bias—graders were trained on a rubric, discussed possible assessments for the same answer, and graded de-identified papers—we find that the traditional grade attribution process produced significant bias.

Considering the difference in student placement on the midterm examFootnote ¹⁰ when assessed by a single grader versus the three-grader average, the mean absolute error (“MAE”) is approximately 22.5. This means that the average student’s rank is 22.5 positions (out of 135 ranked positions for non-missing midterm exams) above or below their “earned” ranking, where the actual rank is the rank from the student’s single assigned grader and the “earned” ranking is the ranking based on the average grade of all three graders.Footnote ¹¹ Rank deviations of this magnitude can ultimately move students multiple grade categories in a class that is curved by the instructor’s choice or university rules.

In the simulation exercise, we repeatedly recreate “new” classes, made up largely of students whose only grade we have access to is the one provided by their assigned teaching assistant. However, for some number of students (the number of bridges), we also have their grades as given from the other two teaching assistants, that is, three grades for each of these bridged students. We then estimate the model above, extract the needed attributes to calculate a score for each student, as well as the relative rank of each student compared to all other students. We compare this estimated rank for each student to their rank in our “gold standard” case: where each student receives grades from all three teaching assistants, and their total grade and rank is determined by the average of these three scores.Footnote ¹² We calculate the MAE and the RMSE for this difference in ranks across this simulation.

We repeat this process 100 times for each number of bridges, which allows us to see how variable our improvement is depending on the students randomly chosen as bridges. We can also thereby roughly establish upper and lower bounds for how much bias can be reduced given a particular number of bridges. We discuss the results of this exercise in the next section.

6 Discussion and Best Practices

In this paper, we show that by creating bridged observations between graders, assessors can severely reduce the bias stemming from grader differences in severity. The reduction can be quite substantial, even at relatively low costs. While the trade-off for any individual instructor will naturally depend on the expected improvements in fairness and the burden of the additional costs, we believe that this is the most economical first step in reducing potential bias.

We should note three qualifications: first, this exercise neither precludes nor eliminates all errors. Regardless of how much of the baseline differences between graders we adjust for, there are still stochastic elements and matters of taste that are hard to model. Second, this algorithm addresses a specific type of error stemming from attributes of the assessors.Footnote ¹⁴ However, we should not expect it to perform (or claim that it performs) equally well across all classrooms.Footnote ¹⁵ Finally, it may be challenging to communicate the process to students unfamiliar with the concepts discussed in this paper and unaware of the bias lurking in more traditional ways of assigning grades. We expect that communication to students is one of the two primary obstacles to implementing this method alongside the implementer’s technical know-how.

For communication, we have produced a simple set of slides that use visualizations of the problem and concrete examples to explain how the bias arises and how this method works to fix it. Alongside an instructor’s guide with common FAQs and citations to more in-depth explanations of the procedures, these slides should ease communication between instructors and students.Footnote ¹⁶

For technical implementation, we have created a simple and straightforward R package that serves as a wrapper for rstan and takes as inputs the bare minimum amount of information from the instructor before outputting a ranking of students. The vignette and package git will have easily reproducible examples so that instructors can become comfortable with implementation before adopting the procedure.

In both cases, we believe that our materials will serve as significant first steps toward making the grader adjustment process a regular part of student assessment. But these resources should not be considered the final word. We hope to start an ongoing conversation on how best to balance student welfare between fairness and ease of understanding the grading process.

The most important contribution of this paper remains the knowledge that it takes an astonishingly small number of bridging observations to dramatically lower bias stemming from having multiple graders. This procedure is a vast improvement from doing nothing to reduce this particular form of bias, which we speculate is the norm in many classroom settings.