A Generalized Thurstonian Paired Comparison Multicriteria Heuristic Model for Peer Evaluation of Individual Performance on IS Team Projects
Julian M. Scher
Scher@adm.njit.edu
Department of Information Systems
College of Computing Sciences
New Jersey Institute of Technology
Newark, New Jersey 07102 USA
Abstract
Information Systems instructors are generally encouraged to introduce team projects into their pedagogy, with a consequential issue of objectively evaluating the performance of each individual team member. The concept of "freeloading" is wellknown for team projects, and for this, and other reasons, a peer review process of team members, by team members, is often advocated. We propose an objective heuristic model for obtaining a scale of individual performance, based upon a generalization of Thurstone's Law of Comparative Judgment, where pair wise comparisons of team member’s performance are elicited with regard to various criteria, and we demonstrate how a scale may be obtained to objectively rate the individual members of each team. A numerical example is provided to illustrate our Generalized Thurstone model’s heuristic methodologies.
Keywords: Thurstone, paired comparison, team projects, peer evaluation, multicriteria, individual performance
1. INTRODUCTION
Team projects should be an inherent goal in the pedagogy of the IS instructor. The justification for incorporating team projects has been succinctly stated by (Steenkamp, 2002):
“The rationale is that once students enter the work environment they will be required to work in teams. Working in a team context challenges team members in a number of ways, such as:
* Teams are composed of individuals with different technical skills, cultural backgrounds, behavioral characteristics, cognitive styles and learning abilities.
* Performance of team members is influenced by the level of teamwork and infield experience, knowledge of the application domain, pressures of schedule, geographical dispersion, fulltime or parttime study.”
The ABETCAC accreditation criteria clearly specify that an accredited IS program must, as part of its objectives, outcomes and assessment, enable all its graduating IS majors to achieve, by the time of their graduation “an ability to function effectively on teams to accomplish a common goal” (ABET, 2008). The issue of proper evaluation of individual effort in a team project is a concurrent dilemma for the IS instructor, who, for instance, needs to be cognizant of the oftenencountered “freeloading” which occurs in team projects (Fox, 2002). Being able to distinguish, identify and measure individual contributions and excellence on a team effort is vital to the success of the pedagogical exercise. The traditional practice of having the instructor review the results of the team project, and then award the identical grade to all members of the team, is problematic, and, as pointed out in (Tu, Lu, 2004), actually encourages and provides incentives for some of the weaker students to “freeload.” While this paper only addresses a peer evaluation methodology to measure relative individual performance in a group project, much has been written in terms of the pedagogical issues relating to optimizing the group project experience for students, and the reader is encouraged to peruse the research recommendations by Tu and Tu(2004b).
One strategy for evaluation of individual performance on project and client teams is that of peer evaluation, where individual students anonymously rate each other (Lewis, 2006). Several methodologies have been suggested for the measurement of individual student performance on teams (Ruble, Hernandez and Amadio, 2004), though there is no universal agreedupon standard. (Tu, Lu, 2004) discuss the issue of truthfulness in peer evaluation rankings of team members, and offer a methodology in peer evaluation so that truthtelling becomes the dominant strategy of individual team members. (Kelley, Sadowski, 2005) found that teams using a peer evaluation instrument in an engineering design graphics course team project functioned better than teams not using a peer evaluation instrument. (Lewis, 2006) suggests that the peer evaluations be done on a weekly basis. On the other hand, there are researchers who have raised doubts as to the usefulness of peer assessments in group projects, such as (Kennedy, 2005), who questions the underlying value of peer assessments, based on his experience with group projects in university computing courses.
In this paper, we shall present a model for a peer evaluation of individual team member performance, based upon some generalizations and extensions of the classic Thurstone’s Law of Comparative Judgment.
The “law” Thurstone created is essentially a measurement model, which requires subjects to make a preference comparison between each of a number of pairs of stimuli with regard to the magnitude of a property, attribute, or attitude. (Thurstone, 1927, 1929, 1959).
2. A GENERALIZED THURSTONE MODEL
Let us assume we have n+1 students on a team, which the instructor has divided the class into project teams or client teams for a particular assignment, or set of assignments. As part of the evaluation process to measure individual student’s performance on his/her team, each individual student will be asked to pairwise compare each of the other n students on his/her team according to a specific set of criteria chosen by the instructor, the number of such criteria to be denoted by m.
Then, since a given student will not be asked to vote in any pair wise comparison involving himself/herself, there will be n*(n1)/2 paired comparisons between different students, for a specific criteria r.
Thurstone (1927) presented a conceptual model for paired comparisons based upon several assumptions:
1. When a stimuli pair is presented to a subject, it will elicit a continuous preference (referred to as a “discriminal process”) for each stimulus.
2. The one stimulus whose value is greater at the moment of the comparison will be the one that is preferred by the subject.
3. The aforementioned preferences are normally distributed in the population.
4. It is assumed that each individual will respond to all of the possible paired comparisons.
Classic Thurstone analysis requires that after the individual students do their pairwise comparisons of other students for each of the m criteria, the results be presented in terms of a frequency matrix F, where f(i,j) denotes the count of those who prefer alternative i to alternative j, where in our case i and j are individual students.
We will extend the original Thurstone model of paired comparisons by assuming that the students will be evaluating each other based upon a set of criteria established by the instructor. The establishment of evaluation criteria for class projects by the instructor is fairly typical in universities – for instance, Chandra (http://tinyurl.com/nuw6ns) utilizes the following 5 criteria in evaluating projects:
* Depth and breadth of research
* Subject Knowledge
* Project Presentation Quality
* Final Project Report
* Original or New Contributions
We will denote by Fr the frequency matrix for criteria r, where r = 1,…,m. Thus, if we have m different criteria which we wish to use to evaluate members of the team, we will then have a total of m frequency matrices.
The individual elements fr(i, j) of the frequency matrix Fr will be the count of those who “prefer” student i over student j according to the r’th criteria.
Quantifying this, we define xi,j(k), the “rating” by student k in terms of his/her preference of student i to student j, for each of the r criteria, where k <> (i,j) and
k = 1,…,n+1:
xi,j(k) =
1 if i is preferred to j by student k
0 if j is preferred to i by student k
where we have the constraints that
xi,j(k) + xj,i(k) = 1 for i <> j and
xi,i(k) = 0 for k = 1,…n+1 .
We also seek transitivity in student k’s rating, i.e., if xa,b(k) = 1 and xb,c(k) = 1, then xa,c(k) = 1. Also, we insist that each student be required to make every comparison, without having any ‘indifferent’ votes.
The elements fr(i, j) of the frequency matrix Fr are then computed as follows:
n+1
fr(i, j) = ??xi,j(k) for i,j = 1,…, n+1
k=1
To illustrate these preliminary concepts with some numerical data, suppose that we have a team of n+1 or 5 students, who we will denote by S1. S2. S3. S4 and S5.
Let us assume that we desire the frequency matrix Fr for a particular criteria (for instance, r=1), and have queried the students to obtain the following pairwise comparisons:
Let student 1’s ratings be as follows:
x2,3(1) = 1
x2,4(1) = 1
x2,5(1) = 0
x3,4(1) = 1
x3,5(1) = 0
x4,5(1) = 0
Let student 2’s ratings be as follows:
x1,3(2) = 0
x1,4(2) = 0
x1,5(2) = 0
x3,4(2) = 1
x3,5(2) = 1
x4,5(2) = 1
Let student 3’s ratings be as follows:
x1,2(3) = 0
x1,4(3) = 0
x1,5(3) = 0
x2,4(3) = 0
x2,5(3) = 1
x4,5(3) = 0
Let student 4’s ratings be as follows:
x1,2(4) = 0
x1,3(4) = 0
x1,5(4) = 0
x2,3(4) = 1
x2,5(4) = 1
x3,5(4) = 1
Let student 5’s ratings be as follows:
x1,2(5) = 0
x1,3(5) = 0
x1,4(5) = 0
x2,3(5) = 1
x2,4(5) = 0
x3,4(5) = 0
We then generate the frequency matrix Fr as presented in Figure 1, noting that we obtain the remaining elements in the frequency matrix Fr by using the fact that
xi,j(k)+ xj,i(k) = 1.
Figure 1: The Frequency Matrix Fr
S1
S2
S3
S4
S5
S1

0
0
0
0
S2
3

3
1
2
S3
3
0

2
2
S4
3
2
1

1
S5
3
1
1
2

The elements in this frequency matrix are computed by the relationship between fr(i, j) and xi,j(k), namely
n+1
fr(i, j) = ??xi,j(k) for i,j = 1,…, n+1
k=1
Thus, for instance,
fr(3,4)= x3,4(1) + x3,4(2) + x3,4(5)
= 1 + 1 + 0 = 2
Also, we need to satisfy f(i,j)+f(j,i) = n1, that is, the number of pairwise comparisons done by the group for any two students will be n1.
With a team of n+1 students, the number of paired comparisons we ask of each student, for each criteria, as previously stated, is n*(n1)/2. For our illustrative example, with n+1 or 5student teams, this involves 4*3/2 or 6 paired comparisons of fellow student teammates for each of the m criteria. The total number of paired comparisons for each of the k students will therefore be m*n*(n1)/2. Typical student team sizes, such as client teams, are often between 4 and 5, and if the instructor seeks to keep the number of different criteria to a small number, such as 3 or 4, then the total number of paired comparisons required of each student in the peer evaluation will be 24 or less.
The second phase of the Thurstone model will be the transformation of the frequency matrices Fr into Probability matrices Pr, where Pr denotes the Probability matrix for the r’th criteria, where r = 1,…,m.
If there are n+1 students, each individual student being asked to make paired comparisons involving each pair of the other n students, then there will be n*(n1)/2 paired comparisons, and the number of students making a paired comparison between i and j, i.e., f(i,j), will be (n+1)2, or (n1) since we omit the two specific students i and j who do not make paired comparisons involving themselves. The elements of the probability matrix, denoted by pr(i,j), are then computed as follows:
pr(i,j) = fr(i, j) / (n1)
We also compute, for each row k in Pr, the sum of the probabilities in row k, which we denote by Vk (k = 1,…,n+1).
The resulting Probability matrix Pr is given in Figure 2.
Figure 2: The Probability Matrix Pr
S1
S2
S3
S4
S5
Vk
S1

0
0
0
0
0.00
S2
1.0

1.0
.33
.67
3.00
S3
1.0
0

.67
.67
2.33
S4
1.0
.67
.33

.33
2.33
S5
1.0
.33
.33
.67

2.33
Following the computation of the Probability matrix, a new matrix is then computed, traditionally called X in the psychometric literature, but for our nomenclature we will refer to it as the Z matrix. The cell values of matrix Z are the standardized normal deviates corresponding to the probabilities given in matrix Pr. Thurstone's Law of Comparative Judgment prescribes that the scale value difference between any two stimuli in a paired comparison assessment is a random variable following a Normal (Gaussian) probability density function. The mean value of this Normal distribution represents the scale value difference between the two stimuli in question.
Transforming the Probability Matrix Pr into the standardized Normal Matrix Zr, where the Z(i,j) values are computed from the N(0,1) tables, and we have approximated the Gaussian distribution with a doubly truncated normal distribution having truncation endpoints at 3 and +3, corresponding to CDF(0) and CDF(1), where CDF(x) represents the cumulative distribution function at point x. While one may obtain precise values for the doubly truncated normal distribution (Johnson and Thermopolous, 2002), there will be no harm in approximating these values by the more widely accessible standardized Normal tables. Invoking the standardized normal tables will yield the Zr matrix of Figure 3.
Figure 3: The Standardized Normal
Matrix Zr (for r=1)
S1
S2
S3
S4
S5
Tk(r)
S1

3
3
3
3
12
S2
3

3
.43
.43
6
S3
3
3

.43
.43
0.86
S4
3
.43
.43

.43
2.57
S5
3
.43
.43
.43

2.57
The last column, Tk(r), of Figure 3 represents the sum of the k’th row’s standardized normal values for the r’th criteria, and so, for each of the m criteria, we have a vector T with k components.
For each of the m criteria, the instructor will assign a weight given by wj, where
j = 1,…,m and the wj are nonnegative
and
??wj = 1 (i.e., the weights constitute a convex combination).
Once we compute the Tk(r) values for all criteria r (r = 1,…,m) for each of the k students (k = 1,…,n+1), we may then compute the scale values Ak as follows:
m
Ak = ??wj Tk(r) for k = 1,…,n+1
J=1
To illustrate the computations of the scale values Ak , let us assume that we have 3 criteria (i.e., m =3) assigned by the instructor:
Criteria(1) = overall quality of work contributed
Criteria(2) = availability and willingness to work with other team members and support the work of the team
Criteria(3) = perceived amount of effort
The instructor believes that criteria(1), the overall quality of work contributed, is twice as important as either of the other two criteria, and that criteria(2) and criteria(3) are equal in importance, which leads us to the following weights:
W1 = .5
W2 = .25
w3 = .25
For simplicity of presentation, we will assume that the previously computed Z matrix was the one generated for criteria(1), and so
T1(1) = 12.0
T2(1) = 6.0
T3(1) = 0.861
T4(1) = 2.5692
T5(1) = 2.5692
We will provide data values for Tk(2) and Tk(3), (for k=1,…5) and not bother the reader with the background details / computations for the associated Z, P and F matrices.
So, let us assume we have for criteria(2)’s Tk(r) values
T1(2) = 5.4
T2(2) = 4.31
T3(2) = 1.8
T4(2) = 1.69 and
T5(2) = 1.2
For criteria(3)’s values, we have
T1(3) = 2.7
T2(3) = 1.2
T3(3) = .9
T4(3) = 1.0 and
T5(3) = 1.4
The “T” matrix will then be:
12.0
6.000
.8616
2.5692
2.5692
5.4
4.31
1.8
1.69
1.2
2.7
1.2
0.9
1.0
1.4
and
A1 = .5*(12) + .25*(5.4) + .25*(2.7)
A1 = 8.025
A2 = .5*(6.0) + .25*(4.31) +.25*(1.2)
A2 = 4.3775
A3 = .5*(.8616) +.25*(1.8) +.25*(.9)
A3 = .2442
A4 =.5*(2.5692) +.25*(1.69) +.25*(1)
A4 = 1.9571
A5 =.5*(2.5692) +.25*(1.2) +.25*(1.4)
A5 = 1.9346
Plotting these points on a scale, we obtain Figure 4.
Figure 4: The 5Student Peer Evaluation Thurstonian Scale of Individual Performance On A Team
Clearly, Student #1 is the least preferred student on the team, as judged by the peer evaluation of the team, and by a significant degree. Student #3 is second in the least preferred category, as evaluated by his/her peers. Student #2’s performance was recognized as being the best on the team. After Student #2, we have Student #4 and Student #5 coming relatively close in the peer evaluation, with Student #4 barely edging out Student #5.
The objective measures we have thus obtained will guide and support the instructor in the difficult task of assigning grades to individual members of this team of five students. While there will be some subjectivity on the part of the instructor in his/her interpretation of these results, our inclination would be to reward Student 2 with the highest grade, an “A,” on his/her individual performance on the team. Student #1 would be awarded the lowest grade, an “F,” for a performance that was clearly recognized as deficient by his/her teammates. Since student #4 and student #5 were viewed positively for their nearequivalent performance, they each should be awarded identical grades, very likely a grade of “B.” For student #3, whose individual performance was viewed as slightly negative in the Thurstonian comparative evaluation, we would be generous and award him/her with a (“gentlemen’s”) “C.”
3. CONCLUSIONS
We have presented a generalized heuristic Thurstonian model for use in peer evaluation of individual performance on team projects. It is based on paired comparisons of individual students, whereby each student will compare pairs of other students according to instructorselected criteria, and the instructor will also select the relative importance of each criteria. Classical Thurstonian concepts are utilized and extended to produce an apropos scale where instructors may review the relative performance of individual students on teams, and observe the resultant scales.
4. ACKNOWLEDGEMENT
We acknowledge a colleague, Distinguished Professor Emeritus Murray Turoff, who well over 35 years ago brought Thurstone’s Law of Comparative Judgment to our attention, and which lay dormant with us for many years, until the need arose for a methodology to assess the individual contributions of students on team projects.
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