Grade performance in statistics: a Bayesian framework.

Author:Fulton, Lawrence V.

    In this paper, we propose a hierarchical Bayesian mixture model to predict future performance and explain variations in the results of an introductory business statistics course exit assessment of Texas State University students. We conceptualize the N students' test scores as clustered in K performance groups where the average grades of the clusters are monotonically ordered indicating relative performance that go from the least to the highest performance group. We refer to these performance groups as grade clusters i. In addition to modeling the average grades in each cluster, we also model the probability that a particular student will belong to any of the K clusters. We note that each semester there exists a group of students who usually do not have a positive probability of passing the course and, therefore, do not show-up for the final assessment exam. The list of zeroes assigned to these students forms our first grade cluster. We propose a mixture model to incorporate the multi-peakedness of the distribution of grades implied by a K that is greater than 1. We use the Harmonic Mean measure of Newton & Raftery (1994) in determining K.

    The hierarchical Bayesian modeling approach allows us to do several things we would not be able to do with a Generalized Linear Model. First, we can use information and make predictions on small sub-groups in our sample. Second, we can investigate a richer structure than is available using any frequentist version of the proposed models. For instance, we can obtain the probability distribution of the average grades in the grade clusters and provide a student with a probability distribution of his/her grade given various covariate values. We can also choose to treat some covariates as nuisance factors, integrate them out, and use only those that a model-user deems relevant while not over/under estimating the uncertainty about the coefficients of the chosen covariates. Furthermore, our model allows a realistic framework in which a student's probabilities of belonging to various clusters are negatively correlated, and we are able to introduce student level variation about the average grades in each cluster.

    We provide a set of methods to help students and instructors to understand the relationship between various covariates and students' probabilistic grade distribution. From the perspective of policy-makers, it would be of interest whether a particular student group seems to perform better or worse than others for fixed values of the other covariates. For example, we can compare probabilistically a particular grade a Hispanic student will obtain compared to a student who is not classified as Hispanic. Our model can be used as a framework to draw a number of inferences as we model the average grade of each grade cluster as well as the probability that a student is going to belong to this cluster.


    In the first sub-section, we discuss the data source used in the analysis. We then describe the mixture model of test scores without any covariates. This allows an intuitive understanding of the general model. Next, we introduce the incorporation of information from the covariates in order to explain the uncertainty about the membership to a particular grade cluster, as well as the average grade in each grade cluster. In the final sub-section, we describe how we select K, the number of grade clusters.

    2.1 Data Source

    Our data consists of information that we obtained from 121 students. We used the Hawkes Learning Systems (HLS) to obtain the number of minutes each student logs in. This measure will serve as a proxy to the amount of time a student puts into class. Note that HLS logs a student out after 5 minutes of inactivity so that we would not have to worry about inactive students with high HLS minutes logged. We obtained age, grade point average (GPA) at the beginning of the semester, Hispanic status, and gender from the university registration system. This study was designated as exempt by the Institutional Review Board. We scale the final exam grade between 0 and 1 and illustrate it in Figure 1. This figure illustrates the multiple peaked nature of the grades. As mentioned above, the first peak of zeroes occur due to the students who are no-shows for the final exam. Note that we determine the number of peaks based on the measure of Harmonic Mean as explained below in the model section.


    The covariates [Y.sub.1] - [Y.sub.5], represent Gender (1-female, 0-male), Age, Hispanic Status (1-Hispanic, 0- otherwise), Overall GPA, and Total Time Spent on the System, respectively. A student whose age (53) was over 8 standard deviations above the mean was removed from further analysis. There is substantial...

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