Worcester Polytechnic Institute Electronic Theses and Dissertations Collection

Title page for ETD etd-042413-163846


Document Typethesis
Author NameLuo, Zhisui
URNetd-042413-163846
TitleA Bayesian Analysis of a Multiple Choice Test
DegreeMS
DepartmentMathematical Sciences
Advisors
  • Balgobin Nandram, Advisor
  • Keywords
  • Dirichlet Process Mixture
  • Markov Chain Monte Carlo
  • Item Response
  • Date of Presentation/Defense2013-04-22
    Availability unrestricted

    Abstract

    In a multiple choice test, examinees gain points based on how many correct responses they got. However, in this traditional grading, it is assumed that questions in the test are replications of each other. We apply an item response theory model to estimate students' abilities characterized by item's feature in a midterm test. Our Bayesian logistic Item response theory model studies the relation between the probability of getting a correct response and the three parameters. One parameter measures the student's ability and the other two measure an item's difficulty and its discriminatory feature. In this model the ability and the discrimination parameters are not identifiable. To address this issue, we construct a hierarchical Bayesian model to nullify the effects of non-identifiability. A Gibbs sampler is used to make inference and to obtain posterior distributions of the three parameters. For a "nonparametric" approach, we implement the item response theory model using a Dirichlet process mixture model. This new approach enables us to grade and cluster students based on their "ability" automatically. Although Dirichlet process mixture model has very good clustering property, it suffers from expensive and complicated computations. A slice sampling algorithm has been proposed to accommodate this issue. We apply our methodology to a real dataset obtained on a multiple choice test from WPI’s Applied Statistics I (Spring 2012) that illustrates how a student's ability relates to the observed scores.

    Files
  • zluo_mathesis.pdf

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