Mathematics Ph.D. Dissertations

Bayesian Model Checking Strategies for Dichotomous Item Response Theory Models

Date of Award

2006

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics/Mathematical Statistics

First Advisor

James Albert (Advisor)

Abstract

Item Response Theory (IRT) models are commonly used in educational and psychological testing. These models are mainly used to assess the latent abilities of examinees and the effectiveness of the test items in measuring this underlying trait. However, model checking in Item Response Theory is still an underdeveloped area. In this dissertation, various model checking strategies from a Bayesian perspective for different Item Response models are presented. In particular, three methods are employed to assess the goodness-of-fit of different IRT models. First, Bayesian residuals and different residual plots are introduced to serve as graphical procedures to check for model fit and to detect outlying items and examinees. Second, the idea of predictive distributions is used to construct reference distributions for different test quantities and discrepancy measures, including the standard deviation of point bi-serial correlations, Bock's Pearson-type chi-square index, Yen's Q1 index, Hosmer-Lemeshow Statistic, Mckinley and Mill's G2 index, Orlando and Thissen's S-G2 and S-X2 indices, Wright and Stone's W-statistic, and the Log-likelihood statistic. The prior, posterior, and partial posterior predictive distributions are discussed and employed. Finally, Bayes factor are used to compare different IRT models in model selection and detection of outlying discrimination parameters. In this topic, different numerical procedures to estimate the Bayes factors for these models are discussed. All of these proposed methods are illustrated using simulated data and Mathematics placement exam data from BGSU.

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