Date of Completion

8-2-2019

Embargo Period

8-1-2020

Keywords

Bayesian, IRT, Marginal Likelihood, Bayes Factor, Model Selection, Monotonicity, g-prior, MCMC

Major Advisor

Ming-Hui Chen

Co-Major Advisor

Xiaojing Wang

Associate Advisor

Dipak K. Dey

Associate Advisor

N/A

Field of Study

Statistics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

Item response theory (IRT) models play a critical role in psychometric studies for the design and analysis of examinations. IRT models mainly consider the relationship among the correctness of items, individual's latent ability, difficulty of each item and other potential factors such as guessing.

In this dissertation, we develop Bayesian modeling methods and model selection techniques under the IRT model framework. For Bayesian model comparison, the Bayes factor is a widely used tool, which requires computation of the marginal likelihoods. For complex models such as the IRT models, the marginal likelihoods are not analytically available. There are a variety of Monte Carlo methods for estimating or computing the marginal likelihoods, though some of them may not be feasible for IRT models due to the high dimensionality of the parameter space. We review several different Monte Carlo methods for marginal likelihood computation under classic IRT models, develop the "best'' implementation of these methods for the IRT models, and apply these methods to a real dataset for comparison of the classic one-parameter IRT model and two-parameter IRT model.

With increasing availability of computerized testing, observations are often collected at irregular and variable time points. We adopt a dynamic IRT model based on the one-parameter IRT model to accommodate this data structure. A hierarchical layer on the dynamic IRT model is built to capture the relationship between the "growth factor" and the characteristics of individuals. We use the Bayes factor to perform variable selection on the covariates linked to the growth, and develop a Monte Carlo approach to compute the Bayes factors for all model pairs using a single Markov chain Monte Carlo (MCMC) output. We also show the model selection consistency of the Bayes factor under certain conditions.

Additionally, to allow more flexibility, we propose a nonparametric model and embed a monotone shape constraint on the mean latent growth trend. Further, we develop a partially collapsed Gibbs sampling algorithm coupled with a reversible jump MCMC technique to sample the dimension-varying parameters from their corresponding posterior distribution.

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