From Bayesian Analysis of Item Response Theory Models Using SAS. Full book available for purchase here.

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1 From Bayesian Analysis of Item Response Theory Models Using SAS. Full book available for purchase here. Contents About this Book...ix About the Authors... xiii Acknowledgments... xv Chapter 1: Item Response Theory... 1 Introduction... 1 Overview of IRT Models and Their Application... 2 Visualization of IRT Models... 3 Organization of the Chapter... 4 Unidimensional IRT Models for Dichotomously Scored Responses... 5 Unidimensional IRT Models for Polytomously Scored Responses... 5 The Graded Response Model... 6 Muraki s Rating Scale Model... 8 The Partial Credit Model and Extensions... 8 The Nominal Response Model Other Testing Effects in Unidimensional IRT Models Locally Dependent Item Sets (Testlets) Rater Effects IRT Models for Multidimensional Response Data Differential Item Functioning and Mixture IRT Models Hierarchical Models Multilevel and Random Effects IRT Models Evaluation of IRT Model Applications Dimensionality Local Independence Form of the IRT Model Speededness Model Fit IRT Model Parameter Estimation Chapter 2: Bayesian Analysis Introduction Elements of Statistical Models and Inference Frequentist and Bayesian Approaches to Statistical Inference Use of Bayesian Analysis to Estimate a Proportion An Example Choosing a Prior Distribution Computing the Posterior and Drawing Inferences Comparing Bayesian and Frequentist Estimates for Proportions... 30

2 iv Bayesian Estimation and the MCMC Method SAS Code for Implementing a Metropolis Sampler to Estimate a Proportion MCMC and the Gibbs Sampler Burn-In and Convergence Informative and Uninformative Prior Distributions Model Comparisons Deviance Information Criterion Bayes Factor Model Fit and Posterior Predictive Model Checks Chapter 3: Bayesian Estimation of IRT Models Using PROC MCMC Introduction Summary of PROC MCMC Statements for IRT Model Estimation MCMC Sampling Algorithms Built-In Distributions Density Plots Arbitrary Distributions PROC MCMC Template for IRT Model Estimation How PROC MCMC Processes the SAS Data Set More on the RANDOM Statement Model Identification Example Use of the Template for the 1-Parameter IRT Model Example Output from Estimating the 1P IRT Model Strategies to Improve Convergence of the Monte Carlo Chain Chain Thinning Parameter Blocking Model Re-parameterizing Analysis of Multiple Chains Gelman-Rubin Test for Convergence Informative, Uninformative, and Hierarchical Priors Informative Priors Uninformative Priors Hierarchical Priors Effect of Different Priors Treatment of Item Parameters as Random Effects Model Fit and Model Comparisons Comparison with WinBUGS The OUTPOST SAS Data Set Autocall Macros for Post-Processing PROC MCMC Data Sets Preliminary Item Analyses Chapter 4: Bayesian Estimation of Unidimensional IRT Models for Dichotomously Scored Items Introduction The 1-Parameter IRT or Rasch Model Use of a Normal Ogive Link Function The 2-Parameter IRT Model... 85

3 v A 2-Parameter IRT Model with a Hierarchical Prior The 3-Parameter IRT Model Comparison of Results Based on MML Estimation Display of Item Response Functions Adding Elements to the Graph: Uncertainty in Item Parameters Adding Elements to the Graph: Observed Results Chapter 5: Bayesian Estimation of Unidimensional IRT Models for Polytomously Scored Items Introduction The Graded Response Model Program Template for the GR Model Options for Specifying Prior Distributions for the Threshold and Intercept Parameters Estimation of the GR Model by Using Method Comparison of Separate and Joint Prior Specifications Output from Estimating the GR Model (Method 1 Prior Specification) Comparison of the Posterior Densities for the Three Prior Specifications Computation of Transformations by Post-Processing Posterior Results Specification of the Likelihood Model with Use of the Table Function Estimation of the One-Parameter Graded Response Model Muraki s Rating Scale Model Estimating the RS-GR Model Output from Estimating the RS-GR Model The Nominal Response Model Estimating the NR Model Output from Estimating the NR Model The Generalized Partial Credit Model Estimating the GPC Model Output from Estimating the GPC Model Comparison of Results Based on MML Estimation Graphs of Item Category Response Functions Graphs of Test Information Functions Chapter 6: IRT Model Extensions Introduction The Bifactor IRT Model Description of the Model Estimation of the Model in PROC MCMC Output from PROC MCMC Other Multidimensional IRT Models The Problem of Label Switching The Testlet IRT Model Description of the Model Estimation of the Model Output from PROC MCMC Hierarchical Models Multilevel IRT Models

4 vi Description of the Multilevel IRT Model Estimation of the Model Output from PROC MCMC Differential Item Functioning Multiple Group and Mixture IRT Models Multiple Group Models for Detecting DIF Mixture IRT Models Chapter 7: Bayesian Comparison of IRT Models Introduction Bayesian Model Comparison Indices Deviance Information Criterion Conditional Predictive Ordinate Computing Model Comparison Statistics in PROC MCMC Example 1: Comparing Models for Dichotomously Scored Items (LSAT Data) DIC Results CPO Results Example 2: Comparing GR and RS-GR Models for Polytomously Scored Items (DASH Item Responses) DIC Results CPO Results Example 3: Comparing a Unidimensional IRT Model and a Bifactor IRT Model DIC Results CPO Results Chapter 8: Bayesian Model-Checking for IRT Models Introduction Different Model-Fit Statistics Test-Level Fit Item-Level Fit Person Fit Posterior Predictive Model Checking The PPMC Method The Posterior Predictive Distribution Discrepancy Measures Test-Level Measures Item-Level Measures Pairwise Measures Person-Fit Measures Evaluation of Model Fit Example PPMC Applications Example 1: Observed and Predicted Test Score Distributions LSAT Data Example 2: Observed and Predicted Item-Test Score Correlations Example 3: Item Fit Plots and Yen s Q1 Measure Example 4: Observed and Predicted Odds Ratio Measure Example 5: Observed and Predicted Yen s Q3 Measure Example 6: Observed and Predicted Person-Fit Statistic

5 vii Example 7: Use of PPMC to Compare Models References Index From Bayesian Analysis of Item Response Theory Models Using SAS, by Clement A. Stone and Xiaowen Zhu. Copyright 2015, SAS Institute Inc., Cary, North Carolina, USA. ALL RIGHTS RESERVED.