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  • Introduction to Bayesian Response Modeling . 1; 1.1 Introduction . .1; 1.1.1 Item Response Data Structures . .3; 1.1.2 Latent Variables . 5; 1.2 Traditional Item Response Models . 6; 1.2.1 Binary Item Response Models . . .7; 1.2.2 Polytomous Item Response Models . . 12; 1.2.3 Multidimensional Item Response Models . 14; 1.3 The Bayesian Approach . .15; 1.3.1 Bayes' Theorem . . 16; 1.3.2 Posterior Inference . . 20; 1.4 A Motivating Example Using WinBUGS . .21; 1.4.1 Modeling Examinees' Test Results . . . . 21; 1.5 Computation and Software . . . . 24; 2 Bayesian Hierarchical Response Modeling . . . . 31; 2.1 Pooling Strength . . . 31; 2.2 From Beliefs to Prior Distributions . . . 33; 2.2.1 Improper Priors . . . 38; 2.2.2 A Hierarchical Bayes Response Model . .39; 2.3 Further Reading . . . 42; 3 Basic Elements of Bayesian Statistics . .45; 3.1 Bayesian Computational Methods . . 45; 3.1.1 Markov Chain Monte Carlo Methods . .46; 3.2 Bayesian Hypothesis Testing . . . 51; 3.2.1 Computing the Bayes Factor .. 54; 3.2.2 HPD Region Testing . . 58; 3.2.3 Bayesian Model Choice . . 59; 4 Estimation of Bayesian Item Response Models . . 67; 4.1 Marginal Estimation and Integrals . . 67; .2 MCMC Estimation . . .. 71; 4.3 Exploiting Data Augmentation Techniques . . 73; 4.3.1 Latent Variables and Latent Responses . 74; 4.3.2 Binary Data Augmentation . . 75; 4.3.3 TIMMS 2007: Dutch Sixth-Graders' Math Achievement . 81; 4.3.4 Ordinal Data Augmentation . .83; 4.4 Identification of Item Response Models . . 86; 4.4.1 Data Augmentation and Identifying Assumptions . . . . . . 87; 4.4.2 Rescaling and Priors with Identifying Restrictions . . . . . 88; 4.5 Performance MCMC Schemes . . . 89; 4.5.1 Item Parameter Recovery . . . 89; 4.5.2 Hierarchical Priors and Shrinkage . . 92; 4.6 European Social Survey: Measuring Political Interest . . . . . . . . . 95; 5 Assessment of Bayesian Item Response Models . . . . . . . . . . . . 107; 5.1 Bayesian Model Investigation . 107; 5.2 Bayesian Residual Analysis . . . . 108; 5.2.1 Bayesian Latent Residuals . . 109; 5.2.2 Computation of Bayesian Latent Residuals . . . . . 109; 5.2.3 Detection of Outliers . . 110; 5.2.4 Residual Analysis: Dutch Primary School Math Test . . . 111; 5.3 HPD Region Testing and Bayesian Residuals . .112; 5.3.1 Measuring Alcohol Dependence: Graded Response Analysis . 116; 5.4 Predictive Assessment . .117; 5.4.1 Prior Predictive Assessment . . . . . 119; 5.4.2 Posterior Predictive Assessment .. 122; 5.5 Illustrations of Predictive Assessment . . .126; 5.5.1 The Observed Score Distribution . . 126; 5.5.2 Detecting Testlet Effects . .. 127; 5.6 Model Comparison and Information Criteria . . . 130; 5.6.1 Dutch Math Data: Model Comparison . .. 131; 5.9 Appendix: CAPS Questionnaire . . 139; 6 Multilevel Item Response Theory Models . .141; 6.1 Introduction: School Effectiveness Research . . . . 141; 6.2 Nonlinear Mixed Effects Models . . 142; 6.3 The Multilevel IRT Model . . . 145; 6.3.1 A Structural Multilevel Model . . 145; 6.3.2 The Synthesis of IRT and Structural Multilevel Models . . . 148; 6.4 Estimating Level-3 Residuals: School E ects . . . . .. . 153; 6.5 Simultaneous Parameter Estimation of MLIRT . . . . 158; 6.6 Applications of MLIRT Modeling . . . 162; 6.6.1 Dutch Primary School Mathematics Test . . . . . 162; 6.6.2 PISA 2003: Dutch Math Data . . . 165; 6.6.3 School Effects in the West Bank: Covariate Error . . . . . . 172; 6.6.4 MMSE: Individual Trajectories of Cognitive Impairment . . . .. 174; 6.9 Appendix: The Expected School Effect . 188; 6.10 Appendix: Likelihood MLIRT Model . . . 190; 7 Random Item Effects Models . .193; 7.1 Random Item Parameters . . .193; 7.1.1 Measurement Invariance . . 194; 7.1.2 Random Item Effects Prior . .195; 7.2 A Random Item Effects Response Model . . .198; 7.2.1 Handling the Clustering of Respondents . .. 203; 7.2.2 Explaining Cross-national Variation . . . . . 203; 7.2.3 The Likelihood for the Random Item E ects Model . . . . 204; 7.3 Identification: Linkage Between Countries . 205; 7.3.1 Identification Without (Designated) Anchor Items . . . . . 206; 7.3.2 Concluding Remarks . . . . . . . . 208; 7.4 MCMC: Handling Order Restrictions . . .209; 7.4.1 Sampling Threshold Values via an M-H Algorithm . . . . . 209; 7.4.2 Sampling Threshold Values via Gibbs Sampling . . . . . . . 211; 7.4.3 Simultaneous Estimation via MCMC. . . . 212; 7.5 Tests for Invariance . . . . . . . 214; 7.6 International Comparisons of Student Achievement . . . . . . 216; 8 Response Time Item Response Models . 227; 8.1 Mixed Multivariate Response Data . . . 227; 8.2 Measurement Models for Ability and Speed . . .228; 8.3 Joint Modeling of Responses and Response Times . . . . . . . . 231; 8.3.1 A Structural Multivariate Multilevel Model . . . . . 232; 8.3.2 The RTIRT Likelihood Model . . . 234; 8.4 RTIRT Model Prior Specifications . . . . . .. 235; 8.4.1 Multivariate Prior Model for the Item Parameters . . . . . 235; 8.4.2 Prior for P with Identifying Restrictions . . . . . . . 236; 8.5 Exploring the Multivariate Normal Structure .. . . . 238; 8.6 Model Selection Using the DIC . . .. 241; 8.7 Model Fit via Residual Analysis . . . . 242; 8.8 Simultaneous Estimation of RTIRT . . 243; 8.9 Natural World Assessment Test . . . .246; 8.12 Appendix: DIC RTIRT Model . . . . . . . . . 254; 9 Randomized Item Response Models . . 255; 9.1 Surveys about Sensitive Topics . . . . 255; 9.2 The Randomized Response Technique . . . 256; 9.2.1 Related and Unrelated Randomized. Response Designs. . . . . . 257; 9.3 Extending Randomized Response Models . . . 258; 9.4 A Mixed E ects Randomized Item Response Model . .. . . . . 259; 9.4.1 Individual Response Probabilities . . .. 259; 9.4.2 A Structural Mixed Effects Model . . . 261; 9.5 Inferences from Randomized Item Response Data . . 262; 9.5.1 MCMC Estimation . . . . 265; 9.5.2 Detecting Noncompliance Behavior . . . 267; 9.5.3 Testing for Fixed-Group Di erences . .. 268; 9.5.4 Model Choice and Fit . . . . . . 270; 9.6 Simulation Study . . . 272; 9.6.1 Different Randomized Response Sampling Designs . . . . . 272; 9.6.2 Varying Randomized Response Design Properties . . . . . . 274; 9.7 Cheating Behavior and Alcohol Dependence . . .275; 9.7.1 Cheating Behavior at a Dutch University . . 275; 9.7.2 College Alcohol Problem Scale . . .279 (xsd:string)
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