An Introduction to the Rasch Model with Examples in R: An Introduction with Examples in R
- Length: 150 pages
- Edition: 1
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2022-05-30
- ISBN-10: 1138710466
- ISBN-13: 9781138710467
- Sales Rank: #2209605 (See Top 100 Books)
This book provides a concise introduction to the topic of the Rasch model, which has become relatively well-known since it was used in the PISA study. In addition to a clear presentation of the underlying theory, the book also offers a practical introduction to fitting Rasch models by means of the freely available statistical software R.
Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Acknowledgment I. Theory 1. Introduction 1.1. The Role of Psychological and Educational Tests 1.2. The Rasch Model and Item Response Theory 1.3. Where You Will Find What in This Book 2. The Rasch Model 2.1. The Data Matrix 2.2. The Item Response Function 2.2.1. Ability and Difficulty 2.2.2. Discrimination 2.2.3. The Logistic Function 2.3. Alternative Representations 2.3.1. Probability of an Incorrect Response 2.3.2. Probability of an Arbitrary Response 2.3.3. Alternative Representation of the Logistic Function 2.3.4. Multiplicative Form 2.4. Properties and Assumptions 2.4.1. Sufficient Statistics 2.4.2. Local Stochastic Independence 2.4.2.1. Items 2.4.2.2. Persons 2.4.3. Specific Objectivity 2.4.4. Unidimensionality 2.4.5. Measurement Scale 2.5. Exercises 3. Parameter Estimation 3.1. Joint Maximum Likelihood Estimation 3.2. Conditional Maximum Likelihood Estimation 3.3. Marginal Maximum Likelihood Estimation 3.4. Bayesian Estimation 3.5. Person Parameter Estimation 3.6. Item and Test Information 3.7. Sample Size Requirements 3.8. Exercises 4. Test Evaluation 4.1. Graphical Assessment 4.1.1. Person Item Map 4.1.2. Empirical ICCs 4.1.3. Graphical Test 4.2. Tests for Item and Person Invariance 4.2.1. Andersen's Likelihood Ratio Test 4.2.2. Martin-Lof Test and Other Approaches for Detecting Multidimensionality 4.2.3. Wald Test 4.2.4. Anchoring 4.2.5. Other Approaches for Detecting DIF 4.2.6. How to Proceed with Problematic Items 4.3. Goodness-of-Fit Tests and Statistics 4.3.1. X2 and G2 Goodness-of-Fit Tests 4.3.2. M2, RMSEA, and SRMSR 4.3.3. Infit and Outfit Statistics 4.3.4. Further Fit Statistics for Items 4.3.5. Fit Statistics for Item Pairs 4.3.6. Fit Statistics for Persons 4.3.7. Nonparametric Goodness-of-Fit Tests 4.3.8. Posterior Predictive Checks 4.4. Separation Indices 4.4.1. Item Separation Index 4.4.2. Person Separation Index 4.5. Evaluation Through Model Comparisons 4.5.1. Models with Additional Item Parameters 4.5.1.1. Two-Parameter Model 4.5.1.2. Three-Parameter Model 4.5.1.3. Four-Parameter Model 4.5.1.4. Sample Size Requirements 4.5.2. Likelihood Ratio Tests 4.5.3. Information Criteria 4.6. Exercises II. Applications 5. Basic R Usage 5.1. Installation of R and Add-On Packages 5.2. Code Editors and RStudio 5.3. Loading and Importing Data 5.4. Getting Information About Persons and Variables 5.5. Addressing Elements in Lists 5.6. Exercises 6. R Package eRm 6.1. Item Parameter Estimation 6.2. Test Evaluation 6.2.1. Person Item Map 6.2.2. Empirical ICCs 6.2.3. Andersen's Likelihood Ratio Test and Graphical Test 6.2.4. Wald Test 6.2.5. Anchoring 6.2.6. Removing Problematic Items 6.2.7. Martin-Lof Test 6.2.8. Item and Person Fit 6.3. Plots of ICCs, Item and Test Information 6.4. Person Parameter Estimation 6.5. Test Evaluation in Small Data Sets 6.6. Exercises 7. R Package mirt 7.1. Model Selection 7.2. Item Parameter Estimates 7.2.1. Illustration via Expected ICCs 7.2.2. Displaying the Estimates 7.3. Evaluating Goodness-of-Fit 7.4. Ability Estimation 7.5. Exercises 8. R Package TAM 8.1. Item Parameter Estimation 8.2. Evaluating Goodness-of-Fit 8.3. Person Parameter Estimation 8.4. Exercises 9. R Interface to Stan 9.1. Stan Models 9.1.1. The data Block 9.1.2. The parameters Block 9.1.3. The transformed parameters Block 9.1.4. The model Block 9.2. Sampling the Posterior Using RStan 9.3. Evaluating Goodness-of-Fit 9.4. Exercises Summary of R Commands for eRm, mirt, and TAM III. Beyond the Rasch Model 10. Extensions to the Rasch Model 10.1. The Linear-Logistic Test Model 10.2. Modeling Differences Between People 10.2.1. The Mixture Rasch Model 10.2.2. Model-Based Recursive Partitioning 10.2.3. Explanatory IRT ‒ The Rasch Model as a Mixed Model 10.3. Multidimensional IRT Models 10.4. Exercises 11. Models for Polytomous Responses 11.1. The Partial Credit Model 11.1.1. CCCs and Threshold Parameters 11.1.2. Alternative Parameterizations 11.1.3. Disordered Thresholds 11.2. The Rating Scale Model 11.3. The Generalized Partial Credit and the Nominal Response Model 11.4. The Graded Response Model 11.5. The Sequential Model 11.6. Sample Size Requirements 11.7. Exercises 11.8. Derivations for the Partial Credit Model 12. Outlook on Special Applications 12.1. Computerized Adaptive Testing 12.2. Test Linking and Equating 12.3. Longitudinal IRT Models 12.4. Exercises Appendices A. Useful Mathematical Formulas A.1. Sums and Products A.2. Exponentials A.3. Logarithms A.4. Differentiation Rules A.4.1. Rules A.4.2. Examples B. Statistical Background B.1. Statistical Estimation B.1.1. The Binomial Distribution B.1.2. Maximum Likelihood Estimation B.1.3. Likelihood for Multiple Observations B.1.4. Bayesian Inference B.1.4.1. Bayes' Rule by Example B.1.4.2. Coin Flipping with a Uniform Prior B.1.4.3. Informative Priors and Beta-Binomial Model B.2. Statistical Testing B.2.1. Tests Based on the X2 Distribution B.2.1.1. X2 Test for Independence B.2.1.2. Goodness-of-Fit and Other X2 Tests B.2.2. Tests Based on the Normal Distribution C. Answers to the End of Chapter Questions C.2. Answers for Chapter 2 C.3. Answers for Chapter 3 C.4. Answers for Chapter 4 C.5. Answers for Chapter 5 C.6. Answers for Chapter 6 C.7. Answers for Chapter 7 C.8. Answers for Chapter 8 C.9. Answers for Chapter 9 C.10. Answers for Chapter 10 C.11. Answers for Chapter 11 C.12. Answers for Chapter 12 References Author Index Index
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