The Theory and Practice of Item Response Theory

Noted for addressing both the "hows" and "whys" of item response theory (IRT), this text has been revised and updated with the latest techniques (multilevel models, mixed models, and more) and software packages. Simple to more complex models are covered in consistently formatted chapters that build sequentially. The book takes the reader from model development through the fit analysis and interpretation phases that would be performed in practice. To facilitate understanding, common datasets are used across chapters, with the examples worked through for increasingly complex models. Exemplary model applications include free (BIGSTEPS, NOHARM, Facets, R packages) and commercial (BILOG-MG, flexMIRT, SAS, WINMIRA, SPSS, SYSTAT) software packages. The companion website provides data files and online-only appendices.
 
 New to This Edition
*Chapter on multilevel models.
*New material on loglinear models, mixed models, the linear logistic trait model, and fit statistics.
*Many additional worked-through examples.
*Updated guidance on software; now includes R, SAS, and flexMIRT.
 

R. J. de Ayala, PhD, is Professor of Educational Psychology at the University of Nebraskan/-/Lincoln. His research interests include psychometrics, item response theory, computerized adaptive testing, applied statistics, and multilevel models. His work has appeared in Applied Psychological Measurement, Applied Measurement in Education, the British Journal of Mathematical and Statistical Psychology, Educational and Psychological Measurement, the Journal of Applied Measurement, and the Journal of Educational Measurement. He is a Fellow of the American Psychological Association’s Division 5: Evaluation, Measurement, and Statistics, and of the American Educational Research Association.
 

Symbols and Acronyms
1. Introduction to Measurement
- Measurement
- Some Measurement Issues
- Item Response Theory
- Classical Test Theory
- Latent Class Analysis
- Summary
2. The One-Parameter Model
- Conceptual Development of the Rasch Model
- The One-Parameter Model
- The One-Parameter Logistic Model and the Rasch Model
- Assumptions Underlying the Model
- An Empirical Data Set: The Mathematics Data Set
- Conceptually Estimating an Individual’s Location
- Some Pragmatic Characteristics of Maximum Likelihood Estimates
- The Standard Error of Estimate and Information
- An Instrument’s Estimation Capacity
- Summary
3. Joint Maximum Likelihood Parameter Estimation
- Joint Maximum Likelihood Estimation
- Indeterminacy of Parameter Estimates
- How Large a Calibration Sample?
- Example: Application of the Rasch Model to the Mathematics Data, JMLE, BIGSTEPS
- Example: Application of the Rasch Model to the Mathematics Data, JMLE, mixRasch
- Validity Evidence
- Summary
4. Marginal Maximum Likelihood Parameter Estimation
- Marginal Maximum Likelihood Estimation
- Estimating an Individual’s Location: Expected A Posteriori
- Example: Application of the Rasch Model to the Mathematics Data, MMLE, BILOG-MG
- Metric Transformation and the Total Characteristic Function
- Example: Application of the Rasch Model to the Mathematics Data, MMLE, mirt
- Summary
5. The Two-Parameter Model
- Conceptual Development of the Two-Parameter Model
- Information for the Two-Parameter Model
- Conceptual Parameter Estimation for the 2PL Model
- How Large a Calibration Sample?
- Metric Transformation, 2PL Model
- Example: Application of the 2PL Model to the Mathematics Data, MMLE, BILOG-MG
- Fit Assessment: An Alternative Approach for Assessing Invariance
- Example: Application of the 2PL Model to the Mathematics Data, MMLE, mirt
- Information and Relative Efficiency
- Summary
6. The Three-Parameter Model
- Conceptual Development of the Three-Parameter Model
- Additional Comments about the Pseudo-Guessing Parameter, Xⱼ
- Conceptual Parameter Estimation for the 3PL Model
- How Large a Calibration Sample?
- Assessing Conditional Independence
- Example: Application of the 3PL Model to the Mathematics Data, MMLE, BILOG-MG
- Fit Assessment: Conditional Independence Assessment
- Fit Assessment: Model Comparison
- Example: Application of the 3PL Model to the Mathematics Data, MMLE, mirt
- Assessing Person Fit: Appropriateness Measurement
- Information for the Three-Parameter Model
- Metric Transformation, 3PL Model
- Handling Missing Responses
- Issues to Consider in Selecting among the 1PL, 2PL, and 3PL Models
- Summary
7. Rasch Models for Ordered Polytomous Data
- Conceptual Development of the Partial Credit Model
- Conceptual Parameter Estimation of the PC Model
- Example: Application of the PC Model to a Reasoning Ability Instrument, MMLE, flexMIRT
- Example: Application of the PC Model to a Reasoning Ability Instrument, MMLE, mirt
- The Rating Scale Model
- Conceptual Parameter Estimation of the RS Model
- Example: Application of the RS Model to an Attitudes Towards Condoms Scale, JMLE, BIGSTEPS
- Example: Application of the PC Model to an Attitudes Towards Condoms Scale, JMLE, mixRasch
- How Large a Calibration Sample?
- Information for the PC and RS Models
- Metric Transformation, PC and RS Models
- Summary
8. Non-Rasch Models for Ordered Polytomous Data
- The Generalized Partial Credit Model
- Example: Application of the GPC Model to a Reasoning Ability Instrument, MMLE, flexMIRT
- Example: Application of the GPC Model to a Reasoning Ability Instrument, MMLE, mirt
- Conceptual Development of the Graded Response Model
- How Large a Calibration Sample?
- Information for Graded Data
- Metric Transformation, GPC and GR Models
- Example: Application of the GR Model to an Attitudes Towards Condoms Scale, MMLE, flexMIRT
- Example: Application of the GR Model to an Attitudes Towards Condoms Scale, MMLE, mirt
- Conceptual Development of the Continuous Response Model
- Summary
9. Models for Nominal Polytomous Data
- Conceptual Development of the Nominal Response Model
- Information for the NR Model
- Metric Transformation, NR Model
- Conceptual Development of the Multiple-Choice Model
- How Large a Calibration Sample?
- Example: Application of the NR Model to a General Science Test, MMLE, mirt
- Summary
10. Models for Multidimensional Data
- Conceptual Development of a Multidimensional IRT Model
- Multidimensional Item Location and Discrimination
- Item Vectors and Vector Graphs
- The Multidimensional Three-Parameter Logistic Model
- Assumptions of the MIRT Model
- Estimation of the M2PL Model
- Information for the M2PL Model
- Indeterminacy in MIRT
- Metric Transformation, M2PL Model
- Example: Calibration of interpersonal engagement instrument, M2PL Model, sirt.noharam
- Obtaining Person Location Estimates
- Example: Calibration of interpersonal engagement instrument, M2PL Model, mirt
- Example: Calibration of interpersonal engagement instrument, M2PL Model, flexMIRT
- Summary
11. Linking and Equating
- Equating Defined
- Equating: Data Collection Phase
- Equating: Transformation Phase
- Example: Application of the Total Characteristic Function Equating Method, EQUATE
- Example: Application of the Total Characteristic Function Equating Method, SNSequate
- Example: Fixed-item and Concurrent Calibration Equating
- Summary
12. Differential Item Functioning
- Differential Item Functioning and Item Bias
- Mantel–Haenszel Chi-Square
- The TSW Likelihood Ratio Test
- Logistic Regression
- Example: DIF Analysis of vocabulary test, SAS CMH
- Example: DIF Analysis of vocabulary test, mantelhaen.test and difR
- Example: DIF Analysis of vocabulary test, SAS proc logistic
- Example: DIF Analysis of vocabulary test, glm and difR
- Summary
13. Multilevel IRT Models
- Multilevel IRT–Two Levels
- Example: Equivalence of the Rasch model and its Multilevel Model Parameterization, proc glimmix
- Example: Rasch model estimation, lme4
- Person-Level Predictors for Items
- Example: Person-Level Predictors for Items–DIF Analysis, proc glimmix
- Example: Person-Level Predictors for Items–DIF Analysis, lme4
- Person-Level Predictors for Respondents
- Example: Person-Level Predictors for Respondents–Nutrition Literacy, proc glimmix
- Example: Person-Level Predictors for Respondents, lme4
- Item-Level Predictors for Items
- Example: Item-Level Predictors for Items - Nutrition Literacy, proc glimmix
- Example: Item-Level Predictors for Items - Nutrition Literacy, lme4
- Multilevel IRT–Three Levels
- Example: Three-Level Model Analysis - Nutrition Literacy, proc glimmix
- Example: Three-level analysis of nutrition literacy data, lme4
Appendix A. Maximum Likelihood Estimation of Person Locations
- Estimating an Individual’s Location: Empirical Maximum Likelihood Estimation
- Estimating an Individual’s Location: Newton’s Method for MLE
- Revisiting Zero Variance Binary Response Patterns
Appendix B. Maximum Likelihood Estimation of Item Locations
Appendix C. The Normal Ogive Models
- Conceptual Development of the Normal Ogive Model
- The Relationship between IRT Statistics and Traditional Item Analysis Indices
- Relationship of the Two-Parameter Normal Ogive and Logistic Models
- Extending the Two-Parameter Normal Ogive Model to a Multidimensional Space
Appendix D. Computerized Adaptive Testing
- A Brief History
- Fixed-Branching Techniques
- Variable-Branching Techniques
- Advantages of Variable-Branching over Fixed-Branching Methods
- IRT-Based Variable-Branching Adaptive Testing Algorithm
Appendix E. Linear Logistic Test Model (LLTM)
- Example of LLTM Calibration using eRm
- Appendix F. Mixture Rasch Model
- Latent Class Analysis
- Mixture Rasch Model
- Example: Application of the mixture Rasch Model to Writing Problem Data, CMLE, WINMIRA
- Example: Application of the mixture Rasch Model to Writing Problem Data, CMLE, psychomix
Appendix G. Miscellanea
- Using Principal Axis for Estimating Item Discrimination
- Infinite Item Discrimination Parameter Estimates
- Example: NOHARM Unidimensional Calibration
- An Approximate Chi-Square Statistic for NOHARM
- Relative Efficiency, Monotonicity, and Information
- FORTRAN Formats
- Odds, Odds Ratios, and Logits
- The Person Response Function
- Linking: A Temperature Analogy Example
- Should DIF Analyses Be Based on Latent Classes?
- The Separation and Reliability Indices
- Dependency in Traditional Item Statistics and Observed Scores
- Conditional Independence using Q₃
- Standalone NOHARM Calibration of interpersonal engagement instrument, M2PL Model
- CFI, GFI, M₂, RMSEA, TLI, and SRMR
- An Introduction to Kernel Equating
- Correspondence between the Rasch model and a Loglinear model
- R Introduction
References
Author Index
Subject Index
About the Author