# The Theory and Practice of Item Response Theory

### R. J. de Ayala

Read the Series Editor’s Note by founding editor David A. Kenny

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

- 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

- Metric Transformation and the Total Characteristic Function

- 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

- Information and Relative Efficiency

- Summary

6. The Three-Parameter Model

- Conceptual Development of the Three-Parameter Model

- Additional Comments about the Pseudo-Guessing Parameter, *X*ⱼ

- Conceptual Estimation for the 3PL Model

- How Large a Calibration Sample?

- Assessing Conditional Independence

- Example: Application of the 3PL Model to the Mathematics Data, MMLE

- 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

- The Rating Scale Model

- Conceptual Estimation of the RS Model

- Example: Application of the RS Model to an Attitudes toward Condom Scale, JMLE

- 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

- Conceptual Development of the Graded Response Model

- How Large a Calibration Sample?

- Example: Application of the GR Model to an Attitudes toward Condom Scale, MMLE

- Information for Graded Data

- Metric Transformation, GPC and GR Models

- Summary

9. Models for Nominal Polytomous Data

- Conceptual Development of the Nominal Response Model

- How Large a Calibration Sample?

- Example: Application of the NR Model to a Science Test, MMLE

- Example: Mixed Model Calibration of the Science Test—NR and PC Models, MMLE

- Example: NR and PC Mixed Model Calibration of the Science Test, Collapsed Options, MMLE

- Information for the NR Model

- Metric Transformation, NR Model

- Conceptual Development of the Multiple-Choice Model

- Example: Application of the MC Model to a Science Test, MMLE

- Example: Application of the BS Model to a Science Test, MMLE

- 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: Application of the M2PL Model, Normal-Ogive Harmonic Analysis Robust Method

- Obtaining Person Location Estimates

- Summary

11. Linking and Equating

- Equating Defined

- Equating: Data Collection Phase

- Equating: Transformation Phase

- Example: Application of the Total Characteristic Function 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

- Summary

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. Miscellanea

- Linear Logistic Test Model (LLTM)

- Using Principal Axis for Estimating Item Discrimination

- Infinite Item Discrimination Parameter Estimates

- Example: NOHARM Unidimensional Calibration

- An Approximate Chi-Square Statistic for NOHARM

- Mixture Models

- Relative Efficiency, Monotonicity, and Information

- FORTRAN Formats

- Example: Mixed Model Calibration of the Science Test—NR and 2PL Models, MMLE

- Example: Mixed Model Calibration of the Science Test—NR and GR Models, MMLE

- 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

References

Author Index

Subject Index