The Theory and Practice of Item Response Theory

R. J. de Ayala

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December 30, 2008
ISBN 9781593858698
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448 Pages
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Read the Series Editor’s Note by founding editor David A. Kenny
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

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, cj

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

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