Introduction to Item Response Theory Models and Applications

1. Introduction

1. Background and Terminology
2. Contents of the Following Chapters
2. Models for Dichotomously-Scored Items

1. Introduction
2. Classical Test theory Models

The Model

Item Parameters and their Estimates

Test Parameters and their Estimates

3. Item Response Theory Models

Introduction

The Normal Ogive Three-Parameter Item Response Theory Model

The Three-Parameter Logistic (3PL) Model

Special Cases: The Two-Parameter and One-Parameter Logistic Models

Relationships Between Probabilities of Alternative Responses

Transformations of Scale

Effects of Changes in Parameters

The Test Characteristic Function

The Item Information Function

The Test Information Function and Standard Errors of Measurement

4. IRT Estimation Methodology

Estimation of Item Parameters

Estimation of Proficiency

Indeterminacy of the Scale in IRT Estimation

5. Summary
3. Analyses of Dichotomously-Scored Item and Test Data

1. Introduction
2. Example Classical Test Theory Analyses with a Small Dataset
3. Test and Item Analyses with a Larger Dataset

CTT Item and Test Analysis Results

4. IRT Item and Test Analysis

IRT Software

Missing Data

Iterative Estimation Methodology

Model Fit

5. IRT Analyses Using PARSCALE

PARSCALE Terminology

Some PARSCALE Options

PARSCALE Item Analysis

PARSCALE Test Analyses

6. IRT Analyses Using flexMIRT

flexMIRT Terminology

Some flexMIRT Options

flexMIRT Item Analyses and Comparisons Between Programs

flexMIRT Test Analyses and Comparisons Between Programs

7. Using IRT Results to Evaluate Items and Tests

Evaluating Estimates of Item Parameters

Evaluating Fit of Models to Items

Evaluating Tests as a Whole or Subsets of Test Items

8. Equating, Linking, and Scaling

Equating

Linking

Scaling

Vertical Scaling

9. Summary
4. Models for Polytomously-Scored Items

1. Introduction
2. The Nature of Polytomously-Scored Items
3. Conditional Probability Forms of Models for Polytomous Items
4. Probability-of-Response Form of the Polytomous Models

The 2PPC Model

The GPC Model

The Graded Response (GR) Model

5. Additional Characteristics of the GPC Model

Effects of Changes in Parameters

Alternative Parameterizations

The Expected Score Function

Functions of Scoring at or Above Categories

Comparison of Conditional Response and P+ Functions

Item Mapping and Standard Setting

The Test Characteristic Function

The Item Information Function

The Item Category Information Function

The Test Information Function

Conditional Standard Errors of Measurement

6. Summary
5. Analyses of Polytomously-Scored Item and Test Data

1. Generation of Example Data
2. Classical Test Theory Analyses

Item Analyses

Test Analyses

3. IRT Analyses

PARSCALE Item Analyses

flexMIRT Item Analyses and Comparisons with PARSCALE

4. Additional Methods of Using IRT Results to Evaluate Items

Evaluating Estimates of Item Parameters

Evaluating Fit of Models to Item Data

Additional Graphical Methods

5. Test Analyses

PARSCALE Test Analyses

flexMIRT Test Analyses

6. Placing the Results from Different Analyses on the Same Scale
7. Summary
6. Multidimensional Item Response Theory Models

1. Introduction
2. The Multidimensional 3PL Model for Dichotomous Items
3. The Multidimensional 2PL Model for Dichotomous Items
4. Is there a Multidimensional 1PL Model for Dichotomous Items
5. Further Comments on MIRT Models

Alternate Parameterizations

Additional Analyses of MIRT Data

6. Noncompensatory MIRT Models
7. MIRT Models for Polytomous Data
8. Summary
7. Analyses of Multidimensional Item Response Data

1. Response Data Generation
2. MIRT Computer Software
3. MIRT and Factor analyses
4. flexMIRT analyses of Example Generated Data

One-dimensional Solution with Two-Dimensional Data

Two-dimensional Solution

5. Summary
8. Overview of More Complex Item Response Theory Models

1. Some More Complex Unidimensional Models

Multigroup Models

Adaptive Testing

Mixture Models

Hierarchical Rater Models

Testlet Models

2. More General MIRT Models: Some Further Reading

Hierarchical Models

3. Cognitive Diagnostic Models
4. Summary

References

Appendix A. Some Technical Background

1. Slope of the 3PL Curve at the Inflection Point where

2. Simplifying Notation for GPC Expressions

3. Some Characteristics of GPC Model Items

Peaks of Response Curves

Crossing Point of Pk and Pk-1

Crossing Point of P0 and P2 for m = 3

Symmetry in the Case of m = 3

Limits of the Expected Score Function

Appendix B. Item Category Information Functions

Appendix C. Item Generating Parameters and Classical and IRT Parameter Estimates

Index