Introduction to Item Response Theory Models and Applications

James E. Carlson received his Ph.D. from the University of Alberta, Canada, specializing in applied statistics. He was professor of education at the universities of Pittsburgh, USA, and Ottawa, Canada. He also held psychometric positions at testing organizations and the National Assessment Governing Board, U. S. Department of Education. He is a former editor of the Journal of Educational Measurement and has authored two book chapters and a number of journal articles and research reports.

Introduction





Background and Terminology



Contents of the Following Chapters




Models for Dichotomously-Scored Items





Introduction



Classical Test theory Models
The Model

Item Parameters and their Estimates

Test Parameters and their Estimates




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




IRT Estimation Methodology
Estimation of Item Parameters

Estimation of Proficiency

Indeterminacy of the Scale in IRT Estimation




Summary




Analyses of Dichotomously-Scored Item and Test Data





Introduction



Example Classical Test Theory Analyses with a Small Dataset



Test and Item Analyses with a Larger Dataset
CTT Item and Test Analysis Results




IRT Item and Test Analysis
IRT Software

Missing Data

Iterative Estimation Methodology

Model Fit




IRT Analyses Using PARSCALE
PARSCALE Terminology

Some PARSCALE Options

PARSCALE Item Analysis

PARSCALE Test Analyses




IRT Analyses Using flexMIRT
flexMIRT Terminology

Some flexMIRT Options

flexMIRT Item Analyses and Comparisons Between Programs

flexMIRT Test Analyses and Comparisons Between Programs




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




Equating, Linking, and Scaling
Equating

Linking

Scaling

Vertical Scaling




Summary




Models for Polytomously-Scored Items





Introduction



The Nature of Polytomously-Scored Items



Conditional Probability Forms of Models for Polytomous Items



Probability-of-Response Form of the Polytomous Models
The 2PPC Model

The GPC Model

The Graded Response (GR) Model




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




Summary




Analyses of Polytomously-Scored Item and Test Data





Generation of Example Data



Classical Test Theory Analyses
Item Analyses

Test Analyses




IRT Analyses
PARSCALE Item Analyses

flexMIRT Item Analyses and Comparisons with PARSCALE




Additional Methods of Using IRT Results to Evaluate Items
Evaluating Estimates of Item Parameters

Evaluating Fit of Models to Item Data

Additional Graphical Methods




Test Analyses
PARSCALE Test Analyses

flexMIRT Test Analyses




Placing the Results from Different Analyses on the Same Scale



Summary




Multidimensional Item Response Theory Models





Introduction



The Multidimensional 3PL Model for Dichotomous Items



The Multidimensional 2PL Model for Dichotomous Items



Is there a Multidimensional 1PL Model for Dichotomous Items



Further Comments on MIRT Models
Alternate Parameterizations

Additional Analyses of MIRT Data




Noncompensatory MIRT Models



MIRT Models for Polytomous Data



Summary




Analyses of Multidimensional Item Response Data





Response Data Generation



MIRT Computer Software



MIRT and Factor analyses



flexMIRT analyses of Example Generated Data
One-dimensional Solution with Two-Dimensional Data

Two-dimensional Solution




Summary




Overview of More Complex Item Response Theory Models





Some More Complex Unidimensional Models
Multigroup Models

Adaptive Testing

Mixture Models

Hierarchical Rater Models

Testlet Models




More General MIRT Models: Some Further Reading
Hierarchical Models




Cognitive Diagnostic Models



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