Introduction to Item Response Theory Models and Applications  book cover
1st Edition

Introduction to Item Response Theory Models and Applications

ISBN 9780367471019
Published October 13, 2020 by Routledge
182 Pages 124 B/W Illustrations

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Book Description

This is a highly accessible, comprehensive introduction to item response theory (IRT) models and their use in various aspects of assessment/testing. The book employs a mixture of graphics and simulated data sets to ease the reader into the material and covers the basics required to obtain a solid grounding in IRT.

Written in an easily accessible way that assumes little mathematical knowledge, Carlson presents detailed descriptions of several commonly used IRT models, including those for items scored on a two-point (dichotomous) scale such as correct/incorrect, and those scored on multiple-point (polytomous) scales, such as degrees of correctness. One chapter describes a model in-depth and is followed by a chapter of instructions and illustrations showing how to apply the models to the reader’s own work.

This book is an essential text for instructors and higher level undergraduate and postgraduate students of statistics, psychometrics, and measurement theory across the behavioral and social sciences, as well as testing professionals.

Table of Contents

  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
    3. The Model

      Item Parameters and their Estimates

      Test Parameters and their Estimates

    4. Item Response Theory Models
    5. 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

    6. IRT Estimation Methodology
    7. Estimation of Item Parameters

      Estimation of Proficiency

      Indeterminacy of the Scale in IRT Estimation

    8. 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
    4. CTT Item and Test Analysis Results

    5. IRT Item and Test Analysis
    6. IRT Software

      Missing Data

      Iterative Estimation Methodology

      Model Fit

    7. IRT Analyses Using PARSCALE
    8. PARSCALE Terminology

      Some PARSCALE Options

      PARSCALE Item Analysis

      PARSCALE Test Analyses

    9. IRT Analyses Using flexMIRT
    10. flexMIRT Terminology

      Some flexMIRT Options

      flexMIRT Item Analyses and Comparisons Between Programs

      flexMIRT Test Analyses and Comparisons Between Programs

    11. Using IRT Results to Evaluate Items and Tests
    12. Evaluating Estimates of Item Parameters

      Evaluating Fit of Models to Items

      Evaluating Tests as a Whole or Subsets of Test Items

    13. Equating, Linking, and Scaling
    14. Equating



      Vertical Scaling

    15. 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
    5. The 2PPC Model

      The GPC Model

      The Graded Response (GR) Model

    6. Additional Characteristics of the GPC Model
    7. 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

    8. Summary

  5. Analyses of Polytomously-Scored Item and Test Data
    1. Generation of Example Data
    2. Classical Test Theory Analyses
    3. Item Analyses

      Test Analyses

    4. IRT Analyses
    5. PARSCALE Item Analyses

      flexMIRT Item Analyses and Comparisons with PARSCALE

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

      Evaluating Fit of Models to Item Data

      Additional Graphical Methods

    8. Test Analyses
    9. PARSCALE Test Analyses

      flexMIRT Test Analyses

    10. Placing the Results from Different Analyses on the Same Scale
    11. 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
    6. Alternate Parameterizations

      Additional Analyses of MIRT Data

    7. Noncompensatory MIRT Models
    8. MIRT Models for Polytomous Data
    9. 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
    5. One-dimensional Solution with Two-Dimensional Data

      Two-dimensional Solution

    6. Summary

  8. Overview of More Complex Item Response Theory Models
    1. Some More Complex Unidimensional Models
    2. Multigroup Models

      Adaptive Testing

      Mixture Models

      Hierarchical Rater Models

      Testlet Models

    3. More General MIRT Models: Some Further Reading
    4. Hierarchical Models

    5. Cognitive Diagnostic Models
    6. Summary


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


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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.


"Carlson’s book is a very clear and well-written introduction to item response theory models that should prove very useful to a wide range of students, instructors, researchers and professionals who want to understand the basics of this useful methodology." -- Lisa L. Harlow, professor of psychology at the University of Rhode Island, USA, and series editor for the Multivariate Applications Series (sponsored by SMEP).

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