Introduction to Item Response Theory Models and Applications

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.

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

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.