Item Response Theory

A complete discussion of fundamental and advanced topics in Item Response Theory written by pioneers in the field

In Item Response Theory, accomplished psychometricians Darrell Bock and Robert Gibbons deliver a comprehensive and up-to-date exploration of the theoretical foundations and applications of Item Response Theory (IRT). Covering both unidimensional and multidimensional IRT, as well as related adaptive test administration of previously calibrated item banks, the book addresses the growing need for understanding of this topic as the use of IRT spreads to other fields.

The first book on the topic that offers a complete and unified treatment of its subject, Item Response Theory prepares researchers and students to understand and apply IRT and multidimensional IRT to fields like education, mental health and marketing. Accessible to first year-graduate students with a foundation in the behavioral or social sciences, basic statistics, and generalized linear models, the book walks readers through everything from the logic of IRT to cutting edge applications of the technique.

Readers will also benefit from the inclusion of:

• A thorough introduction to the foundations of Item Response Theory, including its logic and origins, model-based measurement, psychological scaling, and classical test theory

• An exploration of selected mathematical and statistical results, including points, point sets, and set operations, probability, sampling, and joint, conditional, and marginal probability

• Discussions of unidimensional and multidimensional IRT models, including item parameter estimation with binary and polytomous data

• Analysis of dimensionality, differential item functioning, and multiple group IRT

Perfect for graduate students and researchers studying and working with psychometrics in psychology, quantitative psychology, educational measurement, marketing, and statistics, Item Response Theory will also benefit researchers interested in patient reported outcomes in health research.

Auteur

Darrell Bock is Professor Emeritus at the University of Chicago and is one of the world's leading psychometricians. He coined the term Item Response Theory.

Robert Gibbons is the Blum-Riese Professor of Biostatistics and Pritzker Scholar at the University of Chicago. He is a Fellow of the American Statistical Association, the Royal Statistical Society, and the International Statistical Institute. He is a pioneer of Multidimensional Item Response Theory.

Contenu

Preface xvii

Acknowledgments xix

1 Foundations 1

1.1 The Logic of Item Response Theory 3

1.2 Model-based Data Analysis 4

1.3 Origins 5

1.3.1 Psychometric Scaling 6

1.3.2 Classical Test Theory 9

1.3.3 Contributions fromStatistics 10

1.4 The Population Concept in IRT 11

1.5 Generalizability Theory 14

2 Selected Mathematical and Statistical Results 21

2.1 Points, Point Sets, and Set Operations 21

2.2 Probability 24

2.3 Sampling 25

2.4 Joint, Conditional, and Marginal Probability 26

2.5 Probability Distributions and Densities 28

2.6 Describing Distributions 32

2.7 Functions of RandomVariables 34

2.7.1 Linear Functions 34

2.7.2 Nonlinear Functions 37

2.8 Elements ofMatrix Algebra 37

2.8.1 PartitionedMatrices 41

2.8.2 The Kronecker Product 42

2.8.3 Row and ColumnMatrices 43

2.8.4 Matrix Inversion 43

2.9 Determinants 45

2.10 Matrix Differentiation 45

2.10.1 Scalar Functions of Vector Variables 46

2.10.2 Vector Functions of a Vector Variable 47

2.10.3 Scalar Functions of aMatrix Variable 48

2.10.4 Chain Rule for Scalar Functions of a Matrix Variable 49

2.10.5 Matrix Functions of aMatrix Variable 49

2.10.6 Derivatives of a Scalar Function with Respect to a SymmetricMatrix 50

2.10.7 Second-order Differentiation 52

2.11 Theory of Estimation 53

2.11.1 Analysis of Variance 56

2.11.2 Estimating VarianceComponents 57

2.12 MaximumLikelihoodEstimation (MLE) 59

2.12.1 Likelihood Functions 59

2.12.2 The LikelihoodEquations 60

2.12.3 Examples of Maximum Likelihood Estimation 60

2.12.4 SamplingDistribution of the Estimator 62

2.12.5 The Fisher-scoring Solution of the Likelihood Equations 63

2.12.6 Properties of the Maximum Likelihood Estimator (MLE) 63

2.12.7 Constrained Estimation 64

2.12.8 Admissibility 64

2.13 Bayes Estimation 65

2.14 TheMaximumA Posteriori (MAP) Estimator 68

2.15 Marginal Maximum Likelihood Estimation (MMLE) 69

2.15.1 TheMarginal Likelihood Equations 70

2.15.2 Application in the Normal-Normal Case 72

2.15.3 The EMSolution 75

2.15.4 The Fisher-scoring Solution 75

2.16 Probit and LogitAnalysis 77

2.16.1 ProbitAnalysis 77

2.16.2 LogitAnalysis 79

2.16.3 Logit-linearAnalysis 80

2.16.4 Extension of Logit-linear Analysis to Multinomial Data 82

2.16.4.1 Graded Categories 83

2.16.4.2 NominalCategories 85

2.17 SomeResults fromClassical Test Theory 88

2.17.1 Test Reliability 90

2.17.2 Estimating Reliability 91

2.17.2.1 Bayes Estimation of True Scores 96

2.17.3 When are the Assumptions of Classical Test Theory Reasonable? 97

3 Unidimensional IRT Models 101

3.1 The General IRT Framework 103

3.2 Item ResponseModels 104

3.2.1 DichotomousCategories 105

3.2.1.1 Normal OgiveModel 105

3.2.1.2 2-PLModel 109

3.2.1.3 3-PLModel 111

3.2.1.4 1-PLModel 113

3.2.1.5 Illustration 114

3.2.2 PolytomousCategories 115

3.2.2.1 Graded CategoriesModel 118

3.2.2.2 Illustration 120

3.2.2.3 The NominalCategoriesModel 122

3.2.2.4 Nominal Multiple-Choice Model 130

3.2.2.5 Illustration 132

3.2.2.6 Partial CreditModel 135

3.2.2.7 Generalized Partial Credit Model 136

3.2.2.8 Illustration 136

3.2.2.9 Rating ScaleModels 136

3.2.3 RankingModel 139

4 Item Parameter Estimation - Binary Data 141

4.1 Estimation of Item Parameters Assuming Known Attribute

Values of the Respondents 142

4.1.1 Estimation 143

4.1.1.1 The 1-parameterModel 143 4.1.1.2 The...