# A Course in Item Response Theory and Modeling with Stata

A Course in Item Response Theory and Modeling with Stata, by Tenko Raykov and George A. Marcoulides, is a comprehensive introduction to the concepts of item response theory (IRT) that includes numerous examples using Stata’s powerful suite of IRT commands. The authors’ unique development of IRT builds on the foundations of classical test theory, nonlinear factor analysis, and generalized linear models. The examples demonstrate how to fit many kinds of IRT models, including one-, two-, and three-parameter logistic models for binary items as well as nominal, ordinal, and hybrid models for polytomous items.

Chapters 1 and 2 define item response theory and review the statistical concepts and functions that are used to build item response models.

Chapters 3 and 4 discuss classical test theory, factor analysis, and generalized linear models, which provide the conceptual foundations of item response theory.

Chapters 5 and 6 introduce the fundamentals of item response theory and provide examples to illustrate the concepts.

Chapters 7 and 8 cover model fitting, estimation using maximum likelihood theory, item characteristic curves, and information functions.

Chapters 9 and 10 provide a detailed introduction to instrument construction and differential item functioning.

Chapters 11 and 12 introduce IRT models for nominal and ordinal responses as well as multidimentional IRT models.

A Course in Item Response Theory and Modeling with Stata is an outstanding text both for those who are new to IRT and for those who are familiar with IRT but new to fitting these models in Stata. It is a useful text for IRT courses and for researchers who use IRT.

List of figures
List of tables
Preface
Notation and typography

1 What is item response theory and item response modeling?
1.1 A definition and a fundamental concept of item response theory and item response modeling
1.2 The factor analysis connection
1.3 What this book is, and is not, about
1.4 Chapter conclusion

2 Two basic functions for item response theory and item response modeling and introduction to Stata

2.1 The normal ogive

2.1.1 The normal distribution probability density function
2.1.2 The normal ogive function

2.2 The logistic function and related concepts

2.2.1 Definition, notation, and graph of the logistic function
2.2.2 Invertibility of the logistic function, odds, and logits

2.3 The relationship between the logistic and normal ogive functions and their use to express response probability

2.3.1 Expressing event or response probability in two distinct ways
2.3.2 Alternative response probability as closely related to the logistic function

2.4 Chapter conclusion

3 Classical test theory, factor analysis, and their connections to item response theory

3.1 A brief visit to classical test theory

3.1.1 The classical test theory decomposition (classical test theory equation)
3.1.2 Misconceptions about classical test theory
3.1.3 Binary random variables: Expectation and probability of a prespecified response

3.2 Why classical test theory?
3.3 A short introduction to classical factor analysis

3.3.1 The classical factor analysis model
3.3.2 Model parameters
3.3.3 Classical factor analysis and measure correlation for fixed factor values

3.4 Chapter conclusion

4 Generalized linear modeling, logistic regression, nonlinear factor analysis, and their links to item response theory and item response modeling

4.1 Generalized linear modeling as a statistical methodology for analysis of relationships between response and explanatory variables

4.1.1 The general linear model and its connection to the classical factor analysis model
4.1.2 Extending the linear modeling idea to discrete response variables
4.1.3 The components of a generalized linear model

4.2 Logistic regression as a generalized linear model of relevance for item response theory and item response modeling

4.2.1 Univariate binary logistic regression
4.2.2 Multivariate logistic regression

4.3 Nonlinear factor analysis models and their relation to generalized linear models

4.3.1 Classical factor analysis and its connection to generalized linear modeling
4.3.2 Nonlinear factor analysis models

4.4 Chapter conclusion

5 Fundamentals of item response theory and item response modeling

5.1 Item characteristic curves revisited

5.1.1 What changes across item characteristic curves in a behavioral measurement situation?

5.2 Unidimensionality and local independence

5.2.1 What are the implications of unidimensionality?
5.2.2 A formal definition of local independence
5.2.3 What does it mean to assume local independence in an item response theory setting?

5.3 A general linear modeling property yielding test-free and group-free measurement in item response modeling
5.4 One more look at the logistic function
5.5 The one- and two-parameter logistic models

5.5.1 The two-parameter logistic model
5.5.2 Interpretation of the item parameters in the two-parameter logistic model
5.5.3 The scale of measurement
5.5.4 The one-parameter logistic model
5.5.5 The one-parameter logistic and two-parameter logistic models as nonlinear factor analysis models, generalized linear models, and logistic regression models
5.5.6 Important and useful properties of the Rasch model

5.6 The three-parameter logistic model
5.7 The logistic models as latent variable models and analogs to nonlinear regression models

5.7.1 Item response models as latent variable models
5.7.2 The logistic models as analogs to nonlinear regression models

5.8 Chapter conclusion

6 First applications of Stata for item response modeling
6.1 Reading data into Stata and related activities
6.2 Fitting a two-parameter logistic model
6.3 Testing nested item response theory models and model selection
6.4 Fitting a one-parameter logistic model and comparison with the two-parameter logistic model
6.5 Fitting a three-parameter logistic model and comparison with more parsimonious models
6.6 Estimation of individual subject trait, construct, or ability levels
6.7 Scoring of studied persons
6.8 Chapter conclusion

7 Item response theory model fitting and estimation
7.1 Introduction
7.2 Person likelihood function for a given item set

7.2.1 Likelihood reexpression in log likelihood
7.2.2 Maximum likelihood estimation of trait or ability level for a given person
7.2.3 A brief visit to the general maximum likelihood theory
7.2.4 What if (meaningful) maximum likelihood estimates do not exist?

7.3 Estimation of item parameters

7.3.1 Standard errors of item parameter estimates

7.4 Estimation of item and ability parameters
7.5 Testing and selection of nested item response theory models
7.6 Item response model fitting and estimation with missing data
7.7 Chapter conclusion

8 Information functions and test characteristic curves
8.1 Item information functions for binary items
8.2 Why should one be interested in item information, and where is it maximal?
8.3 What else is relevant for item information?
8.4 Empirical illustration of item information functions
8.5 Test information function
8.6 Test characteristic curve
8.7 The test characteristic curve as a nonlinear trait or ability score transformation
8.8 Chapter conclusion

9 Instrument construction and development using information functions
9.1 A general approach of item response theory application for multi-item measuring instrument construction
9.2 How to apply Lord’s approach to instrument construction in empirical research
9.3 Examples of target information functions for applications of the outlined procedure for measuring instrument construction
9.4 Assumptions of instrument construction procedure
9.5 Discussion and conclusion

10 Differential item functioning
10.1 What is differential item functioning?
10.2 Two main approaches to differential item functioning examination
10.3 Observed variable methods for differential item functioning examination
10.4 Using Stata for studying differential item functioning with observed variable methods
10.5 Item response theory based methods for differential item functioning examination
10.6 Chapter conclusion

Appendix. The Benjamin–Hochberg multiple testing procedure: A brief introduction

11 Polytomous item response models and hybrid models
11.1 Why do we need polytomous items?
11.2 A key distinction between item response theory models with polytomous and dichotomous items
11.3 The nominal response model

11.3.1 An empirical illustration of the nominal response model

11.4 The partial credit and the rating scale models

11.4.1 Partial credit model
11.4.2 Rating scale model

11.5 The generalized partial credit model
11.7 Comparison and selection of polytomous item response models
11.8 Hybrid models
11.9 The three-parameter logistic model revisited
11.10 Chapter conclusion

12 Introduction to multidimensional item response theory and modeling
12.1 Limitations of unidimensional item response theory
12.2 A main methodological principle underlying multidimensional item response theory
12.3 How can we define multidimensional item response theory?
12.4 A main class of multidimensional item response theory models
12.5 Fitting multidimensional item response theory models and comparison with unidimensional item response theory models

12.5.1 Fitting a multidimensional item response theory model
12.5.2 Comparing a multidimensional model with an unidimensional model

12.6 Chapter conclusion

13 Epilogue

References
Author Index

Subject Index 