**List of figures**

**List of tables**

**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

11.6 The graded response 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**