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An Introduction to the Rasch Model with Examples in R offers a clear, comprehensive introduction to the Rasch model along with practical examples in the free, open-source software R.

It is accessible for readers without a background in psychometrics or statistics, while also providing detailed explanations of the relevant mathematical and statistical concepts for readers who want to gain a deeper understanding. Its worked examples in R demonstrate how to apply the methods to real-world examples and how to interpret the resulting output.

In addition to motivating and presenting the Rasch model, the book covers different methods for parameter estimation and for assessing fit and differential item functioning (DIF). While focusing on the Rasch model, it also addresses a variety of other dichotomous and polytomous Rasch and item response theory (IRT) models, such as two-parameter logistic (2PL) and Partial Credit models, and extensions, including mixture Rasch models and computerized adaptive testing (CAT).

Theory is presented in a self-contained way. All necessary mathematical and statistical background is contained in the chapters and appendices. The book also provides detailed, step-by-step instructions for getting started with R and using the eRm, mirt, TAM and rstan packages for fitting Rasch models.

Arvustused

Overall, the book has a lot of great detail and is technically sound. It is also clearly written and at an appropriate level of difficulty. Another big strength is the chapters with R applications. I know my students would love this, and generally this is the type of guidance they want with respect to conducting IRT in R, as opposed to trying to field through the different packages, figure out how they scale latent variables in the package, and more. A final big strength is the focus on fairness as a core underlying issue of model evaluation. From the Introduction chapter and throughout the various other chapters, issues of fairness were put at the forefront. This aligns with the Standards for Educational and Psychological Testing, and in general with modern views of validity and test theory. - Anne Corinne Huggins-Manley. University of Florida

With regard to R, I think it provides a good introduction. The book explains the basic features of R code and then proceeds to give many examples of R in action, which I think it a good approach. Overall, the book is very well-written. Most of the explanations of Rasch model properties or R commands are extraordinary clear. - Leah Feuerstahler, Fordham University Overall, the book has a lot of great detail and is technically sound. It is also clearly written and at an appropriate level of difficulty. Another big strength is the chapters with R applications. I know my students would love this, and generally this is the type of guidance they want with respect to conducting IRT in R, as opposed to trying to field through the different packages, figure out how they scale latent variables in the package, and more. A final big strength is the focus on fairness as a core underlying issue of model evaluation. From the Introduction chapter and throughout the various other chapters, issues of fairness were put at the forefront. This aligns with the Standards for Educational and Psychological Testing, and in general with modern views of validity and test theory. - Anne Corinne Huggins-Manley, University of Florida, USA

With regard to R, I think it provides a good introduction. The book explains the basic features of R code and then proceeds to give many examples of R in action, which I think it a good approach. Overall, the book is very well-written. Most of the explanations of Rasch model properties or R commands are extraordinary clear. - Leah Feuerstahler, Fordham University, USA

"The book An Introduction to the Rasch Model with Examples in R provides a comprehensive compendium of the Rasch model as a cornerstone of psychometric measurement. The authors offer a well-written introduction into the basic concepts of the Rasch model for binary items, with a particular focus on differential item functioning and on fit statistics for model evaluation and item selection. Hands-on examples illustrate the use of R syntax for various packages and demonstrate the interpretation of results for real-life research questions. Moreover, the book contains an up-to-date overview of model extensions to accommodate ordinal response formats and parameter heterogeneity. Throughout the book, the content is accessible to the reader without sacrificing mathematical and statistical rigor. The book therefore proves to be a valuable source for different audiences, including students in introductory or advanced classes on test theory, instructors and interested scholars, as well as applied psychometricians. With its combination of statistical detail and applied perspective, the book will certainly help to pave the way for modern item response theory to be used in practice and to overcome the shortcomings of classical test theory." -Thorsten Meiser, University of Mannheim, Germany

Preface xiii
Acknowledgment xv
I Theory
1(96)
1 Introduction
3(6)
1.1 The Role of Psychological and Educational Tests
3(1)
1.2 The Rasch Model and Item Response Theory
4(2)
1.3 Where You Will Find What in This Book
6(3)
2 The Rasch Model
9(28)
2.1 The Data Matrix
10(2)
2.2 The Item Response Function
12(6)
2.2.1 Ability and Difficulty
12(2)
2.2.2 Discrimination
14(2)
2.2.3 The Logistic Function
16(2)
2.3 Alternative Representations
18(3)
2.3.1 Probability of an Incorrect Response
18(1)
2.3.2 Probability of an Arbitrary Response
19(1)
2.3.3 Alternative Representation of the Logistic Function
20(1)
2.3.4 Multiplicative Form
20(1)
2.4 Properties and Assumptions
21(13)
2.4.1 Sufficient Statistics
21(2)
2.4.2 Local Stochastic Independence
23(1)
2.4.2.1 Items
24(2)
2.4.2.2 Persons
26(2)
2.4.3 Specific Objectivity
28(4)
2.4.4 Unidimensionality
32(1)
2.4.5 Measurement Scale
32(2)
2.5 Exercises
34(3)
3 Parameter Estimation
37(20)
3.1 Joint Maximum Likelihood Estimation
38(1)
3.2 Conditional Maximum Likelihood Estimation
39(4)
3.3 Marginal Maximum Likelihood Estimation
43(2)
3.4 Bayesian Estimation
45(3)
3.5 Person Parameter Estimation
48(2)
3.6 Item and Test Information
50(3)
3.7 Sample Size Requirements
53(1)
3.8 Exercises
54(3)
4 Test Evaluation
57(40)
4.1 Graphical Assessment
59(5)
4.1.1 Person Item Map
59(1)
4.1.2 Empirical ICCs
60(2)
4.1.3 Graphical Test
62(2)
4.2 Tests for Item and Person Invariance
64(8)
4.2.1 Andersen's Likelihood Ratio Test
65(2)
4.2.2 Martin-Lof Test and Other Approaches for Detecting Multidimensionality
67(1)
4.2.3 Wald Test
68(1)
4.2.4 Anchoring
69(2)
4.2.5 Other Approaches for Detecting DIF
71(1)
4.2.6 How to Proceed with Problematic Items
71(1)
4.3 Goodness-of-Fit Tests and Statistics
72(13)
4.3.1 Χ2 and G2 Goodness-of-Fit Tests
72(2)
4.3.2 M2, RMSEA, and SRMSR
74(1)
4.3.3 Infit and Outfit Statistics
75(4)
4.3.4 Further Fit Statistics for Items
79(2)
4.3.5 Fit Statistics for Item Pairs
81(1)
4.3.6 Fit Statistics for Persons
82(1)
4.3.7 Nonparametric Goodness-of-Fit Tests
83(1)
4.3.8 Posterior Predictive Checks
84(1)
4.4 Separation Indices
85(2)
4.4.1 Item Separation Index
85(1)
4.4.2 Person Separation Index
86(1)
4.5 Evaluation Through Model Comparisons
87(7)
4.5.1 Models with Additional Item Parameters
87(1)
4.5.1.1 Two-Parameter Model
87(2)
4.5.1.2 Three-Parameter Model
89(2)
4.5.1.3 Four-Parameter Model
91(1)
4.5.1.4 Sample Size Requirements
91(1)
4.5.2 Likelihood Ratio Tests
92(1)
4.5.3 Information Criteria
93(1)
4.6 Exercises
94(3)
II Applications
97(98)
5 Basic R Usage
99(12)
5.1 Installation of R and Add-On Packages
99(2)
5.2 Code Editors and RStudio
101(1)
5.3 Loading and Importing Data
102(1)
5.4 Getting Information About Persons and Variables
103(5)
5.5 Addressing Elements in Lists
108(2)
5.6 Exercises
110(1)
6 R Package eRm
111(36)
6.1 Item Parameter Estimation
112(7)
6.2 Test Evaluation
119(18)
6.2.1 Person Item Map
120(1)
6.2.2 Empirical ICCs
121(1)
6.2.3 Andersen's Likelihood Ratio Test and Graphical Test
121(6)
6.2.4 Wald Test
127(1)
6.2.5 Anchoring
128(5)
6.2.6 Removing Problematic Items
133(1)
6.2.7 Martin-Lof Test
134(1)
6.2.8 Item and Person Fit
135(2)
6.3 Plots of ICCs, Item and Test Information
137(2)
6.4 Person Parameter Estimation
139(3)
6.5 Test Evaluation in Small Data Sets
142(3)
6.6 Exercises
145(2)
7 R Package mirt
147(18)
7.1 Model Selection
148(2)
7.2 Item Parameter Estimates
150(6)
7.2.1 Illustration via Expected ICCs
150(2)
7.2.2 Displaying the Estimates
152(4)
7.3 Evaluating Goodness-of-Fit
156(4)
7.4 Ability Estimation
160(3)
7.5 Exercises
163(2)
8 R Package TAM
165(8)
8.1 Item Parameter Estimation
166(2)
8.2 Evaluating Goodness-of-Fit
168(2)
8.3 Person Parameter Estimation
170(2)
8.4 Exercises
172(1)
9 R Interface to Stan
173(22)
9.1 Stan Models
174(4)
9.1.1 The data Block
174(2)
9.1.2 The parameters Block
176(1)
9.1.3 The transformed parameters Block
176(1)
9.1.4 The model Block
177(1)
9.2 Sampling the Posterior Using RStan
178(5)
9.3 Evaluating Goodness-of-Fit
183(6)
9.4 Exercises
189(6)
Summary of R Commands for eRm, mirt, and TAM
191(4)
III Beyond the Rasch Model
195(32)
10 Extensions to the Rasch Model
197(8)
10.1 The Linear-Logistic Test Model
197(2)
10.2 Modeling Differences Between People
199(4)
10.2.1 The Mixture Rasch Model
199(1)
10.2.2 Model-Based Recursive Partitioning
200(1)
10.2.3 Explanatory IRT -- The Rasch Model as a Mixed Model
201(2)
10.3 Multidimensional IRT Models
203(1)
10.4 Exercises
203(2)
11 Models for Polytomous Responses
205(18)
11.1 The Partial Credit Model
206(9)
11.1.1 CCCs and Threshold Parameters
209(1)
11.1.2 Alternative Parameterizations
210(4)
11.1.3 Disordered Thresholds
214(1)
11.2 The Rating Scale Model
215(2)
11.3 The Generalized Partial Credit and the Nominal Response Model
217(1)
11.4 The Graded Response Model
217(1)
11.5 The Sequential Model
218(1)
11.6 Sample Size Requirements
219(1)
11.7 Exercises
219(1)
11.8 Derivations for the Partial Credit Model
220(3)
12 Outlook on Special Applications
223(4)
12.1 Computerized Adaptive Testing
223(1)
12.2 Test Linking and Equating
224(1)
12.3 Longitudinal IRT Models
225(1)
12.4 Exercises
226(1)
Appendices
227(62)
A Useful Mathematical Formulas
229(4)
A.1 Sums and Products
229(1)
A.2 Exponentials
230(1)
A.3 Logarithms
230(1)
A.4 Differentiation Rules
231(1)
A.4.1 Rules
231(1)
A.4.2 Examples
232(1)
B Statistical Background
233(22)
B.1 Statistical Estimation
233(1)
B.1.1 The Binomial Distribution
233(2)
B.1.2 Maximum Likelihood Estimation
235(3)
B.1.3 Likelihood for Multiple Observations
238(3)
B.1.4 Bayesian Inference
241(1)
B.1.4.1 Bayes' Rule by Example
241(2)
B.1.4.2 Coin Flipping with a Uniform Prior
243(3)
B.1.4.3 Informative Priors and Beta-Binomial Model
246(2)
B.2 Statistical Testing
248(1)
B.2.1 Tests Based on the Χ2 Distribution
249(1)
B.2.1.1 Χ2 Test for Independence
249(3)
B.2.1.2 Goodness-of-Fit and Other Χ2 Tests
252(1)
B.2.2 Tests Based on the Normal Distribution
253(2)
C Answers to the End of
Chapter Questions
255(34)
C.2 Answers for
Chapter 2
255(1)
C.3 Answers for
Chapter 3
256(3)
C.4 Answers for
Chapter 4
259(3)
C.5 Answers for
Chapter 5
262(3)
C.6 Answers for
Chapter 6
265(9)
C.7 Answers for
Chapter 7
274(7)
C.8 Answers for
Chapter 8
281(2)
C.9 Answers for
Chapter 9
283(3)
C.10 Answers for
Chapter 10
286(1)
C.11 Answers for
Chapter 11
287(1)
C.12 Answers for
Chapter 12
288(1)
References 289(12)
Author Index 301(4)
Index 305
Rudolf Debelak is a Senior Researcher at the University of Zurich, Switzerland. His research interests include psychometrics, with a focus on item response theory, machine learning, and the mathematical and statistical foundations of psychological research methods. Before working in academia, he was employed in the psychological test industry for several years.

Carolin Strobl is a Professor of Psychological Methods at the University of Zurich, Switzerland. Her research spans psychometrics, statistics and machine learning. She has been teaching introductory and advanced courses on statistics and psychometrics for many years and received the 2018 teaching award from her departments student council.

Matthew Zeigenfuse currently works as a data scientist. He spent many years working in academia, researching and teaching cognitive science, psychometrics and Bayesian statistics in both the US and Switzerland.