factor structures, including the two-tier model, the testlet model, and the bifactor model. Estimation is the focus of Chapter 7. Multidimensionality complicates the estimation process, and efficient approaches have only been developed recently. In this chapter, Professor Bonifay provides a general overview of MIRT estimation methods (e.g., adaptive quadrature, Bayesian, and Metropolis-Hastings Robbins-Monro approaches), and points interested readers to advanced resources. Diagnostics are addressed in Chapter 8. These include dimensionality assessment, test- and item-level fit assessment, and model comparisons. The final chapter concludes with some applications under development right now.
A hallmark of the volume is the extensive use of graphical displays. These visualizations are essential to the pedagogy as they provide intuition as to the logic underlying various models and procedures. The graphics are helpful in the discussion of UIRT models, for example, illustrating the parameters (discrimination, difficulty, guessing, and inattention) and the consequences of each for the item response functions and information functions. The graphics are critical to the discussion of MIRT models, as here, we are operating in multidimensional space. Professor Bonifay pairs item response surfaces and contour plots in a way that is extraordinarily useful in illustrating the various MIRT models and clarifying differences between them.
In addition to a well-organized introduction to MIRT models, the volume provides several resources that will be helpful to readers. First, Professor Bonifay has created a Companion Student Study Site at study.sagepub.com/researchmethods/qass/bonifay-multidimensional-item-response-theory-1e that illustrates the procedures and applications discussed in the volume. The Companion Student Study Site leverages a second resource, a freely available irtDemo package developed by Professor Bonifay and a collaborator for the R statistical software environment (Bulus & Bonifay, 2016). Between the two, readers will be able to replicate most of the visualizations in the volume as well as some that cannot be rendered in print, such as the animation of a 360-degree rotating 3-dimensional response surface! Third, the text points to published examples that illustrate the kinds of models and issues discussed in the text. Fourth, basic R programming syntax is included throughout the text to enable readers to conduct their own MIRT analyses. As mentioned, this is an advanced text, but for readers with the appropriate background, it pulls together, organizes, presents the technical literature in an accessible way, and, in doing so, will increase the use of MIRT models in social science generally as well as in educational psychology and evaluation.
—Barbara Entwisle
Series Editor
Acknowledgments
The author would like to acknowledge Dr. Steve Reise and Dr. Li Cai, whose mentorship and instruction contributed substantially to this volume. I am especially grateful to QASS Series Editor Barbara Entwisle for her valuable suggestions and advice, and to SAGE Senior Acquisitions Editor Helen Salmon for her assistance and support. Most important, thank you to Julie for her never-ending patience and encouragement.
Dedicated to my daughter, Wren, who will be disappointed to learn that I did not fulfill her request for a chapter on how to draw animals.
The author and SAGE would like to thank the following reviewers for their feedback:
Maria Pampaka, The University of Manchester
Yan Yan Sheng, Southern Illinois University
Gustavo Gonzalez-Cuevas, Idaho State University
About the Author
Wes Bonifayis Assistant Professor in the Statistics, Measurement, and Evaluation in Education program in the College of Education at the University of Missouri. He earned his PhD in Quantitative Psychology from the University of California, Los Angeles in 2015. His research focuses on the development, evaluation, and application of multidimensional item response theory and other latent variable measurement models. His work has appeared in journals such as Multivariate Behavioral Research and Structural Equation Modeling.
Chapter 1 Introduction
In any statistical modeling scenario, whether the model represents atoms or galaxies or the human brain, it is essential that all variables are measured with optimal precision. Without exact and meticulous measurement, the model may not be an accurate representation of the real-world phenomena under investigation. In the field of psychometrics, the variables in the model are psychological in nature—academic proficiency, personality traits, severity of psychiatric symptoms, and so on. Psychological constructs such as these are inherently complicated and multifaceted, and relatively simple models that only measure a single construct are often insufficient approximations of complex data. As Zhang (2007) noted, “the unidimensionality of a set of items usually cannot be met and most tests are actually multidimensional to some extent” (p. 69). Accordingly, several decades of psychometric research have led to the development of sophisticated models for multidimensional test data, and in recent years, multidimensional item response theory (MIRT) has become a burgeoning topic in psychological and educational measurement. With regard to theoretical development, MIRT is the focus of ongoing research by many leading quantitative methodologists, who are continually supplying the psychometric community with novel and innovative statistical techniques. In terms of application, MIRT has been successfully implemented not only in psychology and education but also in economics, biostatistics, psychiatry, and a number of other scientific disciplines that demand precise measurement of multidimensional psychological constructs.
MIRT is rightly considered to be a cutting-edge statistical technique; indeed, the methodology underlying MIRT can become exceedingly complex, and many leading psychometricians and researchers are actively building upon the foundations of MIRT in increasingly sophisticated ways. As a result, this topic may not receive much attention in an introductory item response theory (IRT) course. In this author’s opinion, however, it is a major misperception to regard MIRT as too advanced or intimidating for inexpert audiences. While MIRT offers many technical challenges, it can certainly be understood and applied by readers who have a firm grounding in unidimensional IRT modeling. As with other titles in the Quantitative Applications in the Social Sciences (QASS) series, the purpose of the book is to present the foundations of an advanced quantitative topic in a palatable and concise format to students, instructors, and researchers. It includes many practical examples and illustrations, along with numerous intuitive and informative figures and diagrams. In addition, many high-quality applied MIRT research articles are cited and discussed throughout the text to demonstrate how the various models and methods are being used in the real world. A particularly useful accompaniment to this volume is the freely available irtDemo package (Bulus & Bonifay, 2016) for the R statistical software environment (R Core Team, 2018). This package was specifically designed to provide students and other learners with a hands-on approach to IRT modeling via a suite of interactive applets. By using these applets, readers can easily manipulate and inspect the complex output produced by several common MIRT models and gain a greater understanding of potentially difficult topics. Furthermore, the irtDemo package can be used to create MIRT figures for use in academic publications; in fact, many of the figures presented in this book were created using irtDemo. This package can be downloaded from the Comprehensive R Archive Network at https://cran.r-project.org/package=irtDemo.
In addition, brief snippets of R code are interspersed throughout the text (with the complete R code included on the Companion Student Study Site at study.sagepub.com/researchmethods/qass/bonifay-multidimensional-item-response-theory-1e) to guide readers in exploring MIRT models, estimating the model parameters, generating plots, and implementing the various procedures and applications discussed throughout this book. The R code is primarily based on the
