Bayesian Psychometric Modeling (Hardback) book cover

Bayesian Psychometric Modeling

By Roy Levy, Robert J. Mislevy

© 2016 – Chapman and Hall/CRC

466 pages | 88 B/W Illus.

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Hardback: 9781439884676
pub: 2016-05-23
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pub: 2017-07-28
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A Single Cohesive Framework of Tools and Procedures for Psychometrics and Assessment

Bayesian Psychometric Modeling presents a unified Bayesian approach across traditionally separate families of psychometric models. It shows that Bayesian techniques, as alternatives to conventional approaches, offer distinct and profound advantages in achieving many goals of psychometrics.

Adopting a Bayesian approach can aid in unifying seemingly disparate—and sometimes conflicting—ideas and activities in psychometrics. This book explains both how to perform psychometrics using Bayesian methods and why many of the activities in psychometrics align with Bayesian thinking.

The first part of the book introduces foundational principles and statistical models, including conceptual issues, normal distribution models, Markov chain Monte Carlo estimation, and regression. Focusing more directly on psychometrics, the second part covers popular psychometric models, including classical test theory, factor analysis, item response theory, latent class analysis, and Bayesian networks. Throughout the book, procedures are illustrated using examples primarily from educational assessments. A supplementary website provides the datasets, WinBUGS code, R code, and Netica files used in the examples.


"This book is a great contribution to the field of Bayesian psychometrics. It provides an excellent introduction to the Bayesian statistical philosophy and the Bayesian way of thinking, with a focus on building statistical models for psychometric analysis. In a clear manner, it describes how Bayesian theory can be used to construct psychometric models and carry out statistical analysis, whilst explaining how to integrate prior knowledge into analysis. It also shows the various profound advantages of the Bayesian approach, and presents a comprehensive toolbox for psychometric data analysis, as opposed to conventional approaches.

This book is highly recommended for graduate students and (applied) researchers, who have a basic understanding of psychometric and statistical theory. The second part of the book contains a wide overview of different psychometric models and theories such as classical test theory, item response theory, latent class analysis, and Bayesian networks. A clear and consistent Bayesian approach introduces these different topics, which are illustrated with educational assessment applications. In addition to several programs in R, the WinBUGS program is also utilised to perform computations, making it possible to directly apply the presented material. Overall, this book provides a thorough and comprehensive overview of psychometric modelling, and truly promotes the use of Bayesian methods."

Jean-Paul Fox, Department of Research Methodology, Measurement and Data Analysis, University of Twente

"Drs. Roy Levy and Robert Mislevy have made several pioneering contributions on the application of Bayesian statistical analysis to educational and psychological measurements, and have now brought their expertise to life in the accessible, up-to-date, and comprehensive book Bayesian Psychometric Modeling. This is a must-read for researchers and practitioners of all levels, from undergraduate students of Psychology or Education to experts on Bayesian psychometrics. A unique aspect of the book is its descriptions of the connections between Bayesian modelling and topics such as evidence-centred design and graphical models, which are the authors’ forte. This book provokes the reader into thinking deeper about the topic and is destined to become a classic for those who are interested in the area."

Sandip Sinharay, Principal Research Scientist, Educational Testing Service (ETS)

"The last couple of decades has seen a widening of the gap between psychometrics as taught in textbooks and psychometrics as practiced in industry-leading agencies, with the latter expanding rapidly due to increasingly sophisticated assessment needs and increasingly accessible computational power. Bayesian Psychometric Modeling helps to close this gap, using a model-based and evidentiary reasoning perspective to frame psychometrics as part of a larger landscape methodologically, statistically, computationally, and philosophically. Levy and Mislevy are that rare combination of outstanding scholars and teachers, and it comes through beautifully in their exemplary treatments of not just standard topics like classical test theory and item response theory, but also of Bayesian inference, graph theory, MCMC estimation, model evaluation, confirmatory factor analysis, multidimensional IRT models, latent class analysis, and Bayesian networks. Truly modern psychometrics demands flexible and integrated thinking, and analytical frameworks to match. This book embodies such thinking and analysis, representing where the leaders are and where the rest of us should be going. And it helps us to get there."

Gregory R. Hancock, Professor and Program Director of Measurement, Statistics and Evaluation, University of Maryland

"This book is excellent; it is the most comprehensive and up-to-date resource for researchers who are studying, or would like to know more about, Bayesian approaches for psychometric and statistical models. It thoroughly covers simple to advanced topics on psychometric models with mathematical and underlying psychometric concepts, with clear illustrations on a wide range of applications using real data in the psychometric field. Not only useful for its theoretical aspects on Bayesian approaches, the technical parts of this text (e.g., annotated software codes that can be easily modified) are also illustrated in an accessible manner for students, applied statisticians, and psychometricians who may want to apply these techniques to their own problems. If you want to use or find out more about Bayesian psychometric approaches, whether it be those at the simplest level or those at the cutting-edge of theoretical research and practical application, then this is the book for you!"

Jaehwa Choi, Assessment, Testing, and Measurement Program, The George Washington University

"I can honestly say that the book Bayesian Psychometric Modeling by Roy Levy and Robert Mislevy represents a tremendous amount of effort, dedication, and care in the service of rigorous professional development for colleagues who need to educate themselves on principled evidentiary reasoning using Bayesian methodologies. While reading, I was constantly amazed how I was (re)learning both foundational and complex statistical concepts while almost "incidentally" expanding my intellectual horizon through reflection on core principles of evidentiary reasoning. Throughout the book, the intellectual thoughtfulness of the authors shines through in innumerable ways, especially in the way they express ideas and structure the presentation of information. They utilize a broad array of diverse representations (textual descriptions, examples, and anecdotes along with formulas, tables, graphics, and code) that are well-suited for their communicative function, often in interesting novel ways that provide true insight into a phenomenon. Thus, rather than serving as just a vehicle for learning about best statistical practices in Bayesian modeling, the book can really serve to develop intellectual stewardship for future voices in educational measurement that transcends this transcends this particular intellectual domain."

André A. Rupp, Educational Testing Service (ETS)

"This book covers a range of models and ideas in psychometrics and should be of interest to many students and researchers."

Andrew Gelman, Departments of Statistics and Political Science, Columbia University

Table of Contents


Overview of Assessment and Psychometric Modeling

Assessment as Evidentiary Reasoning

The Role of Probability

The Role of Context in the Assessment Argument

Evidence-Centered Design

Summary and Looking Ahead

Introduction to Bayesian Inference

Review of Frequentist Inference via Maximum Likelihood

Bayesian Inference

Bernoulli and Binomial Models

Summarizing Posterior Distributions

Graphical Model Representation

Analyses Using WinBUGS

Summary and Bibliographic Note


Conceptual Issues in Bayesian Inference

Relative Influence of the Prior Distribution and the Data

Specifying Prior Distributions

Comparing Bayesian and Frequentist Inferences and Interpretations

Exchangeability, Conditional Independence, and Bayesian Inference

Why Bayes?

Conceptualizations of Bayesian Modeling

Summary and Bibliographic Note

Normal Distribution Models

Model with Unknown Mean and Known Variance

Model with Known Mean and Unknown Variance

Model with Unknown Mean and Unknown Variance



Markov Chain Monte Carlo Estimation

Overview of MCMC

Gibbs Sampling

Metropolis Sampling

How MCMC Facilitates Bayesian Modeling

Metropolis–Hastings Sampling

Single-Component-Metropolis or Metropolis-within-Gibbs Sampling

Practical Issues in MCMC

Summary and Bibliographic Note



Background and Notation

Conditional Probability of the Data

Conditionally Conjugate Prior

Complete Model and Posterior Distribution

MCMC Estimation

Example: Regressing Test Scores on Previous Test Scores

Summary and Bibliographic Note



Canonical Bayesian Psychometric Modeling

Three Kinds of DAGs

Canonical Psychometric Model

Bayesian Analysis

Bayesian Methods and Conventional Psychometric Modeling

Summary and Looking Ahead


Classical Test Theory

CTT with Known Measurement Model Parameters and Hyperparameters, Single Observable (Test or Measure)

CTT with Known Measurement Model Parameters and Hyperparameters, Multiple Observables (Tests or Measures)

CTT with Unknown Measurement Model Parameters and Hyperparameters

Summary and Bibliographic Note


Confirmatory Factor Analysis

Conventional Factor Analysis

Bayesian Factor Analysis

Example: Single Latent Variable (Factor) Model

Example: Multiple Latent Variable (Factor) Model

CFA Using Summary Level Statistics

Comparing DAGs and Path Diagrams

A Hierarchical Model Construction Perspective

Flexible Bayesian Modeling

Latent Variable Indeterminacies from a Bayesian Modeling Perspective

Summary and Bibliographic Note


Model Evaluation

Interpretability of the Results

Model Checking

Model Comparison


Item Response Theory

Conventional Item Response Theory Models for Dichotomous Observables

Bayesian Modeling of Item Response Theory Models for Dichotomous Observables

Conventional Item Response Theory Models for Polytomous Observables

Bayesian Modeling of Item Response Theory Models for Polytomous Observables

Multidimensional Item Response Theory Models

Illustrative Applications

Alternative Prior Distributions for Measurement Model Parameters

Latent Response Variable Formulation and Data-Augmented Gibbs Sampling

Summary and Bibliographic Note


Missing Data Modeling

Core Concepts in Missing Data Theory

Inference under Ignorability

Inference under Nonignorability

Multiple Imputation

Latent Variables, Missing Data, Parameters, and Unknowns

Summary and Bibliographic Note


Latent Class Analysis

Conventional Latent Class Analysis

Bayesian Latent Class Analysis

Bayesian Analysis for Dichotomous Latent and Observable Variables

Example: Academic Cheating

Latent Variable Indeterminacies from a Bayesian Modeling Perspective

Summary and Bibliographic Note


Bayesian Networks

Overview of Bayesian Networks

Bayesian Networks as Psychometric Models

Fitting Bayesian Networks

Diagnostic Classification Models

Bayesian Networks in Complex Assessment

Dynamic Bayesian Networks

Summary and Bibliographic Note



Bayes as a Useful Framework

Some Caution in Mechanically (or Unthinkingly) Using Bayesian Approaches

Final Words

Appendix A: Full Conditional Distributions

Appendix B: Probability Distributions



About the Authors

Roy Levy is an associate professor of measurement and statistical analysis in the T. Denny Sanford School of Social and Family Dynamics at Arizona State University. His primary research and teaching interests include methodological developments and applications of psychometrics and statistics to assessment, education, and the social sciences. He has received awards from the President of the United States, Division D of the American Educational Research Association, and the National Council on Measurement in Education.

Robert J. Mislevy is the Frederic M. Lord Chair in Measurement and Statistics at Educational Testing Service. He was previously a professor of measurement and statistics at the University of Maryland and an affiliated professor of second language acquisition and survey methodology. His research applies developments in statistics, technology, and psychology to practical problems in assessment, including the development of multiple-imputation analysis in the National Assessment of Educational Progress. He is a member of the National Academy of Education and has been a president of the Psychometric Society. He has received awards from the National Council on Measurement in Education and Division D of the American Educational Research Association.

About the Series

Chapman & Hall/CRC Statistics in the Social and Behavioral Sciences

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General