Statistics for Social and Behavioral Sciences - Böcker

Kernel Equating (KE) is a powerful, modern and unified approach to test equating. It is based on a flexible family of equipercentile-like equating functions and contains the linear equating function as a special case. Any equipercentile equating method has five steps or parts. They are: 1) pre-smoothing; 2) estimation of the score-probabilities on the target population; 3) continuization; 4) computing and diagnosing the equating function; 5) computing the standard error of equating and related accuracy measures. KE brings these steps together in an organized whole rather than treating them as disparate problems.KE exploits pre-smoothing by fitting log-linear models to score data, and incorporates it into step 5) above. KE provides new tools for diagnosing a given equating function, and for comparing two or more equating functions in order to choose between them. In this book, KE is applied to the four major equating designs and to both Chain Equating and Post-Stratification Equating for the Non-Equivalent groups with Anchor Test Design.This book will be an important reference for several groups: (a) Statisticians and others interested in the theory behind equating methods and the use of model-based statistical methods for data smoothing in applied work; (b) Practitioners who need to equate tests?including those with these responsibilities in testing companies, state testing agencies and school districts; and (c) Instructors in psychometric and measurement programs. The authors assume some familiarity with linear and equipercentile test equating, and with matrix algebra.Alina von Davier is an Associate Research Scientist in the Center for Statistical Theory and Practice, at Educational Testing Service. She has been a research collaborator at the Universities of Trier, Magdeburg, and Kiel, an assistant professor at the Politechnical University of Bucharest and a research scientist at the Institute for Psychology in Bucharest.Paul Holland holds the Frederic M. Lord Chair in Measurement and Statistics at Educational Testing Service. He held faculty positions in the Graduate School of Education, University of California, Berkeley and the Harvard Department of Statistics. He is a Fellow of the American Statistical Association, the Institute of Mathematical Statistics, and the American Association for the Advancement of Science. He is an elected Member of the International Statistical Institute and a past president of the Psychometric society. He was awarded the (AERA/ACT) E. F.Lindquist Award, in 2000, and was designated a National Associate of the National Academies of Science in 2002. Dorothy Thayer currently is a consultant in the Center of Statistical Theory and Practice, at Educational Testing Service. Her research interests include computational and statistical methodology, empirical Bayes techniques, missing data procedures and exploratory data analysis techniques.

Marginal Models for Dependent, Clustered, and Longitudinal Categorical Data provides a comprehensive overview of the basic principles of marginal modeling and offers a wide range of possible applications. Marginal models are often the best choice for answering important research questions when dependent observations are involved, as the many real world examples in this book show.In the social, behavioral, educational, economic, and biomedical sciences, data are often collected in ways that introduce dependencies in the observations to be compared. For example, the same respondents are interviewed at several occasions, several members of networks or groups are interviewed within the same survey, or, within families, both children and parents are investigated. Statistical methods that take the dependencies in the data into account must then be used, e.g., when observations at time one and time two are compared in longitudinal studies. At present, researchers almost automatically turn to multi-level models or to GEE estimation to deal with these dependencies. Despite the enormous potential and applicability of these recent developments, they require restrictive assumptions on the nature of the dependencies in the data. The marginal models of this book provide another way of dealing with these dependencies, without the need for such assumptions, and can be used to answer research questions directly at the intended marginal level. The maximum likelihood method, with its attractive statistical properties, is used for fitting the models.This book has mainly been written with applied researchers in mind. It includes many real world examples, explains the types of research questions for which marginal modeling is useful, and provides a detailed description of how to apply marginal models for a great diversity of research questions. All these examples are presented on the book's website (www.cmm.st), along with user friendly programs.

This book provides an overview of the developments in the area of Bayesian evaluation of informative hypotheses that took place since the publication of the ?rst paper on this topic in 2001 [Hoijtink, H. Con?rmatory latent class analysis, model selection using Bayes factors and (pseudo) likelihood ratio statistics. Multivariate Behavioral Research, 36, 563–588]. The current state of a?airs was presented and discussed by the authors of this book during a workshop in Utrecht in June 2007. Here we would like to thank all authors for their participation, ideas, and contributions. We would also like to thank Sophie van der Zee for her editorial e?orts during the construction of this book. Another word of thanks is due to John Kimmel of Springer for his con?dence in the editors and authors. Finally, we would like to thank the Netherlands Organization for Scienti?c Research (NWO) whose VICI grant (453-05-002) awarded to the ?rst author enabled the organization of the workshop, the writing of this book, and continuation of the research with respect to Bayesian evaluation of informative hypotheses.

This book begins with a reflection on the history of test design-the core activity of all educational and psychological testing. It then presents a standard language for modeling test design problems as instances of multi-objective constrained optimization. The main portion of the book discusses test design models for a large variety of problems from the daily practice of testing, and illustrates their use with the help of numerous empirical examples. The presentation includes models for the assembly of tests to an absolute or relative target for their information functions, classical test assembly, test equating problems, item matching, test splitting, simultaneous assembly of multiple tests, tests with item sets, multidimensional tests, and adaptive test assembly. Two separate chapters are devoted to the questions of how to design item banks for optimal support of programs with fixed and adaptive tests. Linear Models for Optimal Test Design, which does not require any specific mathematical background, has been written to be a helpful resource on the desk of any test specialist.

This volume gathers together a set of extensions of the Rasch model, one of the most prominent models for measurement in educational research and - cial science developed by Danish mathematician Georg Rasch. The idea for this volume emerged during a meeting of the Psychometric Society in M- terey, CA. At that meeting, friends and colleagues discussed news about the impending retirement of Dr. Jurgen ¨ Rost, an important innovator and m- tor in this ?eld. To recognize Jurgen’s ¨ contributions, we decided to produce a collection of research on extending the Rasch model as well as embedding the Rasch model in more complex statistical models, an area that is receiving broad interest in many ?elds of social sciences at the current time. This collection contains 22 chapters by recognized international experts in the?eld.Thecontributionscovertopicsrangingfromgeneralmodelextensions to application in ?elds as diverse as cognition, personality, organizational and sports psychology, and health sciences and education. The Rasch model is designed for categorical data, often collected as ex- inees’ responses to multiple tasks such as cognitive items from psychological tests or from educational assessments. The Rasch model’s elegant mathema- cal form is suitable for extensions that allow for greater ?exibility in handling complex samples of examinees and collections of tasks from di?erent domains. In these extensions, the Rasch model is enhanced by additional structural - ements that account either for di?erences between diverse populations or for di?erences among observed variables.

He is a past president of the International Association of Survey Statisticians and a past chair of the Survey Research Methods Section of the American Statistical Association.

This edited volume gives a new and integrated introduction to item re­ sponse models (predominantly used in measurement applications in psy­ chology, education, and other social science areas) from the viewpoint of the statistical theory of generalized linear and nonlinear mixed models.

In their preface to the second edition of Test Equating, Scaling, and Linking, Mike Kolen and Bob Brennan (2004) made the following observation: “Prior to 1980, the subject of equating was ignored by most people in the measurement community except for psychometricians, who had responsibility for equating” (p. vii). The authors went on to say that considerably more attention is now paid to equating, indeed to all forms of linkages between tests, and that this increased attention can be attributed to several factors: 1. An increase in the number and variety of testing programs that use multiple forms and the recognition among professionals that these multiple forms need to be linked. 2. Test developers and publishers, in response to critics, often refer to the role of linking in reporting scores. 3. The accountability movement and fairness issues related to assessment have become much more visible. Those of us who work in this field know that ensuring comparability of scores is not an easy thing to do. Nonetheless, our customers—the te- takers and score users—either assume that scores on different forms of an assessment can be used interchangeably or, like the critics above, ask us to justify our comparability assumptions. And they are right to do this. After all, the test scores that we provide have an impact on decisions that affect people’s choices and their future plans. From an ethical point of view, we are obligated to get it right.

"Introduction to Applied Bayesian Statistics and Estimation for Social Scientists" covers the complete process of Bayesian statistical analysis in great detail from the development of a model through the process of making statistical inference. The key feature of this book is that it covers models that are most commonly used in social science research - including the linear regression model, generalized linear models, hierarchical models, and multivariate regression models - and it thoroughly develops each real-data example in painstaking detail.The first part of the book provides a detailed introduction to mathematical statistics and the Bayesian approach to statistics, as well as a thorough explanation of the rationale for using simulation methods to construct summaries of posterior distributions. Markov chain Monte Carlo (MCMC) methods - including the Gibbs sampler and the Metropolis-Hastings algorithm - are then introduced as general methods for simulating samples from distributions. Extensive discussion of programming MCMC algorithms, monitoring their performance, and improving them is provided before turning to the larger examples involving real social science models and data.

The arrival of the computer in educational and psychological testing has led to the current popularity of adaptive testing---a testing format in which the computer uses statistical information about the test items to automatically adapt their selection to a real-time update of the test taker’s ability estimate.

These methods include ways to determine the number of dimensions required to adequately model data, procedures for estimating model parameters, ways to define the space for a MIRT model, and procedures for transforming calibrations from different samples to put them in the same space.

Prove It With Figures

In 1972 a monograph by Cronbach, Gleser, Nanda, and Rajaratnam was published entitled The Dependability of Behavioral Measurements. That book incorporated, systematized, and extended their previous research into what came to be called generalizability theory, which liberalizes classical test theory, in part through the application of analysis of variance proce­ dures that focus on variance components. Generalizability theory is perhaps the most broadly defined measurement model currently in existence, and the Cronbach et al. (1972) treatment of the theory represents a major con­ tribution to psychometrics. However, as Cronbach et al. (1972, p. 3) state, their book is "complexly organized and by no means simple to follow" and, of course, it is nearly 30 years old. In 1983, ACT, Inc. published my monograph entitled Elements of Gen­ eralizability Theory, with a slightly revised version appearing in 1992. That treatment is considerably less comprehensive than Cronbach et al. (1972) but still detailed enough to convey much ofthe richness of the theory and to facilitate its application. However, the 1983/1992 monograph is essen­ tially two decades old, it does not cover multivariate generalizability theory in depth, and it does not incorporate recent developments in statistics that bear upon the estimation of variance components. Also, of course, there have been numerous developments in generalizability theory in the last 20 years.

The goal of this book is to emphasize the formal statistical features of the practice of equating, linking, and scaling. The book encourages the view and discusses the quality of the equating results from the statistical perspective (new models, robustness, fit, testing hypotheses, statistical monitoring) as opposed to placing the focus on the policy and the implications, which although very important, represent a different side of the equating practice. The book contributes to establishing “equating” as a theoretical field, a view that has not been offered often before. The tradition in the practice of equating has been to present the knowledge and skills needed as a craft, which implies that only with years of experience under the guidance of a knowledgeable practitioner could one acquire the required skills. This book challenges this view by indicating how a good equating framework, a sound understanding of the assumptions that underlie the psychometric models, and the use of statistical tests and statistical process control tools can help the practitioner navigate the difficult decisions in choosing the final equating function.  This book provides a valuable reference for several groups: (a) statisticians and psychometricians interested in the theory behind equating methods, in the use of model-based statistical methods for data smoothing, and in the evaluation of the equating results in applied work; (b) practitioners who need to equate tests, including those with these responsibilities in testing companies, state testing agencies, and school districts; and (c) instructors in psychometric, measurement, and psychology programs.

Ordinal Data Modeling is a comprehensive treatment of ordinal data models from both likelihood and Bayesian perspectives. Written for graduate students and researchers in the statistical and social sciences, this book describes a coherent framework for understanding binary and ordinal regression models, item response models, graded response models, and ROC analyses, and for exposing the close connection between these models. A unique feature of this text is its emphasis on applications. All models developed in the book are motivated by real datasets, and considerable attention is devoted to the description of diagnostic plots and residual analyses. Software and datasets used for all analyses described in the text are available on websites listed in the preface.

This book emphasizes the practical side of computer-based testing and presents suggestions, information, and ideas for its actual implementation. Most current references on computer-based testing either cover the topics from a theoretical point of view or present numerous examples of different and often disparate topics that have been assessed using computer-based testing methods.This book is different. It assumes that the reader may have little or no experience in actually implementing a computer-based testing program, even if knowledgeable about the psychometrics and measurement principles that govern it. It provides guidance and information that can be used to make wise, informed decisions, including the type of computer-based test that should be administered, possible cost to examinees, examinee reactions to the test, scoring issues, computer mode effects, the development of innovative item types, the time and cost needed to create and maintain an item pool, and many more.

Public policy research frequently requires using statistics to address important and complex problems in a pragmatic way. In this volume, we describe the rich analytical techniques and substantive applications that typify how statistical thinking has been applied at the RAND Corporation over the past two decades. Our philosophy is to be critical yet constructive. We strive to bridge the gap between being a discerning consumer of analyses, and an applied statistician who uses the available information to advise the policy-maker in the face of substantial uncertainty. Case studies of public policy problems are useful for teaching because they are familiar. Almost everyone knows something abut health insurance, global warming, and capital punishment, to name but a few of the applications covered in this casebook. Each case study has a common format that describes the policy questions, the statistical questions, and the successful and the unsuccessful analytic strategies. Readers should be familiar with basic statistical concepts including sampling and regression. This casebook is designed for statistics courses in areas ranging from economics to health policy to the law at both the advanced undergraduate and graduate levels. Empirical researchers and policy-makers should also find this casebook informative. Sally C. Morton is head of the RAND Statistics Group, a senior statistician in the RAND Health Program, and on the faculty of the UCLA School of Public Health. Her research interests include meta- analysis, the sampling of vulnerable populations, and the use of statistics in health policy. She is an associate editor of the Journal of the American Statistical Association and Statistical Science, has served on the Journal of Computational and Graphical Statistics editorial board, and is a member of the National Academy of Sciences

In social science research, differences among groups or changes over time are a common focus of study. While means and variances are typically the basis for statistical methods used in this research, the underlying social theory often implies properties of distributions that are not well captured by these summary measures. Examples include the current controversies regarding growing inequality in earnings, racial diferences in test scores, socio-economic correlates of birth outcomes, and the impact of smoking on survival and health. The distributional differences that animate the debates in these fields are complex. They comprise the usual mean-shifts and changes in variance, but also more subtle comparisons of changes in the upper and lower tails of distributions. Survey and census data on such attributes contain a wealth of distributional information, but traditional methods of data analysis leave much of this information untapped. In this monograph, we present methods for full comparative distributional analysis.The methods are based on the relative distribution, a nonparametric complete summary of the information required for scale--invariant comparisons between two distributions. The relative distribution provides a general integrated framework for analysis. It offers a graphical component that simplifies exploratory data analysis and display, a statistically valid basis for the development of hypothesis-driven summary measures, and the potential for decomposition that enables one to examine complex hypotheses regarding the origins of distributional changes within and between groups. The monograph is written for data analysts and those interested in measurement, and it can serve as a textbook for a course on distributional methods. The presentation is application oriented,

Expert testimony relying on scientific and other specialized evidence has come under increased scrutiny by the legal system. A trilogy of recent U.S. Supreme Court cases has assigned judges the task of assessing the relevance and reliability of proposed expert testimony. In conjunction with the Federal judiciary, the American Association for the Advancement of Science has initiated a project to provide judges indicating a need with their own expert. This concern with the proper interpretation of scientific evidence, especially that of a probabilistic nature, has also occurred in England, Australia and in several European countries. Statistical Science in the Courtroom is a collection of articles written by statisticians and legal scholars who have been concerned with problems arising in the use of statistical evidence. A number of articles describe DNA evidence and the difficulties of properly calculating the probability that a random individual's profile would "match" that of the evidence as well as the proper way to intrepret the result. In addition to the technical issues, several authors tell about their experiences in court. A few have become disenchanted with their involvement and describe the events that led them to devote less time to this application. Other articles describe the role of statistical evidence in cases concerning discrimination against minorities, product liability, environmental regulation, the appropriateness and fairness of sentences and how being involved in legal statistics has raised interesting statistical problems requiring further research.

As with previous editions, the cases and case studies reflect a broad variety of legal subjects, including antidiscrimination, mass torts, taxation, school finance, identification evidence, preventive detention, handwriting disputes, voting, environmental protection, antitrust, sampling for insurance audits, and the death penalty.

The modeling of item response data is governed by item response theory, also referred to as modern test theory. The usefulness of the Bayesian methodology is illustrated by discussing and applying a range of Bayesian item response models.

Marginal Models for Dependent, Clustered, and Longitudinal Categorical Data provides a comprehensive overview of the basic principles of marginal modeling and offers a wide range of possible applications. Marginal models are often the best choice for answering important research questions when dependent observations are involved, as the many real world examples in this book show.In the social, behavioral, educational, economic, and biomedical sciences, data are often collected in ways that introduce dependencies in the observations to be compared. For example, the same respondents are interviewed at several occasions, several members of networks or groups are interviewed within the same survey, or, within families, both children and parents are investigated. Statistical methods that take the dependencies in the data into account must then be used, e.g., when observations at time one and time two are compared in longitudinal studies. At present, researchers almost automatically turn to multi-level models or to GEE estimation to deal with these dependencies. Despite the enormous potential and applicability of these recent developments, they require restrictive assumptions on the nature of the dependencies in the data. The marginal models of this book provide another way of dealing with these dependencies, without the need for such assumptions, and can be used to answer research questions directly at the intended marginal level. The maximum likelihood method, with its attractive statistical properties, is used for fitting the models.This book has mainly been written with applied researchers in mind. It includes many real world examples, explains the types of research questions for which marginal modeling is useful, and provides a detailed description of how to apply marginal models for a great diversity of research questions. All these examples are presented on the book's website (www.cmm.st), along with user friendly programs.

This book provides an overview of the developments in the area of Bayesian evaluation of informative hypotheses that took place since the publication of the ?rst paper on this topic in 2001 [Hoijtink, H. Con?rmatory latent class analysis, model selection using Bayes factors and (pseudo) likelihood ratio statistics. Multivariate Behavioral Research, 36, 563–588]. The current state of a?airs was presented and discussed by the authors of this book during a workshop in Utrecht in June 2007. Here we would like to thank all authors for their participation, ideas, and contributions. We would also like to thank Sophie van der Zee for her editorial e?orts during the construction of this book. Another word of thanks is due to John Kimmel of Springer for his con?dence in the editors and authors. Finally, we would like to thank the Netherlands Organization for Scienti?c Research (NWO) whose VICI grant (453-05-002) awarded to the ?rst author enabled the organization of the workshop, the writing of this book, and continuation of the research with respect to Bayesian evaluation of informative hypotheses.

Over my nearly forty years of teaching and conducting research in the ?eld of psychometric methods, I have seen a number of major technical advances that respond to pressing educational and psychological measu- mentproblems. Thedevelopmentofcriterion-referencedassessmentwasthe ?rst, beginning in the late 1960s with the important work of Robert Glaser and Jim Popham, in response to the need for assessments that considered candidate performance in relation to a well-de?ned body of knowledge and skills rather than in relation to a norm group. The development of criterion-referenced testing methodology with a focus on decision-theoretic concepts and methods, content validity, standard-setting, and the recog- tionofthemeritsofbothcriterion-norm-referencedandcriterion-referenced assessments has tremendously in?uenced current test theory and testing . The second major advance was the introduction of item response-theory (IRT) and associated models and their applications to replace classical test theory (CTT) and related practices. Beginning slowly in the 1940s and 1950s with the pioneering work of Frederic Lord, Allan Birnbaum, and GeorgRasch,bythe1970sthemeasurementjournalswerefullofimportant research studies describing new IRT models, technical advances in model parameter estimation and model ?t, and research on applications of IRT models to equating, test development, the detection of potentially biased test items, and adaptive testing. The overall goal has been to improve and expand measurement practices by overcoming several shortcomings of cl- sicaltesttheory:dependenceoftest-itemstatisticsandreliabilityestimates on examinee samples, dependence of examinee true score estimates on the particular choices of test items, and the limitation in CTT of modeling ex-viii Foreword aminee performance at the test level rather than at the item level.

This volume gathers together a set of extensions of the Rasch model, one of the most prominent models for measurement in educational research and - cial science developed by Danish mathematician Georg Rasch. The idea for this volume emerged during a meeting of the Psychometric Society in M- terey, CA. At that meeting, friends and colleagues discussed news about the impending retirement of Dr. Jurgen ¨ Rost, an important innovator and m- tor in this ?eld. To recognize Jurgen’s ¨ contributions, we decided to produce a collection of research on extending the Rasch model as well as embedding the Rasch model in more complex statistical models, an area that is receiving broad interest in many ?elds of social sciences at the current time. This collection contains 22 chapters by recognized international experts in the?eld.Thecontributionscovertopicsrangingfromgeneralmodelextensions to application in ?elds as diverse as cognition, personality, organizational and sports psychology, and health sciences and education. The Rasch model is designed for categorical data, often collected as ex- inees’ responses to multiple tasks such as cognitive items from psychological tests or from educational assessments. The Rasch model’s elegant mathema- cal form is suitable for extensions that allow for greater ?exibility in handling complex samples of examinees and collections of tasks from di?erent domains. In these extensions, the Rasch model is enhanced by additional structural - ements that account either for di?erences between diverse populations or for di?erences among observed variables.

We live in the information age. Statistical surveys are used every day to determine or evaluate public policy and to make important business decisions. Correct methods for computing the precision of the survey data and for making inferences to the target population are absolutely essential to sound decision making. Now in its second edition, Introduction to Variance Estimation has for more than twenty years provided the definitive account of the theory and methods for correct precision calculations and inference, including examples of modern, complex surveys in which the methods have been used successfully.The book provides instruction on the methods that are vital to data-driven decision making in business, government, and academe. It will appeal to survey statisticians and other scientists engaged in the planning and conduct of survey research, and to those analyzing survey data and charged with extracting compelling information from such data. It will appeal to graduate students and university faculty who are focused on the development of new theory and methods and on the evaluation of alternative methods. Software developers concerned with creating the computer tools necessary to enable sound decision-making will find it essential.Prerequisites include knowledge of the theory and methods of mathematical statistics and graduate coursework in survey statistics. Practical experience with real surveys is a plus and may be traded off against a portion of the requirement for graduate coursework.This second edition reflects shifts in the theory and practice of sample surveys that have occurred since the content of the first edition solidified in the early 1980’s. Additional replication type methods appeared during this period and have featured prominently in journal publications. Reflecting these developments, the second edition now includes a new major chapter on the bootstrap method of variance estimation. This edition also includes extensive new material on Taylor series methods, especially as they apply to newer methods of analysis such as logistic regression or the generalized regression estimator. An introductory section on survey weighting has been added. Sections on Hadamard matrices and computer software have been substantially scaled back. Fresh material on these topics is now readily available on the Internet or from commercial sources.Kirk Wolter is a Senior Fellow at NORC, Director of the Center for Excellency in Survey Research, and Professor in the Department of Statistics, University of Chicago. He is a Fellow of the American Statistical Association and a Member of the International Statistical Institute. He is a past president of the International Association of Survey Statisticians and a past chair of the Survey Research Methods Section of the American Statistical Association. During the last 35 years, he has participated in the planning, execution, and analysis of large-scale complex surveys andhas provided instruction in survey statistics both in America and around the world.

This edited volume gives a new and integrated introduction to item re­ sponse models (predominantly used in measurement applications in psy­ chology, education, and other social science areas) from the viewpoint of the statistical theory of generalized linear and nonlinear mixed models. Moreover, this new framework aHows the domain of item response mod­ els to be co-ordinated and broadened to emphasize their explanatory uses beyond their standard descriptive uses. The basic explanatory principle is that item responses can be modeled as a function of predictors of various kinds. The predictors can be (a) char­ acteristics of items, of persons, and of combinations of persons and items; they can be (b) observed or latent (of either items or persons); and they can be (c) latent continuous or latent categorical. Thus, a broad range of models can be generated, including a wide range of extant item response models as weH as some new ones. Within this range, models with explana­ tory predictors are given special attention, but we also discuss descriptive models. Note that the 'item responses' that we are referring to are not just the traditional 'test data,' but are broadly conceived as categorical data from a repeated observations design. Hence, data from studies with repeated-observations experimental designs, or with longitudinal designs, mayaIso be modeled. The intended audience for this volume is rather broad.

In their preface to the second edition of Test Equating, Scaling, and Linking, Mike Kolen and Bob Brennan (2004) made the following observation: “Prior to 1980, the subject of equating was ignored by most people in the measurement community except for psychometricians, who had responsibility for equating” (p. vii). The authors went on to say that considerably more attention is now paid to equating, indeed to all forms of linkages between tests, and that this increased attention can be attributed to several factors: 1. An increase in the number and variety of testing programs that use multiple forms and the recognition among professionals that these multiple forms need to be linked. 2. Test developers and publishers, in response to critics, often refer to the role of linking in reporting scores. 3. The accountability movement and fairness issues related to assessment have become much more visible. Those of us who work in this field know that ensuring comparability of scores is not an easy thing to do. Nonetheless, our customers—the te- takers and score users—either assume that scores on different forms of an assessment can be used interchangeably or, like the critics above, ask us to justify our comparability assumptions. And they are right to do this. After all, the test scores that we provide have an impact on decisions that affect people’s choices and their future plans. From an ethical point of view, we are obligated to get it right.

"Introduction to Applied Bayesian Statistics and Estimation for Social Scientists" covers the complete process of Bayesian statistical analysis in great detail from the development of a model through the process of making statistical inference. The key feature of this book is that it covers models that are most commonly used in social science research - including the linear regression model, generalized linear models, hierarchical models, and multivariate regression models - and it thoroughly develops each real-data example in painstaking detail.The first part of the book provides a detailed introduction to mathematical statistics and the Bayesian approach to statistics, as well as a thorough explanation of the rationale for using simulation methods to construct summaries of posterior distributions. Markov chain Monte Carlo (MCMC) methods - including the Gibbs sampler and the Metropolis-Hastings algorithm - are then introduced as general methods for simulating samples from distributions. Extensive discussion of programming MCMC algorithms, monitoring their performance, and improving them is provided before turning to the larger examples involving real social science models and data.