Test Equating, Scaling, and Linking: Methods and Practices / Edition 2

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By providing an introduction to test equating which both discusses the most frequently used equating methodologies and covering many of the practical issues involved, this volume expands upon the coverage of the first edition by providing a new chapter on test scaling and a second on test linking.

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Editorial Reviews

From the Publisher
From the reviews:

"Given the perennial debates about academic standards and grade inflation, it is my view that tools such as those described in this book should be adopted much more widely by the academic community than they are at present. This book provides an excellent overview, and I strongly recommend it." Short Book Reviews of the ISI, April 2005

"I highly recommend this book to everybody who has any interest in equating and linking, be they a student, practitioner, or researcher." Psychometrika, 2006

From the reviews of the second edition:

"This is an extended edition of the book … . It provides an introduction to test equating that both discusses frequently used equating methodologies and covers many of practical issues involved. … Each chapter contains a number of exercises. … It can be used as a textbook for advanced graduate students, entry-level professionals and persons preparing to conduct equating, scaling or linking at the first time. For experienced professionals the book will certainly provide information on recent progress in the field and also a long list of useful references." (Marie Huškova, Zentralblatt MATH, Vol. 1059 (10), 2005)

"Equating is a statistical tool which adjusts for differences between tests which are intended to be similar in difficulty and content. This book describes such tools. … This second edition … has new chapters on test scaling and on test linking. … This book provides an excellent overview, and I strongly recommend it." (D.J. Hand, Short Book Reviews, Vol. 25 (1), 2005)

"With the increasing emphasis on assessment and accountability in education, it would benefit those in academia to have some knowledge of the issues, and their complexities, in assessment. This text provides a reasonable introduction to these topics and suggests there are contgributions to be made by statisticians in this area. Statisticians who collaborate or may collaborate with those in the measurement community, or who perhaps seek to develop a new specialty, would be well served by this book." Thomas R. Boucher, JASA, Vol. 102, No. 478, June 2007

"For psychometricians working in testing organizations or as researchers, this book is a must-have. Most practitioners will not need to know all the methods presented in the book, but when you need to know how to carry out a specific method, you will likely find it here with practicial considerations about evaluating how well the method works. From personal experience, this reviewer has found this book to be most useful at the planning stages of a new testing program or an equating research study when various deisgn and method options are considered." Gary Skaggs, Applied Psychological Measurement, Vol. 30, No. 6, November 2006

"...[T]he first edition of this book represented an important contribution to the field. The second edition takes this level of contribution even further. There is nothing better out there than this book for learning about equating, scaling, and linking. If you have not already done so, it may be time to consider updating your personal equating library." Daniel R. Eignor, Journal of Educational Measurement, Vol. 43, No. 2, Summer 2006

"...[I]t is a joy to read the second edition of Kolen and Brennan's book. They did a meticulous job detailing a complex and challenging topic on test equating, scaling, and linking. The book is well written, easy to read, comprehensive, and technically sound." Chris Chiu, Peggy Carr, Ivy Li, Journal of Educational and Behavioral Statistics, Vol. 32, No. 2, June 2007

"This book is part of the Statistics for Social Science and Public Policy series … . It is mainly written for advanced graduate students in applied statistics, beginning professionals in academic testing agencies, and researchers preparing to conduct equating. … ‘for the reader to understand the principles of equating, scaling, and linking; to be able to conduct equating, scaling, and linking; and to interpret the results in reasonable ways.’ … an excellent volume which will last for long time as a standard textbook on equating." (Seock-Ho Kim, Journal of Applied Statistics, Vol. 34 (10), 2008)

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Product Details

Meet the Author

Michael J. Kolen is a Professor of Educational Measurement at the University of Iowa. Robert L. Brennan is E. F. Lindquist Chair in Measurement and Testing and Director of the Center for Advanced Studies in Measurement and Assessment at the University of Iowa. Both authors are acknowledged experts on test equating, scaling and linking, they have authored numerous publications on these subjects, consult regularly on these topics and they have taught many workshops and courses on equating. Both authors have been President of the National Council on Measurement in Education (NCME), have received an NCME award for Career Contributions to Educational Measurement and have received an NCME award for Outstanding Technical Contributions to Educational Measurement following publication of the first edition of this book.Professor Brennan authored Generalizability Theory published by Springer-Verlag.

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Table of Contents

Contents Preface Notation
1 Introduction and Concepts
1.1 Equating and Related Concepts
1.1.1 Test Forms and Test Specifications
1.1.2 Equating
1.1.3 Processes That Are Related to Equating
1.1.4 Equating and Score Scales
1.1.5 Equating and the Test Score Decline of the 1960s and 1970s
1.2 Equating and Scaling in Practice-A Brief Overview of This Book
1.3 Properties of Equating
1.3.1 Symmetry Property
1.3.2 Same Specifications Property
1.3.3 Equity Properties
1.3.4 Observed Score Equating Properties
1.3.5 Group Invariance Property
1.4 Equating Designs
1.4.1 Random Groups Design
1.4.2 Single Group Design
1.4.3 Single Group Design with Counterbalancing
1.4.4 ASVAB Problems with a Single Group Design
1.4.5 Common-Item Nonequivalent Groups Design
1.4.6 NAEP Reading Anomaly-Problems with Common Items
1.5 Error in Estimating Equating Relationships
1.6 Evaluating the Results of Equating
1.7 Testing Situations Considered
1.8 Preview
1.9 Exercises

2 Observed Score Equating Using the Random Groups Design
2.1 Mean Equating
2.2 Linear Equating
2.3 Properties of Mean and Linear Equating
2.4 Comparison of Mean and Linear Equating
2.5 Equipercentile Equating
2.5.1 Graphical Procedures
2.5.2 Analytic Procedures
2.5.3 Properties of Equated Scores in Equipercentile Equating
2.6 Estimating Observed Score Equating Relationships
2.7 Scale Scores
2.7.1 Linear Conversions
2.7.2 Truncation of Linear Conversions
2.7.3 Nonlinear Conversions
2.8 Equating Using Single Group Designs
2.9 Equating Using Alternate Scoring Schemes
2.10 Preview of What Follows
2.11 Exercises

3 Random Groups-Smoothing in Equipercentile Equating
3.1 A Conceptual Statistical Framework for Smoothing
3.2 Properties of Smoothing Methods
3.3 Presmoothing Methods
3.3.1 Polynomial Log-linear Method
3.3.2 Strong True Score Method
3.3.3 Illustrative Example
3.4 Postsmoothing Methods
3.4.1 Illustrative Example
3.5 Practical Issues in Equipercentile Equating
3.5.1 Summary of Smoothing Strategies
3.5.2 Equating Error and Sample Size
3.6 Exercises

4 Nonequivalent Groups-Linear Methods
4.1 Tucker Method
4.1.1 Linear Regression Assumptions
4.1.2 Conditional Variance Assumptions
4.1.3 Intermediate Results
4.1.4 Final Results
4.1.5 Special Cases
4.2 Levine Observed Score Method
4.2.1 Correlational Assumptions
4.2.2 Linear Regression Assumptions
4.2.3 Error Variance Assumptions
4.2.4 Intermediate Results
4.2.5 General Results
4.2.6 Classical Congeneric Model Results
4.3 Levine True Score Method
4.3.1 Results
4.3.2 First-Order Equity
4.4 Illustrative Example and Other Topics
4.4.1 Illustrative Example
4.4.2 Synthetic Population Weights
4.4.3 Mean Equating
4.4.4 Decomposing Observed Di.erences in Means and Variances
4.4.5 Relationships Among Tucker and Levine Equating Methods
4.4.6 Scale Scores
4.5 Appendix Proof that ó2 s (TX) = ã2 1ó2 s (TV ) Under the Classical Congeneric Model
4.6 Exercises

5 Nonequivalent Groups-Equipercentile Methods
5.1 Frequency Estimation Equipercentile Equating
5.1.1 Conditional Distributions
5.1.2 Frequency Estimation Method
5.1.3 Evaluating the Frequency Estimation Assumption
5.1.4 Numerical Example
5.1.5 Estimating the Distributions
5.2 Braun-Holland Linear Method
5.3 Chained Equipercentile Equating
5.4 Illustrative Example
5.4.1 Illustrative Results
5.4.2 Comparison Among Methods
5.4.3 Practical Issues in Equipercentile Equating with Common Items
5.5 Exercises

6 Item Response Theory Methods
6.1 Some Necessary IRT Concepts
6.1.1 Unidimensionality and Local Independence Assumptions
6.1.2 IRT Models
6.1.3 IRT Parameter Estimation
6.2 Transformations of IRT Scales
6.2.1 Transformation Equations
6.2.2 Demonstrating the Appropriateness of Scale Transformations
6.2.3 Expressing A and B Constants
6.2.4 Expressing A and B Constants in Terms of Groups of Items and/or Persons
6.3 Transforming IRT Scales When Parameters Are Estimated
6.3.1 Designs
6.3.2 Mean/Sigma and Mean/Mean Transformation Methods
6.3.3 Characteristic Curve Transformation Methods
6.3.4 Comparisons Among Scale Transformation Methods
6.4 Equating and Scaling
6.5 Equating True Scores
6.5.1 Test Characteristic Curves
6.5.2 True Score Equating Process
6.5.3 The Newton-Raphson Method
6.5.4 Using True Score Equating with Observed Scores
6.6 Equating Observed Scores
6.7 IRT True Score Versus IRT Observed Score Equating
6.8 Illustrative Example
6.8.1 Item Parameter Estimation and Scaling
6.8.2 IRT True Score Equating
6.8.3 IRT Observed Score Equating
6.8.4 Rasch Equating
6.9 Using IRT Calibrated Item Pools
6.9.1 Common-Item Equating to a Calibrated Pool
6.9.2 Item Preequating
6.9.3 Robustness to Violations of IRT Assumptions
6.10 Equating with Polytomous IRT
6.10.1 Polytomous IRT Models for Ordered Responses
6.10.2 Scoring Function, Item Response Function, and Test Characteristic Curve
6.10.3 Parameter Estimation and Scale Transformation with Polytomous IRT Models
6.10.4 True Score Equating
6.10.5 Observed Score Equating
6.10.6 Example using the Graded Response Model
6.11 Practical Issues and Caveat
6.12 Exercises

7 Standard Errors of Equating
7.1 De.nition of Standard Error of Equating
7.2 The Bootstrap
7.2.1 Standard Errors Using the Bootstrap
7.2.2 Standard Errors of Equating
7.2.3 Parametric Bootstrap
7.2.4 Standard Errors of Smoothed Equipercentile Equating
7.2.5 Standard Errors of Scale Scores
7.2.6 Standard Errors of Equating Chains
7.2.7 Mean Standard Error of Equating
7.2.8 Caveat
7.3 The Delta Method
7.3.1 Mean Equating Using Single Group and Random Groups Designs
7.3.2 Linear Equating Using the Random Groups Design
7.3.3 Equipercentile Equating Using the Random Groups Design
7.3.4 Standard Errors for Other Designs
7.3.5 Approximations
7.3.6 Standard Errors for Scale Scores
7.3.7 Standard Errors of Equating Chains
7.3.8 Using Delta Method Standard Errors
7.4 Using Standard Errors in Practice
7.5 Exercises

8 Practical Issues in Equating
8.1 Equating and the Test Development Process
8.1.1 Test Speci.cations
8.1.2 Characteristics of Common-item Sets
8.1.3 Changes in Test Specifications
8.2 Data Collection: Design and Implementation
8.2.1 Choosing Among Equating Designs
8.2.2 Developing Equating Linkage Plans
8.2.3 Examinee Groups Used in Equating
8.2.4 Sample Size Requirements
8.3 Choosing From Among the Statistical Procedures
8.3.1 Equating Criteria in Research Studies
8.3.2 Characteristics of Equating Situations
8.4 Choosing From Among Equating Results
8.4.1 Equating Versus Not Equating
8.4.2 Use of Robustness Checks
8.4.3 Choosing From Among Results in the Random Groups Design
8.4.4 Choosing From Among Results in the Common-Item Nonequivalent Groups Design
8.4.5 Use of Consistency Checks
8.4.6 Equating and Score Scales
8.4.7 Assessing First and Second Order Equity for Scale Scores
8.5 Importance of Standardization Conditions and Quality Control
8.5.1 Test Development
8.5.2 Test Administration and Standardization Conditions
8.5.3 Quality Control
8.5.4 Reequating
8.6 Conditions Conducive to Satisfactory Equating
8.7 Comparability Issues in Special Circumstances
8.7.1 Comparability Issues with Computer-Based Tests
8.7.2 Comparability of Performance Assessments
8.7.3 Score Comparability with Optional Test Sections
8.8 Conclusion
8.9 Exercises

9 Score Scales
9.1 Scaling Perspectives
9.2 Score Transformations
9.3 Incorporating Normative Information
9.3.1 Linear Transformations
9.3.2 Nonlinear Transformations
9.3.3 Example: Normalized Scale Scores
9.3.4 Importance of Norm Group in Setting the Score Scale
9.4 Incorporating Score Precision Information
9.4.1 Rules of Thumb for Number of Distinct Score Points
9.4.2 Linearly Transformed Score Scales with a Given Standard Error of Measurement
9.4.3 Score Scales with Approximately Equal Conditional Standard Errors of Measurement
9.4.4 Example: Incorporating Score Precision
9.4.5 Evaluating Psychometric Properties of Scale Scores
9.4.6 The IRT è-Scale as a Score Scale
9.5 Incorporating Content Information
9.5.1 Item Mapping
9.5.2 Scale Anchoring
9.5.3 Standard Setting
9.5.4 Numerical Example
9.5.5 Practical Usefulness
9.6 Maintaining Score Scales
9.7 Scales for Test Batteries and Composites
9.7.1 Test Batteries
9.7.2 Composite Scores
9.7.3 Maintaining Scales for Batteries and Composites
9.8 Vertical Scaling and Developmental Score Scales
9.8.1 Structure of Batteries
9.8.2 Type of Domain Being Measured
9.8.3 Definition of Growth
9.8.4 Designs for Data Collection for Vertical Scaling
9.8.5 Test Scoring
9.8.6 Hieronymus Statistical Methods
9.8.7 Thurstone Statistical Methods
9.8.8 IRT Statistical Methods
9.8.9 Thurstone Illustrative Example
9.8.10 IRT Illustrative Example
9.8.11 Statistics for Comparing Scaling Results
9.8.12 Some Limitations of Vertically Scaled Tests
9.8.13 Research on Vertical Scaling
9.9 Exercises

10 Linking
10.1 Linking Categorization Schemes and Criteria
10.1.1 Types of Linking
10.1.2 Mislevy/Linn Taxonomy
10.1.3 Degrees of Similarity
10.2 Group Invariance
10.2.1 Statistical Methods Using Observed Scores
10.2.2 Statistics for Overall Group Invariance
10.2.3 Statistics for Pairwise Group Invariance
10.2.4 Example: ACT and ITED Science Tests
10.3 Additional Examples
10.3.1 Extended Time
10.3.2 Test Adaptations and Translated Tests
10.4 Discussion
10.5 Exercises

11 Current and Future Challenges
11.1 Score Scales
11.2 Equating
11.3 Vertical Scaling
11.4 Linking
11.5 Summary References Appendix A: Answers to Exercises Appendix B: Computer Programs Index

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