The Kernel Method of Test Equating and Notation and Notation The Kernel Method of Test Equating: Theory Data Collection Designs Kernel Equating: Overview Pre smoothing and Estimation of r and s Kernel Equating: Continuization and Equating Kernel Equating: The SEE and the SEED Kernel Equating versus Other Equating Methods The Kernel Method of Test Equating: Applications The Equivalent Groups Design The Single Group Design The Counterbalanced Design The NEAT Design: Chain Equating The NEAT Design: Post Stratification Equating. 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.