Rasch Models. Foundations, Recent Developments, and Applications

Transcription

1 Rasch Models Foundations, Recent Developments, and Applications

2 Gerhard H. Fischer Ivo W. Molenaar Editors Rasch Models Foundations, Recent Developments, and Applications With 19 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

3 Gerhard H. Fischer InstiNt rur Psychologie Universitltt Wien Liebiggasse 5 A lolo Vienna Austria Ivo W. Molenaar Vakgroep Statistiek en Meeuheorie Rijksuniversiteit Groningen Grote Kruisstraat TS Groningen The Netherlands librai)' of Congress Cata!oging-in-Publicarion Data Rasch models: foundations, rectnt developments, and applications I Gerhard H. Fischer, Ivo W. Molenaar, editors. p. em. Papers originally presented at a workshop held at the University of Vienna, Feb , Includes bibliolzl'llohical references and index.es. ISBN- 13: I. Psychology-Mathematical models-congresses. 2. Psychometrics Congresses. 3. Rasch, G. (Georg), I Fischer, Gerhard H. II. Molenaar, Ivo W. BF39.R lso'.i'5195--dc Printed on acid-free paper. C 1995 Springer.Vedag New York, Inc. Softcover reprint of the hardcover I st edition 1995 All rights reserved. This work may not be translated or copied in whole or in part without the written pennission of the publisher (Springer.Vedag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief ex.cerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, tmderruuks, etc., in this publication, even if the former are nol especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Mert:handise Marks Act, may accordingly be used freely by anyone. Production managed by Laura Carbon; manufacturing supervised by Jacqui Ashri. camera-ready copy prepared by the editors. Printed and bound by Edwards Brothers, Inc., Ann Arbor, MI ISBN-13: e-isbn- 13: DOl: /

4 Preface Thirty-two years after the publication of the legendary 'Rasch book' (Rasch, 1960), the rich literature on the Rasch model and its extensions was scattered in journals and many less accessible sources, including 'grey' literature. When asked by students or junior researchers for references to the Rasch model, it was a typical reaction on the part of the editors to state that it was difficult to name one, or just a few; actually, only a whole list of references differing in notation and level of formal abstraction seemed to meet the request in most cases. Therefore, in 1992 the editors decided to invite a number of outstanding authors in the field of Rasch modeling to contribute to a book presenting the current state of knowledge about Rasch models. The aim was not just to collect a number of papers on the subject, rather to produce a well-organized monograph. To this end, a workshop was held in Vienna from 25 to 27 February 1993 in which, after a process of mutual reviewing, drafts of all chapters were read and discussed by all authors, leading to a more systematic organization of the topics treated in unified notation and terminology. (The workshop was sponsored by the University of Vienna; here, the editors would like to express their thanks, in the name of all contributors, both for the financial support and for the hospitality granted.) In spite of the attempts to present a homogeneous monograph, however, it is natural that some individual variations of views and styles of presentation remain, giving expression to the fact that researchers are individuals, and research is a lively subject. It was clear from the beginning that the volume would be published in English, the language of international scientific communication. Since none of the editors and authors is a native English speaker, in spite of all the efforts to produce a correct English text there is no hope that the result will be perfect. We apologize for the lack of idiomatic English, or of elegance of style, wherever it should be missed by the native English reader. This is the price to be paid for the more important goal of reaching the interested audience world-wide. The scholars of item response theory will decide whether the authors and editors of this book should be praised or criticized. Here, however, we would like to mention those who have contributed immensely to the present book project and who might otherwise be forgotten: Ms. Karin Waldherr has typed five and corrected many of the 21 chapters expertly in TEX, and has helped with the list of references and indexes, to say nothing of her many other organizational duties and of her extensive correspondence with the authors. She even pointed out to the editors many errors of all kinds, thus helping to improve the final manuscripts. We would like to thank her heartily for her invaluably loyal service. Thanks also go to Ms. Heidi Glatzmeier, who has spent much time on revisions and authorial corrections of many chapters and on inserting index labels in the

5 vi Preface UTEX files; and to Mag. Elisabeth Seliger for proofreading several chapters. Without the assistance of these ladies it would have been impossible to produce the complicated typescript from which the book is reproduced. Finally, thanks go to the Department of Psychology of the University of Vienna for making the 'ladypower' available to the project. Gerhard H. Fischer Vienna December 12, 1994 Ivo W. Molenaar Groningen

6 Contents Preface Part I: The Dichotomous Rasch Model 1. Some Background for Item Response Theory and the Rasch Model 1. W. Molenaar v 3 2. Derivations of the Rasch Model Estimation of Item Parameters W. Molenaar 4. On Person Parameter Estimation in the Dichotomous Rasch 53 Model H. Hoijtink and A. Boomsma 5. Testing the Rasch Model C. A. W. Glas and N. D. Verhelst 6. The Assessment of Person Fit K. C. Klauer 7. Test Construction from Item Banks E. Timminga and J. J. Adema Part II: Extensions of the Dichotomous Rasch Model 8. The Linear Logistic Test Model 9. Linear Logistic Models for Change 10. Dynamic Generalizations of the Rasch Model N. D. Verhelst and C. A. W. Glas 11. Linear and Repeated Measures Models for the Person Parameters H. Hoijtink

7 viii Contents 12. The One Parameter Logistic Model 215 N. D. Verhelst and C. A. W. Glas 13. Linear Logistic Latent Class Analysis and the Rasch Model 239 A. K. Formann 14. Mixture Distribution Rasch Models 257 J. Rost and M. von Davier Part III: Polytomous Rasch Models and their Extensions 15. Polytomous Rasch Models and their Estimation E. B. Andersen 16. The Derivation of Polytomous Rasch Models 17. The Polytomous Rasch Model within the Class of Generalized Linear Symmetry Models H. Kelderman 18. Tests of Fit for Polytomous Rasch Models C. A. W. Glas and N. D. Verhelst 19. Extended Rating Scale and Partial Credit Models for Assessing Change and I. Ponocny 20. Polytomous Mixed Rasch Models M. von Davier and J. Rost In Retrospect 21. What Georg Rasch Would Have Thought about this Book 383 E. B. Andersen References Author Index Subject Index Abbreviations