Probability Models and Statistical Analyses for Ranking Data PDF
Preview Probability Models and Statistical Analyses for Ranking Data
Lecture Notes in Statistics 80 Edited by 1. Berger, S. Fienberg, J. Gani, K. Krickeberg, 1. OIkin, and B. Singer Michael A. Fligner Joseph S. Verducci (Eds.) Probability Models and Statistical Analyses for Ranking Data Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Michael A. Fligner Joseph S. Verducci Department of Statistics The Ohio State University Columbus, OR 43210 USA Mathematics Subject Classifications: 62-06, 60-06 Library of Congress Cataloging-in-Publication Data Probability models and statistical analyses for ranking data I Michael A. Aigner, Joseph S. Verducci (eds.). p. em. - (Lecture notes in statistics ; SO) Includes bibliographical references. 1. Ranking and selection (Statistics }-Congresses. I. Aigner, Michael A. II. Verducci, Joseph S. ill. Series: Lecture notes in statistics (Springer-Verlag); v. SO. QA27S.75.P76 1992 519.5-dc20 92-30305 Printed on acid-free paper. © 1993 Springer-Verlag New York, Inc. 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To our parents Preface In June of 1990, a conference was held on Probablity Models and Statisti- cal Analyses for Ranking Data, under the joint auspices of the American Mathematical Society, the Institute for Mathematical Statistics, and the Society of Industrial and Applied Mathematicians. The conference took place at the University of Massachusetts, Amherst, and was attended by 36 participants, including statisticians, mathematicians, psychologists and sociologists from the United States, Canada, Israel, Italy, and The Nether- lands. There were 18 presentations on a wide variety of topics involving ranking data. This volume is a collection of 14 of these presentations, as well as 5 miscellaneous papers that were contributed by conference participants. We would like to thank Carole Kohanski, summer program coordinator for the American Mathematical Society, for her assistance in arranging the conference; M. Steigerwald for preparing the manuscripts for publication; Martin Gilchrist at Springer-Verlag for editorial advice; and Persi Diaconis for contributing the Foreword. Special thanks go to the anonymous referees for their careful readings and constructive comments. Finally, we thank the National Science Foundation for their sponsorship of the AMS-IMS-SIAM Joint Summer Programs. Contents Preface vii Conference Participants xiii Foreword xvii 1 Ranking Models with Item Covariates 1 D. E. Critchlow and M. A. Fligner 1.1 Introduction............... 1 1.2 Basic Ranking Models and Their Parameters 2 1.3 Ranking Models with Covariates 8 1.4 Estimation 9 1.5 Example. 11 1.6 Discussion. 14 1.7 Appendix . 15 1.8 References. 16 2 Nonparametric Methods of Ranking from Paired Comparisons 20 H. A. David and D. M. Andrews 2.1 Introduction and Literature Review ............. 20 2.2 The Proposed Method of Scoring . . . . . . . . . . . . . .. 25 2.3 Distribution Theory and Tests of Significance for O:~j = Pij. 29 2.4 Ranking Methods. . 31 2.5 Numerical Example. 31 2.6 References...... 35 3 On the Babington Smith Class of Models for Rankings 37 H. Joe and J. S. Verducci 3.1 Introduction...................... 37 3.2 Alternative Parametrizations and Related Models. 39 3.3 Stochastic Transitivity and Item Preference 42 3.4 Examples and Data Analysis 47 3.5 References................... 51 x Contents 4 Latent Structure Models for Ranking Data 53 M. A. Croon and R. Luijkx 4.1 Introduction......................... 53 4.2 Latent Class Analyses Based on the Bradley-Terry-Luce Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3 Latent Class Analyses Based on a Quasi-independence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Models that Allow for Association Between Choices within the Classes 65 4.5 References ..................... . 73 5 Modelling and Analysing Paired Ranking Data 75 P. D. Feigin 5.1 Introduction ............ . 75 5.2 Two Models ............ . 76 5.3 Estimation and Hypothesis Testing 80 5.4 Analysis of Simulated Data Sets 83 5.5 Analysis of Rogers Data 88 5.6 References ............ . 90 6 Maximum Likelihood Estimation in Mallows's Model U sing Partially Ranked Data 92 L. A. Beckett 6.1 Introduction .......................... . 92 6.2 Notation ............................ . 93 6.3 Maximum Likelihood Estimation Using the EM Algorithm. 95 6.4 Example. 100 6.5 Discussion 104 6.6 References 106 7 Extensions of Mallows' </J Model 108 L. Chung and J. I. Marden 7.1 Introduction ..... . 108 7.2 The General Model .... . 110 7.3 Ties, Partial Rankings .. . 114 7.4 Example: Word Association 117 7.5 Example: APA Voting . 119 7.6 Example: ANOVA ... 128 7.7 Discussion of Contrasts 132 7.8 Appendix 135 7.9 References ....... . 137 Contents xi 8 Rank Correlations and the Analysis of Rank-Based Experimental Designs 140 M. Alvo and P. Cabilio 8.1 Introduction ............... . 140 8.2 Distance Based Measures of Correlation 141 8.3 The Problem of m Rankings ...... . 143 8.4 The Two Sample Problem ....... . 146 8.5 The Problem of m Rankings for a Balanced Incomplete Block Design .......................... . 147 8.6 The Problem of m Rankings for Cyclic Designs ... . 150 8.7 Measuring Correlation Between Incomplete Rankings . 151 8.8 References ........................ . 154 9 Applications of Thurstonian Models to Ranking Data 157 U. Bockenholt 9.1 Introduction ..... 157 9.2 The Ranking Model 158 9.3 Modeling E .... . 160 9.4 Sub populations .. . 161 9.5 Model Estimation and Tests. 163 9.6 Applications. 164 9.7 Discussion. 169 9.8 References .. 169 10 Probability Models on Rankings and the Electoral Process 173 H. Stern 10.1 Introduction .. 173 10.2 Electoral Systems ..... 174 10.3 Models for Permutations. 177 10.4 The American Psychological Association Election. 180 10.5 Simulation Results ..... 183 10.6 Conclusions and Summary. 192 10.7 Acknowledgements 193 10.8 References ......... . 193 11 Permutations and Regression Models 196 P. McCullagh 11.1 Introduction .............. . 196 11.2 Models for Random Permutations. . . 197 11.3 Sufficient Statistics and Log-linear Models 205 11.4 Conclusions 213 11.5 References . . . . . . . . . . . . . . . . . . 214 xii Contents 12 Aggregation Theorems and the Combination of Probabilistic Rank Orders 216 A. A. J. Marley 12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 216 12.2 Notation and Basic Aggregation Theorems. . . . . . . . . . 218 12.3 Specific Multidimensional Ranking and Subset Selection Mod- els and Their Properties . . . . . . . . . . . 227 12.4 Multidimensional Random Variable Models 233 12.5 Conclusion 236 12.6 References . . . . . . . . . . . . . . . . . . . 238 13 A Nonparametric Distance Model for Unidimensional Unfolding 241 R. van Blokland-Vogelesang 13.1 Introduction. . . . . . . . . 241 13.2 Social Choice Theory. . . . 245 13.3 Distance Measures for Rankings . 248 13.4 Strongly Unimodal Distance Models for Rankings . 249 13.5 Generalization of Coombs' and Goodman's Conditions 251 13.6 Equal Results for ML or MNI Criterion ... . . . 254 13.7 Unfolding and Social Choice Theory: Illustrations. 265 13.8 Discussion 271 13.9 References . . . . . . . . . . . . . . . . . . . . . . . 273 Miscellanea Models on Spheres and Models for Permutations 278 P. McCullagh Complete Consensus and Order Independence: Relating Ranking and Choice 284 H. Colonius Ranking From Paired Comparisons by Minimizing Inconsistency 289 E. L. Crow Graphical Techniques for Ranked Data 294 G. L. Thompson Matched Pairs and Ranked Data 299 J. Ye and P. McCullagh
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