Table Of ContentLecture 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
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© 1993 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover 18t edition 1993
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9 S 7 6 5 432 1
ISBN-13: 97S-0-387-97920-5 e-ISBN-13: 978-1-4612-2738-0
DOl: 10.1007/978-1-4612-2738-0
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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
