Probability Models and Statistical Analyses for Ranking Data: (80 Lecture Notes in Statistics)

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.

Table of Contents:
1 Ranking Models with Item Covariates.- 1.1 Introduction.- 1.2 Basic Ranking Models and Their Parameters.- 1.3 Ranking Models with Covariates.- 1.4 Estimation.- 1.5 Example.- 1.6 Discussion.- 1.7 Appendix.- 1.8 References.- 2 Nonparametric Methods of Ranking from Paired Comparisons.- 2.1 Introduction and Literature Review.- 2.2 The Proposed Method of Scoring.- 2.3 Distribution Theory and Tests of Significance for ??ij = pij.- 2.4 Ranking Methods.- 2.5 Numerical Example.- 2.6 References.- 3 On the Babington Smith Class of Models for Rankings.- 3.1 Introduction.- 3.2 Alternative Parametrizations and Related Models.- 3.3 Stochastic Transitivity and Item Preference.- 3.4 Examples and Data Analysis.- 3.5 References.- 4 Latent Structure Models for Ranking Data.- 4.1 Introduction.- 4.2 Latent Class Analyses Based on the Bradley-Terry-Luce Model.- 4.3 Latent Class Analyses Based on a Quasi-independence Model.- 4.4 Models that Allow for Association Between Choices within the Classes.- 4.5 References.- 5 Modelling and Analysing Paired Ranking Data.- 5.1 Introduction.- 5.2 Two Models.- 5.3 Estimation and Hypothesis Testing.- 5.4 Analysis of Simulated Data Sets.- 5.5 Analysis of Rogers Data.- 5.6 References.- 6 Maximum Likelihood Estimation in Mallows’s Model Using Partially Ranked Data.- 6.1 Introduction.- 6.2 Notation.- 6.3 Maximum Likelihood Estimation Using the EM Algorithm.- 6.4 Example.- 6.5 Discussion.- 6.6 References.- 7 Extensions of Mallows’ ? Model.- 7.1 Introduction.- 7.2 The General Model.- 7.3 Ties, Partial Rankings.- 7.4 Example: Word Association.- 7.5 Example: APA Voting.- 7.6 Example: ANOVA.- 7.7 Discussion of Contrasts.- 7.8 Appendix.- 7.9 References.- 8 Rank Correlations and the Analysis of Rank-Based Experimental Designs.- 8.1 Introduction.- 8.2 DistanceBased Measures of Correlation.- 8.3 The Problem of m Rankings.- 8.4 The Two Sample Problem.- 8.5 The Problem of m Rankings for a Balanced Incomplete Block Design.- 8.6 The Problem of m Rankings for Cyclic Designs.- 8.7 Measuring Correlation Between Incomplete Rankings.- 8.8 References.- 9 Applications of Thurstonian Models to Ranking Data.- 9.1 Introduction.- 9.2 The Ranking Model.- 9.3 Modeling ?.- 9.4 Subpopulations.- 9.5 Model Estimation and Tests.- 9.6 Applications.- 9.7 Discussion.- 9.8 References.- 10 Probability Models on Rankings and the Electoral Process.- 10.1 Introduction.- 10.2 Electoral Systems.- 10.3 Models for Permutations.- 10.4 The American Psychological Association Election.- 10.5 Simulation Results.- 10.6 Conclusions and Summary.- 10.7 Acknowledgements.- 10.8 References.- 11 Permutations and Regression Models.- 11.1 Introduction.- 11.2 Models for Random Permutations.- 11.3 Sufficient Statistics and Log-linear Models.- 11.4 Conclusions.- 11.5 References.- 12 Aggregation Theorems and the Combination of Probabilistic Rank Orders.- 12.1 Introduction.- 12.2 Notation and Basic Aggregation Theorems.- 12.3 Specific Multidimensional Ranking and Subset Selection Models and Their Properties.- 12.4 Multidimensional Random Variable Models.- 12.5 Conclusion.- 12.6 References.- 13 A Nonparametric Distance Model for Unidimensional Unfolding.- 13.1 Introduction.- 13.2 Social Choice Theory.- 13.3 Distance Measures for Rankings.- 13.4 Strongly Unimodal Distance Models for Rankings.- 13.5 Generalization of Coombs’ and Goodman’s Conditions.- 13.6 Equal Results for ML or MNI Criterion.- 13.7 Unfolding and Social Choice Theory: Illustrations.- 13.8 Discussion.- 13.9 References.- Miscellanea.- Models on Spheres and Models for Permutations.- Complete Consensus and OrderIndependence: Relating Ranking and Choice.- Ranking From Paired Comparisons by Minimizing Inconsistency.- Graphical Techniques for Ranked Data.- Matched Pairs and Ranked Data.