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Über dieses Buch

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.



Ranking Models with Item Covariates

1. Ranking Models with Item Covariates

Two parametric classes of ranking models are investigated: the Thurstone order statistics models and the Babington Smith models. Both families are natural extensions of commonly used paired comparison models. The concept of an “item parameter” is introduced and studied in the context of each of these classes of models. This distinction between the item parameters and the remaining parameters in a ranking model is useful not only for the general interpretation of model parameters, but also for the specific problem of introducing covariates in these models. Estimation schemes are described for these models, both with and without covariates, and are implemented in an example.
Douglas E. Critchlow, Michael A. Fligner

Nonparametric Methods of Ranking from Paired Comparisons

2. Nonparametric Methods of Ranking from Paired Comparisons

Ranking by row-sum scores in the case of balanced paired-comparison experiments was generalized to unbalanced experiments in David [7]. Statistical properties of the proposed scores and associated tests of significance are developed in Andrews and David [2], where extensions to unbalanced ranked data are also treated. A brief account of this work is given and a possible generalization is introduced and examined. The simple methods here advanced make no assumptions on the pairwise preference probabilities. A secondary aim of this paper is to provide a critical review of competing methods also involving no such assumptions as well as of related methods requiring only mild assumptions. Many of the procedures discussed are illustrated on a worked example.
H. A. David, D. M. Andrews

On the Babington Smith Class of Models for Rankings

3. On the Babington Smith Class of Models for Rankings

In 1950, Babington Smith proposed a general family of probability models for rankings based on a paired comparisons idea. Mallows [9] studied several simple subclasses of the Babington Smith models, but the full class was considered computationaly intractible for practical application at that time. With modern computers, the models are simple to use. With this incentive, we investigate various properties of the Babington Smith models, including their characterization as maximum entropy models, the relationships among different parametrizations of the models, and the conditions under which various forms of stochastic transitivity, unimodality and consensus are obtained. The maximum entropy characterization suggests models that are nested within the Babington Smith models and models that are more general. Computational details for the models are briefly discussed. The models are illustrated with examples where words are ranked in accordance to their perceived degree of association with a target word.
Harry Joe, Joseph S. Verducci

Latent Structure Models for Ranking Data

4. Latent Structure Models for Ranking Data

In this paper several latent structure models for analyzing data that consist of complete or incomplete rankings are discussed. First, attention is given to some latent class extensions of the Bradley-Terry-Luce model for ranking data. Next, various latent class models based on log-linear modeling of ranking data are described. Within this latter family of latent class models, a main distinction is made between models based on the assumption of quasi-independence within the latent classes, and models in which some form of association between the ranking positions is allowed to exist within the classes. All models are applied to a real data set from a large scale cross-national survey on political values.
M. A. Croon, R. Luijkx

Modelling and Analysing Paired Ranking Data

5. Modelling and Analysing Paired Ranking Data

Two models for paired rankings are presented. They describe two different ways in which a post-ranking is related to its pre-ranking for each case (or subject). These models are compared to regression models in order to help motivate their forms. Analysis of paired ranking data is considered in the light of testing for that type of departure from a null model that corresponds to either of the proposed models. The procedure suggested uses a bootstrap method to ascertain the strength of the departure from the null model, and helps one to decide which departure is more strongly indicated. Some analyses of simulated test data sets are described, as well as the analysis of data due to Rogers [7] which is also analysed in Critchlow and Verducci [3].
Paul D. Feigin

Maximum Likelihood Estimation in Mallows’s Model Using Partially Ranked Data

6. Maximum Likelihood Estimation in Mallows’s Model Using Partially Ranked Data

Consider a sample from a population in which each individual is characterized by a ranking on k items, but only partial information about the ranking is available for the individuals in the sample. The problem is to estimate the population distribution of rankings, given the partially ranked data. This paper proposes use of an EM algorithm to obtain maximum likelihood estimates of the parameters in Mallows’s model for the distribution of rankings. Medical applications are discussed where the items are manifestations of a disease or a developmental process, the ranking is the sequence in which they first appear over time, and the partial ranking results from observation of the subjects cross-sectionally or at a few specified times. The methods are illustrated for a longitudinal study of a community population aged 65 years and older, where the signs are self-reporting of impairment in different physical activities.
Laurel A. Beckett

Extensions of Mallows’ ϕ Model

7. Extensions of Mallows’ ϕ Model

Mallows’ ϕ model is a one-parameter exponential family model for vectors of ranks. Fligner and Verducci have extended this model to multistage ranking situations. In this paper we introduce a class of models based on so-called orthogonal contrasts of the objects to be ranked, which we use to analyze three sets of data. The first set, from the GRE, consists of 98 students’ ranking of five words according to their association with the word idea. The second is the American Psychological Association’s 1980 presidential election data. The final set illustrates an approach to rank-based analysis-of-variance.
Lyinn Chung, John I. Marden

Rank Correlations and the Analysis of Rank-Based Experimental Designs

8. Rank Correlations and the Analysis of Rank-Based Experimental Designs

The notion of distance between two permutations is used to provide a unified treatment for various problems involving ranking data. Using the distances defined by Spearman and Kendall, the approach is illustrated in terms of the problem of concordance as well as the problem of testing for agreement among two or more populations of rankers. An extension of the notion of distance for incomplete permutations is shown to lead to a generalization of the notion of rank correlation. Applications are given to the incomplete block design as well as to the class of cyclic designs.
M. Alvo, P. Cabilio

Applications of Thurstonian Models to Ranking Data

9. Applications of Thurstonian Models to Ranking Data

Thurstonian models have proven useful in a wide range of applications because they can describe the multidimensional nature of choice objects and the effects of similarity and comparability in choice situations. Special cases of Thurstonian ranking models are formulated that impose different constraints on the covariance matrix of the objects’ utilities. In addition, mixture models are developed to account for individual differences in rankings. Two estimation procedures, maximum likelihood and generalized least squares, are discussed. To illustrate the approach, data from three ranking experiments are analyzed.
Ulf Böckenholt

Probability Models on Rankings and the Electoral Process

10. Probability Models on Rankings and the Electoral Process

Multicandidate elections with a single winner suggest several questions about the manner in which the preferences of a group of individual voters are aggregated into a single social choice. Obvious examples are the national presidential primaries in the major political parties. However, nonpolitical exercises such as the ranking of job applicants or college football teams provide other examples. If an individual’s preference is viewed as a ranking of the available choices then the literature on probability models for rankings (see the survey by Critchlow, Fligner and Verducci [11]) may be used to analyze methods for combining preferences. Several probability models are used to analyze the results of a five candidate presidential election of the American Psychological Association. In addition, simulated data generated by parametric probability models is used to consider the merits of a variety of voting systems.
Hal Stern

Permutations and Regression Models

11. Permutations and Regression Models

A class of exponential-family models on the set of permutations of k objects or items is described. The null or uniform model gives probability 1/k! to each of the k! possible permutations. The first-order inversion model has as sufficient statistic the k × k matrix listing the number of times that each pair of candidates was ranked in that order, i.e. the number of times that candidate a was preferred over candidate b for all ordered pairs a and b. In the second-order inversion model the sufficient statistic is a similar listing for each ordered triplet of three candidates. Interesting sub-models are identified and used to help in the analysis of the APA election data.
Peter McCullagh

Aggregation Theorems and the Combination of Probabilistic Rank Orders

12. Aggregation Theorems and the Combination of Probabilistic Rank Orders

There are many situations where we wish to combine multiple rank orders or other preference information on a fixed set of options to obtain a combined rank order. Two of the most common applications are determining a social rank order on a set of options from a set of individual rank orders on those options, and predicting (or prescribing) an individual’s overall rank order on a set of options from the rank orders on a set of component dimensions of the options. In this paper, I develop solutions to this class of problems when the rank orders can occur probabilistically. I develop aggregation theorems that are motivated by recent theoretical work on the combination of expert opinions and I discuss various models that have the property that the representations are ‘of the same form’ for both the component and overall rank order probabilities. I also briefly discuss difficulties in actually using such probabilistic ranking models in the social choice situation.
A. A. J. Marley

A Nonparametric Distance Model for Unidimensional Unfolding

13. A Nonparametric Distance Model for Unidimensional Unfolding

The unidimensional unfolding model is placed in the wider context of social choice theory, median procedures and strictly unimodal distance models for rankings. Social choice theory is used to construct a framework for the unfolding model; for example, given single-peaked preference functions for individuals, Simple Majority Rule yields the median ordering as a group consensus ordering. We generalize Coombs’ and Goodman’s (1954) theorems: if the data follow a strictly unimodal distance model, the median ordering is an admissible ordering of the J scale that has the highest probability. This is because the maximum likelihood and the minimum-number-of-inversions criterion yield the same ordering in a strictly unimodal distance model: the mean/modal/median ordering. We prove that the group consensus ordering is transitive and is the modal or median ordering. Also, we prove that the social preference function is unimodal on the J scale in this case.
Rian van Blokland-Vogelesang



Models on Spheres and Models for Permutations

It is shown that the space of permutations is naturally ordered in a circular or spherical manner. By exploiting the geometry of the sample space it is shown that Mallows’s ϕ-model with the Spearman metric is essentially equivalent to the Mallows-Bradley-Terry ranking model, which is essentially equivalent to the von Mises-Fisher model on the sphere. Extensions to bi-polar models are discussed briefly. References
Peter McCullagh

Complete Consensus and Order Independence: Relating Ranking and Choice

Complete Consensus has been introduced as a plausible independence from irrelevant alternatives’ property of probability models on rankings. It is shown here that complete consensus implies order-independence for the choice probabilities of the corresponding random utility models for choice.
Hans Colonius

Ranking From Paired Comparisons by Minimizing Inconsistency

A criterion is presented for the best ranking of items or individuals who have been compared in pairs in an unbalanced fashion. The criterion is to choose the ranking or rankings that minimize the sum over all contestants of the absolute differences between the number of net wins over players ranked above and the number of net losses to players ranked below. A method for reaching the minimum is presented. There are two variations of the criterion. They are illustrated on a small set of 1989 tennis player data.
Edwin L. Crow

Graphical Techniques for Ranked Data

Exploratory graphical methods are critically needed for displaying ranked data. Fully and partially ranked data are functions on the symmetric group of n elements, S n , and on the related coset spaces. Because neither S n nor its coset spaces have a natural linear ordering, graphical methods such as histograms and bar graphs are inappropriate for displaying ranked data. However, a very natural partial ordering on S n and its coset spaces is induced by two reasonable measures of distance: Spearman’s p and Kendall’s τ. Graphical techniques that preserve this partial ordering can be developed to display ranked data and to illustrate related probability density functions by using permutation polytopes. A polytope is the convex hull of a finite set of points in ℜ n−1, and a permutation polytope is the convex hull of the n! permutations of n elements when regarded as vectors in ℜ n (see, for example, Yemelichev, et. al.[9]). This concept is closely related to the observation by McCullagh [7] that the n! elements of S n lie on the surface of a sphere in ℜ n−1.
G. L. Thompson

Matched Pairs and Ranked Data

We consider the Babington Smith and Bradley-Terry Models for ranked data. Both models are based on inversions. In a matched pairs design the pair-specific nuisance parameters are eliminated by a conditioning argument. The conditional likelihood has a form similar to that of a logistic model, so that conditional likelihood computations are straight-forward. An example previously considered by Critchlow and Verducci is analysed using the new method.
Peter McCullagh, Jianming Ye
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