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A Recursive Partitioning Method for the Prediction of Preference Rankings Based Upon Kemeny Distances

Published online by Cambridge University Press:  01 January 2025

Antonio D’Ambrosio Open the ORCID record for Antonio D’Ambrosio [Opens in a new window]  and
Willem J. Heiser
Antonio D’Ambrosio*
Affiliation:
University of Naples Federico II
Willem J. Heiser
Affiliation:
Leiden University
*
Correspondence should be made to Antonio D’Ambrosio, Department of Economics and Statistics, University of Naples Federico II, Via Cinthia, 80126 Naples, Italy. Email: antdambr@unina.it
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Abstract

Preference rankings usually depend on the characteristics of both the individuals judging a set of objects and the objects being judged. This topic has been handled in the literature with log-linear representations of the generalized Bradley-Terry model and, recently, with distance-based tree models for rankings. A limitation of these approaches is that they only work with full rankings or with a pre-specified pattern governing the presence of ties, and/or they are based on quite strict distributional assumptions. To overcome these limitations, we propose a new prediction tree method for ranking data that is totally distribution-free. It combines Kemeny’s axiomatic approach to define a unique distance between rankings with the CART approach to find a stable prediction tree. Furthermore, our method is not limited by any particular design of the pattern of ties. The method is evaluated in an extensive full-factorial Monte Carlo study with a new simulation design.

Keywords

prediction treeskemeny distancepreference rankingsconsensus ranking
Type
Original Paper
Information
Psychometrika , Volume 81 , Issue 3 , September 2016 , pp. 774 - 794
Copyright
Copyright © 2016 The Psychometric Society

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References

Amodio, S., D’Ambrosio, A., & Siciliano, R. (2016). Accurate algorithms for identifying the median ranking when dealing with weak and partial rankings under the Kemeny axiomatic approach. European Journal of Operational Research, 249, 2667–676.CrossRefGoogle Scholar
Ben-Israel, A., & Iyigun, C. (2008). Probabilistic distance clustering. Journal of Classification, 25, 5–26.CrossRefGoogle Scholar
Böckenholt, U. (2001). Mixed-effects analysis of rank-ordered data. Psychometrika, 77, 45–62.CrossRefGoogle Scholar
Bradley, R. A., & Terry, M. A. (1952). Rank analysis of incomplete block designs, I. Biometrika, 39, 324–345.Google Scholar
Breiman, L., Friedman, J. H., Olshen, R. A., & Stone, C. J. (1984). Classification and regression trees, Belmont, CA: Wadsworth International GroupGoogle Scholar
Busing, F. M. T. A. (2009). Some Advances in Multidimensional Unfolding. Doctoral Dissertation, Leiden, The Netherlands: Leiden University.Google Scholar
Busing, FMTA, Groenen, P. J. F., & Heiser, W. J. (2005). Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70, 71–98.CrossRefGoogle Scholar
Busing, FMTA, Heiser, W. J., & Cleaver, G. (2010). Restricted unfolding: Preference analysis with optimal transformations of preferences and attributes. Food Quality and Preference, 21, 82–92.CrossRefGoogle Scholar
Carroll, J. D., & Shepard, R. N., et al.(Eds.), (1972). Individual differences and multidimensional scaling. Multidimensional scaling theory, New York, USA: Seminar Press 105–155.Google Scholar
Chapman, R. G., & Staelin, R. (1982). Exploiting rank order choice set data within the stochastic utility model. Journal of Market Research, 19, 288–301.CrossRefGoogle Scholar
Cheng, W., Hühn, J., & Hüllermeier, E. (2009). Decision Tree and Instance-Based Learning for Label Ranking. Proceedings ICML-2009, 26th International Conference on Machine Learning, pp. 161–168, Montreal.CrossRefGoogle Scholar
Coombs, C. H. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57, 145–158.CrossRefGoogle ScholarPubMed
Coombs, C. H. (1964). A theory of data, New York, USA: WileyGoogle Scholar
Critchlow, D. E. (1985). Metric methods for analyzing partially ranked data, Berlin: SpringerCrossRefGoogle Scholar
Critchlow, D. E., Fligner, M. A., & Verducci, J. S. (1991). Probability models on rankings. Journal of Mathematical Psychology, 35, 294–318.CrossRefGoogle Scholar
Croon, M. A., & De Soete, G., et al. (Eds.), (1989). Latent class models for the analysis of rankings. New developments in psychological choice modeling, North-Holland: Elsevier 99–121.CrossRefGoogle Scholar
D’Ambrosio, A. (2008). Tree-based methods for data editing and preference rankings. Doctoral dissertation. Naples, Italy: Department of Mathematics and Statistics. http://www.fedoa.unina.it/2746/.Google Scholar
D’Ambrosio, A., Amodio, S., & Iorio, C. (2015). Two algorithms for finding optimal solutions of the Kemeny rank aggregation problem for full rankings. Electronic Journal of Applied Statistical Analysis, 8, 2198–213.Google Scholar
Diaconis, P. (1988). Group Representations in Probability and Statistics, Hayward, CA: Institute of Mathematical StatisticsCrossRefGoogle Scholar
De’ath, G. (2002). Multivariate regression trees: A new technique for modeling species-environment relationships. Echology, 83, 41105–1117.Google Scholar
Ditrich, R., Hatzinger, R., & Katzenbeisser, W. (1998). Modelling the effect of subject-specific covariates in paired comparison studies with an application to university rankings. Journal of the Royal Statistical Society C, 47, 511–525.CrossRefGoogle Scholar
Ditrich, R., Katzenbeisser, W., & Hatzinger, R. (2000). The analysis of rank order preference data based on Bradley-Terry Type models. OR Spectrum, 22, 117–134.CrossRefGoogle Scholar
Dusseldorp, E., & Meulman, J. J. (2004). The regression trunk approach to discover treatment covariate interaction. Psychometrika, 69, 3355–374.CrossRefGoogle Scholar
Emond, E. J., & Mason, D. W. (2000), A new technique for high level decision support. ORD project Report PR2000/13 Department of National Defence, Canada.Google Scholar
Emond, E. J., & Mason, D. W. (2002). A new rank correlation coefficient with application to the consensus ranking problem. Journal of Multi-Criteria Decision Analysis, 11, 17–28.CrossRefGoogle Scholar
Feigin, P. D., & Cohen, A. (1978). On a model for concordance between judges. Journal of the Royal Statistical Society, B, 40, 2203–213.CrossRefGoogle Scholar
Fligner, M. A., & Verducci, J. S. (1986). Distance based ranking models. Journal of the Royal Statistical Society, Series B, 48, 359–369.CrossRefGoogle Scholar
Fligner, M. A., & Verducci, J. S. (1988). Multistage rankings models. Journal of the American Statistical Association, 83, 892–901.CrossRefGoogle Scholar
Francis, B., Dittrich, R., Hatzinger, R., & Penn, R. (2002). Analysing partial ranks by using smoothed paired comparison methods: An investigation of value orientation in Europe. Applied Statistics, 51, 319–336.Google Scholar
Fürnkranz, J., & Hüllermeier, E. (2011). Preference learning, Berlin: SpringerCrossRefGoogle Scholar
Gormley, I. C., & Murphy, T. B. (2008). Exploring voting blocs within the Irish electorate: A mixture modeling approach. Journal of the American Statistical Association, 103, 1014–1027.CrossRefGoogle Scholar
Gormley, I. C., & Murphy, T. B. (2008). A mixture of experts model for rank data with applications in election studies. The Annals of Applied Statistics, 4, 21452–1477.Google Scholar
Gross, O. A. (1962). Preferential arrangements. The American Mathematical Monthly, 69, 1–4.CrossRefGoogle Scholar
Hastie, T., Tibshirani, R., & Friedman, J. H. (2009). The Elements of Statistical Learning, New York, USA: SpringerCrossRefGoogle Scholar
Heiser, W. J. (2004). Geometric representation of association between categories. Psychometrica, 69, 4513–545.CrossRefGoogle Scholar
Heiser, W.J., & D’Ambrosio, A. (2013). Clustering and prediction of rankings within a Kemeny distance framework. In B, Lausen, D., Van den Poel, Ultsch, A. (Eds.), Algorithms from and for Nature and Life, Springer series in Studies in Classification, Data Analysis, and Knowledge Organization, 19-31, Springer International Publishing Switzerland.CrossRefGoogle Scholar
Heiser, W. J., & De Leeuw, J. (1981). Multidimensional mapping of preference data. Mathématiques et Sciences Humaines, 19, 39–96.Google Scholar
Inglehart, R. (1977). The silent revolution: Changing values and political styles among Western Publics, Princeton, NJ: Princeton University PressGoogle Scholar
Kemeny, J. G. (1959). Mathematics without numbers. Daedalus, 88, 577–591.Google Scholar
Kemeny, J. G., & Snell, L. (1962). Mathematical models in the social sciences, Boston: Ginn and CompanyGoogle Scholar
Kendall, M. (1948). Rank correlation methods, London: Charles Griffin & Company LimitedGoogle Scholar
Larsen, D. R., & Speckman, C. L. (2004). Multivariate regression trees for analysis of abundance data. Biometrics, 60, 543–459.CrossRefGoogle ScholarPubMed
Lee, P. H., & Yu, P. L. H. (2010). Distance-based tree models for ranking data. Computational Statistics and Data Analysis, 54, 1672–1682.CrossRefGoogle Scholar
Luce, R. D. (1959). Individual choice behavior, New York, USA: WileyGoogle Scholar
Mallows, C. L. (1957). Non-null ranking models, I. Biometrika, 44, 114–130.CrossRefGoogle Scholar
Marden, J. I. (1995). Analyzing and modelling rank data, London: Chapman & HallGoogle Scholar
Meulman, J. J., Van Der Kooij, A. J., Heiser, W. J., & Kaplan, D. (2004). Principal components analysis with nonlinear optimal scaling transformations for ordinal and nominal data. The SAGE handbook of quantitative methodology for the social sciences, Thousand Oaks: Sage 49–70.Google Scholar
Murphy, T. B., & Martin, D. (2003). Mixtures of distance-based models for ranking data. Computational Statistics and Data Analysis, 41, 3645–655.CrossRefGoogle Scholar
Nerini, D., & Ghattas, B. (2007). Classifying densities using functional regression trees: Applications in oceanology. Computational Statistics and Data Analysis, 51, 4984–4993.CrossRefGoogle Scholar
Siciliano, R., & Mola, F. (2000). Multivariate data analysis and modelling through classification and regression trees. Computational Statistics and Data Analysis, 32, 285–301.CrossRefGoogle Scholar
Skrondal, A., & Rabe-Hesketh, S. (2003). Multilevel logistic regression for polytomous data and rankings. Psychometrika, 68, 2267–287.CrossRefGoogle Scholar
Strobl, C., Malley, J., & Tutz, G. (2009). An introduction to recursive partitioning: rationale, application, and characteristics of classification and regression trees, bagging, and random forests. Psychological Methods, 14, 4323–348.CrossRefGoogle ScholarPubMed
Strobl, C., Wickelmaier, F., & Zeileis, A. (2011). Accounting for individual differences in Bradley-Terry models by means of recursive partitioning. Journal of Educational and Behavioral Statistics, 36, 2135–153.CrossRefGoogle Scholar
Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
van Blokland-Vogelesang, R. (1990), Unfolding and group consensus ranking for individual preferences. Unpublished PhD thesis, University of Leiden.Google Scholar
Van Deun, K., Heiser, W. J., & Delbeke, L. (2007). Multidimensional unfolding by nonmetric multidimensional scaling of Spearman distances in the extended permutation polytope. Multivariate Behavioral Research, 42, 103–132.CrossRefGoogle ScholarPubMed
Vermunt, J. K. (2003). Multilevel latent class models. Sociological Methodology, 33, 1213–239.CrossRefGoogle Scholar