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A Thurstonian Ranking Model with Rank-Induced Dependencies

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Abstract

A Thurstonian model for ranks is introduced in which rank-induced dependencies are specified through correlation coefficients among ranked objects that are determined by a vector of rank-induced parameters. The ranking model can be expressed in terms of univariate normal distribution functions, thus simplifying a previously computationally intensive problem. A theorem is proven that shows that the specification given in the paper for the dependencies is the only way that this simplification can be achieved under the process assumptions of the model. The model depends on certain conditional probabilities that arise from item orders considered by subjects as they make ranking decisions. Examples involving a complete set of ranks and a set with missing values are used to illustrate recovery of the objects’ scale values and the rank dependency parameters. Application of the model to ranks for gift items presented singly or as composite items is also discussed.

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References

  • ASHBY, F.G., and ENNIS, D.M. (2002), “A Thurstone-Coombs Model of Concurrent Ratings with Sensory and Liking Dimensions,” Journal of Sensory Studies, 17, 43–59.

    Article  Google Scholar 

  • ASHBY, F.G., and GOTT, R.E. (1988), “Decision Rules in the Perception and Characterization of Multidimensional Stimuli,” Journal of Experimental Psychology: Learning, Memory and Cognition, 14, 33–53.

    Article  Google Scholar 

  • ASHBY, F.G., and LEE, W.W. (1991), “Predicting Similarity and Categorization from Identification,” Journal of Experimental Psychology: General, 120, 150–172.

    Article  Google Scholar 

  • BÖCKENHOLT, U. (1992), “Thurstonian Models for Partial Ranking Data”, British Journal of Mathematical and Statistical Psychology, 43, 31–49.

    Article  Google Scholar 

  • DENNIS, J.E., and SCHNABEL, R.B. (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Englewood Cliffs, NJ: Prentice-Hall.

    MATH  Google Scholar 

  • DE SOETE, G., CARROLL, J.D., and DESARBO, W.S. (1986), “The Wandering Ideal Point Model: A Probabilistic Multidimensional Unfolding Model for Paired Comparisons Data”, Journal of Mathematical Psychology, 30, 28–41.

    Article  MathSciNet  MATH  Google Scholar 

  • DZHAFAROV, E.N. (2006), “On the Law of Regular Minimality: Reply to Ennis”, Journal of Mathematical Psychology, 50, 74–93.

    Article  MathSciNet  MATH  Google Scholar 

  • ENNIS, D.M. (1988), “Confusable and Discriminable Stimuli: Comments on Nosofsky (1986) and Shepard (1986)”, Journal of Experimental Psychology: General, 117, 408–411.

    Article  Google Scholar 

  • ENNIS, D.M. (1992), “Modeling Similarity and Identification When There Are Momentary Fluctuations in Psychological Magnitudes”, in Multidimensional Models of Perception and Cognition, ed. F.G. Ashby, Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • ENNIS, D.M. (1993a), “The Power of Sensory Discrimination Methods”, Journal of Sensory Studies, 8, 353–370.

    Article  Google Scholar 

  • ENNIS, D.M. (1993b), “A Single Multidimensional Model for Discrimination, Identification, and Preferential Choice”, Acta Psychologica, 84, 17–27.

    Article  Google Scholar 

  • ENNIS, D.M. (2006), “Sources and Influence of Perceptual Variance: Comment on Dzhafarov’s Regular Minimality Principle”, Journal of Mathematical Psychology, 50, 66–73.

    Article  MathSciNet  MATH  Google Scholar 

  • ENNIS, D.M., and ASHBY, F.G. (1993), “The Relative Sensitivities of Same-Different and Identification Judgment Models to Perceptual Dependence”, Psychometrika , 58, 257–279.

    Article  Google Scholar 

  • ENNIS, J.M., ENNIS, D.M., YIP, D., and O’MAHONY, M. (1998), “Thurstonian Models for Variants of the Method of Tetrads,” British Journal of Mathematical and Statistical Psychology, 51, 205–215.

    Article  Google Scholar 

  • ENNIS, D.M., and JOHNSON, N.L. (1993), “Thurstone-Shepard Similarity Models as Special Cases of Moment Generating Functions,” Journal of Mathematical Psychology, 37, 104–110.

    Article  MATH  Google Scholar 

  • ENNIS, D.M., and JOHNSON, N.L. (1994), “A General Model for Preferential and Triadic Choice in Terms of Central F Distribution Functions,” Psychometrika, 59, 91–96.

    Article  MATH  Google Scholar 

  • ENNIS, D.M., and MULLEN, K. (1986), “A Multivariate Model for Discrimination Methods,” Journal of Mathematical Psychology, 30, 206–219.

    Article  Google Scholar 

  • ENNIS, D.M., and MULLEN, K., (1992), “A General Probabilistic Model for Triad Discrimination, Preferential Choice, and Two-Alternative Identification,” in Multidimensional Models of Perception and Cognition, ed. F.G. Ashby, Mahwah , NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • ENNIS, D.M., MULLEN, K. and FRITJERS, J.E.R. (1988), “Variants of the Method of Triads: Unidimensional Thurstonian Models,” British Journal of Mathematical and Statistical Psychology, 41, 25–36.

    Article  MATH  Google Scholar 

  • ENNIS, D.M., and O’MAHONY, M. (1995), “Probabilistic Models for Sequential Taste Effects in Triadic Choice,” Journal of Experimental Psychology: Human Perception and Performance, 21, 1–10.

    Article  Google Scholar 

  • ENNIS, D.M., PALEN, J., and MULLEN, K. (1988), “A Multidimensional Stochastic Theory of Similarity,” Journal of Mathematical Psychology, 32, 449–465.

    Article  MathSciNet  MATH  Google Scholar 

  • ENNIS, D.M., and ROUSSEAU, B. (2004), “Motivations for Product Consumption: Application of a Probabilistic Model to Adolescent Smoking,” Journal of Sensory Studies, 19, 107–117.

    Article  Google Scholar 

  • ENNIS, D.M., ROUSSEAU, B., and ENNIS, J.M. (2011), Short Stories in Sensory and Consumer Science, Richmond, VA: The Institute for Perception.

    Google Scholar 

  • HACHER, M.J., and RATCLIFF, R. (1979), “A Revised Table of d′ for m-Alternative Forced Choice,” Perception and Psychophysics, 26, 168–170.

    Article  Google Scholar 

  • MACKAY, D.B., EASLEY, R.F., and ZINNES, J.L. (1995), “A Single Ideal Point Model for Market Structure Analysis,” Journal of Marketing Research, 32, 433–443.

    Article  Google Scholar 

  • MACKAY, D.B., and LILLY, B. (2004), “Percept Variance, Subadditivity and the Metric Classification of Similarity, and Dissimilarity Data, Journal of Classification, 21, 185–206.

    MATH  Google Scholar 

  • MCKEON, J.J. (1961), “Measurement Procedures Based on Comparative Judgment”, unpublished doctoral dissertation, University of North Carolina, Chapel Hill.

    Google Scholar 

  • MULLEN, K., and ENNIS, D.M. (1987), “Mathematical Formulation of Multivariate Euclidean Models for Discrimination Methods,” Psychometrika, 52(2), 235–249.

    Article  MATH  Google Scholar 

  • MULLEN, K., and ENNIS, D.M. (1991), “A Simple Multivariate Probabilistic Model for Preferential and Triadic Choices,” Psychometrika, 56, 69–75.

    Article  Google Scholar 

  • NOSOFSKY, R.M. (1988), “On Exemplar-Based Exemplar Representations: Comment on Ennis (1988),” Journal of Experimental Psychology: General, 117, 412–414.

    Article  Google Scholar 

  • ROUSSEAU, B., and ENNIS, D.M. (2001), “A Thurstonian Model for the Dual Pair (4IAX) Discrimination Method,” Perception and Psychophysics, 63, 1083–1090.

    Article  Google Scholar 

  • ROUSSEAU, B., and ENNIS, D.M. (2002), “The Multiple Dual Pair Method,” Perception and Psychophysics, 64, 1008–1014.

    Article  Google Scholar 

  • SHEPARD, R.N. (1988), “Time and Distance in Generalization and Discrimination: Comment on Ennis (1988),” Journal of Experimental Psychology: General, 117: 415–416.

    Article  Google Scholar 

  • THURSTONE, L.L. (1927), “A Law of Comparative Judgement,” Psychological Review, 34, 273–286.

    Article  Google Scholar 

  • YAO, G., and BÖCKENHOLT, U. (1999), “Bayesian Estimation of Thurstonian Ranking Models Based on the Gibbs Sampler,” British Journal of Mathematical and Statistical Psychology, 52, 79–92.

    Article  Google Scholar 

  • ZINNES, J.L., and GRIGGS, R.A. (1974), “Probabilistic Multidimensional Unfolding Analysis,” Psychometrika, 39, 327–350.

    Article  MathSciNet  MATH  Google Scholar 

  • ZINNES, J.L., and MACKAY, D.B. (1983).,“Probabilistic Multidimensional Scaling: Complete and Incomplete Data,” Psychometrika, 48, 27–48.

    Article  Google Scholar 

  • ZINNES, J.L., and MACKAY, D.B. (1987), “Probabilistic Multidimensional Analysis of Preference Ratio Judgments,” Communication and Cognition, 20, 17–44.

    Google Scholar 

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Correspondence to Daniel M. Ennis.

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Ennis, D.M., Ennis, J.M. A Thurstonian Ranking Model with Rank-Induced Dependencies. J Classif 30, 124–147 (2013). https://doi.org/10.1007/s00357-013-9125-8

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  • DOI: https://doi.org/10.1007/s00357-013-9125-8

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