Advertisement

Psychometrika

, Volume 56, Issue 4, pp 567–587 | Cite as

Was euclid an unnecessarily sophisticated psychologist?

  • Phipps Arabie
Article

Abstract

A survey of the current state of multidimensional scaling using the city-block metric is presented. Topics include substantive and theoretical issues, recent algorithmic developments and their implications for seemingly straightforward analyses, isometries with other metrics, links to graph-theoretic models, and future prospects.

Key words

multidimensional scaling facility planing city-block distances Minkowski metrics rectangular metric rectilinear metric Manhattan metric combination rules 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arabie, P. (1973). Concerning Monte Carlo evaluations of nonmetric scaling algorithms.Psychometrika, 38, 607–608.Google Scholar
  2. Arabie, P., & Boorman, S. A. (1973). Multidimensional scaling of measures of distance between partitions.Journal of Mathematical Psychology, 10, 148–203.Google Scholar
  3. Arabie, P., Carroll, J. D., & DeSarbo, W. S. (1987).Three-way scaling and clustering. Newbury Park, CA: Sage. (Translated into Japanese by A. Okada & T. Imaizumi, 1990. Tokyo: Kyoritsu Shuppan)Google Scholar
  4. Arabie, P., & Hubert, L. (1992). Combinatorial data analysis.Annual Review of Psychology, 43, 169–203.Google Scholar
  5. Arnold, J. B. (1971). A multidimensional scaling study of semantic distance.Journal of Experimental Psychology Monograph, 90(2), 349–372.Google Scholar
  6. Attneave, F. (1950). Dimensions of similarity.American Journal of Psychology, 63, 516–556.Google Scholar
  7. Backhaus, W., Menzel, R., & Kreissl, S. (1987). Multidimensional scaling of color similarity in bees.Biological Cybernetics, 56, 293–304.Google Scholar
  8. Beckman, P. (1971).A history of π (pi) (3rd ed.). New York: St. Martin's Press.Google Scholar
  9. Blaschke, W. (1916). Räumliche Variationsprobleme mit symmetrischer Transversalitätsbedingung [Spatial variations problems with a symmetric transversal condition].Berichten Sächsich Akademie Wissenschaft Leipzig, 68, 50–55.Google Scholar
  10. Blough, D. S. (1972). Recognition by the pigeon of stimuli varying in two dimensions.Journal of the Experimental Analysis of Behavior, 18, 345–367.Google Scholar
  11. Boorman, S. A., & Arabie, P. (1972). Structural measures and the method of sorting. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences. Vol. 1: Theory (pp. 225–249). New York: Seminar Press.Google Scholar
  12. Boorman, S. A., & Olivier, D. C. (1973). Metrics on spaces of finite trees.Journal of Mathematical Psychology, 10, 26–59.Google Scholar
  13. Borg, I. (1983). Scaling: A review of the German scaling literature of the past 15 years.German Journal of Psychology, 7, 63–79.Google Scholar
  14. Borg, I., & Leutner, D. (1983). Dimensional models for the perception of rectangles.Perception & Psychophysics, 34, 257–267.Google Scholar
  15. Borg, I., & Lingoes, J. (1987).Multidimensional similarity structure analysis. New York: Springer-Verlag.Google Scholar
  16. Bortz, J. (1974). Kritische Bemerkungen ueber den Einsatz nicht-euklidischer Metriken im Rahmen der multidimensionalen Skalierung [Critical remarks on the use of non-Euclidean metrics in the context of multidimensional scaling].Archiv fuer Psychologie, 126, 196–212.Google Scholar
  17. Boyd, J. P. (1972). Information distance for discrete structures. In R. N. Shepard, A. K. Romney, & S. B. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences (Vol. I, pp. 213–223). New York: Seminar Press.Google Scholar
  18. Brandeau, M. L., & Chiu, S. S. (1988). Parametric facility location on a tree network with anL p norm cost function.Transportation Science, 22, 59–69.Google Scholar
  19. Buja, A., Logan, B., Reeds, J., & Shepp, L. (1991, June).Garbage in, structure out—the performance of MDS on null data. Paper presented at joint meeting of the Classification Society of North America and the Psychometric Society, New Brunswick, NJ.Google Scholar
  20. Busemann, H. (1950). The foundations of Minkowskian geometry.Commentarii Mathematici Helvetici, 24, 156–186.Google Scholar
  21. Busemann, H. (1955).The geometry of geodesics. New York: Academic Press.Google Scholar
  22. Busemeyer, J. R., & Jones, L. E. (1983). Analysis of multiplicative combination rules when the causal variables are measured with error.Psychological Bulletin, 93, 549–562.Google Scholar
  23. Carroll, J. D. (1976). Spatial, non-spatial and hybrid models for scaling.Psychometrika, 41, 439–463.Google Scholar
  24. Carroll, J. D., & Arabie, P. (1980). Multidimensional scaling. In M. R. Rosenzweig & L. W. Porter (Eds.),Annual review of psychology (Vol. 31, pp. 607–649). Palo Alto, CA: Annual Reviews.Google Scholar
  25. Carroll, J. D., & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via anN-way generalization of “Eckart-Young” decomposition.Psychometrika, 35, 283–319.Google Scholar
  26. Carroll, J. D., & Chang, J. J. (1973). A method for fitting a class of hierarchical tree structure models to dissimilarities data and its application to some “body parts” data of Miller's.Proceedings of the 81st Annual Convention of the American Psychological Association, 8, 1097–1098.Google Scholar
  27. Carroll, J. D., & Wish, M. (1974). Multidimensional perceptual models and measurement methods. In E. C. Carterette & M. P. Friedman (Eds.),Handbook of perception (Vol. 2, pp. 391–447). New York: Academic Press.Google Scholar
  28. Celeux, G., & Govaert, G. (1991). Clustering criteria for discrete data and latent class models.Journal of Classification, 8.Google Scholar
  29. Coombs, C. H. (1964).A theory of data. New York: Wiley.Google Scholar
  30. Cross, D. V. (1965a).Multidimensional stimulus control of the discriminative response in experimental conditioning and psychophysics. (Tech. Rep. No. 05613-4-F(78[d])). Ann Arbor: University of Michigan.Google Scholar
  31. Cross, D. V. (1965b). Metric properties of multidimensional stimulus generalization. In D. I. Mostofsky (Ed.),Stimulus generalization (pp. 72–93). Stanford: Stanford University Press.Google Scholar
  32. Daws, J. T. (1991, June).The analysis of free-sort data: Beyond pairwise cooccurrences. Paper presented at joint meeting of the Classification Society of North America and the Psychometric Society, New Brunswick, NJ.Google Scholar
  33. Daws, J. T., Arabie, P., & Hubert, L. J. (1990, June).A note on expected stress values in nonmetric multidimensional scaling. Paper presented at meeting of the Classification Society of North America, Logan, UT.Google Scholar
  34. de Leeuw, J., & Heiser, W. (1977). Convergence of correction-matrix algorithms for multidimensional scaling. In J. C. Lingoes (Ed.),Geometric representations of relational data: Readings in multidimensional scaling (pp. 735–752). Ann Arbor, MI: Mathesis.Google Scholar
  35. de Leeuw, J., & Heiser, W. (1982). Theory of multidimensional scaling. In P. R. Krishnaiah, & L. N. Kanal (Eds.),Handbook of statistics Vol.2: Classification, pattern recognition and reduction of dimensionality (pp. 285–316). Amsterdam: North-Holland.Google Scholar
  36. De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1987). Least squares algorithms for constructing constrained ultrametric and additive tree representations of symmetric proximity data.Journal of Classification, 4, 155–173.Google Scholar
  37. Dress, A., & Krüger, M. (1987). Parsimonious phylogenetic trees in metric spaces and simulated annealing.Advances in Applied Mathematics, 8, 8–37.Google Scholar
  38. Dunn, T. R., & Harshman, R. A. (1982). A multidimensional scaling model for the size-weight illusion.Psychometrika, 47, 25–45.Google Scholar
  39. Eaton, B. C., & Lipsey, R. G. (1980). The block metric and the law of markets.Journal of Urban Economics, 7, 337–347.Google Scholar
  40. Eisler, H. (1973). The algebraic and statistical tractability of the city block metric.British Journal of Mathematical and Statistical Psychology, 26, 212–218.Google Scholar
  41. Eisler, H., & Lindman, R. (1990). Representations of dimensional models of similarity. In H.-O. Geissler (Ed.),Psychophysical explorations of mental structures (pp. 165–171). Toronto: Hogrefe & Huber.Google Scholar
  42. Ekman, G. (1954). Dimensions of color vision.Journal of Psychology, 38, 467–474.Google Scholar
  43. Evans, M. G. (1991). The problem of analyzing multiplicative composites.American Psychologist, 46, 6–15.Google Scholar
  44. Felfoldy, G. L. (1974). Repetition effects in choice reaction time to multidimensional stimuli.Perception & Psychophysics, 15, 453–459.Google Scholar
  45. Fichet, B. (1987). The role played byL 1 in data analysis. In Y. Dodge (Ed.),Statistical data analysis based on the L 1 -norm and related methods (pp. 185–193). New York: North-Holland.Google Scholar
  46. Fichet, B. (1988).L p spaces in data analysis. In H.-H. Bock (Ed.),Classification and related methods of data analysis (pp. 439–444). Amsterdam: North-Holland.Google Scholar
  47. Fischer, W., & Micko, H. C. (1972). More about metrics of subjective spaces and attention distributions.Journal of Mathematical Psychology, 9, 36–54.Google Scholar
  48. Francis, R. L., & White, J. A. (1974).Facility layout and location: An analytical approach. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  49. Garner, W. R. (1974).The processing of information and structure. Hillsdale, NJ: Erlbaum.Google Scholar
  50. Garner, W. R. (1976). Interaction of stimulus dimensions in concept and choice processes.Cognitive Psychology, 8, 98–123.Google Scholar
  51. Govaert, G. (1984). Classification simultanée de tableaux binaires [Simultaneous classification of binary tables]. In E. Diday, M. Jambu, L. Lebart, J. Pagès, & R. Tomassone (Eds.),Data analysis and informatics, Vol. III (pp. 223–236). Amsterdam: North-Holland.Google Scholar
  52. Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis.Biometrika, 53, 325–338.Google Scholar
  53. Gower, J. C. (1971). A general coefficient of similarity and some of its properties.Biometrics, 27, 857–871.Google Scholar
  54. Gower, J. C., & Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients.Journal of Classification, 3, 5–48.Google Scholar
  55. Green, P. E., & Rao, V. R. (1969). A note on proximity measures and cluster analysis.Journal of Marketing Research, 6, 359–364.Google Scholar
  56. Green, P. E., & Srinivasan, V. (1990, October). Conjoint analysis in marketing: New developments with implications for research and practice.Journal of Marketing, 54, 3–19.Google Scholar
  57. Guénoche, A. (1987). Cinq algorithmes d'approximation d'une dissimilarite par des arbres a distances additives [Five algorithms of dissimilarity approximation by additive trees].Mathématiques et Sciences humaines, 25(98), 21–40.Google Scholar
  58. Hancock, H. (1964).Development of the Minkowski geometry of numbers. New York: Dover. (Original work published 1939)Google Scholar
  59. Hansen, P., Peeters, D., & Thisse, J.-F. (1981). Constrained location and the Weber-Rawls problem.Annals of Discrete Mathematics, 11, 147–166.Google Scholar
  60. Hansen, P., Peeters, D., & Thisse, J.-F. (1982). An algorithm for a constrained Weber problem.Management Science, 11, 1285–1295.Google Scholar
  61. Hansen, P., Perreur, J., & Thisse, J.-F. (1980). Location theory, dominance, and convexity: Some further results.Operations Research, 28, 1241–1250.Google Scholar
  62. Harary, F., Melter, R. A., & Tomescu, I. (1984). Digital metrics: A graph-theoretical approach.Pattern Recognition Letters, 2, 159–163.Google Scholar
  63. Hardzinski, M., & Pachella, R. G. (1977).A psychophysical analysis of complex integrated displays (Tech. Rep. No. 59). Ann Arbor: University of Michigan, Human Performance Center.Google Scholar
  64. Heath, T. L. (1956).The thirteen books of Euclid's Elements. (2nd ed.). New York: Dover.Google Scholar
  65. Heiser, W. J. (1989). The city-block model for three-way multidimensional scaling. In R. Coppi & S. Bolasco (Eds.),Multiway data analysis (pp. 395–404). Amsterdam: North-Holland.Google Scholar
  66. Hewes, A. (1900). Social institutions and the Riemann surface.American Journal of Sociology, 5, 392–403.Google Scholar
  67. Householder, A. S., & Landahl, H. D. (1945).Mathematical biophysics of the central nervous system. Bloomington, IN: Principia Press.Google Scholar
  68. Hubert, L. J. (1984). Statistical applications of linear assignment.Psychometrika, 49, 449–473.Google Scholar
  69. Hubert, L., & Arabie, P. (1985). Comparing partitions.Journal of Classification, 2, 193–218.Google Scholar
  70. Hubert, L., & Arabie, P. (1986). Unidimensional scaling and combinatorial optimization. In J. de Leeuw, W. Heiser, J. Meulman, & F. Critchley (Eds.),Multidimensional data analysis (pp. 181–196). Leiden, The Netherlands: DSWO-Press.Google Scholar
  71. Hubert, L. J., & Arabie, P. (1988). Relying on necessary conditions for optimization: Unidimensional scaling and some extensions. In H.-H. Bock (Ed.),Classification and related methods of data analysis (pp. 463–472). Amsterdam: North-Holland.Google Scholar
  72. Hubert, L. J., & Arabie, P., & Hesson-Mcinnis, M. (1991).Multidimensional scaling in the city-block metric: A combinatorial approach. Manuscript submitted for publication.Google Scholar
  73. Hubert, L. J., & Busk, P. (1976). Normative location theory: Placement in continuous space.Journal of Mathematical Psychology, 14, 187–210.Google Scholar
  74. Joly, S., & Le Calvé, G. (1991). Realisable 0-1 matrices and city block distance. Manuscript submitted for publication.Google Scholar
  75. Jones, L. E. (1982). Construal of social environments: Multidimensional models of interpersonal perception and attraction. In N. Hirschberg & L. G. Humphreys (Eds.),Multivariate applications in the social sciences (pp. 61–85). Hillsdale, NJ: Erlbaum.Google Scholar
  76. Koopman, R. F., & Cooper, M. (1974, March).Some problems with Minkowski distance models in multidimensional scaling. Paper presented at the meeting of the Psychometric Society, Stanford, CA.Google Scholar
  77. Krause, E. F. (1975).Taxicab geometry. Menlo Park, CA: Addison-Wesley.Google Scholar
  78. Kruskal, J. B. (1964a). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 29, 1–27.Google Scholar
  79. Kruskal, J. B. (1964b). Nonmetric multidimensional scaling: A numerical method.Psychometrika, 29, 115–129.Google Scholar
  80. Kruskal, J. B., & Carmone, F. (1970).How to use M-D-SCAL (Version 5M) and other useful information. Cambridge: Marketing Science Institute.Google Scholar
  81. Kruskal, J. B., & Carroll, J. D. (1969). Geometrical models and badness-of-fit functions. In P. R. Krishnaiah (Ed.),Multivariate analysis II (pp. 639–671). New York: Academic Press.Google Scholar
  82. Kruskal, J. B., Young, F. W., & Seery, J. B. (1977).How to use KYST2, a very flexible program to do multidimensional scaling and unfolding. Murray Hill, NJ: AT&T Bell Laboratories.Google Scholar
  83. Landahl, H. D. (1945). Neural mechanisms for the concepts of difference and similarity.Bulletin of Mathematical Biophysics, 7, 83–88.Google Scholar
  84. Larson, R. C., & Sadiq, G. (1983). Facility locations with the Manhattan metric in the presence of barriers to travel.Operations Research, 31, 652–669.Google Scholar
  85. Lashley, K. S. (1942). An examination of the “continuity theory” as applied to discriminative learning.Journal of General Psychology, 26, 241–265.Google Scholar
  86. Le Calvé, G. (1987).L 1-embeddings of a data structure (I, D). In Y. Dodge (Ed.),Statistical data analysis based on the L 1 -norm and related methods (pp. 195–202). New York: North-Holland.Google Scholar
  87. Lew, J. S. (1978). Some counterexamples in multidimensional scaling.Journal of Mathematical Psychology, 17, 247–254.Google Scholar
  88. Lingoes, J. C., & Borg, I. (1983). A quasi-statistical model for choosing between alternative configurations derived from ordinally constrained data.British Journal of Mathematical and Statistical Psychology, 36, 36–53.Google Scholar
  89. Lockhead, G. R., & Pomerantz, J. R. (Eds.)] (in press).The perception of structure. Arlington VA: American Psychological Association.Google Scholar
  90. Lorentz, H. A., Einstein, A., Minkowski, H., & Weyl, H. (1952).The principle of relativity (W. Perrett & G. B. Jeffery, Trans.). New York: Dover. (Original work published 1923)Google Scholar
  91. Melara, R. D. (1989). Similarity relations among synesthetic stimuli and their attributes.Journal of Experimental Psychology: Human Perception and Performance, 15, 212–231.Google Scholar
  92. Mencken, H. L. (1967).The American language. New York: Knopf. (Original work published 1919)Google Scholar
  93. Micko, H. C., & Fischer, W. (1970). The metric of multidimensional psychological spaces as a function of the differential attention to subjective attributes.Journal of Mathematical Psychology, 7, 118–143.Google Scholar
  94. Miller, G. A. (1969). A psychological method to investigate verbal concepts.Journal of Mathematical Psychology, 6, 169–191.Google Scholar
  95. Minkowski, H. (1891). Über die positiven quadratischen Formen und über kettenbruchahnliche Algorithmen [On positive quadratic forms and on algorithms suggesting continued fractions].Journal für die reine und angewandte Mathematik (Crelle's Journal),107, 278–297. (Also available in H. Minkowski, 1967.Gesamelte Abhandlungen, Vol. I. New York: Chelsea)Google Scholar
  96. Minkowski, H. (1908). Space and time. In H. A. Lorentz, A. Einstein, H. Minkowski, & H. Weyl,The principle of relativity (W. Perrett & G. B. Jeffery, Trans., pp. 73–91). New York: Dover. (Original work published 1923)Google Scholar
  97. Minkowski, H. (1957).Diophantische Approximationen [Diophantine approximations]. New York: Chelsea. (Original work published 1907)Google Scholar
  98. Mirkin, B. G., & Chernyi, L. B. (1970). Measurement of the distance between distinct partitions of a finite set of objects.Automation and Remote Control, [Avtomatika i Telemekhanika],1, 786–792.Google Scholar
  99. Muller, J.-C. (1982). Non-Euclidean geographic spaces: Mapping functional distances.Geographical Analysis, 14, 189–203.Google Scholar
  100. Nosofsky, R. M. (1985). Overall similarity and the identification of separable-dimension stimuli: A choice model analysis.Perception & Psychophysics, 38, 414–432.Google Scholar
  101. Nosofsky, R. M. (1986). Attention, similarity, and the identification-categorization relationship.Journal of Experimental Psychology: General, 115, 39–57.Google Scholar
  102. Nosofsky, R. M. (1989). Further tests of an exemplar-similarity approach to relation identification and categorization.Perception & Psychophysics, 45, 279–290.Google Scholar
  103. Nosofsky, R. M. (1992). Similarity scaling and cognitive process models.Annual Review of Psychology, 43, 25–53.Google Scholar
  104. Okada, A., & Imaizumi, T. (1980). Nonmetric method for extended INDSCAL model.Behaviormetrika, 7, 13–22.Google Scholar
  105. Pachella, R. G., Somers, P., & Hardzinski, M. (1981). A psychophysical approach to dimensional integrality. In D. J. Getty & J. H. Howard, Jr. (Eds.),Auditory and visual pattern recognition (pp. 107–126). Hillsdale, NJ: Erlbaum.Google Scholar
  106. Pliner, V. M. (1986). The problem of multivariate metric scaling.Automation and Remote Control, 4, 140–148.Google Scholar
  107. Poole, K. T. (1990). Least squares metric, unidimensional scaling of multivariate linear models.Psychometrika, 55, 123–149.Google Scholar
  108. Ramsay, J. O. (1977). Maximum likelihood estimation in multidimensional scaling.Psychometrika, 42, 241–266.Google Scholar
  109. Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods.Journal of the American Statistical Association, 66, 846–850.Google Scholar
  110. Rao, V. R., & Sabavala, D. J. (1981). Inference of hierarchical choice processes from panel data.Journal of Consumer Research, 8, 85–96.Google Scholar
  111. Richards, J. L. (1988).Mathematical visions: The pursuit of geometry in Victorian England. San Diego: Academic Press.Google Scholar
  112. Richardson, G. D. (1981). The appropriateness of using various Minkowskian metrics for representing cognitive configurations.Environment and Planning A, 13, 475–485.Google Scholar
  113. Richardson, H. W., & Anjomani, A. (1981). The diamond city: The case for rectangular grid models.Socio-Economic Planning Sciences, 15, 295–303.Google Scholar
  114. Robinson, W. S. (1951). A method for chronologically ordering archaeological deposits.American Antiquity, 16, 293–301.Google Scholar
  115. Rosenberg, S. (1982). The method of sorting in multivariate research with applications selected from cognitive psychology and person perception. In N. Hirschberg & L. G. Humphreys (Eds.),Multivariate applications in the social sciences (pp. 117–142). Hillsdale, NJ: Erlbaum.Google Scholar
  116. Rushton, G., & Thill, J. C. (1989). The effect of distance metric on the degree of spatial competition between firms.Environment and Planning A, 21, 499–508.Google Scholar
  117. Russell, B. (1956).An essay on the foundations of geometry. New York: Dover. (Original work published 1897)Google Scholar
  118. Russell, J. A. (1980). A circumplex model of affect.Journal of Personality and Social Psychology, 39, 1161–1178.Google Scholar
  119. Schmalensee, R., & Thisse, J. F. (1986).Perceptual maps and the optimal location of new products (Report No. 86-103). Cambridge, MA: Marketing Science Institute.Google Scholar
  120. Schoenberg, I. J. (1937). On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space.Annals of Mathematics, 38, 787–793.Google Scholar
  121. Schoenberg, I. J. (1938). Metric spaces and positive definite functions.Transactions of the American Mathematical Society, 44, 522–536.Google Scholar
  122. Schönemann, P. H. (1990). Psychophysical maps for rectangles. In H.-O. Geissler (Ed.),Psychophysical explorations of mental structures (pp. 149–164). Toronto: Hogrefe & Huber.Google Scholar
  123. Schönemann, P. H., Dorcey, T., & Kienapple, K. (1985). Subadditive concatenation in dissimilarity judgments.Perception & Psychophysics, 38, 1–17.Google Scholar
  124. Shepard, R. N. (1962a). The analysis of proximities: Multidimensional scaling with an unknown distance function. I.Psychometrika, 27, 125–140.Google Scholar
  125. Shepard, R. N. (1962b). The analysis of proximities: Multidimensional scaling with an unknown distance function. II.Psychometrika, 27, 219–246.Google Scholar
  126. Shepard, R. N. (1964). Attention and the metric structure of the stimulus space.Journal of Mathematical Psychology, 1, 54–87.Google Scholar
  127. Shepard, R. N. (1974). Representation of structure in similarity data: Problems and prospects.Psychometrika, 39, 373–421.Google Scholar
  128. Shepard, R. N. (1978). The circumplex and related topological manifolds in the study of perception. In S. Shye (Ed.),Theory construction and data analysis in the behavioral sciences (pp. 29–80). San Francisco: Jossey-Bass.Google Scholar
  129. Shepard, R. N. (1980a).Internal representation. Unpublished manuscript.Google Scholar
  130. Shepard, R. N. (1980b). Multidimensional scaling, tree-fitting, and clustering.Science, 210, 390–398.Google Scholar
  131. Shepard, R. N. (1986). Discrimination and generalization in identification and classification: Comment on Nosofsky.Journal of Experimental Psychology: General, 115, 58–61.Google Scholar
  132. Shepard, R. N. (1987). Toward a universal law of generalization.Science, 237, 1317–1323.Google Scholar
  133. Shepard, R. N. (in press). Integrality versus separability of stimulus dimensions: From an early convergence of evidence to a proposed theoretical basis. In G. R. Lockhead & J. R. Pomerantz (Eds.),The perception of structure. Arlington VA: American Psychological Association.Google Scholar
  134. Shepard, R. N., & Arabie, P. (1979). Additive clustering: Representation of similarities as combinations of discrete overlapping properties.Psychological Review, 86, 87–123.Google Scholar
  135. Shepard, R. N., & Cermak, G. W. (1973). Perceptual-cognitive explorations of a toroidal set of free-form stimuli.Cognitive Psychology, 4, 351–377.Google Scholar
  136. Sherman, C. R. (1972). Nonmetric multidimensional scaling: A Monte Carlo study of the basic parameters,Psychometrika, 37, 323–355.Google Scholar
  137. Shreider, Y. A. (1974).What is distance? (L. Cohn & H. Edelberg, Trans.). Chicago: University of Chicago Press.Google Scholar
  138. Suppes, P., Krantz, D. M., Luce, R. D., & Tversky, A. (1989).Foundations of measurement: Vol. II. Geometrical, threshold, and probabilistic representations. New York: Academic Press.Google Scholar
  139. Thisse, J.-F., Ward, J. E., & Wendell, R. E. (1984). Some properties of location problems with block and round norms.Operations Research, 32, 1309–1327.Google Scholar
  140. Tong, Y. C., Dowell, R. C., Blamey, P. J., & Clark, G. M. (1983). Two component hearing sensations produced by two-electrode stimulation in the cochlea of deaf patient.Science, 219, 993–994.Google Scholar
  141. Torgerson, W. S. (1958).Theory and methods of scalling. New York: Wiley.Google Scholar
  142. Townes, B. D., & Abbott, R. D. (1974).The psychological interpretation of metrics in ordinal multidimensional scaling (JSAS #604). Washington, DC: American Psychological Association.Google Scholar
  143. Trudeau, R. J. (1987).The Non-Euclidean revolution. Boston: Birkhäuser.Google Scholar
  144. Tversky, A., & Gati, I. (1982). Similarity, separability, and the triangle inequality.Psychological Review, 89, 123–154.Google Scholar
  145. Viegas, J., & Hansen, P. (1985). Finding shortest paths in the plane in the presence of barriers to travel (for anyl p-norm).European Journal of Operational Research, 20, 373–381.Google Scholar
  146. Von Hohenbalken, B., & West, D. S. (1984). Manhattan versus Euclid: Market areas computed and compared.Regional Science and Urban Economics, 14, 19–35.Google Scholar
  147. Ward, J. E., & Wendell, R. E. (1985). Using block norms for location modeling.Operations Research, 33, 1074–1090.Google Scholar
  148. Wiener-Ehrlich, W. K. (1978). Dimensional and metric structures in multidimensional stimuli.Perception & Psychophysics, 24, 399–414.Google Scholar
  149. Wiggins, J. S. (1982). Circumplex models of interpersonal behavior in clinical psychology. In P. C. Kendall & J. N. Butcher (Eds.),Handbook of research methods in clinical psychology (pp. 183–221). New York: Wiley.Google Scholar
  150. Wilkie, D. M. (1989). Evidence that pigeons represent Euclidean properties of space.Journal of Experimental Psychology: Animal Behavior Processes, 15, 114–123.Google Scholar
  151. Wise, J. A., & Mockovak, W. P. (1973). Descriptive modeling of subjective probabilities.Organizational Behavior and Human Performance, 9, 292–306.Google Scholar
  152. Wolfrum, C. (1976a). Zum Auftreten quasiaequivalenter Lösungen bei einer Verallgemeinerung des Skalierungsverfahrens von Kruskal auf metrische Räume mit einer Minkowski-Metrik [On the existence of quasi-equivalent solutions with a generalization of Kruskal's scaling procedure in metric spaces with a Minkowski metric].Archiv für Psychologie, 128, 96–111.Google Scholar
  153. Wolfrum, C. (1976b). Zur Bestimmung eines optimalen Metrikoeffizientenr mit dem Skailerungsverfahren von Kruskal [On the determination of optimal metric coefficientsr with Kruskal's scaling procedure].Zeitschrift für experimentelle und angewandte Psychologie, 23, 339–350.Google Scholar

Reference

  1. Glazer, R., & Nakamoto, K. (1991). Cognitive geometry: An analysis of structure underlying representations of similarity.Marketing Science, 10, 205–228.Google Scholar

Copyright information

© The Psychometric Society 1991

Authors and Affiliations

  • Phipps Arabie
    • 1
  1. 1.Graduate School of ManagementRutgers UniversityNewark

Personalised recommendations