A new nonmetric multidimensional scaling method for sorting data

HIROSHI HOJO

doi:10.4992/psycholres1954.35.129

HIROSHI HOJO

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Keywords: multidimensional scaling (MDS), individual differences scaling (INDSCAL), response models, sorting, face perception

JOURNAL FREE ACCESS

1993 Volume 35 Issue 3 Pages 129-139

DOI https://doi.org/10.4992/psycholres1954.35.129

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Published: November 25, 1993 Received: October 13, 1992 Available on J-STAGE: February 24, 2009 Accepted: July 03, 1993 Advance online publication: - Revised: -
Correction information
Date of correction: February 24, 2009 Reason for correction: - Correction: CITATION Details: Wrong : de Leeuw, J., & Heiser, W. 1977 Convergence of correction-matrix algorithms for multidimensional scaling. In J. C. Lingoes (Ed.), Geometric representations of relational data. Ann Arbor, Mich.: Mathesis Press. Pp. 735-752.
Guttman, L. 1968 A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469-506.
Hojo, H. 1983 A maximum likelihood method for additive clustering and its applications. Japanese Psychological Research, 25, 191-201.
Hojo, H. 1986 Response threshold models for binary ranking and sorting. Behaviormetrika, 20, 1-12.
Hojo, H. 1991 An analysis of three Japanese verbs of breaking: An empirical study of word meaning. Japanese Psychological Research, 33, 64-74.
Kruskal, J. B. 1964a Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-27.
Kruskal, J. B. 1964b Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115-129.
Kunihiro, T. 1970 Imi no shoso (Aspects of meaning). Tokyo: Sanscido. (In Japanese)
Mizuno, R. 1977 Niteiru mono no zahyo (Representation of objects similar to each other in a multidimensional space). In O. Fujita, T. Mori & Y. Isogai (Eds.), Kokoro no jikkenshitsu (Laboratories for psychological researches. Vol. II). Tokyo: Fukumura Shuppan. Pp. 264-282. (In Japanese)
Takane, Y. 1980a Tajigen shakudoho (Multidimensional scaling methods). Tokyo: University of Tokyo Press. (In Japanese)
Takane, Y. 1980b Analysis of categorizing behavior by a quantification method. Behaviormetrika, 8, 75-86.
Takane, Y. 1981 Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, 9-28.
Takane, Y. 1982 Process models for sorting. Proceedings of the 10th Conference of the Behaviormetric Society of Japan at National Language Research Institute, 56-57.
Tone, K. 1981 BASIC. Tokyo: Baifukan. (In Japanese)

Right : de Leeuw, J., & Heiser, W. 1977 Convergence of correction-matrix algorithms for multidimensional scaling. In J. C. Lingoes (Ed.), Geometric representations of relational data. Arm Arbor, Mich.: Mathesis Press. Pp.735-752.
Guttman, L. 1968 A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469-506.
Hojo, H. 1983 A maximum likelihood method for additive clustering and its applications. Japanese Psychological Research, 25, 191-201.
Hojo, H. 1986 Response threshold models for binary ranking and sorting. Behaviormetrika, 20, 1-12.
Hojo, H. 1991 An analysis of three Japanese verbs of breaking: An empirical study of word meaning. Japanese Psychological Research, 33, 64-74.
Kruskal, J. B. 1964a Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-27.
Kruskal, J. B. 19646 Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115-129.
Kunihiro, T. 1970 1mi no shoso (Aspects of meaning). Tokyo: Sanseido.(In Japanese)
Mizuno, R. 1977 Niteiru mono no zahyo (Representation of objects similar to each other in a multidimensional space). In O. Fujita, T. Mori & Y. Isogai (Eds.), Kokoro no jikkenshitsu (Laboratories for psychological researches. Vol.II). Tokyo: Fukumura Shuppan. Pp.264-282.(In Japanese)
Takane, Y. 1980a Tajigen shakudoho (Multidimensional scaling methods). Tokyo: University of Tokyo Press.(In Japanese)
Takane, Y. 1980b Analysis of categorizing behavior by a quantificationmethod. Behaviormetrika, 8, 75-86.
Takane, Y. 1981 Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, 9-28.
Takane, Y. 1982 Process models for sorting. Proceedings of the IOth Conference of the Behaviormetric Society of Japan at National Language Research Institute, 56-57.
Tone, K. 1981 BASIC. Tokyo: Baifukan.(In Japanese)

Date of correction: February 24, 2009 Reason for correction: - Correction: PDF FILE Details: -

Date of correction: February 24, 2009 Reason for correction: - Correction: CITATION Details: Wrong : de Leeuw, J., & Heiser, W. 1977 Convergence of correction-matrix algorithms for multidimensional scaling. In J. C. Lingoes (Ed.), Geometric representations of relational data. Arm Arbor, Mich.: Mathesis Press. Pp.735-752.
Guttman, L. 1968 A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469-506.
Hojo, H. 1983 A maximum likelihood method for additive clustering and its applications. Japanese Psychological Research, 25, 191-201.
Hojo, H. 1986 Response threshold models for binary ranking and sorting. Behaviormetrika, 20, 1-12.
Hojo, H. 1991 An analysis of three Japanese verbs of breaking: An empirical study of word meaning. Japanese Psychological Research, 33, 64-74.
Kruskal, J. B. 1964a Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-27.
Kruskal, J. B. 19646 Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115-129.
Kunihiro, T. 1970 1mi no shoso (Aspects of meaning). Tokyo: Sanseido.(In Japanese)
Mizuno, R. 1977 Niteiru mono no zahyo (Representation of objects similar to each other in a multidimensional space). In O. Fujita, T. Mori & Y. Isogai (Eds.), Kokoro no jikkenshitsu (Laboratories for psychological researches. Vol.II). Tokyo: Fukumura Shuppan. Pp.264-282.(In Japanese)
Takane, Y. 1980a Tajigen shakudoho (Multidimensional scaling methods). Tokyo: University of Tokyo Press.(In Japanese)
Takane, Y. 1980b Analysis of categorizing behavior by a quantificationmethod. Behaviormetrika, 8, 75-86.
Takane, Y. 1981 Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, 9-28.
Takane, Y. 1982 Process models for sorting. Proceedings of the IOth Conference of the Behaviormetric Society of Japan at National Language Research Institute, 56-57.
Tone, K. 1981 BASIC. Tokyo: Baifukan.(In Japanese)

Right : de Leeuw, J., & Heiser, W. 1977 Convergence of correction-matrix algorithms for multidimensional scaling. In J. C. Lingoes (Ed.), Geometric representations of relational data. Ann Arbor, Mich.: Mathesis Press. Pp.735-752.
Guttman, L. 1968 A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469-506.
Hojo, H. 1983 A maximum likelihood method for additive clustering and its applications. Japanese Psychological Research, 25, 191-201.
Hojo, H. 1986 Response threshold models for binary ranking and sorting. Behaviormetrika, 20, 1-12.
Hojo, H. 1991 An analysis of three Japanese verbs of breaking: An empirical study of word meaning. Japanese Psychological Research, 33, 64-74.
Kruskal, J. B. 1964a Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-27.
Kruskal, J. B. 1964b Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115-129.
Kunihiro, T. 1970 Imi no shoso (Aspects of meaning). Tokyo: Sanseido.(In Japanese)
Mizuno, R. 1977 Niteiru mono no zahyo (Representation of objects similar to each other in a multidimensional space). In O. Fujita, T. Mori & Y. Isogai (Eds.), Kokoro no jikkenshitsu (Laboratories for psychological researches. Vol.II). Tokyo: Fukumura Shuppan. Pp.264-282.(In Japanese)
Takane, Y. 1980a Tajigen shakudoho (Multidimensional scaling methods). Tokyo: University of Tokyo Press.(In Japanese)
Takane, Y. 1980b Analysis of categorizing behavior by a quantification method. Behaviormetrika, 8, 75-86.
Takane, Y. 1981 Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, 9-28.
Takane, Y. 1982 Process models for sorting. Proceedings of the 10th Conference of the Behaviormetric Society of Japan at National Language Research Institute, 56-57.
Tone, K. 1981 BASIC. Tokyo: Baifukan.(In Japanese)

Abstract

A multidimensional scaling model to deal with sorting data is developed. The model assumes (a) that stimuli are embedded in a multidimensional space, and (b) that a particular sorting is generated if and only if all intra-cluster distances between stimuli in the space are smaller than all inter-cluster distances. A new parameter estimation method is proposed which yields a configuration that satisfies the above requirement (b) as much as possible for each sorting given as data. This model is extended to the individual differences scaling model. The models are applied to a set of artificial sorting data and two sets of real data.

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