A MAXIMUM LIKELIHOOD METHOD FOR NONMETRIC MULTIDIMENSIONAL SCALING I. THE CASE IN WHICH ALL EMPIRICAL PAIRWISE ORDERINGS ARE INDEPENDENT-THEORY

YOSHIO TAKANE

doi:10.4992/psycholres1954.20.7

A MAXIMUM LIKELIHOOD METHOD FOR NONMETRIC MULTIDIMENSIONAL SCALING

I. THE CASE IN WHICH ALL EMPIRICAL PAIRWISE ORDERINGS ARE INDEPENDENT-THEORY

YOSHIO TAKANE

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JOURNAL FREE ACCESS

1978 Volume 20 Issue 1 Pages 7-17

DOI https://doi.org/10.4992/psycholres1954.20.7

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Published: June 30, 1978 Received: July 28, 1977 Available on J-STAGE: February 24, 2009 Accepted: - Advance online publication: - Revised: -
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Date of correction: February 24, 2009 Reason for correction: - Correction: TITLE Details: Wrong : A MAXIMUM LIKELIHOOD METHOD FOR NONMETRIG MULTIDIMENSIONAL SCALING
Right : A MAXIMUM LIKELIHOOD METHOD FOR NONMETRIC MULTIDIMENSIONAL SCALING

Date of correction: February 24, 2009 Reason for correction: - Correction: ABSTRACT Details: Wrong : A maximum likelihood estimation procedure is developed for nonmetric multidimensional scaling (MDS) which applies to the situation in which all empirical pairwise ordcrings of dissimilarities are assumed to be independent. The proposed method, while formulated within Thustonian framework, does not presuppose initial unidimensional scaling of “observed” distances. Rather, the original nonmertric data (which are the set of empirical ordinal relations on the dissimilarities between stimuli) are directly related to the representation model (which is a distance function of some form) through a single optimization criterion based on the maximum likelihood principle.
Right : A maximum likelihood estimation procedure is developed for nonmetric multidimensional scaling (MDS) which applies to the situation in which all empirical pairwise orderings of dissimilarities are assumed to be independent. The proposed method, while formulated within Thurstonian framework, does not presuppose initial unidimensional scaling of “observed” distances. Rather, the original nonmetric data (which are the set of empirical ordinal relations on the dissimilarities between stimuli) are directly related to the representation model (which is a distance function of some form) through a single optimization criterion based on the maximum likelihood principle.

Date of correction: February 24, 2009 Reason for correction: - Correction: CITATION Details: Wrong : AITCHISON, J., & BROWN, J. A. C. 1963 The lognormal distribution. Cambridge: The Cambridge University Press.
AKAIKE, H. 1976 On entropy maximization principle. Paper presented at the Symposium on Applications of Statistics, Dayton, Ohio.
BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M., & BRUNK, H. D. 1970 Statistical inference under order restrictions. London: Wiley.
BOCK, R. D., & JONES, L. V. 1968 The measurement and prediction of judgment and choice. San Francisco: Holden-Day.
BRADLEY, E. L. 1973 The equivalence of maximum likelihood and weighted least squares estimates in the exponential family. Journal of the American Statistical Association, 68, 199-200.
HUYNH, H., & FELDT, L. S. 1970 Conditions under which mean square ratios in repeated measurements designs have exact F-distributions. Journal of the American Statistical Association, 65, 1582-1589.
JENNRICH, R. I., & MOORE, R. H. 1975 Maximum likelihood estimation by means of nonlinear least squares. Research Bulletin RB_??_7, Educational Testing Service, Princeton, N. J.
KLINGBERG, F. L. 1941 Studies in measurement of the relations among sovereign states. Psychometrika, 6, 335-352.
KRUSKAL, J. B. 1964 Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-29.
MESSICK, S. J. 1956 An empirical evaluation of multidimensional successive intervals. Psychometrika, 21, 367-375.
MOSTELLER, F. 1951 Remarks on the method of paired comparisons. I. The least squares solution assuming equal standard deviations and correlations. Psychometrika, 16, 203-206.
NAKATANI, L. H. 1972 Confusion-choice model for multidimensional psychophysics. Journal of Mathematical Psychology, 9, 104-127.
RAMSAY, J. O. 1976 Two algorithms and various statistical models for multidimensional scaling by maximum likelihood. Unpublished paper, McGill University.
RAMSAY, J. O. 1977 Maximum likelihood es- timation in multidimensional scaling. Psychometrika, 42, 241-266.
RICHARDSON, M. W. 1938 Multidimensional psychophysics. Psychological Bulletin, 35, 659.
SAITO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SHEPARD, R. N. 1962 The analysis of proximities: Multidimensional scaling with unknown distance functions, I and II. Psychometrika, 27, 125-140; 219-246.
SJÖBERG, L. 1967 Successive intervals scaling of paired comparisons. Psychometrika, 32, 297-308.
SUPPES, P., & ZINNES, J. L. 1963 Basic measure- ment theory. In R.. D. Luce et al. (Eds.), Handbook of mathematical psychology, Vol. I. New York: Wiley.
TAKANE, Y. 1977 Statistical procedures for nonmetric multidimensional scaling. Unpublished doctoral dissertation. The University of North Carolina.
TAKANE. Y. 1978 A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent-evaluations. Japanese Psychological Research, 20, (in press).
TORGERSON, W. S. 1952 Multidimesional scaling: I. Theory and method. Psychometrika, 17, 273-286.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
WILKS, S. S. 1962 Mathematical statistics. New York: Wiley.
YOUNG, F. W. 1975 Scaling replicated condi- tional rank-order data. In D. R. Heise (Ed.), Sociological methodology. San Francisco: Jossey-Bass Publishers.
ZINNES, J. L., & GRIGGS, R. A. 1974 Probabilistic multidimensional unfolding analysis. Psychometrika, 39, 327-350.

Right : AITCHISON, J., & BROWN, J. A. C. 1963 The lognormal distribution. Cambridge: The Cambridge University Press.
AKAIKE, H. 1976 On entropy maximization principle. Paper presented at the Symposium on Applications of Statistics, Dayton, Ohio.
BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M., & BRUNK, H. D. 1970 Statistical inference under order restrictions. London: Wiley.
BOCK, R. D., & JONES, L. V. 1968 The measuremerit and prediction of judgment and choice. San Francisco: Holden-Day.
BRADLEY, E. L. 1973 The equivalence of maximum likelihood and weighted least squares estimates in the exponential family. Journal of the American Statistical Association, 68, 199-200.
HUYNH, H., & FELDT, L. S. 1970 Conditions under which mean square ratios in repeated measurements designs haveexact F-distributions. Journal of the American Statistical Association, 65, 1582-1589.
JENNRICH, R. I., & MOORE, R. H. 1975 Maximum likelihood estimation by means of nonlinear least squares, Research Bulletin RB-75-7, Educational Testing Service, Princeton, N. J.
KLINCBERG, F. L. 1941 Studies in measurement of the relations among sovereign states. Psychometrika, 6, 335-352.
KRUSRAL, J. B. 1964 Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-29.
MESSICK, S. J. 1956 An empirical evaluation of multidimensional successive intervals. Psychometrika, 21, 367-375.
MOSTELLER, F.1951 Remarks on the method of paired comparisons. I. The least squares solution assuming equal standard deviations and correlations. Psychometrika, 16, 203-206.
NAKATANI, L. H.1972 Confusion-choice model for multidimensional psychophysics. Journal of Mathematical Psychology, 9, 104-127.
RAMSAY, J. O.1976 Two algorithms and various statistical models for multidimensional scaling by maximum likelihood. paper, McGill University.
RAMSAY, J. O. 1977 Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42, 241-266.
RICHARDSON, M. W. 1938 Multidimensional psychophysics. Psychological Bulletin, 35, 659.
SATTO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SHEPARD, R. N. 1962 The analysis of proximities: Multidimensional scaling with unknown distance functions, I and II. Psychometrika, 27, 125-140.
SHEPARD, R. N. 1962 The analysis of proximities: Multidimensional scaling with unknown distance functions, I and II. Psychometrika, 27, 219-246.
SJÖBERG, L. 1967 Successive intervals scaling of paired comparisons. Psychometrika, 32, 297-308.
SUPPES, P., & ZINNES, J. L. 1963 Basic measurement theory. In R. D. Luce et al.(Eds.), Handbook of mathematical psychology, Vol.I. New York: Wiley.
TAKANE, Y. 1977 Statistical procedures for nonmetric multidimensional scaling. doctoral dissertation. The University of North Carolina.
TAKANE, Y. 1978 A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent-evaluations. Japanese Psrchological Research, 20, (in press).
TORGERSON, W. S. 1952 Multidimesional scaling: I. Theory and method. Psychometrika, 17, 273-286.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
WILKS, S. S. 1962 Mathematical statistics. New York: Wiley.
YOUNG, F. W. 1975 Scaling replicated conditional rank-order data. In D. R. Heise (Ed.), Sociological methodology. San Francisco: Jossey-Bass Publishers.
ZINNES, J. L., & GRIGGS, R. A. 1974 Probabilistic multidimensional unfolding analysis. Psychometrika, 39, 327-350.

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 : AITCHISON, J., & BROWN, J. A. C. 1963 The lognormal distribution. Cambridge: The Cambridge University Press.
AKAIKE, H. 1976 On entropy maximization principle. Paper presented at the Symposium on Applications of Statistics, Dayton, Ohio.
BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M., & BRUNK, H. D. 1970 Statistical inference under order restrictions. London: Wiley.
BOCK, R. D., & JONES, L. V. 1968 The measuremerit and prediction of judgment and choice. San Francisco: Holden-Day.
BRADLEY, E. L. 1973 The equivalence of maximum likelihood and weighted least squares estimates in the exponential family. Journal of the American Statistical Association, 68, 199-200.
HUYNH, H., & FELDT, L. S. 1970 Conditions under which mean square ratios in repeated measurements designs haveexact F-distributions. Journal of the American Statistical Association, 65, 1582-1589.
JENNRICH, R. I., & MOORE, R. H. 1975 Maximum likelihood estimation by means of nonlinear least squares, Research Bulletin RB-75-7, Educational Testing Service, Princeton, N. J.
KLINCBERG, F. L. 1941 Studies in measurement of the relations among sovereign states. Psychometrika, 6, 335-352.
KRUSRAL, J. B. 1964 Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-29.
MESSICK, S. J. 1956 An empirical evaluation of multidimensional successive intervals. Psychometrika, 21, 367-375.
MOSTELLER, F.1951 Remarks on the method of paired comparisons. I. The least squares solution assuming equal standard deviations and correlations. Psychometrika, 16, 203-206.
NAKATANI, L. H.1972 Confusion-choice model for multidimensional psychophysics. Journal of Mathematical Psychology, 9, 104-127.
RAMSAY, J. O.1976 Two algorithms and various statistical models for multidimensional scaling by maximum likelihood. paper, McGill University.
RAMSAY, J. O. 1977 Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42, 241-266.
RICHARDSON, M. W. 1938 Multidimensional psychophysics. Psychological Bulletin, 35, 659.
SATTO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SHEPARD, R. N. 1962 The analysis of proximities: Multidimensional scaling with unknown distance functions, I and II. Psychometrika, 27, 125-140.
SHEPARD, R. N. 1962 The analysis of proximities: Multidimensional scaling with unknown distance functions, I and II. Psychometrika, 27, 219-246.
SJÖBERG, L. 1967 Successive intervals scaling of paired comparisons. Psychometrika, 32, 297-308.
SUPPES, P., & ZINNES, J. L. 1963 Basic measurement theory. In R. D. Luce et al.(Eds.), Handbook of mathematical psychology, Vol.I. New York: Wiley.
TAKANE, Y. 1977 Statistical procedures for nonmetric multidimensional scaling. doctoral dissertation. The University of North Carolina.
TAKANE, Y. 1978 A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent-evaluations. Japanese Psrchological Research, 20, (in press).
TORGERSON, W. S. 1952 Multidimesional scaling: I. Theory and method. Psychometrika, 17, 273-286.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
WILKS, S. S. 1962 Mathematical statistics. New York: Wiley.
YOUNG, F. W. 1975 Scaling replicated conditional rank-order data. In D. R. Heise (Ed.), Sociological methodology. San Francisco: Jossey-Bass Publishers.
ZINNES, J. L., & GRIGGS, R. A. 1974 Probabilistic multidimensional unfolding analysis. Psychometrika, 39, 327-350.

Right : AITCHISON, J., & BROWN, J. A. C. 1963 The lognormal distribution. Cambridge: The Cambridge University Press.
AKAIKE, H. 1976 On entropy maximization principle. Paper presented at the Symposium on Applications of Statistics, Dayton, Ohio.
BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M., & BRUNK, H. D. 1970 Statistical inference under order restrictions. London: Wiley.
BOCK, R. D., & JONES, L. V. 1968 The measurement and prediction of judgment and choice. San Francisco: Holden-Day.
BRADLEY, E. L. 1973 The equivalence of maximum likelihood and weighted least squares estimates in the exponential family. Journal of the American Statistical Association, 68, 199-200.
HUYNH, H., & FELDT, L. S. 1970 Conditions under which mean square ratios in repeated measurements designs have exact F-distributions. Journal of the American Statistical Association, 65, 1582-1589.
JENNRICH, R. I., & MOORE, R. H. 1975 Maximum likelihood estimation by means of nonlinear least squares, Research Bulletin RB-75-7, Educational Testing Service, Princeton, N. J.
KLINGBERG, F. L. 1941 Studies in measurement of the relations among sovereign states. Psychometrika, 6, 335-352.
KRUSKAL, J. B. 1964 Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-29.
MESSICK, S. J. 1956 An empirical evaluation of multidimensional successive intervals. Psychometrika, 21, 367-375.
MOSTELLER, F. 1951 Remarks on the method of paired comparisons. I. The least squares solution assuming equal standard deviations and correlations. Psychometrika, 16, 203-206.
NAKATANI, L. H. 1972 Confusion-choice model for multidimensional psychophysics. Journal of Mathematical Psychology, 9, 104-127.
RAMSAY, J. O. 1976 Two algorithms and various statistical models for multidimensional scaling by maximum likelihood. paper, McGill University.
RAMSAY, J. O. 1977 Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42, 241-266.
RICHARDSON, M. W. 1938 Multidimensional psychophysics. Psychological Bulletin, 35, 659.
SAITO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SHEPARD, R. N. 1962 The analysis of proximities: Multidimensional scaling with unknown distance functions, I and II. Psychometrika, 27, 125-140.
SHEPARD, R. N. 1962 The analysis of proximities: Multidimensional scaling with unknown distance functions, I and II. Psychometrika, 27, 219-246.
SJÖBERG, L. 1967 Successive intervals scaling of paired comparisons. Psychometrika, 32, 297-308.
SUPPES, P., & ZINNES, J. L. 1963 Basic measurement theory. In R.. D. Luce et al.(Eds.), Handbook of mathematical psychology, Vol.I. New York: Wiley.
TAKANE, Y. 1977 Statistical procedures for nonmetric multidimensional scaling. Unpublished doctoral dissertation. The University of North Carolina.
TAKANE, Y. 1978 A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent-evaluations. Japanese Psychological Research, 20,(in press).
TORGERSON, W. S. 1952 Multidimesional scaling: I. Theory and method. Psychometrika, 17, 273-286.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
WILKS, S. S. 1962 Mathematical statistics. New York: Wiley.
YOUNG, F. W. 1975 Scaling replicated conditional rank-order data. In D. R. Heise (Ed.), Sociological methodology. San Francisco: Jossey-Bass Publishers.
ZINNES, J. L., & GRIGGS, R. A. 1974 Probabilistic multidimensional unfolding analysis. Psychometrika, 39, 327-350.

Abstract

A maximum likelihood estimation procedure is developed for nonmetric multidimensional scaling (MDS) which applies to the situation in which all empirical pairwise orderings of dissimilarities are assumed to be independent. The proposed method, while formulated within Thurstonian framework, does not presuppose initial unidimensional scaling of “observed” distances. Rather, the original nonmetric data (which are the set of empirical ordinal relations on the dissimilarities between stimuli) are directly related to the representation model (which is a distance function of some form) through a single optimization criterion based on the maximum likelihood principle.

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