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

YOSHIO TAKANE

doi:10.4992/psycholres1954.20.105

A MAXIMUM LIKELIHOOD METHOD FOR NONMETRIC MULTIDIMENSIONAL SCALING

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

YOSHIO TAKANE

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JOURNAL FREE ACCESS

1978 Volume 20 Issue 3 Pages 105-114

DOI https://doi.org/10.4992/psycholres1954.20.105

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Published: December 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: ABSTRACT Details: Wrong : A maximum likelihood estimation procedure for nonmetric multidimensional scaling (MAXZSCAL-l) described in the previous paper (Takane, 1978) is evaluated using both Monte Carlo and real data. Two Monte Carlo studies are designed each with specific objectives. In the first study various aspects (numerical quality of estimates, robustness, etc.) of solutions are examined as functions of the number of replications per tetrad, the number of observations (the number of tetrads for which observations are made) and the magnitude of discriminal dispersions. The results strongly suggest the importance of replicated observations for obtaining solutions of “good” qualities. In the second Monte Carlo study the effects of a particular type of systematic violations of distributional assumptions are inspected. The estimates of the location parameters (stimulus coordinates) are found to be less susceptible to the kind of distributional violations examined here, while the goodness of fit statistics (the chi-square, the AIC) tend to overestimate the correct dimensionality of the representation space. Two sets of real data are analyzed to demonstrate the advatages of the current procedure, namely in the availability of confidence regions, the availability of the goodness of fit statistics, and the constrained optimization feature for testing hypotheses.
Right : A maximum likelihood estimation procedure for nonmetric multidimensional scaling (MAXSCAL-1) described in the previous paper (Takane, 1978) is evaluated using both Monte Carlo and real data. Two Monte Carlo studies are designed each with specific objectives. In the first study various aspects (numerical quality of estimates, robustness, etc.) of solutions are examined as functions of the number of replications per tetrad, the number of observations (the number of tetrads for which observations are made) and the magnitude of discriminal dispersions. The results strongly suggest the importance of replicated observations for obtaining solutions of “good” qualities. In the second Monte Carlo study the effects of a particular type of systematic violations of distributional assumptions are inspected. The estimates of the location parameters (stimulus coordinates) are found to be less susceptible to the kind of distributional violations examined here, while the goodness of fit statistics (the chi-square, the AIC) tend to overestimate the correct dimensionality of the representation space. Two sets of real data are analyzed to demonstrate the advantages of the current procedure, namely the availability of confidence regions, the availability of the goodness of fit statistics, and the constrained optimization feature for testing hypotheses.

Date of correction: February 24, 2009 Reason for correction: - Correction: AFFILIATION Details: Wrong :
1) Department of Psychology, McGill University

Right :
1) Deportment of Psychology, McGill University

Date of correction: February 24, 2009 Reason for correction: - Correction: CITATION Details: Wrong : MOSTELLLR, F. 1951 Remarks on the method of paired comparisons. I. The least squares solution assuming equal standard deviations and correlations. Psychometrika, 16, 203-206.
SAITO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SCHÖNEMANN, P. H., & CARROLL, R. M. 1970 Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, 35, 245-255.
TAKANE, Y. 1975 Generalized uni- and multidimensional Thurstonian scaling: A maximum likelihood method. Unpublished manuscript. The University of North Carolina.
TAXANE, 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-theory. Japanese Psychological Research, 20, 7-17.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
TVERSRY, A., & Russo, J. E. 1969 Substitutability and similarity in binary choices. Journal of Mathematical Psychology, 6, 1-12.

Right : MOSTELLLR, F. 1951 Remarks on the method of paired comparisons. I. The least squares solution assuming equal standard deviations and correlations. Psychometrika, 16, 203-206.
SAITO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SCHÖNEMANN, P. H., & CARROLL, R. M. 1970 fi tting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, 35, 245-255.
TAKANE, Y. 1978 A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings arc independent-theory. Japanese Psychological Research, 20, 7-17.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
TVERSKY, A., & Russo, J. E. 1969 Substitutability and similarity in binary choices., Journal of Mathematical Psychology, 6, 1-12.

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

Date of correction: February 24, 2009 Reason for correction: - Correction: SUBTITLE Details: [in Japanese]

Date of correction: February 24, 2009 Reason for correction: - Correction: AFFILIATION Details: Wrong :
1) Deportment of Psychology, McGill University

Right :
1) Department of Psychology, McGill University

Date of correction: February 24, 2009 Reason for correction: - Correction: CITATION Details: Wrong : MOSTELLLR, F. 1951 Remarks on the method of paired comparisons. I. The least squares solution assuming equal standard deviations and correlations. Psychometrika, 16, 203-206.
SAITO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SCHÖNEMANN, P. H., & CARROLL, R. M. 1970 fi tting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, 35, 245-255.
TAKANE, Y. 1978 A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings arc independent-theory. Japanese Psychological Research, 20, 7-17.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
TVERSKY, A., & Russo, J. E. 1969 Substitutability and similarity in binary choices., Journal of Mathematical Psychology, 6, 1-12.

Right : 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.
SAITO, T. 1974 Multidimensional Thurstonian scaling and its applications (I). Nippon Univac Soken Kiyo, 4, 87-112.
SCHÖNEMANN, P. H., & CARROLL, R. M. 1970 fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, 35, 245-255.
TAKANE, Y. 1975 Generalized uni-and multidimensional Thurstonian scaling: A maximum likelihood method. Unpublished manuscript. The University of North Carolina.
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-theory. Japanese Psychological Research, 20, 7-17.
TORGERSON, W. S. 1958 Theory and methods of scaling. New York: Wiley.
TVERSKY, A., & RUSSO, J. E. 1969 Substitutability and similarity in binary choices., Journal of Mathematical Psychology, 6, 1-12.

Abstract

A maximum likelihood estimation procedure for nonmetric multidimensional scaling (MAXSCAL-1) described in the previous paper (Takane, 1978) is evaluated using both Monte Carlo and real data. Two Monte Carlo studies are designed each with specific objectives. In the first study various aspects (numerical quality of estimates, robustness, etc.) of solutions are examined as functions of the number of replications per tetrad, the number of observations (the number of tetrads for which observations are made) and the magnitude of discriminal dispersions. The results strongly suggest the importance of replicated observations for obtaining solutions of “good” qualities. In the second Monte Carlo study the effects of a particular type of systematic violations of distributional assumptions are inspected. The estimates of the location parameters (stimulus coordinates) are found to be less susceptible to the kind of distributional violations examined here, while the goodness of fit statistics (the chi-square, the AIC) tend to overestimate the correct dimensionality of the representation space. Two sets of real data are analyzed to demonstrate the advantages of the current procedure, namely the availability of confidence regions, the availability of the goodness of fit statistics, and the constrained optimization feature for testing hypotheses.

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