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Probabilistic Multidimensional Scaling: Complete and Incomplete Data

Published online by Cambridge University Press:  01 January 2025

Joseph L. Zinnes  and
David B. MacKay
Joseph L. Zinnes*
Affiliation:
University of Illinois
David B. MacKay
Affiliation:
Indiana University
*
Requests for reprints should be addressed to Joseph Zinnes, School of Social Science, 220 Lincoln Hall, University of Illinois, Urbana, Illinois 61801.
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Abstract

Simple procedures are described for obtaining maximum likelihood estimates of the location and uncertainty parameters of the Hefner model. This model is a probabilistic, multidimensional scaling model, which assigns a multivariate normal distribution to each stimulus point. It is shown that for such a model, standard nonmetric and metric algorithms are not appropriate.

A procedure is also described for constructing incomplete data sets, by taking into consideration the degree of familiarity the subject has for each stimulus. Maximum likelihood estimates are developed both for complete and incomplete data sets.

Keywords

multidimensional scalingnonmetric scalingmaximum likelihood estimationcomplete and incomplete datanoncentral chi-square approximations
Type
Original Paper
Information
Psychometrika , Volume 48 , Issue 1 , March 1983 , pp. 27 - 48
Copyright
Copyright © 1983 The Psychometric Society

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Footnotes

This research was supported by National Science Grant No. SOC76-20517. The first author would especially like to express his gratitude to the Netherlands Institute for Advanced Study for its very substantial help with this research.

References

Reference Notes

Zinnes, J. L., MacKay, D. B. & Williams, B. A. Incomplete data sets for multidimensional scaling. Unpublished manuscript, 1979.Google Scholar
Kruskal, J. B., Young, F. W. & Seery, J. B. How to use KYST, a very flexible program to do multidimensional scaling and unfolding, Murray Hill, N.J.: Bell Telephone Lab., 1973.Google Scholar
Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, Washington, DC: US Government Printing Office, 1967.Google Scholar
Bechtel, G. G. Multidimensional preference scaling, The Hague: Mouton, 1976.CrossRefGoogle Scholar
Carroll, J. D. Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K. & Nerlove, S. B. (Eds.), Multidimensional Scaling. Vol. I . New York: Seminar Press. 1972, 105–155.Google Scholar
Chandler, J. P. STEPIT—Finds local minima of a smooth function of several parameters. Behavioral Science, 1969, 14, 81–82.Google Scholar
Coombs, C. H., Greenberg, M., & Zinnes, J. L. A double law of comparative judgment for the analysis of preferential choice and similarities data. Psychometrika, 1961, 26, 165–171.CrossRefGoogle Scholar
Graef, F. & Spence, I. Using distance information in the design of large multidimensional scaling experiments. Psychological Bulletin, 1979, 86, 60–66.CrossRefGoogle Scholar
Hefner, R. A. Extensions of the law of comparative judgment to discriminable and multidimensional stimuli. Doctoral Dissertation, University of Michigan, 1958.Google Scholar
IMSL Library Reference Manual. New York: International Mathematical and Statistical Libraries, Inc., 1979.Google Scholar
Kendall, M. G. & Stuart, A. The advanced theory of statistics, Vol. 2, New York: Hafner, 1961.Google Scholar
MacKay, D. B. & Zinnes, J. L. Probabilistic scaling of spatial distance judgments. Geographical Analysis, 1981, 13, 21–37.CrossRefGoogle Scholar
Patnaik, P.B. The non-central chi-square and F-distributions and their applications. Biometrika, 1949, 36, 202–232.Google Scholar
Ramsay, J. O. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182.CrossRefGoogle Scholar
Ramsay, J. O. Maximum likelihood estimation in multidimensional scaling. Psychometrika, 1977, 42, 241–266.CrossRefGoogle Scholar
Richardson, M. W. Multidimensional psychophysics. Psychological Bulletin, 1938, 35, 659–660.Google Scholar
Sankaran, M. On the non-central chi-square distribution. Biometrika, 1959, 46, 235–237.CrossRefGoogle Scholar
Schönemann, P. H. On metric multidimensional unfolding. Psychometrika, 1970, 35, 349–367.CrossRefGoogle Scholar
Sherman, C. R. Nonmetric multidimensional scaling: A Monte Carlo study of the basic parameters. Psychometrika, 1972, 37, 323–355.CrossRefGoogle Scholar
Spence, I. & Domoney, D. W. Single subject incomplete designs for nonmetric multidimensional scaling. Psychometrika, 1974, 39, 469–490.CrossRefGoogle Scholar
Suppes, P. & Zinnes, J. L. Basic measurement theory. In Luce, R. D., Bush, R. R., and Galanter, E. (Eds.), Handbook of Mathematical Psychology, Vol. I . New York: Wiley. 1963, 1–76.Google Scholar
Thurstone, L. L. A law of comparative judgment. Psychological Review, 1927, 34, 273–286.CrossRefGoogle Scholar
Young, F. W. Nonmetric multidimensional scaling: recovery of metric information. Psychometrika, 1970, 35, 455–473.CrossRefGoogle Scholar
Young, F. W. & Cliff, N. Interactive scaling with individual subjects. Psychometrika, 1972, 37, 385–415.CrossRefGoogle Scholar
Young, G. & Householder, A. S. Discussion of a set of points in terms of their mutual distances. Psychometrika, 1938, 3, 19–21.CrossRefGoogle Scholar
Zinnes, J. L. & Griggs, R. A. Probabilistic multidimensional unfolding analysis. Psychometrika, 1974, 39, 327–350.CrossRefGoogle Scholar
Zinnes, J. L. & MacKay, D. B. Multidimensional scaling models: the other side. In Borg, I. (Eds.), Multidimensional Data Representations: When and Why . Ann Arbor, Mich.: Mathesis Press. 1981, 517–542.Google Scholar
Zinnes, J. L. & Wolff, R. P. Single and multidimensional same-different judgments. Journal of Mathematical Psychology, 1977, 16, 30–50.CrossRefGoogle Scholar
Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, Washington, DC: US Government Printing Office, 1967.Google Scholar
Bechtel, G. G. Multidimensional preference scaling, The Hague: Mouton, 1976.CrossRefGoogle Scholar
Carroll, J. D. Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K. & Nerlove, S. B. (Eds.), Multidimensional Scaling. Vol. I . New York: Seminar Press. 1972, 105–155.Google Scholar
Chandler, J. P. STEPIT—Finds local minima of a smooth function of several parameters. Behavioral Science, 1969, 14, 81–82.Google Scholar
Coombs, C. H., Greenberg, M., & Zinnes, J. L. A double law of comparative judgment for the analysis of preferential choice and similarities data. Psychometrika, 1961, 26, 165–171.CrossRefGoogle Scholar
Graef, F. & Spence, I. Using distance information in the design of large multidimensional scaling experiments. Psychological Bulletin, 1979, 86, 60–66.CrossRefGoogle Scholar
Hefner, R. A. Extensions of the law of comparative judgment to discriminable and multidimensional stimuli. Doctoral Dissertation, University of Michigan, 1958.Google Scholar
IMSL Library Reference Manual. New York: International Mathematical and Statistical Libraries, Inc., 1979.Google Scholar
Kendall, M. G. & Stuart, A. The advanced theory of statistics, Vol. 2, New York: Hafner, 1961.Google Scholar
MacKay, D. B. & Zinnes, J. L. Probabilistic scaling of spatial distance judgments. Geographical Analysis, 1981, 13, 21–37.CrossRefGoogle Scholar
Patnaik, P.B. The non-central chi-square and F-distributions and their applications. Biometrika, 1949, 36, 202–232.Google Scholar
Ramsay, J. O. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182.CrossRefGoogle Scholar
Ramsay, J. O. Maximum likelihood estimation in multidimensional scaling. Psychometrika, 1977, 42, 241–266.CrossRefGoogle Scholar
Richardson, M. W. Multidimensional psychophysics. Psychological Bulletin, 1938, 35, 659–660.Google Scholar
Sankaran, M. On the non-central chi-square distribution. Biometrika, 1959, 46, 235–237.CrossRefGoogle Scholar
Schönemann, P. H. On metric multidimensional unfolding. Psychometrika, 1970, 35, 349–367.CrossRefGoogle Scholar
Sherman, C. R. Nonmetric multidimensional scaling: A Monte Carlo study of the basic parameters. Psychometrika, 1972, 37, 323–355.CrossRefGoogle Scholar
Spence, I. & Domoney, D. W. Single subject incomplete designs for nonmetric multidimensional scaling. Psychometrika, 1974, 39, 469–490.CrossRefGoogle Scholar
Suppes, P. & Zinnes, J. L. Basic measurement theory. In Luce, R. D., Bush, R. R., and Galanter, E. (Eds.), Handbook of Mathematical Psychology, Vol. I . New York: Wiley. 1963, 1–76.Google Scholar
Thurstone, L. L. A law of comparative judgment. Psychological Review, 1927, 34, 273–286.CrossRefGoogle Scholar
Young, F. W. Nonmetric multidimensional scaling: recovery of metric information. Psychometrika, 1970, 35, 455–473.CrossRefGoogle Scholar
Young, F. W. & Cliff, N. Interactive scaling with individual subjects. Psychometrika, 1972, 37, 385–415.CrossRefGoogle Scholar
Young, G. & Householder, A. S. Discussion of a set of points in terms of their mutual distances. Psychometrika, 1938, 3, 19–21.CrossRefGoogle Scholar
Zinnes, J. L. & Griggs, R. A. Probabilistic multidimensional unfolding analysis. Psychometrika, 1974, 39, 327–350.CrossRefGoogle Scholar
Zinnes, J. L. & MacKay, D. B. Multidimensional scaling models: the other side. In Borg, I. (Eds.), Multidimensional Data Representations: When and Why . Ann Arbor, Mich.: Mathesis Press. 1981, 517–542.Google Scholar
Zinnes, J. L. & Wolff, R. P. Single and multidimensional same-different judgments. Journal of Mathematical Psychology, 1977, 16, 30–50.CrossRefGoogle Scholar