Hostname: page-component-6bf8c574d5-xtvcr Total loading time: 0 Render date: 2025-02-22T04:50:13.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum Likelihood Additivity Analysis

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane
Yoshio Takane*
Affiliation:
McGill University
*
Requests for reprints should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Avenue Docteur Penfield, Montreal, Quebec, H3A 1B1, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

A maximum likelihood estimation procedure was developed to fit unweighted and weighted additive models to conjoint data obtained by the categorical rating, the pair comparison or the directional ranking method. The scoring algorithm used to fit the models was found to be both reliable and efficient, and the program MAXADD is capable of handling up to 300 parameters to be estimated. Practical uses of the procedure are reported to demonstrate various advantages of the procedure as a statistical method.

Keywords

simple additive modelweighted additive modelmethod of successive categoriesdirectional ranking methodscoring algorithmAIC
Type
Original Paper
Information
Psychometrika , Volume 47 , Issue 3 , September 1982 , pp. 225 - 241
Copyright
Copyright © 1982 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The research reported here was supported by Grant A6394 to the author from the Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Iowa City, Iowa, in May, 1980.

Thanks are due to Jim Ramsay, Justine Sergent and anonymous reviewers for their helpful comments.

Two MAXADD programs which perform the computations discussed in this paper may be obtained from the author.

References

Reference Notes

Takane, Y. Statistical procedures for nonmetric multidimensional scaling. Unpublished doctoral dissertation, The University of North Carolina, 1977.Google Scholar
Jones, C. L. Statistical inference with directional data. Handout for the talk given at the Psychometric Society meeting, Iowa, 1980.Google Scholar

References

Akaike, H. A new look at the statistical model identification. IEEE Transactions on Automatic Control, 1974, 19, 716–723.CrossRefGoogle Scholar
Anderson, N. H. Functional measurement and psychophysical judgment. Psychological Review, 1970, 77, 153–170.CrossRefGoogle ScholarPubMed
Anderson, N. H. Algebraic models in perception. In Carterette, E. C. and Friedman, M. P. (Eds.), Handbook of perception, Vol. II, 1974, New York: Academic Press.Google Scholar
Beals, R., Krantz, D. H. & Tversky, A. A. Foundations of multidimensional scaling. Psychological Review, 1968, 75, 127–142.CrossRefGoogle ScholarPubMed
Bradley, R. A. & Terry, M. E. The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika, 1952, 39, 324–345.Google Scholar
Carroll, J. D. & Chang, J. J. Analysis of individual differences in multidimensional scaling via anN-way generalization of “Eckart-Young” decomposition. Psychometrika, 1970, 35, 283–319.CrossRefGoogle Scholar
Cliff, N. Adverbs as multipliers. Psychological Review, 1959, 66, 27–44.CrossRefGoogle ScholarPubMed
Colonius, H. A new interpretation of stochastic test models. Psychometrika, 1981, 46, 223–225.CrossRefGoogle Scholar
de Leeuw, J., Young, F. W., and Takane, Y. Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika, 1976, 41, 471–503.CrossRefGoogle Scholar
Edgell, S. E., & Geisler, W. S. A set-theoretic random utility model of choice behavior. Journal of Mathematical Psychology, 1980, 21, 265–278.CrossRefGoogle Scholar
Falmagne, J. C. Random conjoint measurement and loudness summation. Psychological Review, 1976, 83, 65–79.CrossRefGoogle Scholar
Falmagne, J. C. Probabilistic choice behavior theory: axioms as constraints in optimization. In Castellan, J. N. Jr., & Restle, F. (Eds.), Cognitive theory, Vol. 3, 1978, New York: Erlbaum Assoc.Google Scholar
Green, P. E., & Rao, V. Conjoint measurement for quantifying judgmental data. Journal of Marketing Research, 1971, 8, 355–363.Google Scholar
Halff, H. M. Choice theories for differentially comparable alternatives. Journal of Mathematical Psychology, 1976, 14, 244–246.CrossRefGoogle Scholar
Hamerle, A. & Tutz, G. Goodness of fit tests for probabilistic measurement models. Journal of Mathematical Psychology, 1980, 21, 153–167.CrossRefGoogle Scholar
Johnson, R. M. Trade-off analysis of consumer values. Journal of Marketing Research, 1974, 11, 121–127.CrossRefGoogle Scholar
Johnson, R. M. A simple method for pairwise monotone regression. Psychometrika, 1975, 40, 163–168.CrossRefGoogle Scholar
Kempler, B. Stimulus correlates of area judgments: a psychophysical developmental study. Developmental Psychology, 1971, 4, 158–163.CrossRefGoogle Scholar
Krantz, D. H. Rational distance functions for multidimensional scaling. Journal of Mathematical Psychology, 1967, 4, 226–245.CrossRefGoogle Scholar
Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. Foundations of measurement. Vol. I, 1971, New York: Academic Press.Google Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J. B. Analysis of factorial experiments by estimating monotone transformations of data. Journal of the Royal Statistical Society, 1965, 27, 251–265.CrossRefGoogle Scholar
Lee, S. Y. & Bentler, P. M. Functional relations in multidimensional scaling. British Journal of Mathematical and Statistical Psychology, 1980, 33, 142–150.CrossRefGoogle Scholar
Luce, R. D. Individual choice behavior: A theoretical analysis, 1959, New York: Wiley.Google Scholar
Nakatani, L. H. Confusion-choice model for multidimensional psychophysics. Journal of Mathematical Psychology, 1972, 9, 104–127.CrossRefGoogle Scholar
Ramsay, J. O. Confidence regions for multidimensional scaling analysis. Psychometrika, 1978, 43, 145–160.CrossRefGoogle Scholar
Ramsay, J. O. Some statistical approaches to multidimensional scaling data. Journal of the Royal Statistical Society, Series A, 1982, in press.CrossRefGoogle Scholar
Restle, F. Psychology of judgment and choice, 1961, New York: Wiley.Google Scholar
Roskam, E. E. Metric analysis of ordinal data in psychology, 1968, Voorschoten, Holland: VAM.Google Scholar
Saito, T. Multidimensional Thurstonian scaling with an application to color metrics. Japanese Psychological Research, 1977, 19, 78–89.CrossRefGoogle Scholar
Stephens, M. A. Multi-sample tests for the Fisher distribution for directions. Biometrika, 1969, 56, 169–181.CrossRefGoogle Scholar
Strauss, D. Choice by features: an extension of Luce's model to account for similarities. British Journal of Mathematical and Statistical Psychology, 1981, 34, 50–61.CrossRefGoogle Scholar
Takane, Y. A maximum likelihood method for nonmetric multidimensional scaling: I. The case in which all empirical pairwise orderings are independent—theory and evaluations. Japanese Psychological Research, 1978, 20, 7–17.CrossRefGoogle Scholar
Takane, Y. Maximum likelihood estimation in the generalized case of Thurstone's model of comparative judgment. Japanese Psychological Research, 1980, 22, 188–196.CrossRefGoogle Scholar
Takane, Y. Multidimensional successive categories scalling: a maximum likelihood method. Psychometrika, 1981, 46, 9–27.CrossRefGoogle Scholar
Takane, Y. The method of triadic combinations: a new treatment and its applications. Behaviormetrika, 1982, 11, 37–48.CrossRefGoogle Scholar
Takane, Y. & Carroll, J. D. Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities. Psychometrika, 1981, 46, 389–405.CrossRefGoogle Scholar
Takane, Y., Young, F. W. & de Leeuw, J. An individual differences additive model: an alternating least squares method with optimal scaling features. Psychometrika, 1980, 45, 183–209.CrossRefGoogle Scholar
Torgerson, W. S. Theory and methods of scaling, 1958, New York: Wiley.Google Scholar
Tversky, A. Intransitivity of preferences. Psychological Review, 1969, 73, 31–48.CrossRefGoogle Scholar
Tversky, A. Elimination by aspects: a theory of choice. Psychological Review, 1972, 79, 281–299.CrossRefGoogle Scholar
Tversky, A. & Krantz, D. H. The dimensional representation and the metric structure of similarity data. Journal of Mathematical Psychology, 1979, 7, 572–596.CrossRefGoogle Scholar
Tversky, A. & Russo, E. Substitutability and similarity in binary choices. Journal of Mathematical Psychology, 1969, 6, 1–12.CrossRefGoogle Scholar
Wallsten, T. S. Using conjoint-measurement models to investigate a theory about probabilistic information processing. Journal of Mathematical Psychology, 1976, 14, 144–185.CrossRefGoogle Scholar
Winsberg, S. & Ramsay, J. O. Monotonic transformations to additivity using splines. Biometrika, 1980, 67, 669–674.CrossRefGoogle Scholar
Young, F. W., de Leeuw, J., & Takane, Y. Regression with qualitative and quantitative variables: an alternating least squares method with optimal scaling features. Psychometrika, 1976, 41, 505–529.CrossRefGoogle Scholar
Zangwill, W. I. Nonlinear programming: a unified approach, 1969, Englewood Cliffs, N. J.: Prentice-Hall.Google Scholar