Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-02-08T00:30:09.587Z Has data issue: false hasContentIssue false
  • English
  • Français

An Individual Differences Additive Model: An Alterating Least Squares Method with Optimal Scaling Features

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane ,
Forrest W. Young  and
Jan de Leeuw
Yoshio Takane*
Affiliation:
McGill University
Forrest W. Young
Affiliation:
University of North Carolina
Jan de Leeuw
Affiliation:
University of Leiden
*
Requests for reprints should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Docteur Penfield Avenue, Montreal, Quebec, H3A 1B1.
Get access Rights & Permissions [Opens in a new window]

Abstract

An individual differences additive model is discussed which represents individual differences in additivity by differential weighting of additive factors. A procedure for estimating the model parameters for various data measurement characteristics is developed. The procedure is evaluated using both Monte Carlo and real data. The method is found to be very useful in describing certain types of developmental change in cognitive structure, as well as being numerically robust and efficient.

Keywords

additivity analysisindividual differencesoptimal scalingconjoint scaling
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 2 , June 1980 , pp. 183 - 209
Copyright
Copyright © 1980 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 work reported here was partly supported by Grant A6394 to the first author by the Natural Sciences and Engineering Research Council of Canada.

References

Reference Notes

de Leeuw, J. Normalized cone regression approach to alternating least squares method. Unpublished manuscript. The University of Leiden, 1977.Google Scholar
Takane, Y. Statistical procedures for nonmetric multidimensional scaling. Unpublished doctoral dissertation. The University of North Carolina, 1977.Google Scholar

References

Anderson, N. H. Failure of additivity in bisection of length. Perception & Psychophysics, 1977, 22, 213–222.CrossRefGoogle Scholar
Anderson, H. H. & Cuneo, D. O. The height + width rule in children's judgments of quantity. Journal of Experimental Psychology: General, 1978, 107, 335–378.CrossRefGoogle Scholar
Carroll, J. D. & Chang, J. J. Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 1970, 35, 283–319.CrossRefGoogle Scholar
Graybill, F. A. Introduction to matrices with applications in statistics, 1969, New York: Wadsworth.Google Scholar
Kempler, B. Stimulus correlates of area judgments: A psychophysical developmental study. Developmental Psychology, 1971, 4, 158–163.CrossRefGoogle Scholar
Krantz, D. H., Luce, R. D., Suppes, P. & Tversky, A. Foundations of measurement, 1971, New York: Academic Press.Google Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, 1964, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J. B. Analysis of factorial experiments by estimating monotone transformations of data. Journal of the Royal Statistical Society, Series B, 1965, 27, 251–265.CrossRefGoogle Scholar
Lawson, C. L., & Hanson, R. J. Solving least squares problems, 1974, Englewood Cliffs, N.J.: Prentice Hall.Google Scholar
de Leeuw, J., Young, F. W. & Takane, Y. Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika, 1976, 41, 471–503.CrossRefGoogle Scholar
Liebert, R. N., Poulos, R. W. & Strauss, G. Developmental psychology, 1974, Englewood Cliffs, N.J.: Prentice Hall.Google Scholar
Nishisato, S. Analysis of categorical data, 1979, Toronto: University of Toronto Press (in press)Google Scholar
Rao, C. R. Linear statistical inference and its applications, 1973, New York: Wiley (1st edition: 1965)CrossRefGoogle Scholar
Roskam, E. E. Metric analysis of ordinal data in psychology, 1968, Voorschoten: Netherlands.Google Scholar
Sayeki, Y. Allocation of importance: An axiom system. Journal of Mathematical Psychology, 1972, 9, 55–65.CrossRefGoogle Scholar
Spence, I. A direct approximation for random rankings stress values. Multivariate Behavioral Research, 1979 (in press).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., Young, F. W. & de Leeuw, J. Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 1977, 42, 7–67.CrossRefGoogle Scholar
Wilkening, F. Combining of stimulus dimensions in children's and adults' judgments of area: An information integration analysis. Developmental Psychology, 1979, 15, 25–33.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
Young, F. W., de Leeuw, J. & Takane, Y. Quantifying qualitative data. In Feger, H. (Eds.), Similarity and choice, 1979, New York: Academic Press.Google Scholar
Zangwill, W. I. Nonlinear programming: A unified approach, 1969, Englewood Cliffs, N.J.: Prentice Hall.Google Scholar