An Individual Differences Additive Model: An Alterating Least Squares Method with Optimal Scaling Features | Psychometrika | Cambridge Core (original) (raw)

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.

References

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 Unpublished manuscript.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 Unpublished doctoral dissertation.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

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