Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-08T09:24:33.007Z Has data issue: false hasContentIssue false
  • English
  • Français

External Analysis with Three-Mode Principal Component Models

Published online by Cambridge University Press:  01 January 2025

Willem A. van der Kloot  and
Pieter M. Kroonenberg
Willem A. van der Kloot*
Affiliation:
Leiden University
Pieter M. Kroonenberg*
Affiliation:
Department of Education, Leiden University
*
Requests for reprints should be sent to Willem A. van der Kloot, Department of Psychology, Hooigracht 15, 2312 KM Leiden, THE NETHERLANDS.
Correspondence regarding the TUCKALS programs should be addressed to Pieter M. Kroonenberg, Department of Education, Postbus 9507, 2300 RA Leiden, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

Through external analysis of two-mode data one attempts to map the elements of one mode (e.g., attributes) as vectors in a fixed space of the elements of the other mode (e.g., stimuli). This type of analysis is extended to three-mode data, for instance, when the ratings are made by more individuals. It is described how alternating least squares algorithms for three-mode principal component analysis (PCA) are adapted to enable external analysis, and it is demonstrated that these techniques are useful for exploring differences in the individuals' mappings of the attribute vectors in the fixed stimulus space. Conditions are described under which individual differences may be ignored. External three-mode PCA is illustrated with data from a person perception experiment, designed after two studies by Rosenberg and his associates whose results were used as external information.

Keywords

multidimensional scalingindividual differencesthree-mode factor analysisperson perception
Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 4 , December 1985 , pp. 479 - 494
Copyright
Copyright © 1985 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

We gratefully acknowledge the assistance of Piet Brouwer in implementing the external analysis options in the TUCKALS programs.

References

Bentler, P. M., Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588–606.CrossRefGoogle Scholar
Benzécri, J. P. (1976). L'Analyse des données. II. L'Analyse des correspondances [Data Analysis. II. Correspondence Analysis 2nd ed.,, Paris: Dunod.Google Scholar
Benzécri, J. P., Benzécri, F. (1980). Pratique de l'analyse des données. I. Analyse des correspondances. Exposé élémentaire [Practical Data Analysis. I. Correspondence Analysis. Introduction], Paris: Dunod.Google Scholar
Bonett, D. G., Bentler, P. M. (1983). Goodness-of-fit procedures for the evaluation and selection of log-linear models. Psychological Bulletin, 93, 149–166.CrossRefGoogle Scholar
Carroll, J. D. (1972). Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K., Nerlove, S. B. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. I: Theory (pp. 105–155). New York: Seminar Press.Google Scholar
Carroll, J. D., Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via anN-way generalization of “Eckart-Young” decomposition. Psychometrika, 35, 283–320.CrossRefGoogle Scholar
Cazes, P. (1982). Note sur les éléments supplémentaires en analyse des correspondances. I. Pratique et utilisation. II. Tableaux multiples [Note on external variables in correspondence analysis. I. Utilization. II. Higher-way tables]. Les Cahiers de l'Analyse des Données, 7, 9–23.Google Scholar
Glaçon, F. (1981). Analyse conjointe de plusieurs matrices de données [Conjoint analysis of several data matrices], France: l'Université Scientifique et Médicinale de Grenoble.Google Scholar
Harshman, R. A. (1970). Foundations of the PARAFAC procedure. UCLA Working Papers in Phonetics, 16, 1–84.Google Scholar
Harshman, R. A. (1972). PARAFAC2: Mathematical and technical notes. UCLA Working Papers in Phonetics, 22, 30–44.Google Scholar
Harshman, R. A., Lundy, M. E. (1984). The PARAFAC model for three-way factor analysis and multidimensional scaling. In Law, H. G., Snyder, C. W. Jr., Hattie, J. A., McDonald, R. P. (Eds.), Research methods for multi-mode data analysis (pp. 122–215). New York: Praeger.Google Scholar
Harshman, R. A., Lundy, M. E. (1984). Data preprocessing and the extended PARAFAC model. In Law, H. G., Snyder, C. W. Jr., Hattie, J. A., McDonald, R. P. (Eds.), Research methods for multi-mode data analysis (pp. 216–284). New York: Praeger.Google Scholar
Jaffrenou, P. A. (1978). Sur l'analyse des familles finies des variables vectorielles: Bases algébrique et application à la description statistique [On the analysis of finite sets of vector variables: Algebraic foundations and applications in descriptive statistics], France: l'Université de Sainte- Etiene.Google Scholar
Kroonenberg, P. M. (1983). Three-mode principal component analysis: Theory and applications, Leiden: DSWO Press.Google Scholar
Kroonenberg, P. M., de Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45, 69–97.CrossRefGoogle Scholar
Kroonenberg, P. M., van der Kloot, W. A., & Brouwer, P. (1983, July). External three-mode principal component analysis. Paper presented at the 3rd European Meeting of the Psychometric Society, Jouy-en-Josas, France.Google Scholar
Kruskal, J. B. (1976). More factors than subjects, tests, and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling. Psychometrika, 41, 281–293.CrossRefGoogle Scholar
Kruskal, J. B. (1977). Three-way arrays: Rank and uniqueness of trilinear decompositions with application to arithmetic complexity and statistics. Linear Algebra and its Applications, 18, 95–138.CrossRefGoogle Scholar
Kruskal, J. B. (1981). Multilinear models for data-analysis. Behaviormetrika, 10, 1–20.CrossRefGoogle Scholar
Kruskal, J. B. (1983). Multilinear methods. In Gnadadesikan, R. (Eds.), Statistical data analysis (pp. 36–62). Providence, RI: American Mathematical Society.Google Scholar
Kruskal, J. B. (1983b). Rank and geometry of three-dimensional matrices. Unpublished manuscript.Google Scholar
Mizère, D. (1981). Analyse d'une cube de données. Decomposition tensorielle et les liens entre procedures de comparaison de tableaux rectangulaires de données [Analysis of three-mode data. Tensor decomposition and relationships between procedures to compare rectangular data matrices], France: l'Université Scientifique et Médicinale de Grenoble.Google Scholar
Ramsay, J. O. (1982). Multiscale II manual, Montreal, Canada: McGill University, Department of Psychology.Google Scholar
Rao, L. R. (1964). The use and interpretation of principal component analysis in applied research. Sankhya, 26(A), 329–358.Google Scholar
Rosenberg, S., Nelson, C., Vivekananthan, P. S. (1968). A multidimensional approach to the structure of personality impressions. Journal of Personality and Social Psychology, 9, 283–294.CrossRefGoogle Scholar
Rosenberg, S., Sedlak, A. (1972). Structural representation of perceived personality trait relationships. In Romney, A. K., Shepard, R. N., Nerlove, S. B. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. II: Applications (pp. 134–163). New York: Seminar Press.Google Scholar
Sands, R., Young, F. W. (1980). Component models for three-way data: ALSCOMP3, an alternating least squares algorithm with optimal scaling features. Psychometrika, 45, 39–67.CrossRefGoogle Scholar
Schiffman, S. S., Reynolds, M. L., Young, F. W. (1981). Introduction to multidimensional scaling: Theory, methods, and applications, New York: Academic Press.Google Scholar
Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed
Tucker, L. R. (1972). Relations between multidimensional scaling and three-mode factor analysis. Psychometrika, 37, 3–27.CrossRefGoogle Scholar
van den Wollenberg, A. (1977). Redundancy analysis: An alternative for canonical correlation analysis. Psychometrika, 42, 207–219.CrossRefGoogle Scholar
van der Kloot, W. A., Kroonenberg, P. M. (1982). Group and individual implicit theories of personality: An application of three-mode principal component analysis. Multivariate Behavioral Research, 17, 471–492.CrossRefGoogle ScholarPubMed
Wiggins, J. S. (1979). A psychological theory of trait-descriptive terms: The interpersonal domain. Journal of Personality and Social Psychology, 37, 395–412.CrossRefGoogle Scholar
Young, F. W., Lewyckyj, R. (1979). ALSCAL-4 user's guide, Carrboro, NC: Data Analysis and Theory Associates.Google Scholar