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Some mathematical notes on three-mode factor analysis

Abstract

The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals.

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References

  1. Bellman, R. R.Introduction to matrix analysis. New York: McGraw-Hill, 1960.

    Google Scholar 

  2. Campbell, D. T. and Fiske, D. W. Convergent and discriminant validation by the multitrait-multimethod matrix.Psychol. Bull., 1959,56, 81–105.

    Google Scholar 

  3. Eckart, C. and Young, G. The approximation of one matrix by another of lower rank.Psychometrika, 1936,1, 211–218.

    Google Scholar 

  4. Levin, J.Three-mode factor analysis. Tech. Rept., Dept. Psychol., Univ. Ill., 1963.

  5. Levin, J. Three-mode factor analysis.Psychol. Bull., 1965,64, 442–452.

    Google Scholar 

  6. MacDuffee, C. C.The theory of matrices. New York: Chelsea, 1946.

    Google Scholar 

  7. Osgood, C. E., Suci, G. J., and Tannenbaum, P. H.The measurement of meaning. Urbana, Illinois: Univ. Ill. Press, 1957.

    Google Scholar 

  8. Tucker, L. R. Implications of factor analysis of three-way matrices for measurement of change. In C. W. Harris (Ed.),Problems in measuring change. Madison, Wis.: Univ. Wis. Press, 1963. Pp. 122–137.

    Google Scholar 

  9. Tucker, L. R. The extension of factor analysis to three-dimensional matrices. In N. Frederiksen and H. Gulliksen (Eds.),Contributions to mathematical psychology. New York: Holt, Rinehart and Winston, 1964. Pp. 109–127.

    Google Scholar 

  10. Tucker, L. R. Experiments in multi-mode factor analysis. InProceedings of the 1964 Invitational Conference on Testing Problems. Princeton, N. J.: Educ. Test. Serv., 1965 pp. 46–57.

    Google Scholar 

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This research was jointly supported by the University of Illinois and the Office of Naval Research under contract Nonr 1834 (39). The author is most appreciative of the careful reading of the manuscript by Dr. Douglas McKie and Dr. Maurice Tatsuoka.

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Tucker, L.R. Some mathematical notes on three-mode factor analysis. Psychometrika 31, 279–311 (1966). https://doi.org/10.1007/BF02289464

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  • DOI: https://doi.org/10.1007/BF02289464

Keywords

  • Principal Component Analysis
  • Public Policy
  • Statistical Theory
  • Mathematical Note
  • Unique Variance