Some mathematical notes on three-mode factor analysis
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.
KeywordsPrincipal Component Analysis Public Policy Statistical Theory Mathematical Note Unique Variance
Unable to display preview. Download preview PDF.
- Bellman, R. R.Introduction to matrix analysis. New York: McGraw-Hill, 1960.Google Scholar
- Campbell, D. T. and Fiske, D. W. Convergent and discriminant validation by the multitrait-multimethod matrix.Psychol. Bull., 1959,56, 81–105.Google Scholar
- Eckart, C. and Young, G. The approximation of one matrix by another of lower rank.Psychometrika, 1936,1, 211–218.Google Scholar
- Levin, J.Three-mode factor analysis. Tech. Rept., Dept. Psychol., Univ. Ill., 1963.Google Scholar
- Levin, J. Three-mode factor analysis.Psychol. Bull., 1965,64, 442–452.Google Scholar
- MacDuffee, C. C.The theory of matrices. New York: Chelsea, 1946.Google Scholar
- Osgood, C. E., Suci, G. J., and Tannenbaum, P. H.The measurement of meaning. Urbana, Illinois: Univ. Ill. Press, 1957.Google Scholar
- 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
- 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
- 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