Some necessary conditions for common-factor analysis
LetR be any correlation matrix of ordern, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrixU2 such thatG = R − U2 is Gramian and of minimum rankr. Lets1 be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatr ≧s1. Two further lower bounds tor are also established that are better thans1. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices inR is small, but the minimum rankr for a GramianG is nevertheless very large compared withn. Heuristic criteria are given for testing the hypothesis that a finiter exists for the infinite universe of content from which the sample ofn observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.
KeywordsLower Bound Public Policy Correlation Matrix Statistical Theory Latent Root
Unable to display preview. Download preview PDF.
- 1.Albert, A. A. The matrices of factor analysis.Proc. Nat. Acad. Sci., 1944,30, 90–95.Google Scholar
- 2.Albert, A. A. The minimum rank of a correlation matrix.Proc. Nat. Acad. Sci.,30, 144–148.Google Scholar
- 3.Bôcher, Maxime. Introduction to Higher Algebra. New York: Macmillan, 1907. (Sixteenth printing, 1952).Google Scholar
- 4.Guttman, Louis. Multiple rectilinear prediction and the resolution into components.Psychometrika, 1940,5, 75–99.Google Scholar
- 5.Guttman, Louis, General theory and methods for matric factoring.Psychometrika, 1944,9, 1–16.Google Scholar
- 6.Guttman, Louis. A new approach to factor analysis: the radex. In Paul F. Lazarsfeld (ed.), Mathematical thinking in the social sciences. The Free Press, 1954.Google Scholar
- 7.Guttman, Louis. Two new approaches to factor analysis. Office of Naval Research, Annual technical report on project Nonr-731(00), 1953.Google Scholar
- 8.Guttman, Louis. Image theory for the structure of quantitative variates.Psychometrika, 1953,19, 277–296.Google Scholar
- 9.Lazarsfeld, Paul F. The logical and mathematical foundation of latent structure analysis. In S. A. Stouffer, et al., Measurement and prediction, Princeton Univ. Press, 1950.Google Scholar
- 10.Rosner, Burt. An algebraic solution for the communalities.Psychometrika, 1948, 13, 181–184.Google Scholar
- 11.Thomson, Godfrey. The factorial analysis of human ability. Univ. London Press, 1950.Google Scholar
- 12.Thurstone, L. L. Multiple-factor analysis. Univ. Chicago Press, 1947.Google Scholar