Abstract
Factorial analysis begins with ann ×n correlation matrixR, whose principal diagonal entries are unknown. If the common test space of the battery is under investigation, the communality of each test is entered in the appropriate diagonal cell. This value is the portion of the test's variance shared with others in the battery. The communalities must be so estimated thatR will maintain the rank determined by its side entries, after the former have been inserted. Previous methods of estimating the communalities have involved a certain arbitrariness, since they depended on selecting test subgroups or parts of the data inR. A theory is presented showing that this difficulty can be avoided in principle. In its present form, the theory is not offered as a practical computing procedure. The basis of the new method lies in the Cayley-Hamilton theorem: Any square matrix satisfies its own characteristic equation.
Similar content being viewed by others
References
Albert, A. A. The minimum rank of a correlation matrix.Proc. National Acad. of Sci., 1944,6, 144–148.
Birkhoff, G. and MacLane, S. A survey of modern algebra. New York: MacMillan, 1941.
MacDuffee, C. C. Vectors and matrices. Ithaca: The Mathematical Assn. of America, 1943.
Thurstone, L. L. Multiple factor analysis. Chicago: Univ. of Chicago Press, 1947.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rosner, B. An algebraic solution for the communalities. Psychometrika 13, 181–184 (1948). https://doi.org/10.1007/BF02289260
Issue Date:
DOI: https://doi.org/10.1007/BF02289260