Abstract
Methods are developed for factoring an arbitrary rectangular matrixS of rankr into the formFP, whereF hasr columns andP hasr rows. For the statistical problem of factor analysis,S may be the score matrix of a population of individuals on a battery of tests. ThenF is a matrix of factor loadings,P is a matrix of factor scores, andr is the number of factor variates. (As in current procedures, there remains a subsequent problem of rotation of axes and interpretation of factors, which is not discussed here.) Methods are also developed for factoring an arbitrary Gramian matrixG of rankr into the formFF′, whereF hasr columns andF′ denotesF transposed. For the statistical problem of factor analysis,G may be the matrix of intercorrelations,R, of a battery of tests, with unity, communalities, or other parameters in the principal diagonal.R is proportional toSS′, and it is shown thatS can be factored by factoringR. This may usually be the most economical procedure in practice; it should not be overlooked, however, thatS can be factored directly. The general methods build up anF (andP) in as many stages as desired; as many factors as may be deemed computationally practical can be extracted at a time. Perhaps it will usually be found convenient to extract not more than three factors at a time. Current procedures, like the centroid and principal axes, are special cases of a general method presented here for extracting one factor at a time.
Similar content being viewed by others
References
Aitken, A. C. Determinants and matrices. Edinburgh: Oliver and Boyd, 1939.
Duncan, W. J., Frazer, R. A., and Collar, A. R. Elementary matrices. England: Univ. Cambridge Press, 1938. Chap. IV.
Guttman, Louis. Properties of Gramian matrices. Chapter Two inThe prediction of quantitative variates by factor analysis. Doctoral dissertation, Univ. Minnesota, 1942.
Holzinger, Karl J. and Harman, Harry H. Factor analysis. Chicago: Univ. Chicago Press, 1941.
Horst, Paul. A method of factor analysis by means of which all coordinates of the factor matrix are given simultaneously.Psychometrika, 1937,2, 225–236.
Hotelling, Harold. Analysis of a complex of statistical variables into principal components.J. educ. Psychol., 1933,24, 417–441, 498–520.
Hotelling, Harold. Simplified calculation of principal components.Psychometrika, 1936,1, 27–35.
Householder, Alston S. and Young, Gale. Matrix approximation and latent roots,Amer. Math. Monthly, 1938,45, 165–171.
Thurstone, Louis Leon. The vectors of mind. Chicago: Univ. Chicago Press, 1935.
Wedderburn, J. H. M. Lectures on matrices. Amer. Math. Soc. Colloquium Publications, Vol. XVII, 1934.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Guttman, L. General theory and methods for matric factoring. Psychometrika 9, 1–16 (1944). https://doi.org/10.1007/BF02288709
Issue Date:
DOI: https://doi.org/10.1007/BF02288709