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On Weighted Procrustes and Hyperplane Fitting in Factor Analytic Rotation

Published online by Cambridge University Press:  01 January 2025

William Meredith*
Affiliation:
University of California at Berkeley
*
Requests for reprints should be sent to William Meredith, Department of Psychology, University of California, Berkeley, California 94720.

Abstract

A weighted collinearity criterion for Procrustean rotation is developed, and it is shown that special cases with respect to the choice of weights and a vector norm are forms of hyperplane fitting, classical oblique Procrustes, etc.; a family of Procrustean transformation procedures is thereby generated. Numerical illustrations utilizing the Holzinger-Swineford data are presented.

Type
Original Paper
Copyright
Copyright © 1977 The Psychometric Society

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References

Reference Notes

Jöreskog, K. G. On rotation to a specified simple structure, 1965, Princeton, N. J.: Educational Testing Service.Google Scholar
Tucker, L. R. Fitting of factor analytic hyperplanes by a personal probability function. Topics in Factor Analysis II, (ONR Contract Report 00014-67-A 0305-0003, NR 150-304). U. S. Navy, 1970.Google Scholar
Tucker, L. R. A method for synthesis of factor analysis studies, 1951, Washington, D. C.: Department of the Army.CrossRefGoogle Scholar

References

Browne, M. W. On oblique Procrustes rotation. Psychometrika, 1967, 32, 125132.CrossRefGoogle ScholarPubMed
Browne, M. W. Oblique rotation to a partially specified target. British Journal of Mathematical and Statistical Psychology, 1972, 25, 207212.CrossRefGoogle Scholar
Browne, M. W. Orthogonal rotation to a partially specified target. British Journal of Mathematical and Statistical Psychology, 1972, 25, 115120.CrossRefGoogle Scholar
Browne, M. W., and Kristof, W. On the oblique rotation of a factor matrix to a specified pattern. Psychometrika, 1969, 34, 237248.CrossRefGoogle Scholar
Graybill, F. A. Introduction to matrices with applications in statistics, 1969, Belmont, CA: Wadsworth.Google Scholar
Green, B. F. The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 1952, 17, 429440.CrossRefGoogle Scholar
Gruvaeus, G. T. A general approach to Procrustes pattern rotation. Psychometrika, 1970, 35, 493505.CrossRefGoogle Scholar
Harris, C. W. Some Rao-Guttman relationships. Psychometrika, 1962, 27, 247263.CrossRefGoogle Scholar
Harris, C. W., and Kaiser, H. F. Oblique factor analytic solutions by orthogonal transformations. Psychometrika, 1964, 29, 347362.CrossRefGoogle Scholar
Holzinger, K. J. and Swineford, F.A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, 1939, 48.Google Scholar
Horst, A. P. A non-graphical method for transforming an arbitrary factor matrix into a simple structure factor matrix. Psychometrika, 1941, 6, 7999.CrossRefGoogle Scholar
Horst, A. P. Matrix Algebra for Social Scientists, 1963, New York: Holt, Rinehart & Winston.Google Scholar
Horst, P. Factor analysis of data matrices, 1965, New York: Holt, Rinehart & Winston.Google Scholar
Hurley, J. R., and Cattell, R. B. The Procrustes program: Producing direct rotation to test a hypothesized factor structure. Behavioral Science, 1962, 7, 258262.CrossRefGoogle Scholar
Jöreskog, K. G. Statistical estimation in factor analysis, 1963, Stockholm: Almquist and Wiksell.Google Scholar
Jöreskog, K. G., and Goldberger, A. S. Factor analysis by generalized least squares. Psychometrika, 1972, 37, 243260.CrossRefGoogle Scholar
Kaiser, H. F., and Caffrey, J. Alpha factor analysis. Psychometrika, 1965, 30, 114.CrossRefGoogle ScholarPubMed
Kendall, M. G., and Stuart, A. The advanced theory of statistics, 2nd ed., London: Griffin, 1963.Google Scholar
Lawley, D. N., and Maxwell, A. E. Factor transformation methods. British Journal of Statistical Psychology, 1964, 17, 97103.CrossRefGoogle Scholar
Lawley, D. N., and Maxwell, A. E. Factor Analysis as a Statistical Method, 1971, New York: American Elsevier.Google Scholar
Lingoes, J. C., and Schonemann, P. H. Alternative measures of fit for the Schonemann-Carroll matrix fitting algorithm. Psychometrika, 1974, 39, 423427.CrossRefGoogle Scholar
Mosier, C. I. Determining a simple structure when loadings for certain tests are known. Psychometrika, 1939, 4, 149162.CrossRefGoogle Scholar
Mulaik, S. A. The foundations of factor analysis, 1972, New York: McGraw-Hill.Google Scholar
Schonemann, P. H. The generalized solution of the orthogonal Procrustes problem. Psychometrika, 1966, 31, 116.CrossRefGoogle Scholar
Schonemann, P. H., and Carroll, R. M. Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, 1970, 35, 245255.CrossRefGoogle Scholar
Tucker, L. R. A rotational method based upon the mean principal axis of a sub-group of tests. Psychological Bulletin, 1940, 37, 578578.Google Scholar
Tucker, L. R. A semi-analytical method of factorial rotation to simple structure. Psychometrika, 1944, 9, 4368.CrossRefGoogle Scholar