Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-08T09:09:19.232Z Has data issue: false hasContentIssue false
  • English
  • Français

A Semi-Orthogonal Dependent Factor Solution

Published online by Cambridge University Press:  01 January 2025

Olgierd R. Porebski
Olgierd R. Porebski*
Affiliation:
University of New South Wales*
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper presents a general framework for the dependent factor method in which judgmental as well as analytic criteria may be employed. The procedure involves a semi-orthogonal transformation of an oblique solution comprising a number of reference factors and a number of experimental factors (composites). It determines analytically the residual factors in the experimental field keeping the reference field constant. The method is shown to be a generalization of the multiple factor procedure [Thurstone, 1947] in so far as it depends on the use of generalized inverse in deriving partial structure from total pattern. It is also shown to provide an example of the previously empty category (Case III) of the Harris-Kaiser generalization [1964]. A convenient computational procedure is provided. It is based on an extension of Aitken's [1937] method of pivotal condensation of a triple-product matrix to the evaluation of a matrix of the form H − VA−1U' (for a nonsingular A).

Keywords

Public PolicyStatistical TheoryGeneral FrameworkAnalytic CriterionExperimental Factor
Type
Original Paper
Information
Psychometrika , Volume 33 , Issue 4 , December 1968 , pp. 451 - 468
Copyright
Copyright © 1968 Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Now at the University of Ottawa

References

Aitken, A. C. The evaluation of a certain triple-product matrix. Proceedings of the Royal Society, Edinburgh, 1937, 172–181.Google Scholar
Anderson, T. W. An introduction to multivariate statistical analysis, New York: Wiley, 1958.Google Scholar
Bartlett, M. S. Internal and external factor analysis. British Journal of Psychology, Statistical Section, 1948, 1, 73–81.CrossRefGoogle Scholar
Burt, C. Group factor analysis. British Journal of Psychology, Statistical Section, 1950, 3, 40–75.CrossRefGoogle Scholar
Carroll, J. B. An analytic solution for approximating simple structure in factor analysis. Psychometrika, 1953, 18, 23–38..CrossRefGoogle Scholar
Cattell, R. B. Factor analysis, New York: Harper, 1952.Google Scholar
Cattell, R. B. and Dickman, K. A. Dynamic model of physical influences demonstrating the necessity of oblique simple structure. Psychological Bulletin, 1962, 59, 389–400.CrossRefGoogle ScholarPubMed
DuBois, P. H. Multivariate correlational analysis, New York: Harper, 1957.Google Scholar
Dwyer, P. S. and MacPhail, M. S. Symbolic matrix derivatives. Annals of Mathematical Statistics, 1948, 19, 517–534.CrossRefGoogle Scholar
Eyseneck, H. J. Criterion analysis. An application of the hypothetico-deductive method to factor analysis. Psychological Review, 1950, 57, 38–53.CrossRefGoogle Scholar
Guilford, J. P. and Zimmerman, W. A. Some variable-sampling problems in the rotation of axes in factor analysis. Psychological Bulletin, 1963, 60, 289–301.CrossRefGoogle ScholarPubMed
Guttman, L. General theory and methods for matrix factoring. Psychometrika, 1944, 9, 1–16.CrossRefGoogle Scholar
Guttman, L. Image theory for the structure of quantitative variates. Psychometrika, 1953, 18, 277–296.CrossRefGoogle Scholar
Guttman, L. Multiple group methods for common-factor analysis: their basis, computation, and interpretation. Psychometrika, 1954, 19, 39–45.Google Scholar
Harman, H. H. The square root method and multiple group methods of factor analysis. Psychometrika, 1954, 19, 39–45.CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, Chicago: Univ. of Chicago Press, 1960.Google Scholar
Harris, C. W. Some Rao-Guttman relationships. Psychometrika, 1962, 27, 247–263.CrossRefGoogle Scholar
Harris, C. W. Some problems in the description of change. Educational and Psychological Measurement, 1962, 22, 303–319.CrossRefGoogle Scholar
Harris, C. W. Problems in measuring change, Madison, Wisc.: Univ. of Wisc. Press, 1963.Google Scholar
Harris, C. W. Some recent developments in factor analysis. Educational and Psychological Measurement, 1964, 24, 193–206.CrossRefGoogle Scholar
Harris, C. W. and Kaiser, H. F. Oblique factor analytic solutions by orthogonal transformations. Psychometrika, 1964, 29, 347–362.CrossRefGoogle Scholar
Hilderbrand, F. B. Methods of applied mathematics. Prentice Hall, 1965.Google Scholar
Holzinger, K. J. A simple method of factor analysis. Psychometrika, 1944, 9, 257–262.CrossRefGoogle Scholar
Horst, P. A method of factor analysis by means of which all co-ordinates of the factor matrix are given simultaneously. Psychometrika, 1937, 2, 225–377.CrossRefGoogle Scholar
Horst, P. Factor analysis of data matrices, New York: Holt, Rinehart and Winston, 1965.Google Scholar
Hotelling, H. Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 1933, 24, 417–441.CrossRefGoogle Scholar
Hotelling, H. Relations between two sets of variates. Biometrika, 1936, 28, 321–377.CrossRefGoogle Scholar
Householder, A. S. The theory of matrices in numerical analysis, New York: Blaisdell, 1964.Google Scholar
Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis. Psychometrika, 1958, 23, 187–200.CrossRefGoogle Scholar
Kaiser, H. F. Psychometric approaches to factor analysis, Princeton, New Jersey: Educational Testing Service, 1964.Google Scholar
Kaiser, H. F. and Caffrey, J. Alpha factor analysis. Psychometrika, 1965, 30, 1–14.CrossRefGoogle ScholarPubMed
Lawley, D. N. and Maxwell, A. E. Factor analysis as a statistical method, London: Butterworths, 1963.Google Scholar
Lord, F. M. Further problems in the measurement of growth. Educational and Psychological Measurement, 1958, 18, 437–451.CrossRefGoogle Scholar
Pease, M. C. III. Methods of matrix algebra, New York: Academic Press, 1965.Google Scholar
Penrose, R. A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society, 1955, 51, 406–413.CrossRefGoogle Scholar
Porebski, O. R. Discriminatory and canonical analysis of technical college data. British Journal of Mathematical and Statistical Psychology, 1966, 19, 215–236.CrossRefGoogle Scholar
Rao, C. R. Estimation and tests of significance in factor analysis. Psychometrika, 1955, 20, 93–111.CrossRefGoogle Scholar
Rao, C. R. A note on generalized inverse of a matrix with applications to problems in mathematical statistics. Journal of the Royal Statistical Society, 1962, 24, 152–158.CrossRefGoogle Scholar
Rao, C. R. Linear statistical inference and its applications, New York: Wiley, 1965.Google Scholar
Seal, H. L. Multivariate statistical analysis for biologists, London: Methuen, 1966.Google Scholar
Spearman, C. The abilities of man, New York: MacMillan, 1927.Google Scholar
Thomson, G. H. Weighting for battery reliability and prediction. British Journal of Psychology, 1940, 30, 357–366.Google Scholar
Thomson, G. H. The factorial analysis of human ability, New York: Houghton Mifflin, 1951.Google Scholar
Thurstone, L. L. A multiple group method of factoring the correlation matrix. Psychometrika, 1945, 10, 73–78.CrossRefGoogle Scholar
Thurstone, L. L. Multiple factor analysis, Chicago: Univ. of Chicago Press, 1947.Google Scholar
Tucker, L. R. An inter-battery method of factor analysis. Psychometrika, 1958, 23, 111–136.CrossRefGoogle Scholar
Turnbull, H. W. and Aitken, A. C. An introduction to the theory of canonical matrices, London: Blackie, 1932.Google Scholar
Vernon, P. E. The structure of human abilities, New York: Wiley, 1961.Google Scholar
Wilks, S. S. Moment generating operators for determinants of product moments in samples from a normal system. Annals of Mathematics, 1934, 35, 312–340.CrossRefGoogle Scholar