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Structural Analysis of Covariance and Correlation Matrices

Published online by Cambridge University Press:  01 January 2025

Karl G. Jöreskog
Karl G. Jöreskog*
Affiliation:
University of Uppsala
*
Requests for reprints should be addressed to Karl G. Jöreskog, Department of Statistics, University of Uppsala, P.O. Box 513, S-751 20 UPPSALA, Sweden
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Abstract

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.

Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.

Keywords

covariance structure analysisfactor analysisvariance componentspath analysissimplexcircumplex
Type
Original Paper
Information
Psychometrika , Volume 43 , Issue 4 , December 1978 , pp. 443 - 477
Copyright
Copyright © 1978 The Psychometric Society

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Footnotes

1978 Psychometric Society Presidential Address.

This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitled Structural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.

References

Reference Notes

Gruveaus, G. & Jöreskog, K. G. A computer program for minimizing a function of several variables, 1970, Princeton, N. J.: Educational Testing Service.Google Scholar
Jöreskog, K. G. & Sörbom, D. Some regression models useful in the measurement of change, 1974, Uppsala, Sweden: Department of Statistics, University of Uppsala.Google Scholar

References

Aitken, A. C. On least squares and the linear combination of observations. Proceedings of the Royal Society of Edinburgh, 1934, 55, 42–28.CrossRefGoogle Scholar
Anderson, T. W. Some stochastic process models for intelligence test scores. In Arrow, K. J., Karlin, S. & Suppes, P. (Eds.), Mathematical methods in the social sciences, 1959. Stanford, Calif.: Stanford University Press. 1960, 205–220.Google Scholar
Anderson, T. W. The use of factor analysis in the statistical analysis of multiple time series. Psychometrika, 1963, 28, 1–25.CrossRefGoogle Scholar
Anderson, T. W. Statistical inference for covariance matrices with linear structure. In Krishnaiah, P. R. (Eds.), Multivariate analysis–II. New York: Academic Press. 1969, 55–66.Google Scholar
Archer, C. O. & Jennrich, R. I. Standard errors for rotated factor loadings. Psychometrika, 1973, 38, 581–592.CrossRefGoogle Scholar
Bock, R. D. Components of variance analysis as a structural and discriminal analysis for psychological tests. British Journal of Statistical Psychology, 1960, 13, 151–163.CrossRefGoogle Scholar
Bock, R. D. & Bargmann, R. E. Analysis of covariance structures. Psychometrika, 1966, 31, 507–534.CrossRefGoogle ScholarPubMed
Browne, M. W. Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 1974, 8, 1–24.Google Scholar
Browne, M. W. The analysis of patterned correlation matrices by generalized least squares. British Journal of Mathematical and Statistical Psychology, 1977, 30, 113–124.CrossRefGoogle Scholar
Campbell, D. T. & Fiske, D. W. Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 1959, 56, 81–105.CrossRefGoogle ScholarPubMed
Fletcher, R. & Powell, M. J. D. A rapidly convergent descent method for minimization. The Computer Journal, 1963, 6, 163–168.CrossRefGoogle Scholar
French, J. W. The description of aptitude and achievement tests in terms of rotated factors. Psychometric Monographs, 5, 1951.Google Scholar
Graybill, F. A. An introduction to linear statistical models (Vol. 1), 1961, New York: McGraw-Hill.Google Scholar
Guilford, J. P. The structure of intellect. Psychological Bulletin, 1956, 53, 267–293.CrossRefGoogle ScholarPubMed
Guttman, L. A new approach to factor analysis: The radex. In Lazarsfeld, P. F. (Eds.), Mathematical thinking in the social sciences. New York: Columbia University Press. 1954, 258–348.Google Scholar
Harman, H. H. Modern factor analysis, 1967, Chicago: The University of Chicago Press.Google Scholar
Hauser, R. M. & Goldberger, A. S. The treatment of unobservable variables in path analysis. In Costner, H. L. (Eds.), Sociological Methodology 1971. London: Jossey-Bass. 1971, 81–117.Google Scholar
Hendrickson, A. E. & White, P.O. Promax: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 1964, 17, 65–70.CrossRefGoogle Scholar
Humphreys, L. G. The fleeting nature of college academic success. Journal of Educational Psychology, 1968, 59, 375–380.CrossRefGoogle ScholarPubMed
Jennrich, R. I. Standard errors for obliquely rotated factor loadings. Psychometrika, 1973, 38, 593–604.CrossRefGoogle Scholar
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis. Psychometrika, 1967, 32, 443–482.CrossRefGoogle Scholar
Jöreskog, K. G. A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 1969, 34, 183–202.CrossRefGoogle Scholar
Jöreskog, K. G. A general method for analysis of covariance structures. Biometrika, 1970, 57, 239–251.CrossRefGoogle Scholar
Jöreskog, K. G. Estimation and testing of simplex models. British Journal of Mathematical and Statistical Psychology, 1970, 23, 121–145.CrossRefGoogle Scholar
Jöreskog, K. G. Analysis of covariance structures. In Krishnaiah, P. R. (Eds.), Multivariate analysis–III. New York: Academic Press, Inc.. 1973, 263–285.CrossRefGoogle Scholar
Jöreskog, K. G. Analyzing psychological data by structural analysis of covariance matrices. In Atkinson, R. C., Krantz, D. H., Luce, R. D. & Suppes, P. (Eds.), Contemporary developments in mathematical psychology–Volume II. San Francisco: W. H. Freeman & Co.. 1974, 1–56.Google Scholar
Jöreskog, K. G. Factor analysis by least squares and maximum likelihood methods. In Enslein, K., Ralston, A. & Wilf, M. S. (Eds.), Statistical methods for digital computers. New York: Wiley. 1977, 125–165.Google Scholar
Jöreskog, K. G. Structural equation models in the social sciences: Specification, estimation and testing. In Krishnaiah, P. R. (Eds.), Applications of statistics. Amsterdam: North Holland Publishing Co.. 1977, 265–286.Google Scholar
Jöreskog, K. G. Statistical estimation of structural models in longitudinal-developmental investigations. In Nesselroade, J. R. & Baltes, P. B. (Eds.), Longitudinal research in the behavioral sciences: Design and analysis, 1978, New York: Academic Press.Google Scholar
Jöreskog, K. G. & Goldberger, A. S. Factor analysis by generalized least squares. Psychometrika, 1972, 37, 243–260.CrossRefGoogle Scholar
Jöreskog, K. G. & Goldberger, A. S. Estimation of a model with multiple indicators and multiple causes of a single latent variable. Journal of the American Statistical Association, 1975, 10, 631–639.Google Scholar
Jöreskog, K. G. & Sörbom, D. Statistical models and methods for analysis of longitudinal data. In Aigner, D. J. & Goldberger, A. S. (Eds.), Latent variables in socio-economic models. Amsterdam: North Holland Publishing Co.. 1977, 235–285.Google Scholar
Jöreskog, K. G. & Sörbom, D. LISREL IV–A general computer program for estimation of a linear structural equation system by maximum likelihood methods, 1978, Chicago: National Educational Resources.Google Scholar
Jöreskog, K. G. & Sörbom, D. EFAP II-Exploratory factor analysis program, 1979, Chicago: National Educational Resources.Google Scholar
Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis. Psychometrika, 1958, 23, 187–200.CrossRefGoogle Scholar
Kristof, W. On the theory of a set of tests which differ only in length. Psychometrika, 1971, 36, 207–225.CrossRefGoogle Scholar
Lawley, D. N. & Maxwell, A. E. Factor analysis as a statistical method, 2nd ed., New York: American Elsevier, 1971.Google Scholar
Lord, F. M. A study of speed factors in tests and academic grades. Psychometrika, 1956, 21, 31–50.CrossRefGoogle Scholar
Lord, F. M. & Novick, M. E. Statistical theories of mental test scores, 1968, Reading: Addison-Wesley Publishing Co..Google Scholar
Miller, D. M. & Lutz, M. V. Item design for an inventory of teaching practices and learning situations. Journal of Educational Measurement, 1966, 3, 53–61.CrossRefGoogle Scholar
McDonald, R. P. Testing pattern hypotheses for covariance matrices. Psychometrika, 1974, 39, 189–20.CrossRefGoogle Scholar
McDonald, R. P. Testing pattern hypotheses for correlation matrices. Psychometrika, 1975, 40, 253–255.CrossRefGoogle Scholar
Mukherjee, B. N. Likelihood ratio tests of statistical hypotheses associated with patterned covariance matrices in psychology. British Journal of Mathematical and Statistical Psychology, 1970, 23, 120–120.CrossRefGoogle Scholar
Rao, C. R. Linear statistical inference and its applications, 2nd ed., New York: Wiley, 1973.CrossRefGoogle Scholar
Rock, D. A., Werts, C. E., Linn, R. L. & Jöreskog, K. G. A maximum likelihood solution to the errors in variables and errors in equations model. Journal of Multivariate Behavioral Research, 1977, 12, 187–197.CrossRefGoogle Scholar
Thurstone, L. L. Primary mental abilities. Psychometric Monographs, 1, 1938.Google Scholar
Thurstone, L. L. Multiple factor analysis, 1947, Chicago: University of Chicago Press.Google Scholar
Warren, R. D., White, J. K. & Fuller, W. A. An errors in variables analysis of managerial role performance. Journal of the American Statistical Association, 1974, 69, 886–893.CrossRefGoogle Scholar
Werts, C. E. & Linn, R. L. Path analysis: Psychological examples. Psychological Bulletin, 1970, 67, 193–212.CrossRefGoogle Scholar
Werts, C. E., Linn, R. L. & Jöreskog, K. G. Reliability of college grades from longitudinal data. Educational and Psychological Measurement, 1978, 38, 89–95.CrossRefGoogle Scholar
Wheaton, B., Muthén, B., Alwin, D. & Summers, G. Assessing reliability and stability in panel models. In Heise, D. R. (Eds.), Sociological Methodology 1977, 1977, San Francisco: Jossey-Bass.Google Scholar
Wiley, D. E., Schmidt, W. H. & Bramble, W. J. Studies of a class of covariance structure models. Journal of the American Statistical Association, 1973, 68, 317–323.CrossRefGoogle Scholar
Wright, S. The method of path coefficients. The Annals of Mathematical Statistics, 1934, 5, 161–215.CrossRefGoogle Scholar