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Psychometrika

, Volume 43, Issue 4, pp 443–477 | Cite as

Structural analysis of covariance and correlation matrices

  • Karl G. Jöreskog
Article

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.

Key words

covariance structure analysis factor analysis variance components path analysis simplex circumplex 

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Reference notes

  1. Gruveaus, G. & Jöreskog, K. G.A computer program for minimizing a function of several variables (Res. Bull. 70-14). Princeton, N. J.: Educational Testing Service, 1970.Google Scholar
  2. Jöreskog, K. G. & Sörbom, D.Some regression models useful in the measurement of change (Res. Rep. 74-3). Uppsala, Sweden: Department of Statistics, University of Uppsala, 1974.Google Scholar

References

  1. Aitken, A. C. On least squares and the linear combination of observations.Proceedings of the Royal Society of Edinburgh, 1934–35,55, 42–28.Google Scholar
  2. Anderson, T. W. Some stochastic process models for intelligence test scores. In K. J. Arrow, S. Karlin & P. Suppes (Eds.),Mathematical methods in the social sciences, 1959. Stanford, Calif.: Stanford University Press, 1960, 205–220.Google Scholar
  3. Anderson, T. W. The use of factor analysis in the statistical analysis of multiple time series.Psychometrika, 1963,28, 1–25.Google Scholar
  4. Anderson, T. W. Statistical inference for covariance matrices with linear structure. In P. R. Krishnaiah (Ed.),Multivariate analysis—II. New York: Academic Press, 1969, 55–66.Google Scholar
  5. Archer, C. O. & Jennrich, R. I. Standard errors for rotated factor loadings.Psychometrika, 1973,38, 581–592.Google Scholar
  6. 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.Google Scholar
  7. Bock, R. D. & Bargmann, R. E. Analysis of covariance structures.Psychometrika, 1966,31, 507–534.Google Scholar
  8. Browne, M. W. Generalized least squares estimators in the analysis of covariance structures.South African Statistical Journal, 1974,8, 1–24. (Reprinted in D. J. Aigner & A. S. Goldberger (Eds.),Latent variables in socio-economic models. North Holland Publishing Co., 1977, 203–226.)Google Scholar
  9. Browne, M. W. The analysis of patterned correlation matrices by generalized least squares.British Journal of Mathematical and Statistical Psychology, 1977,30, 113–124.Google Scholar
  10. Campbell, D. T. & Fiske, D. W. Convergent and discriminant validation by the multitrait-multimethod matrix.Psychological Bulletin, 1959,56, 81–105.Google Scholar
  11. Fletcher, R. & Powell, M. J. D. A rapidly convergent descent method for minimization.The Computer Journal, 1963,6, 163–168.Google Scholar
  12. French, J. W. The description of aptitude and achievement tests in terms of rotated factors.Psychometric Monographs, 5, 1951.Google Scholar
  13. Graybill, F. A.An introduction to linear statistical models (Vol. 1). New York: McGraw-Hill, 1961.Google Scholar
  14. Guilford, J. P. The structure of intellect.Psychological Bulletin, 1956,53, 267–293.Google Scholar
  15. Guttman, L. A new approach to factor analysis: The radex. In P. F. Lazarsfeld (Ed.),Mathematical thinking in the social sciences. New York: Columbia University Press, 1954, 258–348.Google Scholar
  16. Harman, H. H.Modern factor analysis. Chicago: The University of Chicago Press, 1967.Google Scholar
  17. Hauser, R. M. & Goldberger, A. S. The treatment of unobservable variables in path analysis. In H. L. Costner (Ed.),Sociological Methodology 1971. London: Jossey-Bass, 1971, 81–117.Google Scholar
  18. 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.Google Scholar
  19. Humphreys, L. G. The fleeting nature of college academic success.Journal of Educational Psychology, 1968,59, 375–380.Google Scholar
  20. Jennrich, R. I. Standard errors for obliquely rotated factor loadings.Psychometrika, 1973,38, 593–604.Google Scholar
  21. Jöreskog, K. G. Some contributions to maximum likelihood factor analysis.Psychometrika, 1967,32, 443–482.Google Scholar
  22. Jöreskog, K. G. A general approach to confirmatory maximum likelihood factor analysis.Psychometrika, 1969,34, 183–202.Google Scholar
  23. Jöreskog, K. G. A general method for analysis of covariance structures.Biometrika, 1970,57, 239–251. (a)Google Scholar
  24. Jöreskog, K. G. Estimation and testing of simplex models.British Journal of Mathematical and Statistical Psychology, 1970,23, 121–145. (b)Google Scholar
  25. Jöreskog, K. G. Analysis of covariance structures. In P. R. Krishnaiah (Ed.),Multivariate analysis—III. New York: Academic Press, Inc., 1973, 263–285.Google Scholar
  26. Jöreskog, K. G. Analyzing psychological data by structural analysis of covariance matrices. In R. C. Atkinson, D. H. Krantz, R. D. Luce & P. Suppes (Eds.),Contemporary developments in mathematical psychology—Volume II. San Francisco: W. H. Freeman & Co., 1974, 1–56.Google Scholar
  27. Jöreskog, K. G. Factor analysis by least squares and maximum likelihood methods. K. Enslein, A. Ralston & M. S. Wilf (Eds.),Statistical methods for digital computers. New York: Wiley, 1977, 125–165. (a)Google Scholar
  28. Jöreskog, K. G. Structural equation models in the social sciences: Specification, estimation and testing. In P. R. Krishnaiah (Ed.),Applications of statistics. Amsterdam: North Holland Publishing Co., 1977, 265–286. (b)Google Scholar
  29. Jöreskog, K. G. Statistical estimation of structural models in longitudinal-developmental investigations. In J. R. Nesselroade & P. B. Baltes (Eds.),Longitudinal research in the behavioral sciences: Design and analysis. New York: Academic Press, 1978, in press.Google Scholar
  30. Jöreskog, K. G. & Goldberger, A. S. Factor analysis by generalized least squares.Psychometrika, 1972,37, 243–260.Google Scholar
  31. 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
  32. Jöreskog, K. G. & Sörbom, D. Statistical models and methods for analysis of longitudinal data. In D. J. Aigner & A. S. Goldberger (Eds.),Latent variables in socio-economic models. Amsterdam: North Holland Publishing Co., 1977, 235–285.Google Scholar
  33. Jöreskog, K. G. & Sörbom, D.LISREL IV—A general computer program for estimation of a linear structural equation system by maximum likelihood methods. Chicago: National Educational Resources, 1978.Google Scholar
  34. Jöreskog, K. G. & Sörbom, D.EFAP II-Exploratory factor analysis program. Chicago: National Educational Resources, 1979, in press.Google Scholar
  35. Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis.Psychometrika, 1958,23, 187–200.Google Scholar
  36. Kristof, W. On the theory of a set of tests which differ only in length.Psychometrika, 1971,36, 207–225.Google Scholar
  37. Lawley, D. N. & Maxwell, A. E.Factor analysis as a statistical method (2nd ed.). New York: American Elsevier, 1971.Google Scholar
  38. Lord, F. M. A study of speed factors in tests and academic grades.Psychometrika, 1956,21, 31–50.Google Scholar
  39. Lord, F. M. & Novick, M. E.Statistical theories of mental test scores. Reading: Addison-Wesley Publishing Co., 1968.Google Scholar
  40. 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.Google Scholar
  41. McDonald, R. P. Testing pattern hypotheses for covariance matrices.Psychometrika, 1974,39, 189–20.Google Scholar
  42. McDonald, R. P. Testing pattern hypotheses for correlation matrices.Psychometrika, 1975,40, 253–255.Google Scholar
  43. 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.Google Scholar
  44. Rao, C. R.Linear statistical inference and its applications (2nd ed.). New York: Wiley, 1973.Google Scholar
  45. 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.Google Scholar
  46. Thurstone, L. L. Primary mental abilities.Psychometric Monographs, 1, 1938.Google Scholar
  47. Thurstone, L. L.Multiple factor analysis. Chicago: University of Chicago Press, 1947.Google Scholar
  48. 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.Google Scholar
  49. Werts, C. E. & Linn, R. L. Path analysis: Psychological examples.Psychological Bulletin, 1970,67, 193–212.Google Scholar
  50. Werts, C. E., Linn, R. L. & Jöreskog, K. G. Reliability of college grades from longitudinal data.Educational and Psychological Measurement, 1978,38, 89–95.Google Scholar
  51. Wheaton, B., Muthén, B., Alwin, D. & Summers, G. Assessing reliability and stability in panel models. In D. R. Heise (Ed.),Sociological Methodology 1977. San Francisco: Jossey-Bass, 1977.Google Scholar
  52. 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.Google Scholar
  53. Wright, S. The method of path coefficients.The Annals of Mathematical Statistics, 1934,5, 161–215.Google Scholar

Copyright information

© Psychometric Society 1978

Authors and Affiliations

  • Karl G. Jöreskog
    • 1
  1. 1.Department of StatisticsUniversity of UppsalaUppsalaSweden

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