Abstract
Structural models that yield circumplex inequality patterns for the elements of correlation matrices are reviewed. Particular attention is given to a stochastic process defined on the circle proposed by T. W. Anderson. It is shown that the Anderson circumplex contains the Markov Process model for a simplex as a limiting case when a parameter tends to infinity.
Anderson's model is intended for correlation matrices with positive elements. A replacement for Anderson's correlation function that permits negative correlations is suggested. It is shown that the resulting model may be reparametrzed as a factor analysis model with nonlinear constraints on the factor loadings. An unrestricted factor analysis, followed by an appropriate rotation, is employed to obtain parameter estimates. These estimates may be used as initial approximations in an iterative procedure to obtain minimum discrepancy estimates.
Practical applications are reported.
Similar content being viewed by others
References
Amemiya, Y., & Anderson, T. W. (1990). Asymptotic chi-square tests for a large class of factor analysis models.Annals of Statistics, 18, 1453–1463.
Anderson, T. W. (1960). Some stochastic process models for intelligence test scores. In K. J. Arrow, S. Karlin & P. Suppes (Eds.),Mathematical methods in the social sciences (pp. 205–220), Stanford, CA: Stanford University Press.
Anderson, T. W., & Amemiya, Y. (1988). The asymptotic normal distribution of estimators in factor analysis under general conditions.Annals of Statistics, 16, 759–771.
Bentler, P. M., & Weeks, D. G. (1980). Linear structural equations with latent variables.Psychometrika, 45, 289–308.
Browne, M. W. (1982) Covariance structures. In D. M. Hawkins (Ed.),Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.
Browne, M. W., & Cudeck, R. (1992). Alternative ways of assessing model fit.Sociological Methods and Research, 21, 230–258.
Browne, M. W., & Du Toit, S. H. C. (1992). Automated fitting of nonstandard models.Multivariate Behavioral Research, 27, 269–300.
Browne, M. W., & Shapiro, A. (1988). Robustness of normal theory methods in the analysis of linear latent variate models.British Journal of Mathematical and Statistical Psychology, 14, 193–208.
Beuhring, T., & Cudeck, R. (1985).Development of the culture fair interest inventory: Experimental junior version (Final Report on Project No. 85/1). Pretoria, South Africa: Human Sciences Research Council.
Cudeck, R. (1986). A note on structural models for the circumplex.Psychometrika, 51, 143–147.
Fraser, C., & McDonald, R. P. (1988). COSAN: Covariance structure analysis.Multivariate Behavioral Research, 23, 263–265.
Guttman, L. (1954). A new approach to factor analysis: The radex. In P. F. Lazarsfeld (Ed.),Mathematical thinking in the social sciences (pp. 258–348). New York: Columbia University Press.
Hartmann, W. M. (1992).The CALIS procedure: Extended user's guide. Cary, NC: SAS Institute.
Jöreskog, K. G. (1963).Statistical estimation in factor analysis: A new technique and its foundation. Stockholm: Almqvist & Wiksell.
Jöreskog, K. G. (1970). Estimation and testing of simplex models.British Journal of Mathematical and Statistical Psychology, 23, 121–145.
Jöreskog, K. G. (1974). Analyzing psychological data by structural analysis of covariance matrices. In D. H. Krantz, R. C. Atkinson, R. D. Luce, & P. Suppes (Eds.),Contemporary developments in mathematical psychology (Vol. 2, pp. 1–56). San Francisco: W. H. Freeman.
Jöreskog, K. G. (1977). Structural equation models in the social sciences: Specification estimation and testing. In P. R. Krishnaiah (Ed.),Applications of statistics (pp. 265–287). Amsterdam: North Holland.
Jöreskog, K. G., & Sörbom, D. (in press).Lisrel 8 User's Guide. Chicago, IL: Scientific Software.
McArdle, J. J., & McDonald, R. P. (1984). Some algebraic properties of the Reticular Action Model for moment structures.British Journal of Mathematical and Statistical Psychology, 37, 234–251.
McDonald, R. P. (1980). A simple comprehensive model for the analysis of covariance structures: Some remarks on applications.British Journal of Mathematical and Statistical Psychology, 33, 161–183.
Schönemann, P. H. (1970). Fitting a simplex symmetrically.Psychometrika, 35, 1–21.
Steiger, J. H., & Lind, J. C. (1980, May).Statistically based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa City, IA.
Steiger, J. H., Shapiro, A., & Browne, M. W. (1985). On the multivariate asymptotic distribution of sequential chi-square statistics.Psychometrika, 50, 253–264.
van den Wollenberg, A. (1978). Nonmetric representation of the radex in its factor pattern parametrization. In S. Shye (Ed.),Theory construction and data analysis in the behavioral sciences (pp. 326–349). San Francisco: Jossey-Bass.
Whittaker, E. T., & Watson, G. N. (1969).A course of modern analysis. Cambridge: Cambridge University Press.
Wiggins, J. S. (1979). A psychological taxonomy of trait descriptive terms.Journal of Personality and Social Psychology, 37, 395–412.
Wiggins, J. S., Steiger, J. H., & Gaelick, L. (1981). Evaluating circumplexity in personality data.Multivariate Behavioral Research, 16, 263–289.
Young, F. W., & Hamer, R. M. (1987).Multidimensional scaling: History, theory and applications. Hillsdale, NJ: Erlbaum.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Browne, M.W. Circumplex models for correlation matrices. Psychometrika 57, 469–497 (1992). https://doi.org/10.1007/BF02294416
Issue Date:
DOI: https://doi.org/10.1007/BF02294416