This paper demonstrates the feasibility of using the penalty function method to estimate parameters that are subject to a set of functional constraints in covariance structure analysis. Both types of inequality and equality constraints are studied. The approaches of maximum likelihood and generalized least squares estimation are considered. A modified Scoring algorithm and a modified Gauss-Newton algorithm are implemented to produce the appropriate constrained estimates. The methodology is illustrated by its applications to Heywood cases in confirmatory factor analysis, quasi-Weiner simplex model, and multitrait-multimethod matrix analysis.
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Lee, S. Y. & Bentler, P. M.Some Asymptotic properties of constrained generalized least squares estimation in covariance structure models. Manuscript submitted for publication, 1979.
Aitchison, J. & Silvey, S. C. Maximum likelihood estimation of parameters subject to restraints.Annals of Mathematical Statistics, 1958,29, 813–828.
Bentler, P. M. Multistructure statistical model applied to factor analysis.Multivariate Behavioral Research, 1976,11, 3–25.
Bentler, P. M. & Lee, S. Y. Statistical aspects of a three-mode factor analysis model.Psychometrika, 1978,43, 343–352.
Bilodeau, E. A. Prediction of complex task proficiency by means of component responses.Perception and Motor Skills, 1961,12, 299–306.
Bock, R. D. & Bargmann, R. E. Analysis of covariance structure.Psychometrika, 1966,31, 507–534.
Carroll, C. W. The created response surface technique for optimizing nonlinear, restrained systems.Operation Research, 1961,9, 169–184.
Clarke, M. R. B. A rapidly convergent method for maximum likelihood factor analysis.British Journal of Mathematical and Statistical Psychology, 1970,23, 43–52.
Fiacco, A. V. & McCormick, G. P.Nonlinear programming: Sequential unconstrained minimization technique. New York: Wiley, 1968.
Guttman, L. A new approach to factor analysis: The radex. In P. F. Lazarsfeld (ed.),Mathematical Thinking in the Social Sciences. New York: Columbia Press, 1954.
Harman, H. H.Modern Factor Analysis (3rd ed.). Chicago: University of Chicago Press, 1976.
Jennrich, R. I. & Sampson, P. F. Application of stepwise regression to nonlinear estimation.Technometrics, 1968,40, 63–72.
Jennrich, R. I. & Robinson, S. M. A Newton-Raphson algorithm for maximum likelihood factor analysis.Psychometrika, 1969,34, 111–123.
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis.Psychometrika, 1967,32, 443–482.
Jöreskog, K. G. A general approach to confirmatory maximum likelihood factor analysis.Psychometrika, 1969,34, 183–202.
Jöreskog, K. G. Estimation and testing of simplex models.British Journal of Mathematical and Statistical Psychology, 1970,23, 121–145. (a).
Jöreskog, K. G. A general method for analysis of covariance structure.Biometrika, 1970,57, 239–251. (b).
Jöreskog, K. G. & Goldberger, A. S. Factor analysis by generalized least squares.Psychometrika, 1972,37, 243–260.
Jöreskog, K. G. & Sörbom, D.LISREL IV: Estimation of linear structural equation systems by maximum likelihood method. Chicago: National Education Resources. 1976.
Lawler, E. E. The multitrait-multirater approach to measuring managerial job performance.Journal of Applied Psychology, 1967,51, 369–381.
Lee, S. Y. & Jennrich, R. I. A study of algorithms for covariance structure analysis with specific comparisons using factor analysis.Psychometrika, 1979,43, 99–113.
Luenberger, D. G.Introduction to linear and nonlinear programming. Reading, Ma: Addison-Wesley, 1973.
Martin, J. K. & McDonald, R. P. Bayesian estimation in unrestricted factor analysis: A treatment for Heywood cases.Psychometrika, 1975,40, 505–517.
Maxwell, E. A. Recent trends in factor analysis.Journal of the Royal Statistical Society, Series A, 1961,124, 49–59.
McDonald, R. P. A simple comprehensive model for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 1978,31, 59–72.
Ostrom, T. M. Relationship between affect, behavior, and cognition.Journal of Experimental Social Psychology, 1969,5, 12–30.
Rao, C. R. Estimation and tests of significance in factor analysis.Psychometrika, 1955,20, 93–111.
Schmitt, N. Path analysis of multitrait-multimethod matrices.Applied Psychology Measurement, 1978,2, 157–173.
Silvey, S. D. The Lagrangian multiplier test.Annals of Mathematical Statistics, 1959,30, 389–407.
van Driel, O. P. On various causes of improper solutions in maximum likelihood factor analysis.Psychometrika, 1978,43, 225–243.
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.
The author is indebted to several anonymous reviewers for creative suggestions for improvement of this paper. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.
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Lee, SY. Estimation of covariance structure models with parameters subject to functional restraints. Psychometrika 45, 309–324 (1980). https://doi.org/10.1007/BF02293906
- confirmatory factor analysis
- covariance structure analysis
- equality constraints
- Gauss-Newton algorithm
- Heywood case
- inequality constraints
- penalty function
- quasi-Weiner simplex model
- Scoring algorithm