Abstract
A method is presented for constructing a covariance matrix Σ*0 that is the sum of a matrix Σ(γ0) that satisfies a specified model and a perturbation matrix,E, such that Σ*0=Σ(γ0) +E. The perturbation matrix is chosen in such a manner that a class of discrepancy functionsF(Σ*0, Σ(γ0)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ0 as minimizer and a prespecified minimum δ A matrix constructed in this way seems particularly valuable for Monte Carlo experiments as the covariance matrix for a population in which the model does not hold exactly. This may be a more realistic conceptualization in many instances. An example is presented in which this procedure is employed to generate a covariance matrix among nonnormal, ordered categorical variables which is then used to study the performance of a factor analysis estimator.
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References
Browne, M. W. (1969). Fitting the factor analysis model.Psychometrika, 34, 375–394.
Browne, M. W. (1974). Generalized least-squares estimators in the analysis of covariance structures.South African Statistical Journal, 8, 1–24.
Gill, P. E., Murray, W., & Wright, M. H. (1981).Practical optimization. San Diego, CA: Academic Press.
Graham, A. (1981).Kronecker products and matrix calculus, with applications. London: Ellis Horwood.
Hakstian, A. R., Rogers, W. T., & Cattell, R. B. (1982). The behavior of number-of-factors rules with simulated data.Multivariate Behavioral Research, 17, 193–219.
Huynh, H. (1978). Some approximate tests for repeated measurement designs.Psychometrika, 43, 161–175.
Jöreskog, K. G. (1973). A general method for estimating a linear structural equation system. In A. S. Goldberger & O. D. Duncan (Eds.),Structural equation models in the social sciences (pp. 85–112). New York: Seminar Press.
Kano, Y. (1986). Conditions for consistency of estimators in covariance structure models.Journal of the Japanese Statistical Society, 16, 75–80.
Laughlin, J. E. (1979). A Bayesian alternative to least squares and equal weighting coefficients in regression.Psychometrika, 44, 271–288.
Muthén, B., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables.British Journal of Mathematical and Statistical Psychology, 38, 171–189.
Nel, D. G. (1980). On matrix differentiation in statistics.South African Statistics Journal, 14, 137–193.
Shapiro, A., & Browne, M. W. (1988). On the asymptotic bias of estimators under parameter drift.Statistics and Probability Letters, 7, 221–224.
Thisted, R. A. (1988).Elements of statistical computing. New York: Chapman & Hall.
Tucker, L. R., Koopman, R. F., & Linn, R. L. (1969). Evaluation of factor analytic research procedures by means of simulated correlation matrices.Psychometrika, 34, 421–459.
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We are grateful to Alexander Shapiro for suggesting the proof of the solution in section 2.
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Cudeck, R., Browne, M.W. Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value. Psychometrika 57, 357–369 (1992). https://doi.org/10.1007/BF02295424
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DOI: https://doi.org/10.1007/BF02295424