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Quantifying Adventitious Error in a Covariance Structure as a Random Effect

Published online by Cambridge University Press:  01 January 2025

Hao Wu  and
Michael W. Browne
Hao Wu*
Affiliation:
Boston College
Michael W. Browne
Affiliation:
The Ohio State University
*
Correspondence should be made to HaoWu, Department of Psychology, Boston College, Chestnut Hill, MA02467, USA. Email: hao.wu.5@bc.edu
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Abstract

We present an approach to quantifying errors in covariance structures in which adventitious error, identified as the process underlying the discrepancy between the population and the structured model, is explicitly modeled as a random effect with a distribution, and the dispersion parameter of this distribution to be estimated gives a measure of misspecification. Analytical properties of the resultant procedure are investigated and the measure of misspecification is found to be related to the root mean square error of approximation. An algorithm is developed for numerical implementation of the procedure. The consistency and asymptotic sampling distributions of the estimators are established under a new asymptotic paradigm and an assumption weaker than the standard Pitman drift assumption. Simulations validate the asymptotic sampling distributions and demonstrate the importance of accounting for the variations in the parameter estimates due to adventitious error. Two examples are also given as illustrations.

Keywords

covariance structureadventitious errormisspecificationPitman driftRMSEArandom effect

Information

Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 3 , September 2015 , pp. 571 - 600
Copyright
Copyright © 2014 The Psychometric Society

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References

Billingsley, P. (1999). Convergence of probability measures. New York: Wiley.CrossRefGoogle Scholar
Briggs, N.E., & MacCallum, R.C. (2003). Recovery of weak common factors by maximum likelihood and ordinary least squares estimation. Multivariate Behavioral Research, 38, 25–56.CrossRefGoogle ScholarPubMed
Browne, M.W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.Google Scholar
Browne, M.W. (1984). Asymptotically distribution free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Browne, M.W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods and Research, 21(2), 230–258.CrossRefGoogle Scholar
Chen, C-F. (1979). Bayesian inference for a normal dispersion matrix and its application to stochastic multiple regression analysis. Journal of the Royal Statistical Society, Series B, 41, 235–248.CrossRefGoogle Scholar
Cheung, MW-L. (2012). Constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16, 267–294.CrossRefGoogle Scholar
Chun, S.Y., & Shapiro, A. (2009). Normal versus noncentral Chi square asymptotics of misspecified models. Multivariate Behavioral Research, 44, 803–827.CrossRefGoogle ScholarPubMed
Costa, P.T., & McCrae, R.R. (1992). (NEO PI-R) professional manual. Odessa, FL: Psychological Assessment Resources, Inc.Google Scholar
Cudeck, R., & Henly, S.J. (1991). Model selection in covariance structures and the "problem" of sample size: A clarification. Psychological Bulletin, 109, 512–519.CrossRefGoogle ScholarPubMed
Gupta, A.K., & Nagar, D.K. (1999). Matrix variate distributions. London: Chapman & Hall/CRC.Google Scholar
Lawley, D.N., & Maxwell, A.E. (1971). Factor analysis as a statistical method. New York: Elsevier.Google Scholar
Kennedy, M.C., & O’Hagan, A. (2001). Bayesian calibration of computer models. Journal of Royal Statistical Society, Series B, 63(3), 425–464.CrossRefGoogle Scholar
MacCallum, R.C. (2003). Working with imperfect models. Multivariate Behavior Research, 38, 113–139.CrossRefGoogle ScholarPubMed
MacCallum, R.C. (2011a). Exploring uncertainty in structural equation modeling, presented at the American Psychological Association Annual Convention, Washington, DC.Google Scholar
MacCallum, R.C. (2011b). A brief introduction to uncertainty quantification, presented at the annual meeting of Society of Multivariate Experimental Psychology, Norman, OK.Google Scholar
MacCallum, R.C., Browne, M.W., & Sugawara, H.M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1, 130–149.CrossRefGoogle Scholar
MacCallum, R.C., & Tucker, L.R. (1991). Representing sources of errors in a common factor model. Psychological Bulletin, 109, 502–511.CrossRefGoogle Scholar
Neale, M.C., & Miller, M.B. (1997). The use of likelihood based confidence intervals in genetic models. Behavior Genetics, 27, 113–120.CrossRefGoogle ScholarPubMed
Resnick, S.I. (2001). A probability path. Boston: Birkhäuser.Google Scholar
Roux, J. J. J., & Becker, P. J., (1984). On prior inverted Wishart distribution. Department of Statistics and Operations Research, University of South Africa, Pretoria, Research Report No. 2.Google Scholar
Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (a unified approach). South African Statistical Journal, 17, 33–81.Google Scholar
Shapiro, A. (1984). A note on consistency of estimators in the analysis of moment structures. British Journal of Mathematical and Statistical Psychology, 37, 84–88.CrossRefGoogle Scholar
Shapiro, A. (1985). Asymptotic equivalence of minimum discrepancy function estimators to GLS estimators. South African Statistical Journal, 19, 73–81.Google Scholar
Shapiro, A. (2007). Statistical inference in moment structures. In S.-Y. Lee (Ed.), Hand Book of Latent Variable and Related Models (pp. 229–260).CrossRefGoogle Scholar
Steiger, J. H., & Lind, J. C., (1980). Statistically based tests for the number of factors. Presented at the annual meeting of the Psychometric Society, Iowa City, IA.Google Scholar
Swain, A.J. (1975). A class of factor analysis estimation procedures with common asymptotic sampling properties. Psychometrika, 40, 315–335.CrossRefGoogle Scholar
Thurstone, L.L., & Thurstone, T.G. (1941). Factorial studies of intelligence. Psychometric monographs, No. 2. Chicago: Univ. Chicago Press.Google Scholar
Trucano, T.G., Swiler, L.P., Igusa, T., Oberkampf, W.L., & Pilch, M. (2006). Calibration, validation and sensitivity analysis: What’s what. Reliability Engineering and System Safety, 91, 1331–1357.CrossRefGoogle Scholar
Tucker, L.R. (1958). An inter-battery method of factor analysis. Psychometrika, 23, 111–136.CrossRefGoogle Scholar
Tucker, L.R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38, 1–10.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Wardorp, L.J., Grasman, RPPP, & Huizenga, H.M. (2006). Goodness of fit and confidence intervals of approximate models. Journal of Mathematical Psychology, 50, 203–213.CrossRefGoogle Scholar
White, H. (1980). Nonlinear regression on cross-section data. Econometrika, 48, 721–746.CrossRefGoogle Scholar
White, H. (1981). Consequences and detections of misspecified nonlinear regression models. JASA, 76, 419–433.CrossRefGoogle Scholar
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrika, 50, 126–150.CrossRefGoogle Scholar
Wu, H., & Neale, M.C. (2012). Adjusted confidence intervals for a bounded parameter. Behavior Genetics, 42, 886–898.CrossRefGoogle ScholarPubMed
Yuan, K-H. (2008). Noncentral chi-square versus normal distributions in describing the likelihood ratio statistic: The univariate case and its multivariate implication. Multivariate Behavioral Research, 43, 109–136.CrossRefGoogle ScholarPubMed
Yuan, K-H, Hayashi, K., & Bentler, P.M. (2007). Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses. Journal of Multivariate Analysis, 98, 1262–1282.CrossRefGoogle Scholar