Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-08T16:28:33.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimating Structural Equation Models Using James–Stein Type Shrinkage Estimators

Published online by Cambridge University Press:  01 January 2025

Elissa Burghgraeve Open the ORCID record for Elissa Burghgraeve [Opens in a new window] ,
Jan De Neve Open the ORCID record for Jan De Neve [Opens in a new window]  and
Yves Rosseel Open the ORCID record for Yves Rosseel [Opens in a new window]
Elissa Burghgraeve*
Affiliation:
Ghent University
Jan De Neve
Affiliation:
Ghent University
Yves Rosseel
Affiliation:
Ghent University
*
Correspondence should be made to Elissa Burghgraeve, Department of Data Analysis, Ghent University, Henri Dunantlaan 1, Ghent, Belgium. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a two-step procedure to estimate structural equation models (SEMs). In a first step, the latent variable is replaced by its conditional expectation given the observed data. This conditional expectation is estimated using a James–Stein type shrinkage estimator. The second step consists of regressing the dependent variables on this shrinkage estimator. In addition to linear SEMs, we also derive shrinkage estimators to estimate polynomials. We empirically demonstrate the feasibility of the proposed method via simulation and contrast the proposed estimator with ML and MIIV estimators under a limited number of simulation scenarios. We illustrate the method on a case study.

Keywords

regression calibrationmeasurement errorshrinkage estimator
Type
Theory and Methods
Information
Psychometrika , Volume 86 , Issue 1 , March 2021 , pp. 96 - 130
Copyright
Copyright © 2021 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bentler, P. M. (1968). Alpha-maximized factor analysis (alphamax): Its relation to alpha and canonical factor analysis. Psychometrika. 33 (3), 335– 345. CrossRefGoogle ScholarPubMed
Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley. CrossRefGoogle Scholar
Bollen, K. A. (1995). Structural equation models that are nonlinear in latent variables: A least-squares estimator. Sociological Methodology. 25 223– 251. CrossRefGoogle Scholar
Bollen, K. A. (1996). An alternative two stage least squares (2SLS) estimator for latent variable equations. Psychometrika. 61 (1), 109– 121. CrossRefGoogle Scholar
Bollen, K. A. (2018). Model implied instrumental variables (MIIVs): An alternative orientation to structural equation modeling. Multivariate Behavioral Research. 54, 1– 16. Google ScholarPubMed
Carroll, R. J., Ruppert, D, Stefanski, L. A., & Crainiceanu, C. M. (2006). Measurement error in nonlinear models. 2 New York: Chapman and Hall. CrossRefGoogle Scholar
Efron, B. (2011). Tweedie’s formula and selection bias. Journal of the American Statistical Association. 496 (106), 1602– 1614. CrossRefGoogle Scholar
Efron, B., & Morris, C. (1973). Stein’s estimation rule and its competitors—An empirical Bayes approach. Journal of the American Statistical Association. 68 (341), 117– 130. Google Scholar
Efron, B., & Morris, C. (1975). Data analysis using Stein’s estimator and its generalizations. Journal of the American Statistical Association. 70 (350), 311– 319. CrossRefGoogle Scholar
Fisher, Z., Bollen, K., Gates, K., & Rönkkö, M. (2019). MIIVsem: Model implied instrumental variable (MIIV) estimation of structural equation models. R package version, 0.5.4. Google Scholar
Fuller, W. A. (1987). Measurement error models. New York: Wiley. CrossRefGoogle Scholar
Fuller, W. A., & Hidiroglou, M. A. (1978). Regression estimation after correcting for attenuation. Journal of the American Statistical Association. 73 (361), 99– 104. CrossRefGoogle Scholar
Gleser, L. J. (1990). Improvements of the naive approach to estimation in nonlinear errors-in-variables regression models. Contemporary Mathematics. 112 99– 114. CrossRefGoogle Scholar
Harman, H. H. (1976). Modern factor analysis. Chicago: University of Chicago Press. Google Scholar
James, W., & Stein, C. (1961). Estimation with quadratic loss. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 361–379). Google Scholar
Jöreskog, K. G., & Sörbom, D. (1993). LISREL8: User’s reference guide. Lincolnwood: Scientific Software International. Google Scholar
Kano, Y. (1990). Noniterative estimation and the choice of the number of factors in exploratory factor analysis. Psychometrika. 55 (2), 277– 291. CrossRefGoogle Scholar
Kelley, T. L. (1947). Fundamentals of statistics. Cambridge: Harvard University Press. Google Scholar
Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interactive effect of latent variables. Psychological Bulletin. 96 (1), 201– 210. CrossRefGoogle Scholar
Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2005). Applied linear statistical models. 5 New York: McGraw-Hill/Irwin. Google Scholar
R Core Team (2018). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. Google Scholar
Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software. 48 (2), 1– 36. CrossRefGoogle Scholar
Sargan, J. D. (1958). The estimation of economic relationships using instrumental variables. Econometrica. 26 393– 415. CrossRefGoogle Scholar
Shi, J., Feng, J., & Song, W. (2018). Estimation in linear regression with Laplace measurement error using Tweedie-type formula. Journal of Systems Science and Complexity. 32 1– 20. Google Scholar
Spearman, C. (1904). ‘General intelligence’, objectively determined and measured. The American Journal of Psychology. 15 (2), 201– 293. CrossRefGoogle Scholar
Spearman, C. (1927). The ability of man. London: Macmillan. Google Scholar
Warren, R. D., White, J. K., & Fuller, W. A. (1974). An errors-in-variables analysis of managerial role performance. Journal of the American Statistical Association. 69 (348), 886– 893. CrossRefGoogle Scholar
Whittemore, A. S. (1989). Errors-in-variables regression using Stein estimates. The American Statistician. 43 (4), 226– 228. CrossRefGoogle Scholar
Yalcin, I., & Amemiya, Y. (2001). Nonlinear factor analysis as a statistical method. Statistical Science. 16 (3), 275– 294. Google Scholar