Least square estimation of non-linear structural models

  • Reinhard Oldenburg Augsburg University, Germany
Keywords: Structural equation model, Simulation study, Nonlinear regression model, error estimation


A new method for estimating a wide class of structural equation models (SEM) is proposed and evaluated. A weighted least squares approach is used that estimates parameters and latent variables. This new approach is flexible enough to handle non-linear and non-smooth models and allows us to model various constraints. The method includes various strategies to deal with the problem of choosing weights. The principle strengths and weaknesses of this approach are discussed, and simulation studies are performed to reveal the problems and potential of this approach.


Adachi, K. Some contributions to data-fitting factor analysis. J. Jpn. Soc. Comp. Statist 2012, 25, 197.

Adachi, K. Matrix-based Introduction to Multivariate Data Analysis; Springer: New York, 2020.

Arbuckle, J. Full information estimation in the presence of incomplete data. In Advanced Structural Equation Modeling: Issues and Techniques; Marcoulides, G.; Schumacker, R., Eds.; Lawrence Erlbaum Associates: Mahwah, 1996; pp. 243–277.

Bollen, K.A. Structural Equations with Latent Variables; John Wiley: Hoboken, 1989.

Cho, G.; Hwang, H. Structured Factor Analysis: A Data Matrix-Based Alternative Approach to Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal 2022, 0, 1–14.

Devlieger, I.; Mayer, A.; Rosseel, Y. Hypothesis Testing Using Factor Score Regression: A Comparison of Four Methods. Educational and Psychological Measurement 2016, 76, 741–770.

DiStefano, V.; Zhu, M.; Mindril, D. Understanding and Using Factor Scores: Considerations for the Applied Researcher. Practical Assessment, Research & Evaluation 2009, 14, 20.

Esposito Vinzi, V.; Chin, W.W.; Henseler, J.; Wang, H.E. Handbook of Partial Least Squares; Springer: New York, 2010.

Fisher, R.A. On the ’probable error’ of a coefficient of correlation deduced from a small sample. Metron 1921, 1.

Glaister, P. Least Squares Revisited. The Mathematical Gazette 2001, 85, 104.

Gras, R. Statistical implicative anaylsis; Springer: New York, 2008.

Guo, R.; Zhu, H.; Chow, S.M.; Ibrahim, J.G. Bayesian Lasso for Semiparametric Structural Equation Models. Biometrics 2012, 68, 567–577.

Hoyle, R.H.E. Handbook of Structural Equation Modeling; The Guilford Press: New York, 2012.

Hwang, H.; Takane, Y.; Jung, K. Generalized Structured Component Analysis with Uniqueness Terms for Accommodating Measurement Error. Frontiers in Psychology 2017, 8, 2137. https://doi.org/10.3389/fpsyg.2017.02137.

Hwang, H.; Cho, G.; Jung, K.; Lee, S.; et al. An approach to structural equation modeling with both factors and components: Integrated generalized structured component analysis. Psychological Methods 2020. https://doi.org/10.1037/met0000336.

Ivanov, A.V. Asymptotic Theory of Nonlinear Regression; Kluwer: Dordrecht, 1997.

Kelava, A.; Werner, C.S.; Schermelleh-Engel, K.; Moosbrugger, H.; Zapf, D.; Ma, Y.; Cham, H.; Aiken, L.S.; West, S.G. Advanced Nonlinear Latent Variable Modeling: Distribution Analytic LMS and QML Estimators of Interaction and Quadratic Effects. Structural Equation Modeling 2011, 18, 465–491.

Kelava, A.; Brandt, H. Estimation of nonlinear latent structural equation models using the extended unconstrained approach. Review of Psychology 2009, 16, 123–131.

Lancaster, T. The incidental parameter problem since 1948. Journal of Econometrics 2000, 95, 391–413.

Lee, S.Y.; Song, X.Y. Basic and Advanced Bayesian Structural Equation Modeling; Wiley, 2012.

Marcoulides, G.; Schumacker, R., Eds. Advanced Structural Equation Modeling: Issues and Techniques; Psychology Press, 1996. https://doi.org/10.4324/9781315827414.

Merkle, E.C.; Rosseel, Y. blavaan: Bayesian Structural Equation Models via Parameter Expansion. Journal of statistical software 2015, 85, 4.

Neumann, I.e.a. Modeling and assessing mathematical competence over the lifespan. Journal for educational research online 2013, 2, 80–109. https://doi.org/10.25656/01:8426.

Oldenburg, R. Do fuzzy-logic non-linear models provide a benefit for the modelling of algebraic competency? International Journal of Research in Education Methodology 2022, 13, 1-10. https://doi.org/10.24297/ijrem.v13i.9198.

Oldenburg, R. Structural Equation Modeling – Comparing Two Approaches. The Mathematica Journal 2020.

Philips, P.C.B. On the Consistency of Nonlinear FIML. Econometrica 1982, 50, 5.

Rosseel, Y. lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software 2012, 48, 1–36.

Rosseel, Y.; Loh, W.W. A structural after measurement approach to structural equation modeling. Psychological Methods 2022. https://doi.org/10.1037/met0000503.

Schumacker, R.E.; Marcoulides, G.A. Interaction and nonlinear effects in structural equation modeling; Lawrence Erlbaum Associates: Mahwah, NJ, 1998.

Seber, G.A.F.; Wild, C.J. Nonlinear Regression; John Wiley: New York, 1988.

Umbach, N.; Naumann, K.; Brandt, H.; Kelava, A. Fitting Nonlinear Structural Equation Models in R with Package nlsem. Journal of Statistical Software 2017, 77, 7.

Unkel, S.; Trendafilov, N.T. Simultaneous Parameter Estimation in Exploratory Factor Analysis: An Expository Review.

International Statistical Review 2010, 78, 363–382.

Yung, Y.F.; Yuan, K.H. Bartlett Factor Scores: General Formulas and Applications to Structural Equation Models. In New Developments in Quantitative Psychology; E., M.R.; van der Ark, L.A.; Bolt, D.M.; Woods, C.M., Eds.; Springer: New York, 2013.

Zadeh, L.A. Fuzzy Sets. Information and Control 1965, 8, 338–353.

How to Cite
Oldenburg, R. (2023). Least square estimation of non-linear structural models. Statistics, Optimization & Information Computing, 12(2), 281-297. https://doi.org/10.19139/soic-2310-5070-1868
Research Articles