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Efficient Likelihood Estimation of Generalized Structural Equation Models with a Mix of Normal and Nonnormal Responses

Published online by Cambridge University Press:  01 January 2025

Nicholas J. Rockwood Open the ORCID record for Nicholas J. Rockwood [Opens in a new window]
Nicholas J. Rockwood*
Affiliation:
Loma Linda University
*
Correspondence should be made to Nicholas J. Rockwood, Division of Interdisciplinary Studies, School of Behavioral Health, Loma Linda University, 11065 Campus St., Loma Linda, CA92350, USA. Email: [email protected]; URL: https://www.njrockwood.com
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Abstract

A maximum likelihood estimation routine is presented for a generalized structural equation model that permits a combination of response variables from various distributions (e.g., normal, Poisson, binomial, etc.). The likelihood function does not have a closed-form solution and so must be numerically approximated, which can be computationally demanding for models with several latent variables. However, the dimension of numerical integration can be reduced if one or more of the latent variables do not directly affect any nonnormal endogenous variables. The method is demonstrated using an empirical example, and the full estimation details, including first-order derivatives of the likelihood function, are provided.

Keywords

structural equation modelinglatent variable modelinggeneralized linear modelsmaximum likelihood estimation
Type
Theory and Methods
Information
Psychometrika , Volume 86 , Issue 2 , June 2021 , pp. 642 - 667
Copyright
Copyright © 2021 The Psychometric Society

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References

Asparouhov, T., Masyn, K., & Muthen, B. (2006). Continuous time survival in latent variable models. In Proceedings of the joint statistical meeting in Seattle (pp. 180-187).Google Scholar
Bartholomew, D.J., Knott, M. (1999). Kendall’s library of statistics 7: Latent variable models and factor analysis, London: Edward ArnoldGoogle Scholar
Bauer, D.J. (2003). Estimating multilevel linear models as structural equation models. Journal of Educational and Behavioral Statistics, 28 (2), 135–167CrossRefGoogle Scholar
Bauer, D.J. (2005). A semiparametric approach to modeling nonlinear relations among latent variables. Structural Equation Modeling, 12 (4), 513–535CrossRefGoogle Scholar
Bentler, P. M. (2004). Eqs 6: Structural equations program manual, Encino: Multivariate SoftwareGoogle Scholar
Bianconcini, S. et al.. (2014). Asymptotic properties of adaptive maximum likelihood estimators in latent variable models. Bernoulli, 20 (3), 1507–1531CrossRefGoogle Scholar
Cai, L. (2010). High-dimensional exploratory item factor analysis by a Metropolis–Hastings Robbins–Monro algorithm. Psychometrika, 75 (1), 33–57CrossRefGoogle Scholar
Collins, S. E., Witkiewitz, K., Larimer, M. E. (2011). The theory of planned behavior as a predictor of growth in risky college drinking. Journal of Studies on Alcohol and Drugs, 72 (2), 322–332CrossRefGoogle ScholarPubMed
Curran, P. J. (2003). Have multilevel models been structural equation models all along?. Multivariate Behavioral Research, 38 (4), 529–569CrossRefGoogle ScholarPubMed
du Toit, S. H., Cudeck, R. (2009). Estimation of the nonlinear random coefficient model when some random effects are separable. Psychometrika, 74 (1), 65–82CrossRefGoogle Scholar
Eddelbuettel, D. (2013). Seamless R and C++ integration with Rcpp, New York: SpringerCrossRefGoogle Scholar
Eddelbuettel, D., Sanderson, C. (2014). RcppArmadillo: Accelerating R with high-performance C++ linear algebra. Computational Statistics and Data Analysis, 71, 1054–1063CrossRefGoogle Scholar
Fisher, R. A. (1925). Theory of statistical estimation. In Mathematical proceedings of the Cambridge Philosophical Society (Vol. 22, pp. 700–725).CrossRefGoogle Scholar
Fox, J., Nie, Z., & Byrnes, J. (2017). sem: Structural equation models [Computer software manual]. Retrieved from https://CRAN.R-project.org/package=sem (R package version 3.1-9)Google Scholar
Jöreskog, K. G., Sörbom, D. (1996). Lisrel 8: User’s reference guide, Skokie: Scientific Software InternationalGoogle Scholar
Lambert, D. (1992). Zero-in ated Poisson regression, with an application to defects in manufacturing. Technometrics, 34 (1), 1–14CrossRefGoogle Scholar
Liu, H. (2007). Growth curve models for zero-in ated count data: An application to smoking behavior. Structural Equation Modeling: A Multidisciplinary Journal, 14 (2), 247–279CrossRefGoogle Scholar
Liu, Q., Pierce, D. A. (1994). A note on Gauss–Hermite quadrature. Biometrika, 81 (3), 624–629Google Scholar
Moustaki, I., Knott, M. (2000). Generalized latent trait models. Psychometrika, 65 (3), 391–411CrossRefGoogle Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49 (1), 115–132CrossRefGoogle Scholar
Muthén, B. O., & Asparouhov, T. (2008). Growth mixture modeling: Analysis with non-Gaussian random effects. In G. Fitzmaurice, M. Davidian, G. Verbeke, & G. Molenberghs (Eds.), Longitudinal data analysis (pp. 143–165). Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
Muthén, B., & Asarouhov, T. (2011). Beyond multilevel regression modeling: Multilevel analysis in a general latent variable framework. In J. Hox & J. K. Roberts (Eds.), Handbook of advanced multilevel analysis (pp. 15-40). New York, NY: Taylor and Francis.Google Scholar
Muthén, B. O. (2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29 (1), 81–117CrossRefGoogle Scholar
Muthén, B. O., Muthén, L. K., Asparouhov, T. (2017). Regression and mediation analysis using mplus, Los Angeles, CA: Muthén & MuthénGoogle Scholar
Muthén, L. K., Muthén, B. (2017). Mplus version 8 [Computer software manual], Los Angeles, CA: Muthén & MuthénGoogle Scholar
Nash, J. C. (2014). On best practice optimization methods in R. Journal of Statistical Software, 60 (2), 1–14CrossRefGoogle Scholar
Neale, M. C., Hunter, M. D., Pritikin, J. N., Zahery, M., Brick, T. R., Kirkpatrick, R. M., Boker, S. M. (2016). OpenMx 2.0: Extended structural equation and statistical modeling. Psychometrika, 81 (2), 535–549CrossRefGoogle ScholarPubMed
Pek, J., Sterba, S. K., Kok, B. E., Bauer, D. J. (2009). Estimating and visualizing nonlinear relations among latent variables: A semiparametric approach. Multivariate Behavioral Research, 44 (4), 4070–4436CrossRefGoogle ScholarPubMed
Pinheiro, J. C., Bates, D. M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4 (1), 12–35CrossRefGoogle Scholar
R Core Team. (2019). R: A language and environment for statistical computing [Computer software manual]. Vienna, Austria. Retrieved from https://www.R-project.org/Google Scholar
Rabe-Hesketh, S., Skrondal, A. (2007). Multilevel and latent variable modeling with composite links and exploded likelihoods. Psychometrika, 72 (2), 123CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., Pickles, A. (2002). Reliable estimation of generalized linear mixed models using adaptive quadrature. The Stata Journal, 2 (1), 1–21CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., Pickles, A. (2004). Generalized multilevel structural equation modeling. Psychometrika, 69 (2), 167–190CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., Pickles, A. (2005). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128 (2), 301–323CrossRefGoogle Scholar
Rockwood, N. J. (2020). Maximum likelihood estimation of multilevel structural equation models with random slopes for latent covariates. Psychometrika, 85 (2), 275–300CrossRefGoogle ScholarPubMed
Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. version 0.5–12 (BETA). Journal of Statistical Software, 48 (2), 1–36CrossRefGoogle Scholar
Skrondal, A., Rabe-Hesketh, S. (2003). Multilevel logistic regression for polytomous data and rankings. Psychometrika, 68 (2), 267–287CrossRefGoogle Scholar
Skrondal, A., Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models, Boca Raton: CRC PressCrossRefGoogle Scholar
Skrondal, A., Rabe-Hesketh, S. (2009). Prediction in multilevel generalized linear models. Journal of the Royal Statistical Society: Series A (Statistics in Society), 172 (3), 659–687CrossRefGoogle Scholar
StataCorp. (2019a). Stata statistical software: Release 16. StataCorp.Google Scholar
StataCorp. (2019b). Stata structural equation modeling reference manual release 16. StataCorp LLC.Google Scholar
Wang, L. (2010). IRT-ZIP modeling for multivariate zero-in ated count data. Journal of Educational and Behavioral Statistics, 35 (6), 671–692CrossRefGoogle Scholar
Zheng, X., Rabe-Hesketh, S. (2007). Estimating parameters of dichotomous and ordinal item response models with gllamm. The Stata Journal, 7 (3), 313–333CrossRefGoogle Scholar