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Maximum Likelihood Estimation of Multilevel Structural Equation Models with Random Slopes for Latent Covariates

Published online by Cambridge University Press:  01 January 2025

Nicholas J. Rockwood
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, CA 92350, USA. Email: nrockwood@llu.edu; URL: https://njrockwood.com
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Abstract

A maximum likelihood estimation routine for two-level structural equation models with random slopes for latent covariates is presented. Because the likelihood function does not typically have a closed-form solution, numerical integration over the random effects is required. The routine relies upon a method proposed by du Toit and Cudeck (Psychometrika 74(1):65–82, 2009) for reformulating the likelihood function so that an often large subset of the random effects can be integrated analytically, reducing the computational burden of high-dimensional numerical integration. The method is demonstrated and assessed using a small-scale simulation study and an empirical example.

Keywords

multilevel SEMrandom effectsrandom slopesmaximum likelihood estimation

Information

Type
Theory and Methods
Information
Psychometrika , Volume 85 , Issue 2 , June 2020 , pp. 275 - 300
Copyright
Copyright © 2020 The Psychometric Society

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References

Asparouhov, T., & Muthén, B. (2019a). Latent variable centering of predictors and mediators in multilevel and time-series models. Structural Equation Modeling: A Multidisciplinary Journal, 26 (1), 119– 142. CrossRefGoogle Scholar
Asparouhov, T., & Muthén, B. (2019b). Latent variable interactions using maximum-likelihood and Bayesian estimation for single- and two-level models. Mplus Web Notes: No. 23. Google Scholar
Bentler, P. M. (2004). Eqs 6: Structural equations program manual. Encino: Multivariate software. Google Scholar
Bollen, K. A. (1989). Structural equations with latent variables. Hoboken: Wiley. CrossRefGoogle Scholar
Cai, L. (2010). A two-tier full-information item factor analysis model with applications. Psychometrika, 75 (4), 581612. CrossRefGoogle Scholar
Carpenter, B., Hoffman, M. D., Brubaker, M., Lee, D., Li, P., & Betancourt, M. (2015). The stan math library: Reverse-mode automatic differentiation in c++. arXiv preprint arXiv:1509.07164. Google Scholar
Cronbach, L. J. (1976). Research on classrooms and schools: Formulation of questions, design and analysis. Stanford, CA: Stanford University Evaluation Consortium. Google Scholar
Cudeck, R. (2005). Fitting psychometric models with methods based on automatic differentiation. Psychometrika, 70 (4), 599– 617. CrossRefGoogle Scholar
Cudeck, R., Harring, J. R., & du Toit, S. H. (2009). Marginal maximum likelihood estimation of a latent variable model with interaction. Journal of Educational and Behavioral Statistics, 34 (1), 131– 144. CrossRefGoogle Scholar
du Toit, S. H., & Cudeck, R. (2009). Estimation of the nonlinear random coefficient model when some random effects are separable. Psychometrika, 74 (1), 6582. CrossRefGoogle Scholar
du Toit, S. H., & du Toit, M. (2008). Multilevel structural equation modeling. In Handbook of multilevel analysis (pp. 435–478). Springer. CrossRefGoogle Scholar
Eddelbuettel, D. (2013). Seamless R and C++ integration with Rcpp. New York: Springer. ISBN 978-1-4614-6867-7. https://doi.org/10.1007/978-1-4614-6868-4. CrossRefGoogle Scholar
Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. Psychological Methods, 12 (2), 121– 138. 17563168 CrossRefGoogle Scholar
Gibbons, R. D. Hedeker, D. R. (1992). Full-information item bi-factor analysis. Psychometrika, 57 (3), 423– 436. CrossRefGoogle Scholar
Goldstein, H., & McDonald, R. P. (1988). A general model for the analysis of multilevel data. Psychometrika, 53 (4), 455– 467. CrossRefGoogle Scholar
Griewank, A., & Walther, A. (2008). Evaluating derivatives: Principles and techniques of algorithmic differentiation (105). Philadelphia: SIAM. CrossRefGoogle Scholar
Hallquist, M. N., & Wiley, J. F. (2018). MplusAutomation: An R package for facilitating large-scale latent variable analyses in Mplus. Structural Equation Modeling: A Multidisciplinary Journal, 25 (4), 621– 638. CrossRefGoogle Scholar
Jöreskog, K. G., & Sörbom, D. (1996). Lisrel 8: User’sreference guide. Lincolnwood: Scientific Software International. Google Scholar
Lee, S. -Y. (1990). Multilevel analysis of structural equation models. Biometrika, 77 (4), 763– 772. CrossRefGoogle Scholar
Liang, J., & Bentler, P. M. (2004). An EM algorithm for fitting two-level structural equation models. Psychometrika, 69 (1), 101– 122. CrossRefGoogle Scholar
Lüdtke, O., Marsh, H. W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthén, B. (2008). The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies. Psychological Methods, 13 (3), 203– 229. 18778152 CrossRefGoogle Scholar
Maas, C. J., & Hox, J. J. (2005). Sufficient sample sizes for multilevel modeling. Methodology, 1 (3), 86– 92. CrossRefGoogle Scholar
Marsh, H. W., Lüdtke, O., Nagengast, B., Trautwein, U., Morin, A. J., Abduljabbar, A. S. Köller, O et al. (2012). Classroom climate and contextual effects: Conceptual and methodological issues in the evaluation of group-level effects. Educational Psychologist, 47 (2), 106–124. CrossRefGoogle Scholar
McDonald, R. P. (1993). A general model for two-level data with responses missing at random. Psychometrika, 58 (4), 575– 585. CrossRefGoogle Scholar
McDonald, R. P., & Goldstein, H. (1989). Balanced versus unbalanced designs for linear structural relations in two-level data. British Journal of Mathematical and Statistical Psychology, 42 (2), 215– 232. CrossRefGoogle Scholar
Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations modeling. Psychological Methods, 10 (3), 259– 284. 16221028 CrossRefGoogle ScholarPubMed
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49 (1), 115– 132. CrossRefGoogle Scholar
Muthén, B. O. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54 (4), 557– 585. CrossRefGoogle Scholar
Muthén, B. O., & Satorra, A. (1989). Multilevel aspects of varying parameters in structural models. In Multilevel analysis of educational data, (pp. 87–99). Elsevier. CrossRefGoogle Scholar
Muthén, L. K., & Muthén, B. (2017). Mplus version 8 [Computer software manual]. Los Angeles, CA: Muthén & Muthén. Google Scholar
Nash, J. C. (2014). On best practice optimization methods in R. Journal of Statistical Software, 60 (2), 1– 14. CrossRefGoogle Scholar
Neale, M. C., Hunter, M. D., Pritikin, J. N. Zahery, M. Brick, T. R. Kirkpatrick, R. M. et al. (2016). OpenMx 2.0: Extended structural equation and statistical modeling. Psychometrika, 81 (2), 535– 549. 25622929 CrossRefGoogle ScholarPubMed
OECD. (2003). Programme for International Student Assessment 2003. Retrieved February 1, 2019 from http://www.oecd.org/pisa/data/database-pisa2003.htm. Google Scholar
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– 35. CrossRefGoogle Scholar
Preacher, K. J., Zhang, Z., & Zyphur, M. J. (2016). Multilevel structural equation models for assessing moderation within and across levels of analysis. Psychological Methods, 21 (2), 189– 205. 26651982 CrossRefGoogle ScholarPubMed
Preacher, K. J., Zyphur, M. J., & Zhang, Z. (2010). A general multilevel SEM framework for assessing multilevel mediation. Psychological Methods, 15 (3), 209– 233. 20822249 CrossRefGoogle ScholarPubMed
R Core Team. (2019). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. https://urldefense.proofpoint.com/v2/url?u=https-3A__www.r2Dproject.org&d=DwIGaQ&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=cijxKIUfIjh6xB35XSxKelnSNfz2185wGO_qFr-DFH8&m=2tlyHFkIA11Yzt64XP7lrKUV1F_N4EqYjSlNwvys8zE&s=yIxHEeyUO9whFYdd4zujlILP0486_iGL1mIlAxlRsc&e=. Google 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– 21. CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004). Generalized multilevel structural equation modeling. Psychometrika, 69 (2), 167– 190. 10.1007/bf02295939 CrossRefGoogle 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– 323. CrossRefGoogle Scholar
Rijmen, F. (2009). An efficient EM algorithm for multidimensional IRT models: Full information maximum likelihood estimation in limited time (Tech. Rep.). Princeton, NJ: ETS Research Report (RR0903).Google Scholar
Rijmen, F. (2010). Formal relations and an empirical comparison among the bi-factor, the testlet, and a second-order multidimensional irt model. Journal of Educational Measurement, 47 (3), 361– 372. CrossRefGoogle Scholar
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– 36. CrossRefGoogle Scholar
Schmidt, W. H. (1969). Covariance structure analysis of the multivariate random effects model (Unpublished doctoral dissertation). Department of Education: University of Chicago.Google Scholar
Shin, Y., & Raudenbush, S. W. (2012). A latent cluster-mean approach to the contextual effects model with missing data. Journal of Educational and Behavioral Statistics, 35 (1), 26– 53. CrossRefGoogle Scholar
Stan Development Team. (2020). RStan: The R interface to Stan. R package version 2.19.3. https://urldefense.proofpoint.com/v2/url?u=https-3A__mc-2Dstan.org&d=DwIGaQ&c=vh6FgFnduejNhPPD0fl_yRaSfZy8CWbWnIf4XJhSqx8&r=cijxKIUfIjh6xB35XSxKelnSNfz2185wGO_qFr-DFH8&m=2tlyHFkIA11Yzt64XP7lrKUV1F_N4EqYjSlNwvys8zE&s=w3Alv-F1vraydtUj39bGIhlI9AU2loF1hvQZLAAB16w&e=.Google Scholar
Stapleton, L. M., & Johnson, T. L. (2019). Models to examine the validity of cluster-level factor structure using individual-level data. Advances in Methods and Practices in Psychological Science, 2 (3), 312– 329. CrossRefGoogle Scholar
Stapleton, L. M., Yang, J. S., & Hancock, G. R. (2016). Construct meaning in multilevel settings. Journal of Educational and Behavioral Statistics, 41 (5), 481– 520. CrossRefGoogle Scholar
StataCorp. (2005). Stata statistical software: Release 15 College Station, TX: StataCorp LLC. Google Scholar