Maximum likelihood estimation of multilevel structural equation models with random slopes for latent covariates. (English) Zbl 1458.62275

Summary: 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 S. H. C. du Toit and R. Cudeck [Psychometrika 74, No. 1, 65–82 (2009; Zbl 1284.62778)] 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.


62P15 Applications of statistics to psychology
62F12 Asymptotic properties of parametric estimators


Zbl 1284.62778
Full Text: DOI


[1] Asparouhov, T.; Muthén, B., Latent variable centering of predictors and mediators in multilevel and time-series models, Structural Equation Modeling: A Multidisciplinary Journal, 26, 1, 119-142 (2019)
[2] 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.
[3] Bentler, PM, Eqs 6: Structural equations program manual (2004), Encino: Multivariate software, Encino
[4] Bollen, KA, Structural equations with latent variables (1989), Hoboken: Wiley, Hoboken · Zbl 0731.62159
[5] Cai, L., A two-tier full-information item factor analysis model with applications, Psychometrika, 75, 4, 581-612 (2010) · Zbl 1208.62183
[6] 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.
[7] Cronbach, LJ, Research on classrooms and schools: Formulation of questions, design and analysis (1976), Stanford, CA: Stanford University Evaluation Consortium, Stanford, CA
[8] Cudeck, R., Fitting psychometric models with methods based on automatic differentiation, Psychometrika, 70, 4, 599-617 (2005) · Zbl 1306.62401
[9] Cudeck, R.; Harring, JR; du Toit, SH, Marginal maximum likelihood estimation of a latent variable model with interaction, Journal of Educational and Behavioral Statistics, 34, 1, 131-144 (2009)
[10] du Toit, SH; Cudeck, R., Estimation of the nonlinear random coefficient model when some random effects are separable, Psychometrika, 74, 1, 65-82 (2009) · Zbl 1284.62778
[11] du Toit, S. H., & du Toit, M. (2008). Multilevel structural equation modeling. In Handbook of multilevel analysis (pp. 435-478). Springer.
[12] Eddelbuettel, D. (2013). Seamless R and C++ integration with Rcpp. New York: Springer. ISBN 978-1-4614-6867-7. 10.1007/978-1-4614-6868-4. · Zbl 1283.62001
[13] Enders, CK; Tofighi, D., Centering predictor variables in cross-sectional multilevel models: A new look at an old issue, Psychological Methods, 12, 2, 121-138 (2007)
[14] Gibbons, RD; Hedeker, DR, Full-information item bi-factor analysis, Psychometrika, 57, 3, 423-436 (1992) · Zbl 0760.62097
[15] Goldstein, H.; McDonald, RP, A general model for the analysis of multilevel data, Psychometrika, 53, 4, 455-467 (1988) · Zbl 0718.62158
[16] Griewank, A.; Walther, A., Evaluating derivatives: Principles and techniques of algorithmic differentiation (2008), Philadelphia: SIAM, Philadelphia · Zbl 1159.65026
[17] Hallquist, MN; Wiley, JF, MplusAutomation: An R package for facilitating large-scale latent variable analyses in Mplus, Structural Equation Modeling: A Multidisciplinary Journal, 25, 4, 621-638 (2018)
[18] Jöreskog, KG; Sörbom, D., Lisrel 8: User’sreference guide (1996), Lincolnwood: Scientific Software International, Lincolnwood
[19] Lee, S-Y, Multilevel analysis of structural equation models, Biometrika, 77, 4, 763-772 (1990) · Zbl 0711.62045
[20] Liang, J.; Bentler, PM, An EM algorithm for fitting two-level structural equation models, Psychometrika, 69, 1, 101-122 (2004) · Zbl 1306.62467
[21] Lüdtke, O.; Marsh, HW; Robitzsch, A.; Trautwein, U.; Asparouhov, T.; Muthén, B., The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies, Psychological Methods, 13, 3, 203-229 (2008)
[22] Maas, CJ; Hox, JJ, Sufficient sample sizes for multilevel modeling, Methodology, 1, 3, 86-92 (2005)
[23] Marsh, HW; Lüdtke, O.; Nagengast, B.; Trautwein, U.; Morin, AJ; Abduljabbar, AS; Köller, O., Classroom climate and contextual effects: Conceptual and methodological issues in the evaluation of group-level effects, Educational Psychologist, 47, 2, 106-124 (2012)
[24] McDonald, RP, A general model for two-level data with responses missing at random, Psychometrika, 58, 4, 575-585 (1993) · Zbl 0826.62100
[25] McDonald, RP; Goldstein, H., Balanced versus unbalanced designs for linear structural relations in two-level data, British Journal of Mathematical and Statistical Psychology, 42, 2, 215-232 (1989) · Zbl 0718.62168
[26] Mehta, PD; Neale, MC, People are variables too: Multilevel structural equations modeling, Psychological Methods, 10, 3, 259-284 (2005)
[27] Muthén, B., A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators, Psychometrika, 49, 1, 115-132 (1984)
[28] Muthén, BO, Latent variable modeling in heterogeneous populations, Psychometrika, 54, 4, 557-585 (1989)
[29] Muthén, B. O., & Satorra, A. (1989). Multilevel aspects of varying parameters in structural models. In Multilevel analysis of educational data (pp. 87-99). Elsevier.
[30] Muthén, LK; Muthén, B., Mplus version 8 [Computer software manual] (2017), Los Angeles, CA: Muthén & Muthén, Los Angeles, CA
[31] Nash, JC, On best practice optimization methods in R, Journal of Statistical Software, 60, 2, 1-14 (2014)
[32] Neale, MC; Hunter, MD; Pritikin, JN; Zahery, M.; Brick, TR; Kirkpatrick, RM, OpenMx 2.0: Extended structural equation and statistical modeling, Psychometrika, 81, 2, 535-549 (2016) · Zbl 1345.62162
[33] OECD. (2003). Programme for International Student Assessment 2003. Retrieved February 1, 2019 from http://www.oecd.org/pisa/data/database-pisa2003.htm.
[34] Pinheiro, JC; Bates, DM, Approximations to the log-likelihood function in the nonlinear mixed-effects model, Journal of Computational and Graphical Statistics, 4, 1, 12-35 (1995)
[35] Preacher, KJ; Zhang, Z.; Zyphur, MJ, Multilevel structural equation models for assessing moderation within and across levels of analysis, Psychological Methods, 21, 2, 189-205 (2016)
[36] Preacher, KJ; Zyphur, MJ; Zhang, Z., A general multilevel SEM framework for assessing multilevel mediation, Psychological Methods, 15, 3, 209-233 (2010)
[37] 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=.
[38] Rabe-Hesketh, S.; Skrondal, A.; Pickles, A., Reliable estimation of generalized linear mixed models using adaptive quadrature, The Stata Journal, 2, 1, 1-21 (2002)
[39] Rabe-Hesketh, S.; Skrondal, A.; Pickles, A., Generalized multilevel structural equation modeling, Psychometrika, 69, 2, 167-190 (2004) · Zbl 1306.62484
[40] Rabe-Hesketh, S.; Skrondal, A.; Pickles, A., Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects, Journal of Econometrics, 128, 2, 301-323 (2005) · Zbl 1336.62079
[41] 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).
[42] Rijmen, F., 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 (2010)
[43] Rosseel, Y., Lavaan: An R package for structural equation modeling and more. version 0.5-12 (BETA), Journal of Statistical Software, 48, 2, 1-36 (2012)
[44] Schmidt, W. H. (1969). Covariance structure analysis of the multivariate random effects model (Unpublished doctoral dissertation). Department of Education: University of Chicago.
[45] Shin, Y.; Raudenbush, SW, A latent cluster-mean approach to the contextual effects model with missing data, Journal of Educational and Behavioral Statistics, 35, 1, 26-53 (2010)
[46] 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=.
[47] Stapleton, LM; Johnson, TL, 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 (2019)
[48] Stapleton, LM; Yang, JS; Hancock, GR, Construct meaning in multilevel settings, Journal of Educational and Behavioral Statistics, 41, 5, 481-520 (2016)
[49] StataCorp. (2005). Stata statistical software: Release 15. College Station, TX: StataCorp LLC.
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