Estimating Multilevel Structural Equation Models with Random Slopes with Laplace and Variational Approximations

Multilevel structural equation models (MSEMs) are an important statistical approach to analyze hierarchically nested data. While Bayesian methods are commonly used to estimate the MSEM parameters, maximum likelihood (ML) approaches are less often employed because of the computational challenges in the required numerical integration. Building on Rockwood’s reformulation of the MSEM, we investigate two computationally efficient approximation methods to the likelihood function—the Laplace approximation (LA) and the extended variational approximation (EVA)—by comparing their performance against Gauss-Hermite (GH) quadrature and a Bayesian approach in two simulation studies. Results demonstrate that LA and EVA provide accurate parameter estimates with substantially shorter computation times than GH, especially for a more complex model. Furthermore, LA and EVA showed almost no bias, good convergence rates, and appropriate coverage, particularly with larger sample sizes. Altogether, these findings suggest that LA and EVA are promising approaches for ML estimation of MSEMs with random slopes.