Multilevel Structural Equation Modeling

In conventional structural equation models, all latent variables and indicators vary between units (typically subjects) and are assumed to be independent across units. The latter assumption is violated in multilevel settings where units are nested in clusters, leading to within-cluster dependence. Different approaches to extending structural equation models for such multilevel settings are examined. The most common approach is to formulate separate within-cluster and between-cluster models. An advantage of this set-up is that it allows software for conventional structural equation models to be ‘tricked’ into estimating the model. However, the standard implementation of this approach does not permit cross-level paths from latent or observed variables at a higher level to latent or observed variables at a lower level, and does not allow for indicators varying at higher levels. A multilevel regression (or path) model formulation is therefore suggested in which some of the response variables and some of the explanatory variables at the different levels are latent and measured by multiple indicators. The Generalized Linear Latent and Mixed Modeling (GLLAMM) framework allows such models to be specified by simply letting the usual model for the structural part of a structural equation model include latent and observed variables varying at different levels. Models of this kind are applied to the U.S. sample of the Program for International Student Assessment (PISA) 2000 to investigate the relationship between the school-level latent variable ‘teacher excellence’ and the student-level latent variable ‘reading ability’, each measured by multiple ordinal indicators.