People are variables too: multilevel structural equations modeling

The article uses confirmatory factor analysis (CFA) as a template to explain didactically multilevel structural equation models (ML-SEM) and to demonstrate the equivalence of general mixed-effects models and ML-SEM. An intuitively appealing graphical representation of complex ML-SEMs is introduced that succinctly describes the underlying model and its assumptions. The use of definition variables (i.e., observed variables used to fix model parameters to individual specific data values) is extended to the case of ML-SEMs for clustered data with random slopes. Empirical examples of multilevel CFA and ML-SEM with random slopes are provided along with scripts for fitting such models in SAS Proc Mixed, Mplus, and Mx. Methodological issues regarding estimation of complex ML-SEMs and the evaluation of model fit are discussed. Further potential applications of ML-SEMs are explored.