Have Multilevel Models Been Structural Equation Models All Along?

Abstract

A core assumption of the standard multiple regression model is independence of residuals, the violation of which results in biased standard errors and test statistics. The structural equation model (SEM) generalizes the regression model in several key ways, but the SEM also assumes independence of residuals. The multilevel model (MLM) was developed to extend the regression model to dependent data structures. Attempts have been made to extend the SEM in similar ways, but several complications currently limit the general application of these techniques in practice. Interestingly, it is well known that under a broad set of conditions SEM and MLM longitudinal "growth curve" models are analytically and empirically identical. This is intriguing given the clear violation of independence in growth modeling that does not detrimentally affect the standard SEM. Better understanding the source and potential implications of this isomorphism is my focus here. I begin by exploring why SEM and MLM are analytically equivalent methods in the presence of nesting due to repeated observations over time. I then capitalize on this equivalency to allow for the extension of SEMs to a general class of nested data structures. I conclude with a description of potential opportunities for multilevel SEMs and directions for future developments.