Unbelievably Fast Estimation of Nested Multilevel Structural Equation Models

We introduce relational SEM, an adaptation of structural equation modeling to relational databases. Relational SEM is a superset of the mixed model and multilevel SEM. In addition, we introduce Rampart, a new computational strategy for frequently encountered relational SEM models with all continuous indicators. Rampart is inspired by the fact that the multivariate normal density is transparent to orthogonal rotation. Well suited to big data, Rampart becomes more effective as the size of the data set increases. When data are strictly nested then there are usually fewer variables in the upper level connected to many more variables in the lower levels. A regression from teacher skill to student performance has this characteristic. In such a model, under typical conditions, a rotation can be applied to eliminate all but one of the links from teacher to student with a corresponding rotation applied to the observations. This transformation leaves the likelihood function unchanged, but offers a major benefit: dramatically increased independence in the model implied covariance matrix. Rampart requires strictly nested structure and identical sub-models. Rampart can be applied locally to the part of a model that meets these criteria. Rampart is implemented in OpenMx. OpenMx is free and open software that runs on all major operating systems.