Multiple Group Alignment for Exploratory and Structural Equation Models

The multiple group alignment methodology is adapted to the general structural equation model. This includes models with cross-loadings, covariates, and structural relations among the factors. A group-specific model for the factors can be estimated even when measurement invariance does not hold, including groups-specific factor means, intercepts, and variances. The methodology is also extended to the weighted least squares estimation method to accommodate models with continuous, binary, and ordered variables. The alignment method is further extended to multiple group EFA and ESEM models. This is accomplished by combining the alignment loss function with the rotation loss function. We obtain an EFA/ESEM model with group-specific factor means, variance/covariance, rotation, and measurement model parameters (loadings and indicator intercepts). Simulation studies and two empirical examples are used for illustration purposes.