Scale-Invariance, Equivariance and Dependency of Structural Equation Models

Data in social and behavioral sciences typically contain measurement errors and do not have predefined metrics. Structural equation modeling (SEM) is widely used for the analysis of such data, where the scales of the manifest and latent variables are often subjective. This article studies how the model, parameter estimates, their standard errors (SEs), and the corresponding z-statistics are affected by the scales of the manifest and latent variables. Analytical and empirical results show that (1) the normal-distribution-based likelihood ratio statistic is scale-invariant with respect to scale changes of manifest and latent variables as well as to anchor change of latent variables; (2) the normal-distribution-based maximum likelihood (NML) parameter estimates are scale-equivariant with respect to scale-change of manifest and latent variables as well as to anchor change of latent variables; (3) standard errors (SEs) following the NML method are parallel-scale-equivariant with respect to scale changes of the manifest and latent variables; and (4) the z-statistics are scale-invariant with respect to scale changes of the manifest and latent variables. However, only (1) and (2) hold if latent variables are rescaled by changing anchors. Nevertheless, parameters that are not directly related to latent variables with changing anchors are still scale-equivariant and their z-statistics are still scale-invariant. The results are expected to advance understanding of SEM analysis, and also facilitate result interpretation and comparison across studies as in meta analysis.