Local Fit Evaluation of Structural Equation Models Using Graphical Criteria

Evaluation of model fit is critically important for every structural equation model and sophisticated methods have been developed for this task. Among them are the χ2 goodness-of-fit test, decomposition of the χ2, derived measures like the popular RMSEA or CFI, or inspection of residuals or modification indices. Many of these methods provide a global approach to model fit evaluation: A single index is computed that quantifies the fit of the entire SEM to the data. In contrast, graphical criteria like d-separation or trek-separation allow to derive implications that can be used for local fit evaluation, an approach that is hardly ever applied. We provide an overview of local fit evaluation from the viewpoint of SEM practitioners. In the presence of model misfit, local fit evaluation can potentially help in pinpointing where the problem with the model lies. For models that do fit the data, local tests can identify the parts of the model that are corroborated by the data. Local tests can also be conducted before a model is fitted at all, and they can be used even for models that are globally under-identified. We discuss appropriate statistical local tests, and provide applied examples. We also present novel software in R that automates this type of local fit evaluation.