Assessing the Size of Model Misfit in Structural Equation Models

When a statistically significant mean difference is found, the magnitude of the difference is judged qualitatively using an effect size such as Cohen's d. In contrast, in a structural equation model (SEM), the result of the statistical test of model fit is often disregarded if significant, and inferences are drawn using "close" models retained based on point estimates of sample statistics (goodness-of-fit indices). However, when a SEM cannot be retained using a test of exact fit, all substantive inferences drawn from it are suspect. It is therefore important to determine the size of the model misfit. Standardized residual covariances and residual correlations provide standardized effect sizes of the misfit of SEM models. They can be summarized using the Standardized Root Mean Squared Residual (SRMSR) and the Correlation Root Mean Squared Residual (CRMSR) which can be used as overall effect sizes of the misfit. Statistical theory is provided that allows the construction of confidence intervals and tests of close fit based on the SRMSR and CRMSR. It is hoped that the use of standardized effect sizes of misfit will help reconcile current practices in SEM and elsewhere in statistics.