On congruence and incongruence of measures of fit in structural equation modeling

Guidelines to evaluate the fit of structural equation models can only offer meaningful insights to the extent that they apply equally to a wide range of situations. However, a number of previous studies found that statistical power to reject a misspecified model increases and descriptive fit-indices deteriorate when loadings are high, thereby inappropriately panelizing high reliability indicators. Based on both theoretical considerations and empirical simulation studies, we show that previous results only hold for a particular definition and a particular type of model error. At a constant degree of misspecification (as measured through the minimum of the fit-function), statistical power to reject a wrong model and noncentrality based fit-indices (such as the root-mean squared error of approximation; RMSEA) are independent of loading magnitude. If the degree of model error is controlled through the average residuals, higher loadings are associated with increased statistical power and a higher RMSEA when the measurement model is misspecified, but with decreased power and a lower RMSEA when the structural model is misspecified. In effect, inconsistencies among noncentrality and residual based fit-indices can provide information about possible sources of misfit that would be obscured when considering either measure in isolation. (PsycINFO Database Record