On Congruence and Incongruence of Measures of Fit in Structural Equation Modeling

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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.

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