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This study examines the unscaled and scaled root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker-Lewis index (TLI) of diagonally weighted least squares (DWLS) and unweighted least squares (ULS) estimators in structural equation modeling with ordered categorical data. We show that the number of categories and threshold values for categorization can unappealingly impact the DWLS unscaled and scaled fit indices, as well as the ULS scaled fit indices in the population, given that analysis models are misspecified and that the threshold structure is saturated. Consequently, a severely misspecified model may be considered acceptable, depending on how the underlying continuous variables are categorized. The corresponding CFI and TLI are less dependent on the categorization than RMSEA but are less sensitive to model misspecification in general. In contrast, the number of categories and threshold values do not impact the ULS unscaled fit indices in the population.
Xia et al. (Mon,) studied this question.