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There has been relatively little systematic investigation of the effect of the number of variables on measures of model fit in structural equation modeling. There is conflicting evidence as to whether measures of fit tend to improve or decline as more variables are added to the model. We consider 3 different types of specification error: minor factors, 2-factor models, and method errors. Using a formal method based on the noncentrality parameter (NCP), we find that root mean squared error of approximation (RMSEA) tends to improve regardless of the type of specification error and that the comparative fit index (CFI) and Tucker-Lewis Index (TLI), generally, though not always, tend to worsen as the number of variables in the model increases. The formal method that we develop can be used to investigate other measures of fit and other types of misspecification.
Kenny et al. (Tue,) studied this question.