The Relationship Between Root Mean Square Error of Approximation and Model Misspecification in Confirmatory Factor Analysis Models

The fit index root mean square error of approximation (RMSEA) is extremely popular in structural equation modeling. However, its behavior under different scenarios remains poorly understood. The present study generates continuous curves where possible to capture the full relationship between RMSEA and various “incidental parameters,” such as factor loadings and model size, for different types of misspecification. Population RMSEA is studied, removing the influence of sampling fluctuations and making the findings directly applicable to tests of close fit and not-close fit, which require the specification of a population cutoff value. Confirmatory factor analysis models are studied. The results introduce many new findings, including that RMSEA is often insensitive to multiple omitted cross-loadings and to clusters of correlated residuals, that it sometimes behaves counterintuitively as a function of model size, and that it is insensitive to the underlying number of latent factors when a model with one factor is fit.