Examining the effect of missing data on RMSEA and CFI under normal theory full-information maximum likelihood

Normal theory full-information maximum likelihood (FIML) is a common estimation technique for incomplete data in structural equation modeling (SEM). However, it is not commonly known that approximate fit indices (AFIs) can be distorted, relative to their complete data counterparts, when FIML is used to handle missing data. In this article, we show that two most popular AFIs, the root-mean-square error of approximation (RMSEA) and the comparative fit index (CFI) often approach different population values under FIML estimation when missing data are present. By deriving the FIML fit function for incomplete data and showing that it is different from the usual maximum likelihood (ML) fit function for complete data, we provide a mathematical explanation for this phenomenon. We also present several analytic examples as well as the results of two large sample simulation studies to illustrate how AFIs change with missing data.