Tetrad Fit Index for Factor Analysis Models

Traditional fit indices used in the context of factor analysis are based on the objective function of the Maximum Likelihood (ML), or modified ML, estimates of the free parameters. Therefore, these indices are an indication of how well the fitted model describes the observed correlation matrix. However, this these indices do not provide a direct assessment of the validity of the assumed causal relations between the latent and observed variables. The objective of this study is to propose a tetrad fit index (TFI) which reflects how well the assumed causal relations in the model are reflected in the data. The TFI is defined as the complement of the average of the root-mean-squared difference between the tetrads of the observed correlation matrix and the correlation matrix implied by a fitted factor analytic model. A preliminary simulation study provides initial evidence in favor of using the TFI instead of other traditional fit indices to identify the correct factor model in comparison to concurrent models.