Covariance-Based Structural Equation Modeling in the Journal of Advertising: Review and Recommendations

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Covariance-Based Structural Equation Modeling in the Journal of Advertising: Review and Recommendations

Hair, Babin, and Krey (2017) propose the Theoretical Fit Index (TFI) as a heuristic for assessing the theoretical or structural model (TM) fit, composed of the inter-relationships among latent constructs, relative to the best possible fitting theoretical, recursive model. The index is premised on the fact that the confirmatory factor model (CFA) provides the upper bound (best) for the fit of the theoretical model. The index is proposed as:

Additionally, the authors propose an adjustment for parsimony to the TFI in the form of the Adjusted Theoretical Fit Index (ATFI). The adjustment involves the ratio of degrees of freedom (DF) for the CFA and TM models, which was incorrectly expressed in the article (the ratio of DF is incorrectly transposed). The correct expression of the ATFI is:

For example, a CFA with 100 degrees of freedom and a CFI of 0.98, associated with a TM with 110 degrees of freedom and a CFI of 0.95 would yield an ATFI of:

However, with a more parsimonious theoretical model yielding the same fit, as in a TM with 125 degrees of freedom and the same CFI as above, the ATFI would be:

The TFI and ATFI work as badness of fit indicators, meaning a larger value means a relatively worse fit. In this case, the ATFI takes into account the complexity of the model by suggesting the more parsimonious model has a better fit of 0.025 compared to the model with more paths estimated, and therefore fewer degrees of freedom, of 0.028.