Structural Equation Modeling:What does it mean to be fit?

What a Model-Fit actually means

A good-fitting model is one that is reasonably consistent with the data and is required before interpreting the causal paths of the structural model. It must be kept in kind that a good-fitting model is not necessarily a valid model. It must be carefully examined to determine if one has a good model as well as a good-fitting model. Also it is important to realize that one might obtain a good-fitting model, yet it is still possible to improve the model and remove specification error. Finally, having a good-fitting model does not prove that the model is correctly specified. Conversely, it should be noted that a model all of whose parameters are statistically significant can be from a poor fitting model.

In Technical Terms

Technically speaking model fit determines the degree to which the sample variance covariance data fit the structural equation model. Model fit criteria commonly used are chi-square (x2), the goodness-of-fit index (GFI), the adjusted goodness-of-fit index (AGFI), and the root-mean-square residual (RMR).

Chi-Square (X2)

A significant X2 value relative to the degrees of freedom indicates that the observed and implied (estimated) variance-covariance matrices differ. Statistical significance indicates the probability that this difference is due to sampling variation. A nonsignificant X2 value indicates that the two matrices are similar, indicating that the implied theoretical model significantly reproduces the sample variance-covariance relationships in the matrix.

Goodness-of-Fit Index (GFI) and Adjusted Goodness-of-Fit Index (AGFI)

The goodness-of-fit index (GFI) is based on the ratio of the sum of the squared differences between the observed and reproduced matrices to the observed variances, thus allowing for scale. The GFI measures the amount of variance and covariance in S that is predicted by the reproduced matrix E. The adjusted goodness-of-fit (AGFI) index is adjusted for the degrees of freedom of a model relative to the number of variables. The GFI and AGFI indices can be used to compare the fit of two different models with the same data or compare the fit of a single model using different data, such as separate data sets for males and females.

Root-Mean-Square Residual Index (RMR)

The RMR index uses the square root of the mean-squared differences between matrix elements. It is used to compare the fit of two different models with the same data.