Structural Equation Modelling: How much is fit enough?
Major Indices
There are some major indicators which determine the fitness of a SEM. Main indices are: goodness-of-fit index (GFI), adjusted goodness-of-fit index (AGFI), root mean squared residual (RMR) and root mean squared error of approximations (RMSEA).
Goodness-of-fit indices ranges from 0 to 1 and values exceeding 0.9 indicate a good fit. Adjusted goodness of fit is differentiated from regular goodness of fit in that it adjusts for degrees of freedom in the particular model. The range for AGFI is also 0 to 1, with values larger than 0.9 indicating a good fit. Root mean squared residual (RMR) is calculated from the residuals and ranges from 0 to 1 with a significantly good fit indicated if the value is less than 0.05. Root mean square error of approximation (RMSEA) is a measure of fit that ‘could be expected if the model were estimated from the entire population, not just the samples drawn for estimation’. As with RMR, the indicator that a reasonable goodness of fit has been found is when the value is less than 0.05.
Other Indices
Comparative or relative fit refers to a situation where two or more models are compared to see which one provides the best fit to the data. One of the models will be the total independence or null hypothesis model where there is known to be a poor fit. The comparative fit index (CFI) is the primary measurement here and ranges from 0 to 1 with values above 0.9 considered to indicate a good fit. The non-normed fit index (and the associated normed fit index (NFI)) are fairly robust measures unless the sample size is small (small here meaning less than the recommended sample size of over 200), in which case the result can be difficult to interpret. Parsimonious fit is about finding a balance between adding more parameters to give a better fit and shaving these parameters for better statistical validity. Parsimonious normed fit indices and parsimonious goodness of fit have ranges 0 to 1, with results approaching 1 (0.90 or higher) indicating a parsimonious fit.
