Structural equation modeling (SEM) provides various fit indices to test the appropriateness of the SEM model. To assist with meeting the acceptable cut-off level and to determine the model fit, the cut-off criteria for the fit indices are summarized in Table 3.2.
Table 3.2: Cut-off criteria for fit indices
Sources: adapted from Hair et al. (2010); Schreiber, Stage, King, Nora, and Barlow (2006)
No. Measure Fit criteria Definition
1. Chi-square (χ2) Non-significant (χ2) at least p-value > 0.05
The fundamental measure used in SEM to quantify
the differences between the observed and estimated covariance matrices.
2. Normed fit chi-square (χ2/df [degree of freedom])
Ratio 2: 1 or 3: 1 A measure of absolute fit and parsimony.
3. GFI (Goodness-of-Fit Index) Value > 0.95 indicates a good fit, and 0.90–0.95 an adequate fit
A fit statistic that is less sensitive to sample size and indicates how well the specified model
reproduces the covariance matrix among the indicator items (i.e. the similarity of the observed and estimated covariance matrices).
4. AGFI (Adjusted Goodness- of-Fit Index)
Value > 0.95 indicates a good fit, and 0.90–0.95 an adequate fit
An index representing goodness- of-fit for
the degree of freedom.
5. RMSEA (root mean square error of approximation)
Values < 0.05 indicate an adequate fit
A measure that attempts to correct for the tendency of the X2 test statistic to reject models with large samples or large numbers of observed variables.
6. NFI (Normed Fit Index) Values > 0.95 indicate a good fit, and 0.90–0.95 an adequate fit
A ratio of the differences in the X2 value for the fitted model and a null model divided by the X2 value for the null model.
7. CFI (Comparative Fit Index) Values > 0.95 indicate a good fit, and 0.90–0.95 an adequate fit
An incremental fit index that is an improved version of the Normed Fit Index (NFI).
8. TLI (Tucker–Lewis Index) Values > 0.95 indicate a good fit, and 0.90–0.95 an adequate fit
A comparative index between proposed and null models adjusted based on degrees of freedom.
9 IFI (Incremental Fit Index) Values > 0.95 indicate a good fit, and 0.90–0.95 an adequate fit
An index interpreted similarly to TLI and CFI.
Absolute fit indices
To begin with, absolute fit indices determine a direct measurement of how well the proposed model reproduces the observed data or fits the sample data.
The chi-square (x2) is mostly used to evaluate the overall model fit. It also assesses the magnitude of discrepancy between the sample and fitted covariance matrices by a good model fit which would indicate an insignificant result at the 0.05 threshold (Hooper, Coughlan, & Mullen, 2008). However, while chi-square is still popular for a fit statistic test, some limitations are apparent. Firstly, it assumes a multivariate normality and observes deviation from normality that probably contributes to a model rejection although the model was properly specified. Secondly, it is relatively sensitive to the size of the sample, meaning that the chi-square statistic almost always rejects the model which is using a large sample size; in contrast, when the sample size is small, the chi-square statistic lacks power. However, to minimize the restrictiveness of the chi-square’s response to sample size, the relative/normed chi-square (x2/df) statistic is used to measure. Although no consensus exists on an acceptable ratio, recommendations range from 2 to 5 (Wheaton, Muthén, Alwin, & Summers, 1977).
Furthermore, applying the root mean square error of approximation (RMSEA) takes into account the error of approximation in the population. It is used to measure how well the model, with unknown but optimally chosen parameter values, fits the population covariance matrix. It explicitly tries to correct for both model complexity and sample size by including each in its computation. According to Byrne (2001), a value less than 0.05 represents a better fit, while a value higher than 0.08 can be considered to indicate reasonable errors of approximation in the population, and above 0.10 indicates a poor fit. In addition, the Goodness-of-Fit Index (GFI) is used to assess the model’s overall goodness of fit, that is, how well the specified model reproduces the covariance matrix among the indicator items. Included with the Adjusted Goodness- of-Fit Index (AGFI), it adjusts the GFI value based upon the degrees of freedom. These GFI and AGFI statistics range from 0 to 1, with larger samples increasing in value, and a higher value representing a better fit. Thus, a value of 0.9 or greater is considered to be a good fit.
Incremental fit indices
Incremental fit indices are comparative or relative fit indices (Hooper, Coughlan, & Mullen, 2008). The Normed Fit Index (NFI) is used to measure the model fit in relation to its complexity, so high values would represent a better fit (Hair et al., 2010). The recommended value greater than 0.9 shows a good fit. Moreover, the Comparative Fit Index (CFI) is a revised form of the NFI that takes into account the sample size and performs well even when the sample size is small (Byrne, 2001).
It ranges from 0 to 1 with the higher value considered as a better fit, so the recommended value for a good fit should be 0.9 or greater (Byrne, 2001).
Parsimony fit indices
The Parsimony Goodness-of-Fit Index (PGFI) and the Parsimonious Normed Fit Index (PNFI) have been developed for a less rigorous theoretical model that paradoxically provides better fit indices (Hooper, Coughlan, & Mullen, 2008).
The PGFI is an adjusted GFI based on the loss of degrees of freedom while the PNFI is an adjusted NFI based on the loss of degrees of freedom. No threshold levels have been recommended for these indices, but it is strongly recommended that these parsimony fit indices are used in tandem with other measures of goodness-of-fit. In addition, the Akaike Information Criterion (AIC) or the consistent version of AIC (CAIC) are generally used when comparing non-nested or non-hierarchical models that are estimated with the same data and indices. A smaller value is recommended for these indices which is normally between 0 and 1 on the scale.