SEM provides a range of fit indices to assess the overall fit of the entire structural model. GOF indicates how well the specified model reproduces the covariance matrix among the indicator items. The basic and most commonly-used fit index reported is the chi-square (χ2) statistic. With 212 samples analysed in this study, this approach is considered appropriate to be used (Kline, 1998). The difference in the covariance matrices is the key value in assessing the GOF of any SEM model. SEM estimation procedures such as a MLE produce parameter estimates that mathematically minimize this difference in the specified model. A χ2 test provides a statistical test of the resulting difference.
It is represented mathematically by the following equation:
χ2 = (N - 1)(S - ∑k)
Where N = overall sample size;
S = observed sample covariance matrix;
∑k = SEM estimated covariance matrix.
The SEM estimated covariance matrix is influenced by how many parameters are free to be estimated (the k in ∑k), so the model degrees of freedom (df) also influence the χ2 GOF test. The df for an analysis of a covariance structure model is determined by:
df = ẵ [(p)(p + 1)] – k
Where p = total number of observed variables k = number of estimated (free) parameters
With the χ2 GOF test, the smaller the p-value, the greater the chance that observed sample and SEM estimated covariance matrices are not equal. Thus, with SEM, we do not want the p-value for the χ2 test to be small (or significant). If theory is to be supported by the test, the small χ2is needed (and corresponding large p-value; i.e.
>0.05), that indicates no statistically significant difference between the matrices.
Another problem with χ2 is that the more complex the model, the bigger the χ2 will be and the more likely it is that the specified model will be rejected (Holmes-Smith, 2005). For this reason, a “normed” χ2 is sometimes used where χ2 is divided by the df (χ2/df) for the model to give a χ2 measure per df. The acceptable level for normed χ2 should be greater than 1.0 but smaller than 2.0 (although values between 2.0 to 3.0 indicate a reasonably good fit). Values of less than 1.0 indicate overfit (Holmes- Smith, 2005).
Other commonly-used fit indices are:
- Goodness of Fit Index (GFI)
- The GFI is an early attempt to produce a fit statistic that was less sensitive to sample size. The possible range of GFI values is 0 to 1 with higher
values indicating better fit. The common threshold value for GFI (as well as AGFI) is more than 0.95, although values greater than 0.9 also indicate reasonable fit (see Table 3.3 for detail fit values).
- Root Mean Square Residual (RMR) and Standardized Root Mean Square Residual (SRMR)
- The RMR is an average of the residuals between individual observed and estimated covariance and variance terms. SRMR is the alternative statistic based on residuals. It is a standardized value of RMR and thus is more useful for comparing fit across models. Lower RMR and SRMR value represent better fit. RMR and SRMR are sometimes known as badness- of-fit measures in which high values are indicative of poor fit.
- RMR should be less than 0.05 (Holmes-Smith, 2005).
- Root Mean Square Error of Approximation (RMSEA)
- RMSEA is a measure that attempts to correct for the tendency of the χ2 GOF test statistics to reject models with large samples or a large number of observed variables. Lower RMSEA values indicate better fit. Like the SRMR and RMSR, it is a badness-of-fit index. Typically, values of below 0.05 indicate the most acceptable models (although values between 0.05 and 0.08 indicate reasonable fit) (Holmes-Smith, 2005).
- Comparative Fit Index (CFI)
- The CFI is an incremental fit index that is an improved version of the NFI. The values range between 0 and 1, with higher values indicating better fit (>0.90) (Holmes-Smith, 2005). Because the CFI has many desirable properties including its relative, but not complete, insensitivity to model complexity, it is among the most widely used indices.
- Normed Fit Index (NFI)
- The NFI is one of the original incremental fit indices. It is a ratio of the difference in the χ2 value for the fitted model and null model divided by the χ2 value for the null model. It ranges between 0 and 1 and a model with perfect fit would produce an NFI of 1.
- Model Parsimony
- The more parameters added to a model the more sample specific the model becomes and less likely it is that the different sample could support such a highly specific model (Holmes-Smith, 2005). The more parsimonious the model, the more likely it is that the model could generalised to the population. Thus, the “best” model is the model with the smallest model parsimony fit measure. Some functions used to measure model parsimony are Akaike Information Criterion (AIC) and Consistent Akaike Information Criterion (CAIC).
According to Hair et al. (2006), multiple fit indices should be used to assess a model’s GOF which include:
- The χ2 value and the associated df
- One absolute fit index (i.e., GFI, RMSEA, or SRMR) - One incremental fit index (i.e., CFI or NFI, etc.) - One goodness-of-fit index (i.e., GFI, CFI, NFI, etc.) - One badness-of-fit index (RMSEA, SRMR, etc.)
The ultimate goal for any of these fit indices is to assist the researcher in discriminating between acceptably and unacceptably specified models. Academic journals are replete with SEM results citing a 0.90 value on key indices, such as CFI, NFI, or GFI, as indicating an acceptable model (Hair et al., 2006). Hair et al. (2006) provides some guidelines for using fit indices in different situations (see Table 3.3).
The guidelines are primarily on simulation research that considers different sample sizes, model complexity, and degrees of error in model specification. One key point across the results is that, simpler models and smaller samples should be subject to stricter evaluation than are more complex models with larger samples.