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The model with the best fit, however, appears to be the bi-factor model, which has the highest TLI and the lowest AIC, BIC and RMSEA. The correlated four-factor model was also a better fit than the higher-order factor model (О”О§ 2 = 8.33, p < 0.05). The four-factor model has a CFI, RMSEA and TLI which indicate reasonable fit. It is clear that the one, two and three factor solutions are inadequate. Model fit statistics are presented in Table 2. It has also been suggested that the Tucker Lewis Index (TLI), the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), which contain a penalty for model complexity, should be used when comparing higher-order and bi-factor models however, it should be noted that the very nature of these indices works in favour of a bi-factor model. The fit statistics of the higher-order model became progressively worse than the bi-factor model.

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Gignac demonstrated this effect in a series of simulations in which the correlations between subtests became increasingly unequal so that the general-to-specific ratio became proportionately larger. The higher-order factor model requires the general factor and specific factor loadings to be equal, whereas the bi-factor model does not contain this constraint.

Furthermore, Morgan, Hodge, Wells and Watkins provided some support for such a bias in a simulation study that found that, вЂњwhen samples were generated from a true higher-order factor structure, approximate fit statistics tended to identify the bi-factor solution as best fitting.вЂќ More recently, Gignac has tried to explain the bias in terms of the proportionality constraint.

Recently, the bi-factor approach has been criticised for an inherent statistical bias that makes it more likely to have acceptable fit statistics than the higher-order factor model.

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