Factor Analyses Of Mental Abilities

Oblique VS Hierarchical Model Fit

Something to consider:

If hierarchical g models perform equivalently in confirmatory factor analysis to oblique models where group factor correlations aren't explained by g, then this basically means the group factor correlations pass Spearman's vanishing-tetrads test.

In other words, hierarchical g models don't have to display superior CFA performance to vindicate g theory, it's actually a strong confirmation of g theory if they merely display equivalent CFA model fit. If hierarchical models have a significantly but negligibly-worse model fit, then g theory is a negligibly-incomplete picture of the structure of mental abilities.

When we survey the confirmatory factor-analytic literature, this is the picture we see it painting [1, p.148; 2, pp.5 & 7; 3, p.129; 4, table 3, models 2 & 3; & 5, p.89, table 4-5]. In the Woodcock-Johnson Revised edition, they used to base their rejection of g on differences in model fit, but the equivalence of model fit is evident even in the WJ-R standardization data [5, p.89, table 4-5]; fit differences range from non-existant to negligible. They don't actually run significance tests for differences in model fit, they just note that the hierarchical g fit is "lower". The difference in rmr is negligible however (.045 VS .053), and if we look at the factor correlations on the next page and run Spearman's vanishing-tetrads test, the mean absolute tetrad difference is only .0396, which is even stronger support for g than in Spearman's original paper.

One possible reaction to seeing this is to say “SeE?!?!?! fAcToR aNaLySiS iS aBLe To PoSiT iNfiNiTeLy-MaNy DiFfErEnT MoDeLs Of EqUiVaLeNt FiT!!!!!!” Such an objection is missing the forest for the trees. If the oblique model which best satisfies simple structure passes Spearman's vanishing-tetrads test, g is an objectively necessary feature for describing your dataset; the g-loadings of the group factors imply factor correlations which satisfy Thurstonian simple structure, at least in the oblique sense.

Now, there are cases where people will posit only two group factors (say, GF VS GC), thereby precluding the possibility of doing this test. Whenever this happens, we know from the CHC research program, objectively, that it is the multiple abilities theorists who are at fault for positing too few group factors rather than the g theorists who are at fault for positing too many. The most-comprehensive taxonomy of test content in the domain of mental abilities simply demands enough group factors to reasonably accommodate Spearman's test; the collinearity needed to posit fewer just isn't there.

This stated, it isn't the end of the world to have to compete against oblique models in bad datasets like these. The superiority of bifactor models to both hierarchical-g models and non-hierarchical oblique models is arguably an even-harder condemnation than the fact that g models outright achieve the univariate equivalent of the satisfaction of simple structure. Comparing bifactor models and oblique models is the proper way to evaluate g theory, giving both proper opportunity to capitalize upon what's unique about their respective underlying theories in order to try to achieve superior model fit to their respective alternatives.

In non-hierarchical models, group factors are allowed to correlate in the way that maximizes their ability to satisfy Thurstonian simple structure (i.e. correlations between group factors are minimized with the restriction that the correlations between group factors must be able to account for any examples of subtests correlating substantially with more than one group factor). This freedom that group models afford to the factor correlations allows the subtests to correlate with their respective group factors as highly as possible; higher than would be possible in hierarchical models during scenarios where the tetrads test fails.

Bifactor models on the other hand relax the proportionality constraint imposed in hierarchical models. Basically, If g only affects subtest performance by means of affecting some group factor, this is equivalent to imposing a constraint where a subtest's g-loading can't differ from what's implied by how the subtest loads on the group factor. Hierarchical models demand that all subtests on a given group factor be identical to each other in terms of the ratio of their g-loading to their s-loading, whereas bifactor models have no such unreasonable requirement (i.e. bifactor models aren't a strawman of g-theory the way hierarchical models are, as they actually allow for g + s to be things which exist at the group factor level of the hierarchy and for which subtests can vary in loading for reasons other than measurement error and subtest specificity).

Really though, the superiority of bifactor models is just the cherry on top. Again, if there aren't enough group factors to test the degree to which g satisfies simple structure, then it's the test battery that's the problem. Ironically, the existance of a large diversity of mental abilities necessitates g.

Addendum: Spearman’s Vanishing-Tetrads Test Explained:

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