Is CHC a hierarchical or a Bi-factorial model of intelligence?

This is a very theoretical post that I hope will help me argue that some of the critique of psychologists' emphasis on broad abilities over general intelligence is unjustified.

According to the excellent paper by Beaujean ,(2015)there are two theoretical approaches that conceptualize the relations between g, broad cognitive abilities/group factors and narrow cognitive abilities/subtests: the British approach (bi-factor model) and the American approach (hierarchical model).

The American approach is shown in figure 1 (click to enlarge).

Figure 1

As is evident from the figure, this approach sees intelligence as a hierarchical structure.  The approach works "bottom up": at the lowest level are subtests which measure narrow abilities.  Higher order factors (broad abilities like  fluid ability, comprehension – knowledge) are formed from clusters of subtests that measure a common essence (and that have common variance).  Every subtest is a measure of the broad ability to which it belongs and also measures something unique.  If enough broad abilities/group factors are present, and they share enough variance, then their correlations can be factor analyzed to find higher-order factors.  The apex of this higher-order factor model often containes a single factor – general intelligence (g).  Each broad ability is a measure of g and also measures something unique.  Because g is formed out of the broad abilities, in this model the broad abilities and g are dependent on each other.  

Thus, in the American approach, which sees intelligence as a hierarchical structure, broad cognitive abilities are formed before g and precede it.  g is formed out of the broad abilities and expresses their common variance, their correlation, their common component.  Thus, g is secondary to the broad abilities.

In a hierarchical model the common variance of the subtests does not contribute directly to the formation of g because g is formed out of the broad abilities (not the narrow ones).  And vice versa:  g does not have a direct influence on the subtests.  It affects broad abilities directly and the broad abilities affect subtests.

In a hierarchical model, the main difference between g and broad abilities is their place in the hierarchy:  g is superior to the broad abilities and represents an entity that is more abstract than broad abilities.   

The British approach is presented in figure 2.

Figure 2

As you can see in the figure, g is formed directly from the subtests, and it expresses the common component of all subtests.  Each subtest is a measure of g and also measures something unique.  The broad abilities are estimated from the covariance remaining among the subtests after accounting for g; they reflect what is common among a group of subtests (after accounting for g).  There still remains in each subtest a component which is not common to the g or to the broad ability – a unique element that this subtest measures, which is specific to the narrow ability.  Since this approach begins with the g, and g precedes the broad abilities, this approach works "top down".

This model is a bi-factor model: one factor is g and the other factor is the broad abilities.  In a bi-factor model, the factors are independent of each other.  g is not dependent on the broad abilities.  A change in performance in specific subtests can affect specific broad abilities but not g.  A change in g will affect performance in the subtests directly, but it will not necessarily affect all broad abilities and so on.  General intelligence in this model directly affects the subtests (not through the broad abilities, as happens in the hierarchical model).

In a bi-factor model, the main difference between g and broad abilities is in the breath of influence.  General intelligence affects all subtests, while each broad ability affects only the subtests that measure it.

What's the significance of all this?

Since g in a hierarchical model and in a bi-factor model is created out of different sets of relations between variables, the general intelligence score derived by each model will usually be different.  One reason for this could be the composite score extremity effect (but I'm not sure.  This is only my guess and I haven't found basis for it in the literature yet).  Suppose a child has a low score, say 6, in each subtest.  Because of the composite score extremity effect, his broad ability scores will be even lower – say 5.  In a hierarchical model, since general intelligence is formed out of the broad abilities, the composite score extremity effect will work here as well. The child's g score will be even lower than the broad ability scores, say 4.  In a bi – factor model, the composite score extremity effect will work separately on g and on the broad abilities.  When a child scores 6 on all subtests, his g score will be lower than 6. The question is how low will it be, and whether it will be lower in a bifactor or in a hierarchical model.

The broad ability composites will also be different in the two models.  In a bi – factor model, a broad ability reflects the common essence of all subtests that comprise it, after accounting for g.  In a hierarchical model, a broad ability reflects the common essence of the subtests that comprise it; g is optional but is not always formed.  Only if there are enough broad abilities with enough common variance can g be formed.  If g is formed in the model, each broad ability can be said to measure both g and something unique.

Does this mean that in a bi-factor model the broad abilities are "cleaner" – reflecting more of the unique thing they measure and less of the g?  Does this mean that in a bi –factor model the broad abilities are more independent of each other?  That there is less correlation between them? That each broad ability in a bi-factor model reflects a skill that is more unique than in a hierarchical model?

Why is this interesting?   What are the consequences of the differences between the models?  Is CHC a hierarchical model of intelligence or a bi-factor model of intelligence?

Prof. McGill and his colleagues are attempting to prove that after accounting for g, broad abilities explain only a very small amount of incremental variance.  For example, McGill found that the general ability score explains 67% of the variance in reading comprehension at the age of 17; broad abilities explain only 10% of the variance in reading comprehension at this age.  Given these results, McGill suggests that "a more circumspect appraisal of the importance of CHC dimensions in relationship to the development of reading skills may be needed in the professional literature" (McGill, 2017).

McGill and Busse (2015) found similar results:  they re-analyzed the data of 6-19 year old children from the norming sample of the WJ3 test.  The children in this analysis were also given the WJ3 achievement battery.  McGill and Busse found that general intelligence explained between 29% (in math calculations) and 56% (in reading comprehension) of the variance in achievement.  Broad abilities explained between 2% (in mathematical reasoning) and 23% (in oral expression) of the incremental variance, over and above g.  It was comprehension knowledge that predicted 23% of achievement in oral expression (but oral expression tests are also measures of comprehension knowledge…).  The other broad abilities did not explain more than 5% of the variance in reading/writing/math over and above general intelligence.  The contribution of broad abilities was significant but very small.

However, McGill and his colleagues conducted these studies from a bi – factor orientation, and thus entered g first into the factor analysis.  McGill and Busse (2015) write that reverse entry of the independent variables (entering the broad ability clusters first), would result in the clusters accounting for approximately the same variance proportions that were attributed to the g. Consequently, the g would provide little incremental prediction. Order of entry arbitrarily determines whether scores such as the g mean everything or nothing. But they argue that "order of entry is not an arbitrary process and must be determined a priori according to expected theoretical relationships between the variables and causal priority. Contemporary intelligence theory (e.g., CHC) and the WJ III COG structural model support entering the GIA-E before the clusters because the cluster scores are both theoretically and statistically subordinate to the GIA-E" (GIA-E is the general intelligence score).

I think that they may be mistaken twice:

The first mistake is that if indeed g is hierarchically higher than broad abilities, we are in the domain of a hierarchical model.  In a hierarchical model it is befitting to enter g to the analysis last  -  not first.

The second mistake:  when g is entered into the analysis first, one works within a bi-factor model framework (not within a hierarchical one).  Beaujean(2015)  as well as Benson et al (2018) think that Carroll's model is bi-factorial, not hierarchical.  Since Carroll's model is one of the bases for the CHC model, they conclude that the CHC model is bi-factorial.  CHC is an integration of the models of Carroll and Cattell – Horn.  But in Cattell – Horn's model there is no g at all!  Cattell – Horn's model comprises of broad and narrow abilities and is a hierarchical model.  Thus, there are good reasons to think that the CHC model is hierarchical and not bi –factorial.  If it is hierarchical, it should be built "bottom up".   In a hierarchical model, broad abilities precede g, and they should be entered into the analysis first.

When researchers take a hierarchical model approach, they see that broad abilities do explain substantial variance in achievement, as was found by, for example, McGrew and Wendling (2010).

Beaujean, A. A. (2015). John Carroll’s views on intelligence: Bi-factor vs. higher-order models. Journal of Intelligence, 3(4), 121-136.‏   http://www.mdpi.com/2079-3200/3/4/121/htm

Benson, N. F., Beaujean, A. A., McGill, R. J., & Dombrowski, S. C. (2018). Revisiting Carroll's survey of factor-analytic studies: Implications for the clinical assessment of intelligence. Psychological assessment.‏

McGill, R. J. (2017). Re (Examining) Relations between CHC Broad and Narrow Cognitive Abilities and Reading Achievement. Journal of Educational and Developmental Psychology, 7(1), 265.‏ http://www.ccsenet.org/journal/index.php/jedp/article/viewFile/66066/36510

McGill, R. J., & Busse, R. T. (2015). Incremental validity of the WJ III COG: Limited predictive effects beyond the GIA-E. School Psychology Quarterly, 30(3),https://pdfs.semanticscholar.org/f5b5/d70077a1b7747a31bbcd5fb7b7dfcc38c2a3.pdf

McGrew, K. S., & Wendling, B. J. (2010). Cattell–Horn–Carroll cognitive‐achievement relations: What we have learned from the past 20 years of research. Psychology in the Schools, 47(7), 651-675.