# Multiple-stratum intelligence models

My question concerns the mathematics of multiple-stratum models, about which I've not found much that's quantitative, so excuse my inventing my own notation to explain my thoughts.

In models such as Cattell-Horn-Carroll, intelligence components are arranged in, say, 3 to 4 strata. I'm going to invert the usual numbering of these strata so I can summarise my current understanding of these models, regardless of the total number of strata. In other words:

• I'm putting $g$ on its own in "Stratum 1". (I realise it's usually in stratum III in a 3-stratum model.)
• Stratum 2 components are then functions of $g$ and noise terms; for example, they might be approximated in factor analysis as $x_i=l_i g +\varepsilon_i$.
• Stratum 3 components are expressed in terms of stratum 2 components in a similar way, viz. $y_j=l_{ji}x_i+\eta_j=l_{ji}l_i g+l_{ji}\varepsilon_i+\eta_j$ with implicit summation over repeated indices. (I think you can see now why I've inverted the stratum order.) The noise term will still be Normal if the $\varepsilon_i$ and $\eta_j$ are Normal and independent. That's a big if, mind.
• We can repeat this in a fourth stratum if the model calls for it (as for example occurs in g-VPR), viz. $z_k=m_{kj}y_j+\theta_k=m_{kj}l_{ji}l_i g+m_{kj}l_{ji}\varepsilon_i+m_{kj}\eta_j+\theta_k$.

My question is this: what distinguishes the contents of the various strata? In the above linear special case the $x_i,\,y_j,\,z_k$ all have essentially the same form, a multiple of $g$ plus a noise term (and a more complicated model would still have them each reduce to a noisy function of $g$). My suspicion is that stratum delineation comes down to nesting non-linear transformations and/or noise terms that don't have stable distributions.

You are making two mistakes, one in relation to the identification of the factors or what they represent (and factor analyzes have to be studied a lot by correlations with the tasks that each factor represents). And two, the relationships between the factors.

You do not take into account that each factor would represent very different abilities or processes (for example, the closing speed and the sequential reasoning is very different), the factors can be independent of this way, there does not have to be a linear relationship and in case if correlation exists, it will be correlation for a specific task, which leads me to indicate that it is proposing wrong relationships about completely different abilities or processes. This should clarify and is the model to start studying the factorial analysis:

x1 = a1.F1 + a2.F2 ... + b1.S1 + c1.E1

x2=...

x3=...

x1 (score on the task of a specific person) = a1 (factorial weight of common factor). F1 (common factor) + a2.F2 .... + b1 .S (specific factor) + c (factorial weight of the error factor) + E1 (error)

(In the majority of analyzes, an error for a specific factor is not studied)

• Can you recommend any references that go through this mathematics in detail? – J.G. Feb 2 '18 at 18:55
• The book that I would most recommend is in Spanish language is part of a collection of an editorial on statistics books, very interesting but I´m sure in the library of your faculty you have excellent books. – hexadecimal Feb 4 '18 at 1:49
• @hexadecimal - What is that book you recommend? – Chris Rogers Mar 5 '18 at 8:35