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.
