Comment by rbehrends

> For instance, OP's definition H = Var[G] / Var[P] seems to bypass the issues you mentioned:

No, this is exactly the definition I am talking about. The problem is that while theoretically you could work with Var(G)/Var(P) even if Cov(G, E) ≠ 0, studies are not designed to capture that.

In fact, the standard ACE model [1] used in twin studies explicitly assumes among other things that there is no gene-environment correlation. This means that it gets silently added to one or more of the ACE components; not because of any ill intentions, but simply because if you included covariance, the resulting system of equations would be underdetermined and could not be solved [2].

But to make matters worse, gene-environment correlation/interaction itself is disproportionately absorbed by the A and C components rather than E. All this can lead to inflated heritability estimates.

And to clarify, I am not making any pronunciations about how much relevance or magnitude that effect has; for all I know, this could in the end be a minor effect. My point here is that there is a lot of mathematical handwaving going on with very limited testability of the modeling.

[1] https://en.wikipedia.org/wiki/ACE_model

[2] If you want to be precise, you need to actually distinguish between gene-environment correlation and interaction and use P = G + E + (G x E), but that makes the system even more underdetermined, because now we have both Cov(G, E) and Var(G x E) to worry about.