Steven Byrnes comments on Heritability: Five Battles

[Steven Byrnes]

Thanks so much! I think your response has some mistakes, but I think you had the right big picture, especially the idea of bounding between 0 and ¼.

Here’s my attempt, inspired by reading what you wrote + your two sources (the Zuk et al. 2012 supplementary info section 1, and the book Genetics and Analysis of Quantitative Traits pp 81ff):

Assume no assortative mating and no gene-environment interaction. Normalize the variance of the trait across the population to 1 (so “fraction of variance due to X” is the same as “variance due to X”). Then we have:

where

• = contribution to variance from unique environmental effects

• = contribution to variance from common-between-siblings environmental effects

• = a genetic contribution to variance, involving an interaction of additive effects at i different loci, and dominance effects at j different loci (with the convention for nonsense combinations like ).

We split the last term into:

• An additive contribution

• A non-additive contribution

Importantly, is the fraction of population variation explained by a PGS, in the limit of perfect PGS measurement (infinite sample size, and measuring every last rare variant, structural variant, copy number variant, and whatever else).

Next:

Now split up the additive vs non-additive parts:

Subtract to get:

Key step: the coefficients in the sum are all between 0 and ¼. Thus:

Now substitute (from above), and we finally get the following:

Reassuringly, this agrees with both of the intuitive guesses I suggested above: assuming , then for , we get , i.e. the PGS would (in the limit of perfect PGS measurement) work almost perfectly, with no missing heritability; and for , we get , i.e. the PGS cannot possibly work at all.