I am reading the classic Weir and Cockerham 1984 paper about $F_{ST}$ estimation. At the beginning (first page, right column), they define 3 statistics.

  • $F$ is the correlation of genes within individuals ("inbreeding")

  • $\theta$ is the correlation of genes of different individuals in the same population ("coancestry")

  • $f$ is the correlation of genes within individuals within populations.

They also state that the 3 statistics are related by

$$f = (F-\theta)(1-\theta)$$

I don't quite understand those 3 statistics and especially I don't understand why this relationship holds true. Can you help me with that?

  • 1
    $\begingroup$ I find the 1984 paper to be a bit dense in some places. I don't have time at the moment to explicate this, but I found Bruce Weir's book very illuminating. You don't need to read it straight through. There are two or three relevant chapters. However, for your question, a good population genetics textbook such as Hartl and Clark might be sufficient. $\endgroup$
    – Mars
    Oct 29 '15 at 5:49

I am a bit wobbly on the subject, but I think the most important bit of information is that they are re-parametrising Wright's (1951) hierarchical analysis of variation, "F-statistics," "hierarchical partitioning of variation," or "population parameters," depending on whom you ask. The parameters correspond as follows (on the bottom of p.1358): Fit=F, Fis=f, Fst=θ.

The relationship arises given some assumptions. Crucially here, if Fis (or f) is a measure of departure from Hardy-Weinberg Principle, and all populations identically depart from HWP, then Fit = 1 - Hi/Ht. It follows that, 1 - Fit = Hi/Ht. As well, we can rewrite this, so that, Hi/Ht = (Hi/Hs)(Hs/Ht).

Together, you can (maybe) see that, 1-Fit = (1−Fis)(1−Fst). Substituting, 1-F = (1-f)(1-θ).

(I realise that this is not a complete answer, but you can rearrange it with some algebra to get the Weir&Cockerham equation, I think).

[Update Oct 25, 2016]: it eventually yields f = (F-θ)/(1-θ). I think the posted question (above) contains a typo--specifically a missing division operator. Perhaps someone missed the stroke on a typewriter in the original paper?

  • $\begingroup$ Thanks for the answer. Solving $1-F(1-f)(1-\theta)$ for $f$ gives $f=\frac{\theta-F}{\theta-1}$. $\endgroup$
    – Remi.b
    Aug 6 '16 at 17:17

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