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Question on Effective Population Size

Daniel Weinreich dmw at MCZ.HARVARD.EDU
Mon Dec 4 16:10:30 EST 1995

```On 1 Dec 1995, henry harpending wrote:

> We want the variance of a gene frequency in the offspring to look like
>
>          x(1-x)/2N_e
>
> where x is the gene frequency in the parents.  This offspring gene frequency
> is the sum of the frequency in  gametes from parental males, with variance
> x(1-x)/2N_m, and of the frequency in gametes from parental females, with
> variance x(1-x)/2N_f.  Add these, divide by two, and you get the expression
> given by Li and Graur.  Note the assumption that the parental frequency is
> the same in both sexes.

That's a new derivation for me; I quite like it especially since it
highlights the implicit assumption of equal x's in the two sexes.  It
differs from the "what's going on the the grandparent's generation"
analysis in finding the Variance Effective Size, rather than the
Inbreeding Effective Size.  But in this case the difference comes to
nothing.  (S'pose the gene were X-linked...?)

With respect and apologies to my better, I offer the following minor
correction, more or less cribbing directly from Crow & Kimura (p. 358).

x' is the mean of the male contribution and the female contribution, so
the variance in x' is 1/4 the sum of the male and female variance, since
in general, Var(X/2 + Y/2) = 1/4*[Var(X) + Var(Y)] (assuming X & Y are
independent).  That gives

x(1-x)/2N_e = 1/4 [x(1-x)/2N_m + x(1-x)2N_f]

and solving for for N_e gives the required equation.  It's also clear from
this how to proceed if the x's are not equal in the two sexes, though it's
messy!

Regards, Dan.

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