Re: Underestimating 'r'
- From: an588@xxxxxxxxxxxxxxxxxxx (Catherine Woodgold)
- Date: Sat, 22 Oct 2005 13:02:09 -0400 (EDT)
(name_and_address_supplied@xxxxxxxxxxx) writes:
>
> Let Dzbar = "delta z-bar" = "the change in the mean value of trait z in
> the population, due to natural selection". Let w = individual fitness.
> Let wbar = mean fitness of the population. let Cov[X,Y] be the
> covariance between the random variables X and Y. Let Reg[X,Y] be the
> least squares regression of X on Y. Let Var[X] be the variance in X.
>
> Dzbar = Cov[w/wbar, z] (Price's theorem)
>
> Cov[X,Y] = Reg[X,Y]Var[Y]
>
> => Cov[w/wbar] = Reg[w/wbar, z]Var[z]
>
> => Dzbar = Reg[w/wbar, z]Var[z]
>
> The change in the mean value of a trait in the population, due to
> natural selection, is the product of the selection differential
> Reg[w/wbar, z] on that trait and the variation in that trait Var[z].
> Since Var[z] > 0 if there is any variation, then if we assume there is
> some variation in z, the mean value of the trait will be selected to
> increases when
>
> Reg[w/wbar, z] > 0
>
> Now,
>
> Reg[x,y] = Reg[x, y | y2] + Reg[x, y2 | y] Reg[y2, y]
>
> where we allow for another predictor variable y2, and where "|" denotes
> "conditional on". Then we have the change in the mean trait value due
> to selection is positive when:
>
> Reg[w/wbar , z | Z] + Reg[w/wbar, Z | z] Reg[Z, z] > 0
>
> Lets interpret z as the trait value for our focal individual, and Z as
> the trait value for her social partners.
>
> Reg[w/wbat, z | Z] is the impact of our individual's own actions on her
> own relative fitness, holding fixed the actions of her social partners.
> Lets call this the personal "cost" of the individual's actions, -c.
>
> Reg[w/wbar, Z | z] is the impact of the social partners' actions on the
> focal individual's relative fitness, holding fixed her actions. Lets
> call this the 'benefit', b.
>
> And Reg[Z, z] tell us how the behaviour of social partners varies with
> one's own behaviour. Lets call this r, for short.
>
> So, our condition for when our social action will evolve is:
>
> -c + b r > 0
>
> That is just about as general a derivation of Hamilton's rule that you
> will find, for a simple single class model. It is easily extended to
> multiple classes -- see Price 1970 on how to weight the covariance by
> class reproductive values -- giving the same result. We find that r is
> fundamentally a regression measure.
>
> For an explicitly genetical model, we can write the regression in terms
> of probabilities of identity in state:
>
> r = ((Prob of picking gene from actor and recipient and them being the
> same)- (population average))/((prob of picking two genes with
> replacement from actor and them being the same)-(population average))
>
> For a rare gene, this simply the ratio of IBD for actor-recipient and
> IBD actor-actor. Since we are typically looking for ESSs, it is the
> behaviour of a rare gene that we are ultimately interested in, hence
> the focus on IBD.
I've looked at this post several times and seem to have
gotten a little closer to understanding it, but still
not very far. It looks interesting, though.
I suppose z is a real number associated with each
individual, such as: height, or amount of pigment in
the hair, etc. I suppose dzbar is the average amount
of z in one generation minus the average amount of z
in the previous generation.
I've done linear regression in the past, and I just skimmed the
Wikipedia article for regression, but I don't know whether
Reg[X,Y] is a number, or a two-element vector (slope and
intercept), or some other mathematical object.
If X and Y are expressed as vectors, then I think
Cov(X,Y) is:
sum(xi yi)/sqrt((sum(xi xi) sum(yi yi))
or something like that. The definition for
random variables will be something similar.
Hmm, I think I'm starting to understand. Price's
theorem looks as if it may be relatively obvious once
I get the definitions clear in my mind.
Oh, well, I'm starting to fade out. Maybe I'll
get further with it another day.
--
Cathy Woodgold
http://www.ncf.ca/~an588/par_home.html
We are all Iraqis now.
.
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- Underestimating 'r'
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