"Guy Hoelzer"  wrote in message
news:cbdlb2$2mpm$1{at}darwin.ediacara.org...
> in article cbc9qk$276c$1{at}darwin.ediacara.org, Perplexed in Peoria at
> jimmenegay{at}sbcglobal.net wrote on 6/23/04 9:06 AM:
> > "Guy Hoelzer"  wrote in message
> > news:cbaj57$1kj0$1{at}darwin.ediacara.org...
> >> in article cba42o$1eub$1{at}darwin.ediacara.org, Perplexed in Peoria at
> >> jimmenegay{at}sbcglobal.net wrote on 6/22/04 1:16 PM:
> >>> "Guy Hoelzer"  wrote in message
> >>> news:cb7rtf$mn0$1{at}darwin.ediacara.org...
> >>>> in article car9d1$2lm6$1{at}darwin.ediacara.org,
Perplexed in Peoria at
> >>>> jimmenegay{at}sbcglobal.net wrote on 6/16/04 10:15 PM:
> >>>> Whatever method is
> >>>> used, the result has to be an "r" with the
following property:
Randomly
> >>>> choose one of the two genes at any locus in the
donor. Suppose that
the
> >>>> frequency of this allele in the general population is
"p". Now,
randomly
> >>>> choose one of the two genes at the same locus in the
recipient.  It
must be
> >>>> the case that the probability that the two randomly
selected genes
are
> >>>> identical is (r + (1 - r)p).  That is, there is a
probability r that
they
> >>>> are identical IBD, but if not, then there is still a
probability p
that
> >>>> they are identical for other reasons - because the
allele is fairly
common
> >>>> in the population.
> >>>> [snip]
> >> Well, "r" can never be known and can only be
estimated with an
> >> underestimation bias.  The degree of the bias depends on the amount of
> >> unknown inbreeding in the common ancestry of two individuals.
> > [snip]
> >> Putting aside the fact that
> >> we rarely have information going back more than one generation, the
greater
> >> issue has to do with inbreeding.  The parameter "r"
can take on any
value in
> >> the face of inbreeding.  For example, full sibs actually have
0.5 >= r
<= 1.
> >> In fact, it is very common to have levels of inbreeding that make our
naïve
> >> estimates of "r" significantly lower than they
actually are.  Truth be
told,
> >> r=1 for every pair of individuals (even from different species) if you
were
> >> to consider all the data (and ignore mutation/divergence).  Do you have
a
> >> justifiable rule for how ignorant we ought to be when we try to
estimate "r"
> >> in order to make the false estimate useful in understanding kin
selection in
> >> nature?
> >
> > It is clear that you still don't understand what "r"
is.  The history of
the
> > population, and the fact that it may be inbred in the recent or distant
> > past is totally irrelevant.  The only way inbreeding could be relevant
would
> > be if the population routinely breeds with close relatives for reasons
other
> > than small population.
> >
> > Forget IBD for a moment.  Let p be the frequency of the altruistic
allele
> > in the general population.  Let P be the frequency of the allele in the
> > recipients.  Define r to be that value which satisfies the equation
> >    P = r + (1-r)p
> > That is, r represents, in a mathematically wierd way, the degree to
which
> > P exceeds p.
>
> As I said before, I like this way of thinking about the model, at least in
> some ways.  I think you have internalized Maynard-Smith's version of
> Hamilton's model.  Those two great thinkers thought about things in very
> different ways, IMHO.  I am not sure that Hamilton ever verified that the
> Maynard-Smith version was consistent with his thinking.  IMHO Hamilton
> thought of "r" primarily as a measure of genealogical
relationship, which
is
> not what it means in your equation above.

I'm not sure how Hamilton thought of "r".  But someone else on this thread
seemed pretty convinced that WDH thought it was a regression coefficient.
If I understand things correctly, that is exactly what the "r" in
my formula
is.

> > If you wish to understand why inbreeding is not important, perform
> > the following thought experiment.  Imagine a population derived
> > from a single breeding pair which has grown to a population of
> > 64 with the population doubling each generation.  But, to make
> > sure that we have variation for altruism, make both of the original
> > pair heterozygous.  Assume that mating is random and monogamous.
> > "p" is 1/2.  Assume the altruism is directed to full
sibs.  I think
> > that you will find that r is not much larger than .5 and certainly
> > less than .6.  Or, for variety, start with p = 3/4 or 1/4.
> > "r" still will be less than .6 AFAICS
>
> You assumed something like random mating in an exponentially growing
> population, which basically the same as assuming that inbreeding is not
> occurring.

Inbreeding is not occurring now, though it did in the F1 and F2
generations.

Yes, I am trying to show that in an inbred population, which is now
mating randomly, the fact of the past inbreeding makes little difference
in whether "r" calculated by IBD matches "r" calculated
by my formula.
I thought that I was responding to what you wrote above.  Apparently,
I misinterpreted what (still) seems to be a clear claim that the
degree to which the population is inbred must be taken into account
before assuming that a truncated IBD calculation is a good estimator
of "r".

> It is certainly not surprising to conclude that inbreeding does
> not affect "r" when you assume that inbreeding does not occur.  To
> illustrate why inbreeding IS important, consider a hypothetical population
> in which sib-sib mating is the norm.  Such a population quickly loses its
> heterozygosity and becomes constituted by families filled with altruists
and
> families lacking altruists.  Now P>>p, and your "r" value is
correspondingly
> much higher.  So "r" is sensitive to inbreeding.

I never said it wasn't.  To quote myself a few lines above: "The only way
inbreeding could be relevant would be if the population routinely breeds
with close relatives for reasons other than small population."

I'm still not sure I understand the differences between Malecot's IBD
and Wright's coefficient of relationship in populations with recent
inbreeding.  I also don't know which is the best estimator of "r".
However, I am not yet convinced that one of these genealogical
parameters, even if truncated after N generations, say, doesn't
still give an estimate of "r" that is valid to within 1 / 2^N.
You are welcome to try to convince me.  Obviously, the true IBD "r"
for full sibs will be greater than 1/2 in your incestuous population,
as will the truncated IBD "r" and the real (my formula) "r".  But
I don't quite know how to calculate any of them.

> >>>>> [snip]
> >>>>> Or, if like McGinn, you have an intuition that
geneological history
cannot
> >>>>> be causal in this situation, ignore the IBD
above.  "r" is simply a
> >>>>> measure of how much more likely than
"p" it is that the two genes
are
> >>>>> identical for whatever reason.  The key thing is
that the formula (r
+
> >>>>> (1-r)p) gives the probability that the alleles
are "shared".
> >>>>>
> >>>> Hmm.  There are some things about this formulation
that I like, and
some
> >>>> problems I see.  Can you please save me a little
research time and
tell us
> >>>> where you come by the formula (r + (1-r)p)?  Is this your
interpretation of
> >>>> Hamilton, or has it been published?
> >>>
> >>> It is a straightforward interpretation of the verbal
explanation given
in
> >>> Maynard Smith's "Evolutionary Genetics" (2nd ed. p169)
> >>>
> >>>   Now we can picture the genome of the recipient as
consisting of two
> >>>   parts:
> >>>   1. a fraction r containing genes IBD to genes in the actor; and
> >>>   2. a fraction (1-r) consisting of genes that are a random sample
> >>>      of genes in the population.
> >>
> >> This is a very familiar modeling trick.  The same thing is done when
modeling
> >> inbreeding for other purposes.  It is, however, just a trick that makes
the
> >> math work out easily.  The flaw becomes clear when you recognize that
any
> >> random sample of the gene pool will potentially contain gene copies
that are
> >> IBD with the target, so the fractions are not mutually exclusive. Given
your
> >> definition of "r", "(1-r)" must be the
fraction of the recipient's
genome
> >> containing genes that are NOT IBD, which is different from
"a random
sample
> >> of genes in the population."
> >
> > I think that my argument above including the phrase "forget about IBD
> > for a moment" addresses this concern.
>
> I agree that this second way in which you defined your parameters is more
> logical.  However, that does not validate the way you defined them at
first,
> which I still argue had a logical flaw.  In fact, I think these two
> definition sets are inconsistent with one another, so you should decide
> which one you want to use and stick with it.

I wish you had given that advise to Hamilton.  ;-)

Even in Maynard Smith's formulation, there is not the logical flaw that
you claim.  The fraction "r" does not contain ALL of the IBD genes.  Only
enough of them that the "1-r" fraction looks like the general population
in ITS content of IBD genes.

Another way of looking at this is that the fraction "r" contains
approximately all of the genes that are recently IBD, and those
genes that are IBD due to events in the distant past (a history you
share with the general population) are in the (1-r) fraction.
Of course, this fuzzy language cannot be part of the definition,
it is part of a fuzzy "empirical" observation.


> >> [snip]  BTW, have
> >> you agreed in the past that all of this goes out the window for
deterministic
> >> reasons when there is only one copy of the allele around, because then
> >> altruism only costs the allele fitness points?
> >
> > I understand what you are saying.  Clearly, the only altruist in the
> > population cannot also be a recipient, and hence can't be more fit than
the
> > rest of the population.  [snip]

> So I guess that you concede that the validity of Hamilton's rule depends
> "p", at least at this singular point.  I suspect you would
even concede
that
> the effect of "p" would be observed at very low values of
"p" when there
is
> more than one copy of the altruism allele around.  [snip]

Actually, if "r" is calculated by my formula, I don't need to concede
anything.  My formula gives the value of "r" as zero (or actually -1/N)
when there is only one carrier of the allele.  So Hamilton's rule still
works.

What you are actually proving is that the assumption that r is independent
of p is an impossible assumption to actually carry out when Dr Hoelzer is
busy constructing counter-examples.


> [snip] I think the assumption
> of constant "p" values within subpopulations subverts the
problem I was
> pointing to.  Indeed, that assumption is inconsistent with Hamilton's
model,
> which is about predicting changes in "p".

My understanding of the jargon of pop gen is that an assumption that "p"
is constant in a model which attempts to predict changes in "p" is
known as a "weak selection assumption".  It is, IIUIC, pretty standard.


> [snip]
> So your "r" is critically dependent upon population structure.  Right?

Of course.  It is very dependent on it if the altruism is dispersed
indiscriminately to neighbors.  It is less dependent if the altruism
is based on kin recognition, but even here, it is hard to be nice
to your kin if an ecologist hostile to Hamilton has rearranged the
population so that you can't find your kin.  :-)

> Didn't you once argue that Hamilton's rule did not depend upon population
> structure?

I don't recall doing so.  I would be extremely reluctant to do so now,
given your creativity in postulating structures.  However, I will note
that having "r" dependent on the population structure is a different
matter than having the Rule dependent.


> [snip]
> What do you think of my "flipside" argument above regarding systems
> constrained to sib-sib mating?

You want to know what I *honestly* think of it?  ;-)

I can't offer an opinion until someone clears up the Wright/Malecot
confusion for me.  But I strongly suspect that it makes absolutely
no difference to the applicability of the Rule when "r" is calculated
using my formula.  It may make a difference as to whether IBD
is a good estimator of "r".
---
þ RIMEGate(tm)/RGXPost V1.14 at BBSWORLD * Info{at}bbsworld.com

---
 * RIMEGate(tm)V10.2áÿ* RelayNet(tm) NNTP Gateway * MoonDog BBS
 * RgateImp.MoonDog.BBS at 6/25/04 6:32:11 PM