Re: Underestimating 'r'
- From: "Jim McGinn" <jimmcginn@xxxxxxxxx>
- Date: Fri, 7 Oct 2005 13:59:41 -0400 (EDT)
Perplexed in Peoria wrote:
> Catherine Woodgold wrote:
> > Perplexed in Peoria writes:
<snip>
> Dr. Felsenstein has several times tried demonstrate
> to McGinn that the rule works in some special cases, but he was
> deterred from finishing the job by McGinn's snarling. If you have
> to see the numbers before you believe, those postings of Felsenstein
> are probably worth a look.
Predictably Joe produced nonsense numbers to support a
nonsense argument.
<snip>
> > I don't see it that way; and can you define what you
> > mean by "more closely related"?
Catherine snarls too. (Let's see what kind of
non-answer PiP has for us today.)
> Well, 'more closely related' is ambiguous. It might refer to distance
> in a genealogical tree - "Siblings are more closely related than cousins".
> Or it might refer to degree of similarity of genomes.
>
> And, of course, there are a variety of ways in which either of these
> two concepts might be turned into numbers. And even more ways in which
> the concept of a comparison between two comparisons can be quantified.
PiP is setting you up for obfuscation. Watch, he
will not answer the question.
> Now, it turns out (expect McGinn to scoff at this) that there is a way
> of quantifying 'more closely related' genealogically that gives almost
> exactly the same answer (in most cases) as a way of quantifying 'more
> closely related) in terms of genetic similarity.
Yes, and as you will see it requires one to make
abundant use of the human ability to suspend
disbelief.
> Here is how it works. Start with genetic similarity. Choose your favorite
> allele at some locus of the donor. Ask what is the probability that
> an identical allele will be found at that locus in a second individual.
> More precisely, since the second individual has two genes at that locus,
> we randomly choose one of the two and ask for the probability that it
> is identical to our favorite allele. Also, more precisely, I should
> point out that that second individual was also chosen randomly -
> either from the set of the donor's recipients, or from the general
> population.
We know that for any two members of the same
species that upwards of 99% of the time they
will both have the allele. Less than 1% of
the time only one of them will have the allele.
(We also know that using genealogical relatedness,
genes IBD = R, that the closest it can ever be
[excluding incest] is 50%. So PiP has already
lost the argument. Not that he will ever admit
it.)
> Clearly, the probability will be low if the allele is rare in the population,
> but the probability will be high if the gene is common. Say that the
> gene has a frequency p in the population. Say that the probability we
> are looking for is Q. I claim that there is a number r with the property
> that Q = r + (1-r)p. What is more, I claim that this number r will be
> the same for all alleles at all loci in the genome.
>
> A good heuristic way of looking at this is to say that our second individual's
> genome consists of two parts - a part of size r and a part of size 1-r.
> The r sized part is identical to the donor. The (1-r) sized part looks
> like the general population.
Clearly, nothing stated in these two paragraphs
has anything to do with the question Cathy asked.
It's as if he's stating, 1 + 2 = 3.
> Ok, so much for genetic similarity. Now look at genealogical relatedness.
> Now we come up with an r much more directly. Now ask yourself - what
> is the formula for Q given r and p. Once again, we get Q = r + (1-r)p.
> Or at least that is the case if you assume that the population is not
> very inbred so that p consists mostly of genes that are not IBD to the
> donor in our truncated genealogical tree.
Ta da: 1 + 2 = 3 still! PiP's explanation reminds
me of an elementary school trick where they ask
you to chose a number between 1 and 100 but keep
it to yourself. They then ask you to add x,
subtract y, multiply it by z, and finally, subtract
the number you originally chose. Your final result
is, q. Ta da.
> So, to answer your original question 'can you define what you mean by
> "more closely related"?' - my answer is 'Take your pick'.
>
> Of course, as McGinn will point out, nothing I have said here justifies
> Hamilton's rule.
Correct.
> It only makes it a bit less mysterious that one metric
> of IBD relatedness can be used interchangeably with one particular
> metric of genetic similarity.
No, it didn't even do that. Once again, Pip,
you only manages to completely avoid the issue.
Jim
.
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