Re: Hamilton's rule

I may have bit off more than I can chew in one or two
earlier posts. I implied or claimed that I had a simple
and a complicated algebraic solution. Well, I started
typing up the simple one, and it didn't come out quite
the same as it had on paper. (I had never actually
finished working out the complicated one.)

I ended up coming full circle and finding myself
face-to-face with the question of how to define
identical-by-descent again.

Then I figured: if I wanted to be able to model small
populations, maybe I should get rid of the infinite-population
assumption earlier in the calculations. (Duh.)

So I started thinking about how to do that, and
haven't gotten very far at all.

Here's a complexity:

Suppose you have a population of N individuals, where
N is not large enough to be treated as infinite. N might
be around 20, for example.

Suppose the actual average rate of allele A in the population
is p. Suppose an individual has AA. Now, right away, you
have to realize that the rate of A in the rest of the population
will not average out to p, (unless p is 1), because the rest
of the population has to have a bit less than p so that
when it's averaged with this one individual, it will all
work out to p.

That's not that hard, so far.

But then you can begin to consider the set containing that
individual and that individual's relatively close relatives.
Again, the ones who are not in that set would have a lower
rate than p. And you begin to run into the problem of
where to draw the line between relatives and non-relatives.

Well, maybe that can be sidestepped somehow. But consider
this: Suppose we know that a certain other individual is
a sibling of the AA individual. What is the expected value
of the rate of A in that individual?

That is not a very easy question to answer! (Or, to put
it another way, I haven't answered it yet.)

I've realized that you can't just assume that there's a
50% probability that a certain gene inherited from the
mother in the AA individual was also inherited by the
other individual. Rather, the probability has to be
considered to be something other than 50%, using all the
available information.

This becomes more obvious when you consider certain
specific examples:

Suppose the average rate of A in the 20 individuals is
1/20. Then what is the probability that one of the A
homologs in the AA individual was also inherited by
the particular sibling in question?

Suppose the AA individual has 10 siblings in the population
of 20, and the rate of A in the population is 3/40.
Then, given one of the siblings, what is the probability
that the sibling has an A allele?
--
Cathy Woodgold
http://www.ncf.ca/~an588/par_home.html
We are all Iraqis now.

.

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