Re: Units of IBD
- From: "Perplexed in Peoria" <jimmenegay@xxxxxxxxxxxxx>
- Date: Wed, 5 Oct 2005 01:00:33 -0400 (EDT)
"Jim McGinn" <jimmcginn@xxxxxxxxx> wrote in message news:dhv9kr$1m6k$1@xxxxxxxxxxxxxxxxxxxxxx
>
> Perplexed in Peoria wrote:
> > Jim McGinn wrote:
> >
> > > IBD does have units. Use them.
>
> <snip>
>
> > I'm going to venture a guess. Let me know if I guess correctly.
> > The "units" that you have in mind are "IBD-alleles-at-the-locus
> > over all-alleles-at-the-locus".
>
> Good. So you are cognizant of the units/dimensions.
> And all that subterfuge you presented earlier is, well,
> subterfuge.
>
> [innocuous rearrangement here]
> > That is the only sense I can
> > make out of your claim that all probabilities have units.
>
> Gee, this time it only took you about a week and a
> half to answer the question honestly.
As I am sure others will also point out, there are some severe
technical problems with your position that probabilities have
units. For example, recall my question as to what are the "units"
of the probability (1/6) of throwing a seven with two dice.
You would presumably say that the units are "sevens per throw".
Similarly, you would calculate the probability of throwing an eleven
as "1/18 elevens per throw. Lets add these two probabilities to get
the probability of 'craps'. The answer is 2/9 and the units are
presumably "sevens or elevens per throw".
Hmmm. Is that the rule for adding dimensional quantities? Lets try
it with velocities. Suppose we have a fish swimming downstream at
a velocity of 3 (units of feet per second) and the stream is moving
with a velocity of 12 (units of centimeters/second). So if we use
the same rule that was used for probabilities, we would calculate the
total velocity of the fish as 15 (units of 'feet or centimeters per
second').
As most people understand dimensional analysis, you simply are not
allowed to add or subtract two quantities unless their dimensions
match EXACTLY. McGinn style 'extended dimensional analysis' (in
which probabilities have dimensions) follows no such rules.
You are imagining a 'proof' that Hamilton must have made a mistake
that runs something like this. Hamilton started with a model
of population genetics in which "alleles IBD at the locus" just
doesn't show up as a factor. After some algebra, he ends up with
a formula "rb>c" in which "alleles IBD at the locus" does appear
in the units of r. How can that be?
Well, I suspect that the loophole in your sketch proof is the fact
that probabilities don't add by the same rules as velocities do.
In fact, "alleles IBD" + "alleles identical but not by descent" +
"alleles not identical" add together in a way such that the units
all cancel out and you are left with simply "alleles". But in
order to see that this is actually the way Hamilton avoids your
'proof', you are going to have to sit down and work through the
algebra. USING A TEXTBOOK!
> [end innocuous rearrangement]
> IBD (identical by descent) indicates the
> number of genes (in this instance I'm using the term
> gene to indicate alleles at locuses [which is really
> the only rationally consistent usage of gene]) that
> have been copied through osmosis
"Osmosis"? No comment. I think I will just skip this diversion.
> from a specific
> individual *and* that are identical (not either
> variations or mutations). It does *not* indicate the
> number of genes that are identical.
>
> Now see if you can do one of the following:
>
> 1) Explain why Hamilton assumed genes IBD of two
> individuals represents the relatedness (R) of these
> two individuals.
He didn't, if you continue to think that "relatedness" means
degree of genetic similarity. But if "relatedness" is taken
to mean something like 'degree of closeness in a genealogical
tree' then it is fairly obvious that the probability of IBD
is a natural measure of 'relatedness'. Hamilton chose this
metric of relatedness rather than some other natural metric
because this metric results in a simple formula "rb>c".
It is important to keep that word 'probability'. The probability
that any particular gene of mine is shared IBD by my brother
is 0.5. However, it is possible, due to the random nature of
meiosis, that the fraction of our genes shared IBD is anywhere
between 0.0 and 1.0. When you add in the fraction of genes that
we shar identical but not by descent, you get up near 99%. But
this is simply irrelevant.
> 2) Explain Hamilton's rule using only the term "genes
> IBD" in place of R (relatedness).
>
> If you can do either of these I will send you a check
> for $10,000.
Buy a textbook. The author has already received royalties
exceeding $10,000 and you only lose $50-100. Use the money
that you saved to buy some more textbooks. Who knows, you
might actually learn something.
.
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