"John Edser"  wrote in message
news:cev3s6$17m5$1{at}darwin.ediacara.org...
>
>
>  "Perplexed in Peoria" 
>
> > > I don't care HOW you measure relatedness,
> > > how can ANY measure of relatedness be
> > > <0 within ANY science of biology?
>
> > JM:-
> > This is explained without much math in:
> > "A geometric view of relatedness", Alan Grafen, 1985
> > http://users.ox.ac.uk/~grafen/cv/oseb.pdf
> > You don't even need to know what a regression coefficient is.
>
> JE:-
> Thank you for this reference. I have
> read it and remain happy to debate its
> content in another thread.

Ok.  Your move.

> In BIOLOGICAL TERMS relatedness is
> just genes held in in common.
> How this can be measured remains a
> problem. However the concept determines
> what is to be measured, the measuring
> system does NOT determine the concept.

Agreed.  But ...

We don't put concepts into mathematical "laws" such as
Hamilton's rule.  We put measurements in.  If we somehow
came up with a measuring system for the concept of
relatedness that was guaranteed to always give us a
non-negative measurement, then perhaps we could reformulate
Hamilton's rule to work with that method of measuring
relatedness.  But the rule would then be more complicated
algebraicly.

As Grafen explains, the use of the regression coefficient
method of measuring r, rather than the IBD method of
measuring r, gives us a nice, easy to use formula for
Hamilton's rule - "rb>c".  If we used IBD as our method
of measuring the concept of relatedness, then we would
have to use a more complicated rule - perhaps "(r-1/N)b>c"
where N is the population size, though even that wouldn't
be exactly right.

Arguing, based purely on philosophy, that Hamilton's rule
can't be right because it uses a measuring scheme that
allows negative relatedness is silly - just as silly as
complaining about a physics law, framed to work with
Centigrade temperatures, because that allows the temperature
measurement to be negative when the real temperature concept
should always be positive.

> Hamilton's rule attempts to
> measure relatedness using a probability.

True only if you ASSUME that IBD is the intended concept
to be measured.

> IBD relatedness is only the probability that
> you inherit a gene from a defined biological
> source that is the same gene (i.e. is
> not a mutation of that gene). Thus, -IBD
> relatedness is nonsense. [snip]

Agreed.  IBD relatedness is always positive (or zero if you
allow truncation of the calculation, as my wife points out).
IBD is never negative, and neither Hamilton, NAS, nor
myself has ever said it could be negative.

> A regression coefficient is just a statistical
> rework of reality just like any simple average is.

Agreed.  A good analogy.

> If the average total fitness is 2.67, then absolutely
> NOBODY can be average because the fraction is
> BIOLOGICAL NONSENSE unless it is reworked into
> some kind of a representation of a BIOLOGICAL
> REALITY.

Agreed that an individual fitness has to be an integer.
But I disagree that an average of integers is going
to be nonsense.  Sometimes it is the average value that
is biologically causal rather than the integer individual
value.  An example:

Litter size in pigs is an integer.  A pig never gives
birth to a litter of 3.57 piglets.  But it might well
be the case that one population of pigs has an average
litter size of 3.57 while another has an average litter
size of 3.87.  Now, suppose we are trying to develop
a theory relating the size of mammary fat deposits in
pre-pubescent female pigs to litter size.  Is it the
integer litter size of a particular litter that is
causal here.  I think not.  Instead, it is the average
litter size of the recent ancestors of the pig that
created the selective environment for the adaptation
of mammary fat deposits.

Yes, integer litter sizes may be the ultimate biological
reality, but averages can be causal in biology, and hence
measurements of averages can (in fact MUST) appear in
biological laws.  To deny this may make philosophical
sense, but it does not make biological sense.

> The applies to any measure of -r.
> It has to be reworked into zero or a positive number.

My argument applies even more strongly to r.  The actual
individual IBD r between donor and recipient is not
causal in Hamilton's rule.  No part of the donor - not
his brain, nor his hormones, nor his biochemistry - is
making a calculation of IBD relationship before "deciding"
whether to act altruistically.  It is Nature herself that
"makes the calculation" and Nature is only concerned with
averages.

Or, to look at it another way, what really matters is whether
the recipient actually shares the gene or not - not the
probability that the gene is shared.  The recipient can
share either 0, 1, or 2 copies of the gene.  Looked at in
this way, it is integers that are causal, rather than
probabilities.  Causal, not of behavior, but of changes in
gene "populations".  However, I would still claim that what
matters in evolution is the long term trends, not the
individual fluctuations, and thus we are back to probabilities
and averages.  Besides, looking at it in this way is
gene-level thinking, and I understand you are skeptical of
gene-level thinking.

As far as objections to the concept of a negative relatedness
goes, I have already given my argument above.  The important
issue for deciding upon a measurement system for "relatedness"
is not how well it captures the CONCEPT of relatedness.  It
is in how well that measurement, as used in a "law", matches
with empirical reality.

You may have a case for arguing that Hamilton's r should not
be called "relatedness" if r is measured as Grafen suggests.
But you can't argue on philosophical grounds that r should not
be measured that way.  How r should be measured for use in the
rule ought to be an empirical question, if the rule is taken
to be a candidate law of nature.  Or, if Hamilton's rule is
taken to be a derived theorem from a model, then the question
is a mathematical one. So argue that r should be IBD, if you wish,
by pointing that no clear-cut examples of Hamilton spite have
been found in nature.  Or, point out a mistake in the derivation
of the rule from the model.  Don't claim that you are the only
person interested in this topic who appreciates biological
reality.  There are many professional biologists, including
Hamilton himself, who have spent a lot more time than you or I
out in the field, developing an appreciation of biological
reality far more intimate than ours.  (Though I may be presuming
here.  I know you are now a professional musician, but I don't
know if you used to be a field biologist.)
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