Hardy-Weinberg law - Page 2

"Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in message
news:cb2fdq$21bh$1@darwin.ediacara.org...
> Tim Tyler wrote:
> > Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote or
> > quoted:
> The theorem was derived for an infinite population.
>
> > E.g. see:
> >
> > http://library.thinkquest.org/19926/java/tour/06.htm?-
> > tqskip1=1
> >
> > Alas, this page expresses the law in terms of an
> > infinite population :-(
> >
> > A disasterous error - IMO - since talking about gene
> > frequencies in an infinite population is a sign of
> > mathematical ignorance.
> >
> No, it's a simplifying assumption. I think to accuse Hardy
> in particular of mathematical ignorance deserves, well, a
> non-mathematician's apology.

Huzzah!

> > Popularisers should make explicit the behaviour is what
> > happens as the population size tends towards infinity -
> > and not attempt to pass it off as an effect in an
> > infinite population.
>
> But it is - in finite populations, you get an excess of
> homozygotes, as any student of population genetics
> should know.

I am a student of population genetics who is foolishly
attempting to proceed without purchasing a good textbook.
Your claim surprised me, so I tried to check it. I used a
population of 4 consisting of two heterozygotes and both
kinds of homozygotes. Hermaphrodites. I performed all
crosses except the selfings with four offspring per cross.
(I am assuming that the finite population effect arises from
the prohibited selfings.) The problem is that I ended up
with an excess of heterozygotes. What am I doing wrong?

On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
<bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote:

>Tim Tyler wrote:
>> Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote or
>> quoted:
>>
>>>Jim McGinn wrote:
>>>
>>>>jason@kalavinka.freeserve.co.uk (friend) wrote in
>>>>message news:<car9d7$2lsg$1@darwin.ediacara.org>...
>>>
>>
>>>>>Pardon my ignorance, but I have only just discovered
>>>>>this law.
>>>>
>>
>> [...]
>>
>>
>>>>Many of the more popular myths of the current paradigm
>>>>of evolutionary biology pivot off a kind of rhetorical
>>>>trick. Specifically the trick involves employing a word
>>>>that has more than one meaning in an argument (or
>>>>special case) to achieve the illusion of scientific
>>>>validity. This is *all* that's going on with the Hardy-
>>>>Weinberg, socalled, Law. And you hit the nail on the
>>>>head with respect to which word is the "pivot" with
>>>>respect to how this rhetorical trick is manifested in
>>>>Hardy-Weinberg: randomness.
>>>
>>>Wierd. The Hardy-Weinberg law is deterimistic: there is
>>>no randomness in it.
>>
>>
>> The Hardy-Weinberg law is normally stated in a form that
>> refers to a large population where mating is random.
>>
>The theorem was derived for an infinite population.
>
>> E.g. see:
>>
>> http://library.thinkquest.org/19926/java/tour/06.htm?t-
>> qskip1=1
>>
>> Alas, this page expresses the law in terms of an infinite
>> population :-(
>>
>> A disasterous error - IMO - since talking about gene
>> frequencies in an infinite population is a sign of
>> mathematical ignorance.
>>
>No, it's a simplifying assumption. I think to accuse Hardy
>in particular of mathematical ignorance deserves, well, a
>non-mathematician's apology.
>
>> Popularisers should make explicit the behaviour is what
>> happens as the population size tends towards infinity -
>> and not attempt to pass it off as an effect in an
>> infinite population.
>
>But it is - in finite populations, you get an excess of
>homozygotes, as any student of population genetics
>should know.

This above statement doesn't sound right to me? It is
known that if the matings are other than random there
will be an excess of homozygotes (over a Hardy-Weinberg
equilibrium prediction with random matings) but this is
even usually best to see when the populations are large.
Small populations (with random matings) are expected to
diverge from a precise Hardy-Weinberg equilibrium simply
from the sampling effects of the small size but I don't
recall any bias to this divergence (more or fewer
homozygotes than a Hardy-Weinberg prediction). If the
sampling (mating) is truly random, I don't see how you
could predict a direction (excess of homozygotes)?
William L Hunt

>
>Bob
>
>--
>Bob O'Hara
>
>Dept. of Mathematics and Statistics
>P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
> Helsinki Finland Telephone: +358-9-191 23743 Mobile:
> +358 50 599 0540 Fax: +358-9-191 22 779 WWW:
> http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
> Results - EEB: http://www.jnr-eeb.org

Perplexed in Peoria wrote:
> "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in
> message news:cb2fdq$21bh$1@darwin.ediacara.org...
>
>>Tim Tyler wrote:
>>
>>>Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote or
>>>quoted:
>>
>>The theorem was derived for an infinite population.
>>
>>
>>>E.g. see:
>>>
>>> http://library.thinkquest.org/19926/java/tour/06.htm?t-
>>> qskip1=1
>>>
>>>Alas, this page expresses the law in terms of an infinite
>>>population :-(
>>>
>>>A disasterous error - IMO - since talking about gene
>>>frequencies in an infinite population is a sign of
>>>mathematical ignorance.
>>>
>>
>>No, it's a simplifying assumption. I think to accuse Hardy
>>in particular of mathematical ignorance deserves, well, a
>>non-mathematician's apology.
>
>
> Huzzah!
>
>
>>>Popularisers should make explicit the behaviour is what
>>>happens as the population size tends towards infinity -
>>>and not attempt to pass it off as an effect in an
>>>infinite population.
>>
>>But it is - in finite populations, you get an excess of
>>homozygotes, as any student of population genetics
>>should know.
>
>
> I am a student of population genetics who is foolishly
> attempting to proceed without purchasing a good textbook.
> Your claim surprised me, so I tried to check it. I used a
> population of 4 consisting of two heterozygotes and both
> kinds of homozygotes. Hermaphrodites. I performed all
> crosses except the selfings with four offspring per cross.
> (I am assuming that the finite population effect arises
> from the prohibited selfings.) The problem is that I ended
> up with an excess of heterozygotes. What am I doing wrong?
>
You're not using a good textbook. :-)

You need to include the selfings, otherwise you skew the
genotype freqencies.

Bob

Tim Tyler wrote:
> Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote
> or quoted:
>
>>Tim Tyler wrote:
>>
>>>Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote or
>>>quoted:
>>>
>>>>Jim McGinn wrote:
>>>>
>>>>>jason@kalavinka.freeserve.co.uk (friend) wrote in
>>>>>message news:<car9d7$2lsg$1@darwin.ediacara.org>...
>
>
>>>>>>Pardon my ignorance, but I have only just discovered
>>>>>>this law.
>>>
>>>[...]
>>>
>>>
>>>>>Many of the more popular myths of the current paradigm
>>>>>of evolutionary biology pivot off a kind of rhetorical
>>>>>trick. Specifically the trick involves employing a word
>>>>>that has more than one meaning in an argument (or
>>>>>special case) to achieve the illusion of scientific
>>>>>validity. This is *all* that's going on with the Hardy-
>>>>>Weinberg, socalled, Law. And you hit the nail on the
>>>>>head with respect to which word is the "pivot" with
>>>>>respect to how this rhetorical trick is manifested in
>>>>>Hardy-Weinberg: randomness.
>>>>
>>>>Wierd. The Hardy-Weinberg law is deterimistic: there is
>>>>no randomness in it.
>>>
>>>The Hardy-Weinberg law is normally stated in a form that
>>>refers to a large population where mating is random.
>>
>>The theorem was derived for an infinite population.
>>
>>
>>>E.g. see:
>>>
>>> http://library.thinkquest.org/19926/java/tour/06.htm?t-
>>> qskip1=1
>>>
>>>Alas, this page expresses the law in terms of an infinite
>>>population :-(
>>>
>>>A disasterous error - IMO - since talking about gene
>>>frequencies in an infinite population is a sign of
>>>mathematical ignorance.
>>
>>No, it's a simplifying assumption. I think to accuse Hardy
>>in particular of mathematical ignorance deserves, well, a
>>non-mathematician's apology.
>
>
> I never said the problem was at Hardy's end.
>
In fairness, Hardy does suppose "that the numbers are fairly
large". But, as we know know, in a finite population, there
will be an excess of homozygotes (because of inbreeding), so
H-W isn't correct (but a reasonable approximation if the
population is large).

>
>>>Popularisers should make explicit the behaviour is what
>>>happens as the population size tends towards infinity -
>>>and not attempt to pass it off as an effect in an
>>>infinite population.
>>
>>But it is - in finite populations, you get an excess of
>>homozygotes, as any student of population genetics
>>should know.
>
>
> Any mention of gene frequencies in an infinite population
> is nonsense - as I stated originally.
>
> You can't talk about a fraction of an infinite population
> having a trait. You would get different results for that
> fraction depending on how you enumerated through the
> population.
>
I don't understand what you mean, but by that argument, you
can't even define a fraction or a probability.

> It's like claiming that half the integers are even.
>
Err, they are. There are just rather a lot of them.

> Such statements are total mathematical gibberish.
>
> What *can* be said is that the fraction of the set
> of integers from to N that are even tends to 0.5 -
> as N => oo.
>
So what happens when N is infinity?

> No serious mathematician can talk about fractions of
> infinite sets and expect to be taken seriously.

But they do. It's how probability is defined as a concept. I
have a colleague who even wrote mathematical papers about
fractions of uncountable sets.

Infinity is a difficult concept (I know - there are lots of
it I don't understand), so I think one should be cautious
about making any pronouncements on it unless one is sure
about what mathematics does and does not say on the subject.

Bob

--
Bob O'Hara Department of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: www.jnr-eeb.org

"Perplexed in Peoria" <jimmenegay@sbcglobal.net> wrote in message news:<cb5h28$2un7$1@darwin.ediacara.org>...
> "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote in
> message news:cb2fdq$21bh$1@darwin.ediacara.org...
> > But it is - in finite populations, you get an excess of
> > homozygotes, as any student of population genetics
> > should know.
>
> Dooohh! Ignore my other post. Reality gives an excess of
> heterozygotes. The Hardy-Weinberg "law" gives an excess of
> homozygotes.
>

To clarify, are we assuming no selfing here? If not, I'm
struggling to see why you get the result that you do.

William L Hunt wrote:
> On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
> <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote:
>
>
>>Tim Tyler wrote:
>>
>>>Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote or
>>>quoted:
>>>
>>>
>>>>Jim McGinn wrote:
>>>>
>>>>
>>>>>jason@kalavinka.freeserve.co.uk (friend) wrote in
>>>>>message news:<car9d7$2lsg$1@darwin.ediacara.org>...
>>>>
>>>>>>Pardon my ignorance, but I have only just discovered
>>>>>>this law.
>>>>>
>>>[...]
>>>
>>>
>>>
>>>>>Many of the more popular myths of the current paradigm
>>>>>of evolutionary biology pivot off a kind of rhetorical
>>>>>trick. Specifically the trick involves employing a word
>>>>>that has more than one meaning in an argument (or
>>>>>special case) to achieve the illusion of scientific
>>>>>validity. This is *all* that's going on with the Hardy-
>>>>>Weinberg, socalled, Law. And you hit the nail on the
>>>>>head with respect to which word is the "pivot" with
>>>>>respect to how this rhetorical trick is manifested in
>>>>>Hardy-Weinberg: randomness.
>>>>
>>>>Wierd. The Hardy-Weinberg law is deterimistic: there is
>>>>no randomness in it.
>>>
>>>
>>>The Hardy-Weinberg law is normally stated in a form that
>>>refers to a large population where mating is random.
>>>
>>
>>The theorem was derived for an infinite population.
>>
>>
>>>E.g. see:
>>>
>>> http://library.thinkquest.org/19926/java/tour/06.htm?t-
>>> qskip1=1
>>>
>>>Alas, this page expresses the law in terms of an infinite
>>>population :-(
>>>
>>>A disasterous error - IMO - since talking about gene
>>>frequencies in an infinite population is a sign of
>>>mathematical ignorance.
>>>
>>
>>No, it's a simplifying assumption. I think to accuse Hardy
>>in particular of mathematical ignorance deserves, well, a
>>non-mathematician's apology.
>>
>>
>>>Popularisers should make explicit the behaviour is what
>>>happens as the population size tends towards infinity -
>>>and not attempt to pass it off as an effect in an
>>>infinite population.
>>
>>But it is - in finite populations, you get an excess of
>>homozygotes, as any student of population genetics
>>should know.
>
>
> This above statement doesn't sound right to me? It is
> known that if the matings are other than random there
> will be an excess of homozygotes (over a Hardy-Weinberg
> equilibrium prediction with random matings) but this is
> even usually best to see when the populations are large.
> Small populations (with random matings) are expected to
> diverge from a precise Hardy-Weinberg equilibrium simply
> from the sampling effects of the small size but I don't
> recall any bias to this divergence (more or fewer
> homozygotes than a Hardy-Weinberg prediction). If the
> sampling (mating) is truly random, I don't see how you
> could predict a direction (excess of homozygotes)?

I don't know which textbooks you have to hand, I have
Futuyma's "Evolutionary Biology" (2nd ed. from 1986), and in
Chapter 5 ("Population Structure and Genetic Drift") he has
a section called "Population Size, Inbreeding, and Genetic
Drift" where he shows that any finite population will become
inbred, which means a reduction in heterozygosity. I'm sure
the same thing is in Hartl & Clarke. Look out for equations
like H_t = H_0 (1-1/2N)^t.

In essence, any finite population will become inbred over
time (at least to some extent), and this increases
homozygosity.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org

Hi Bill,

in article cb7rte$mmb$1@darwin.ediacara.org, William L Hunt
at wlhunt@earthlink.net wrote on 6/21/04 4:44 PM:

>>> Popularisers should make explicit the behaviour is what
>>> happens as the population size tends towards infinity -
>>> and not attempt to pass it off as an effect in an
>>> infinite population.
>>
>> But it is - in finite populations, you get an excess of
>> homozygotes, as any student of population genetics
>> should know.
>
> This above statement doesn't sound right to me? It is
> known that if the matings are other than random there
> will be an excess of homozygotes (over a Hardy-Weinberg
> equilibrium prediction with random matings) but this is
> even usually best to see when the populations are large.

Non-random mating does not necessarily produce an excess of
homozygotes relative to HW expectations. Non-random mating
can produce any pattern you can imagine. Negative
assortative mating (e.g., rare male mating advantage), for
example generates an excess of heterozygotes. A scheme of
non-random mating can even be engineered to generate HW
genotype proportions while reducing variance in the
distribution of outcomes relative to random mating.

Cheers,

Guy

"Name And Address Supplied" <name_and_address_supplied@hotmail.com> wrote in
message news:cb7rtc$mj2$1@darwin.ediacara.org...
> "Perplexed in Peoria" <jimmenegay@sbcglobal.net> wrote
> in message
news:<cb5h28$2un7$1@darwin.ediacara.org>...
> > "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote
> > in message news:cb2fdq$21bh$1@darwin.ediacara.org...
> > > But it is - in finite populations, you get an excess
> > > of homozygotes,
as
> > > any student of population genetics should know.
> >
> > Dooohh! Ignore my other post. Reality gives an excess of
heterozygotes.
> > The Hardy-Weinberg "law" gives an excess of homozygotes.
> >
>
> To clarify, are we assuming no selfing here? If not, I'm
> struggling to see why you get the result that you do.

Whoops, I was confused as to whether I was responding to NAS
or BOH. I'll watch for BOH's answer as to why finite
populations affect HW in his response to Bill Hunt.

"Name And Address Supplied" <name_and_address_supplied@hotmail.com> wrote in
message news:cb7rtc$mj2$1@darwin.ediacara.org...
> "Perplexed in Peoria" <jimmenegay@sbcglobal.net> wrote
> in message
news:<cb5h28$2un7$1@darwin.ediacara.org>...
> > "Anon." <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote
> > in message news:cb2fdq$21bh$1@darwin.ediacara.org...
> > > But it is - in finite populations, you get an excess
> > > of homozygotes,
as
> > > any student of population genetics should know.
> >
> > Dooohh! Ignore my other post. Reality gives an excess of
heterozygotes.
> > The Hardy-Weinberg "law" gives an excess of homozygotes.
> >
>
> To clarify, are we assuming no selfing here? If not, I'm
> struggling to see why you get the result that you do.
>

I am assuming no selfings. Thus, in a finite population, I
get results at variance with Hardy-Weinberg. But, if I
include the selfings, I get results identical with HW.

It might help if you would give a short explanation of
why you wrote: "in finite populations, you get an excess
of homozygotes, as any student of population genetics
should know."

Perhaps I should know, but I don't.

Bob O'Hara <bob.ohara@nospam.helsinki.fi> wrote or quoted:
> Tim Tyler wrote:
> > Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote or
> > quoted:
> >>Tim Tyler wrote:
> >>>Anon. <bob.ohara@sod.off.spammers.helsinki.fi> wrote or
> >>>quoted:

> >>>>Wierd. The Hardy-Weinberg law is deterimistic: there
> >>>>is no randomness in it.
> >>>
> >>>The Hardy-Weinberg law is normally stated in a form
> >>>that refers to a large population where mating is
> >>>random.
> >>
> >>The theorem was derived for an infinite population.
> >>
> >>>E.g. see:
> >>>
> >>> http://library.thinkquest.org/19926/java/tour/06.htm-
> >>> ?tqskip1=1
> >>>
> >>>Alas, this page expresses the law in terms of an
> >>>infinite population :-(
> >>>
> >>>A disasterous error - IMO - since talking about gene
> >>>frequencies in an infinite population is a sign of
> >>>mathematical ignorance.
> >>
> >>No, it's a simplifying assumption. I think to accuse
> >>Hardy in particular of mathematical ignorance deserves,
> >>well, a non-mathematician's apology.
> >
> > I never said the problem was at Hardy's end.
>
> In fairness, Hardy does suppose "that the numbers are
> fairly large". But, as we know know, in a finite
> population, there will be an excess of homozygotes
> (because of inbreeding), so H-W isn't correct (but a
> reasonable approximation if the population is large).

I suggested explaing using a limit - not a mere finite
population.

> >>>Popularisers should make explicit the behaviour is what
> >>>happens as the population size tends towards infinity -
> >>>and not attempt to pass it off as an effect in an
> >>>infinite population.
> >>
> >>But it is - in finite populations, you get an excess of
> >>homozygotes, as any student of population genetics
> >>should know.
> >
> > Any mention of gene frequencies in an infinite
> > population is nonsense - as I stated originally.
> >
> > You can't talk about a fraction of an infinite
> > population having a trait. You would get different
> > results for that fraction depending on how you
> > enumerated through the population.
>
> I don't understand what you mean, but by that argument,
> you can't even define a fraction or a probability.

Fractions have nothing to do with infinite sets.

> > It's like claiming that half the integers are even.
>
> Err, they are. There are just rather a lot of them.

No, there aren't.

There are an infinite number of even numbers.

There are an infinite number of odd numbers.

Divide infinity by infinity and the result is indeterminate.

> > Such statements are total mathematical gibberish.
> >
> > What *can* be said is that the fraction of the set of
> > integers from to N that are even tends to 0.5 - as N
> > => oo.
>
> So what happens when N is infinity?

The fraction is undefined.

> > No serious mathematician can talk about fractions of
> > infinite sets and expect to be taken seriously.
>
> But they do.

No - not unless the fractions are "zero" or "one".

> It's how probability is defined as a concept.

Probability is defined as a mathematical limit, as N
approaches infinity.

That uses a limit as a finite set increases in size - not a
fraction of an infinite set.

E.g. see:

http://www.wordiq.com/definition/Probability

> I have a colleague who even wrote mathematical papers
> about fractions of uncountable sets.

If you can show me, I should be able to tell you if they
contain the fallacy under discussion.

Probably he doesn't do that at all - and instead uses a
limit.

> Infinity is a difficult concept (I know - there are lots
> of it I don't understand), so I think one should be
> cautious about making any pronouncements on it unless one
> is sure about what mathematics does and does not say on
> the subject.

How is that relevant?

Are you suggesting I don't know what I am talking about?

That is not the case.
--
__________
|im |yler http://timtyler.org/ tim@tt1lock.org Remove
lock to reply.

Hi Bob,

I snipped the following out of a longer post.

in article cb7rta$mfq$1@darwin.ediacara.org, Bob O'Hara at
bob.ohara@NOSPAM.helsinki.fi wrote on 6/21/04 4:44 PM:

> But, as we know know, in a finite population, there will
> be an excess of homozygotes (because of inbreeding), so
> H-W isn't correct (but a reasonable approximation if the
> population is large).

I am not sure how extensive you intended this claim to be,
but I think it is generally false. The finiteness of
populations does not cause an excess of homozygosity
relative to HWC expectations. Similarly, there is no reason
to expect the excess of homozygosity to grow with decreasing
population size, nor is there a reason to expect the degree
of inbreeding to increase relative to that which would occur
in random mating as population size decreases. The HWC model
merely predicts the way in which existing allelic diversity
is translated into genotypic diversity. It makes no
assertions about the extent of allelic diversity you are
likely to find in a population, which is strongly influenced
by population size.

Cheers,

Guy

> In essence, any finite population will become inbred over
> time (at least to some extent), and this increases
> homozygosity.

Boy, are you going to feel foolish after you get a good
night's sleep and review what you have written :-)

On Tue, 22 Jun 2004 20:16:27 +0000 (UTC), "Anon."
<bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote:

>William L Hunt wrote:
>> On Sat, 19 Jun 2004 22:40:58 +0000 (UTC), "Anon."
>> <bob.ohara@SOD.OFF.Spammers.helsinki.fi> wrote:

>>>>Popularisers should make explicit the behaviour is what
>>>>happens as the population size tends towards infinity -
>>>>and not attempt to pass it off as an effect in an
>>>>infinite population.

>>>:-BOH
>>>But it is - in finite populations, you get an excess of
>>>homozygotes, as any student of population genetics
>>>should know.
>>
>>:-WLH
>> This above statement doesn't sound right to me? It is
>> known that if the matings are other than random there
>> will be an excess of homozygotes (over a Hardy-Weinberg
>> equilibrium prediction with random matings) but this is
>> even usually best to see when the populations are
>> large. Small populations (with random matings) are
>> expected to diverge from a precise Hardy-Weinberg
>> equilibrium simply from the sampling effects of the
>> small size but I don't recall any bias to this
>> divergence (more or fewer homozygotes than a Hardy-
>> Weinberg prediction). If the sampling (mating) is truly
>> random, I don't see how you could predict a direction
>> (excess of homozygotes)?
>
>:-BOH
>I don't know which textbooks you have to hand, I have
>Futuyma's "Evolutionary Biology" (2nd ed. from 1986), and
>in Chapter 5 ("Population Structure and Genetic Drift") he
>has a section called "Population Size, Inbreeding, and
>Genetic Drift" where he shows that any finite population
>will become inbred, which means a reduction in
>heterozygosity. I'm sure the same thing is in Hartl &
>Clarke. Look out for equations like H_t = H_0 (1-1/2N)^t.
>
>In essence, any finite population will become inbred over
>time (at least to some extent), and this increases
>homozygosity.
>
>Bob
>

The discussion was of a Hardy-Weinberg equilibrium and your
original statement "in finite populations, you get an
excess of homozygotes", I took to refer to an excess over
that predicted by HW. Apparently you were referring to
something else that has nothing to do with Hardy-Weinberg
equilibrium. This threw me off and I suspect it may have
for some others also. Guy Hoelzer also responds to this in
an above thread. Yes, if the population is small, many more
loci will reach fixation (homozygote) to one allele or the
other and there is less genetic diversity. This has nothing
to do with Hardy-Weinberg. Hardy-Weinbery only speaks to
loci where there are still two alleles present in the
population at some frequency and it predicts what the
distribution will be. It originally sounded like you were
saying was that if there is a frequency of p=.5 for allele
'A' and q=.5 for allele 'a', that in a large population you
would expect the distribution to be a Hardy-Weinberg
AA=.25, Aa = .50, aa=.25 but for some small population
size, "you" might expect it show an "excess of
homozygotes", such as AA=.30, Aa=.40, aa=.30. I now am not
sure what you meant? Possibly what confused me is that in
large widespread populations one doesn't really expect to
see a precise Hardy-Weinberg equilibrium. I expect there
will always be some "isolation-by-distance" in the mating
choices that will actually cause an excess in the
homozygotes above that predicted by HW (not a random mating
condition). This is detectable in the human population
blood group alleles but the measurement is only slightly
more homozygotes than an HW prediction. William L Hunt

>--
>Bob O'Hara
>
>Dept. of Mathematics and Statistics
>P.O. Box 4 (Yliopistonkatu 5) FIN-00014 University of
> Helsinki Finland Telephone: +358-9-191 23743 Mobile:
> +358 50 599 0540 Fax: +358-9-191 22 779 WWW:
> http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
> Results - EEB: http://www.jnr-eeb.org

Tim Tyler wrote:
> Bob O'Hara <bob.ohara@nospam.helsinki.fi> wrote or quoted:
>
>>Tim Tyler wrote:
>>
>>>>>Popularisers should make explicit the behaviour is what
>>>>>happens as the population size tends towards infinity -
>>>>>and not attempt to pass it off as an effect in an
>>>>>infinite population.
>>>>
>>>>But it is - in finite populations, you get an excess of
>>>>homozygotes, as any student of population genetics
>>>>should know.
>>>
>>>Any mention of gene frequencies in an infinite population
>>>is nonsense - as I stated originally.
>>>
>>>You can't talk about a fraction of an infinite population
>>>having a trait. You would get different results for that
>>>fraction depending on how you enumerated through the
>>>population.
>>
>>I don't understand what you mean, but by that argument,
>>you can't even define a fraction or a probability.
>
>
> Fractions have nothing to do with infinite sets.
>
But there are an infinite number of fractions, so they have
at least that to do with infinite sets.

>
>>>It's like claiming that half the integers are even.
>>
>>Err, they are. There are just rather a lot of them.
>
>
> No, there aren't.
>
> There are an infinite number of even numbers.
>
> There are an infinite number of odd numbers.
>
> Divide infinity by infinity and the result is
> indeterminate.
>
If there are an equal number of even and odd numbers, then
half of the numbers must be even.

This must be true because for every even number, I can add
1 and get an odd number. Conversely for every odd number I
can add 1 and get an even number. Hence, by the operation
of adding 1, I can produce an even number for every odd
number and vice versa. Ergo, half of all numbers are even,
and half are odd.

I find this sort of proof preferable to throwing my hands up
in defeat.

<snip>
>>>No serious mathematician can talk about fractions of
>>>infinite sets and expect to be taken seriously.
>>
>>But they do.
>
>
> No - not unless the fractions are "zero" or "one".
>
Rubbish, unless you're denying the existence of fractions.
Fractions are fractions of an infinite set, because there is
an infinite number of numbers between 0 and 1 (proof: take
the reciprocal of every positive integer).

>
>>It's how probability is defined as a concept.
>
>
> Probability is defined as a mathematical limit, as N
> approaches infinity.
>
> That uses a limit as a finite set increases in size - not
> a fraction of an infinite set.
>
> E.g. see:
>
> http://www.wordiq.com/definition/Probability
>
This doesn't show that probability is defined as a
limit - the nearest you get is in the section
"Probability in mathematics", where they use "one
approach" to give an interpretation - essentially, the
frequentist approach. Note that when they discuss
Kolmonogorov's definition of probability as a measure,
they make no mention of any limits.

>
>>I have a colleague who even wrote mathematical papers
>>about fractions of uncountable sets.
>
>
> If you can show me, I should be able to tell you if they
> contain the fallacy under discussion.
>
> Probably he doesn't do that at all - and instead uses
> a limit.
>
This was (I think - my copy is at home) the paper:

E. Arjas & E. Nummelin & R.L. Tweedie: Semi-Markov processes
on a general state space -theory and quasi-stationarity.
J. Aust. Math. Soc. (Series A) 30 (1980): 187 - 200.

>
>>Infinity is a difficult concept (I know - there are lots
>>of it I don't understand), so I think one should be
>>cautious about making any pronouncements on it unless one
>>is sure about what mathematics does and does not say on
>>the subject.
>
>
> How is that relevant?
>
You're trying to argue about the use of infinity. I'm
pointing out that one should be careful when doing this.
This seems relevant.

> Are you suggesting I don't know what I am talking about?
>
> That is not the case.

Your evidence for this is?

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
E.A. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org

Guy Hoelzer wrote:
> Hi Bob,
>
> I snipped the following out of a longer post.
>
> in article cb7rta$mfq$1@darwin.ediacara.org, Bob O'Hara at
> bob.ohara@NOSPAM.helsinki.fi wrote on 6/21/04 4:44 PM:
>
>
>>But, as we know know, in a finite population, there will
>>be an excess of homozygotes (because of inbreeding), so
>>H-W isn't correct (but a reasonable approximation if the
>>population is large).
>
>
> I am not sure how extensive you intended this claim to be,
> but I think it is generally false. The finiteness of
> populations does not cause an excess of homozygosity
> relative to HWC expectations. Similarly, there is no
> reason to expect the excess of homozygosity to grow with
> decreasing population size, nor is there a reason to
> expect the degree of inbreeding to increase relative to
> that which would occur in random mating as population size
> decreases.

OK, at the very least I should have specified random mating
(which is specified under H-W as well). Under these
conditions, a smaller population size means a greater chance
of mating with a close relative
(i.e. inbreeding), which leads to the increase in inbreeding
over time (at least in the simple models).

On the point that the expected excess of homozygosity
growing with decreasing population size, I agree. What will
happen, under random mating (in a closed population etc.
etc.), is that the rate of increase in homozygosity over
time will increase.

How all this relates to the real world is another matter -
this all started off with a discussion about the H-W model,
rather than about reality.

The HWC model merely predicts the way in which
existing allelic
> diversity is translated into genotypic diversity. It makes
> no assertions about the extent of allelic diversity you
> are likely to find in a population, which is strongly
> influenced by population size.
>
Yes, we're in agreement here - there's too many other things
that can affect the population structure (see the last 80
years or so of population genetics for references!).

And all this is _before_ we start discussing selection!

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
i.f. Box 4 (Yliopistonkatu 5) FIN-00014 University of
Helsinki Finland Telephone: +358-9-191 23743 Mobile:
+358 50 599 0540 Fax: +358-9-191 22 779 WWW:
http://www.RNI.Helsinki.FI/~boh/ Journal of Negative
Results - EEB: http://www.jnr-eeb.org