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Population genetic question

Joe Felsenstein joe at evolution.genetics.washington.edu
Sun Nov 26 11:32:38 EST 1995

In article <48voe7$l56 at sirio.cineca.it>,
Stefano Leonardi <stefano at eagle.bio.unipr.it> wrote:
>In population genetics F(st) can be considered the "among" populations
>component of genetic variation.
>I would like to know if there is any "mathematical" or simply
>"logical" relationship between F(st) parameter and results of
>discriminant analysis performed on allele frequency data.
[description of the same populations studied by Fst and by discriminant
analysis omitted here]
>Did these two analyses reveal the same pattern? Is the second result
>simply a consequence of the first result?

I have cross-posted this to bionet.population-bio, where it would
be more appropriate, except that that group has been abandoned to people who
want (inappropriately) to discuss the politics of overpopulation, not the
biology of natural populations.  There ought to be a post on its real subject
to that group at least once a month or so!

There is a relationship between Fst and discrimination, if the differentiation
between the populations is due to genetic drift.  In that case the
differentiation is expected to be spread equally (but in practice not, owing
to randomness of the drift) among loci, among alleles, and among pairs of
populations.  The formulas for the expected confidence level in the
differentiation involve Fst all right, but also involve the degrees of
freedom and the sample sizes.

Without developing those formulas here, one ought to make the point that
the apparent discrepancy DiStefano saw could be explained without difficulty.
He saw 100% discrimination between all three pairs of populations.  With
any level of Fst, if there is a big enough sample size, you will get nearly
100% discrimination.  So we might assume the sample size was big enough here
to get successful discrimination among populations even in the case with the
smallest of the three Fst's.

There is a literature on this that should be cited.  It was a controversy
(mostly) in American Naturalist in 1978.   Richard Spielmann and
Peter Smouse proposed the use of the amount of statistical discrimintion
between populations as a measure of differentiation, and was answered by
Richard Lewontin and others who pointed out that this was a function of sample
size, and such a measure ought not be.  It is essentially the same question
raised by DiStefano.

Some references are listed here.  I have missed a few but they will refer to
others that aren't in this list:

Smouse, P. E.  and R. S. Spielman. 1977.  How allocation of individuals depends
   on genetic differences among populations. pp. 255-260 in Human Genetics,
   Proc. Fifth International Congress of Human Genet ics, ed. S. Armendares and
   R. Lisker. Excerpta Medica, Amsterdam and Oxford.
Lewontin, R. C.  1978.  Single and multiple-locus measures of genetic distance
   between groups.  Amer. Naturalist  112: 1138-1139.
Mitton, J. B.  1978.  Measurement of differentiation: reply to Lewontin,
   Powell, and Taylor.   Amer. Naturalist  112: 1142-1144.
Powell, J. R.  and C. E. Taylor.  1978.  Are human races "substantially"
   different genetically?  Amer. Naturalist  112: 1139-1142.

Joe Felsenstein         joe at genetics.washington.edu     (IP No.
 Dept. of Genetics, Univ. of Washington, Box 357360, Seattle, WA 98195-7360 USA

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