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Encyclopedia > Ewens's sampling formula

In population genetics, Ewens's sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once in the sample, and a2 alleles represented twice, and so on, is Population genetics is the study of the distribution of and change in allele frequencies under the influence of the four evolutionary forces: natural selection, genetic drift, mutation and migration. ... Gametes (in Greek: γαμέτες) —also known as sex cells, germ cells, or spores—are the specialized cells that come together during fertilization (conception) in organisms that reproduce sexually. ... This stylistic schematic diagram shows a gene in relation to the double helix structure of DNA and to a chromosome (right). ... The word locus (plural loci) is Latin for place. In biology, a locus is the position of a gene (or other significant sequence) on a chromosome. ... The word probability derives from the Latin probare (to prove, or to test). ... An allele is any one of a number of alternative forms of the same gene occupying a given locus (position) on a chromosome. ...

for some positive number θ, whenever a1, ..., an is a sequence of nonnegative integers such that

The phrase "under certain conditions", used above, must of course be made precise. The assumptions are (1) the sample size n is small by comparison to the size of the whole population, and (2) the population is in statistical equilibrium under mutation and genetic drift and the role of selection at the locus in question is negligible, and (3) every mutant allele is novel. This article is about mutation in biology, for other meanings see: mutation (disambiguation). ... Genetic drift is a mechanism of evolution that acts in concert with natural selection to change the characteristics of species over time. ...


This is a probability distribution on the set of all partitions of the integer n. Among probabilists and statisticians it is often called the Ewens distribution. In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...


When θ = 0, the probability is 1 that all n genes are the same. When θ = 1, then the distribution is precisely that of the integer partition induced by a uniformly distributed random permutation. As the probability that no two of the n genes are the same approaches 1.


This family of probability distributions enjoys the property that if after the sample of n is taken, m of the n gametes are chosen without replacement, then the resulting probability distribution on the set of all partitions of the smaller integer m is just what the formula above would give if m were put in place of n.


References

  • Warren Ewens, "The sampling theory of selectively neutral alleles", Theoretical Population Biology, volume 3, pages 87—112, 1972.
  • J.F.C. Kingman, "Random partitions in population genetics", Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, volume 361, number 1704, 1978.

  Results from FactBites:
 
Ewens's sampling formula - Wikipedia, the free encyclopedia (278 words)
In population genetics, Ewens's sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a
The assumptions are (1) the sample size n is small by comparison to the size of the whole population, and (2) the population is in statistical equilibrium under mutation and genetic drift and the role of selection at the locus in question is negligible, and (3) every mutant allele is novel.
This is a probability distribution on the set of all partitions of the integer n.
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