Abstract
The sample frequency spectrum is an informative and frequently employed approach for summarizing DNA variation data. Under the standard neutral model the expectation of the sample frequency spectrum has been derived by at least two distinct approaches. One relies on using results from diffusion approximations to the Wright-Fisher Model. The other is based on Pólya urn models that correspond to the standard coalescent model. A new proof of the expected frequency spectrum is presented here. It is a proof by induction and does not require diffusion results and does not require the somewhat complex sums and combinatorics of the derivations based on urn models.
Highlights
A useful summary description of DNA sequence variation in samples from a population is the sample frequency spectrum
The sample frequency spectrum for a sample of size n, is a vector, fsiðnÞg; i 1⁄4 1; . . . ; n À 1, where sðinÞ is the number of polymorphic sites at which there are i copies of the mutant allele in the sample
This sample frequency spectrum has been the basis for numerous estimators and test statistics for analyzing population genetics data. (See for example section 6.4 of Charlesworth and Charlesworth [1].) Under the standard infinite-sites neutral model with constant diploid population size, N, it is well known that: Eðsði nÞ Þ
Summary
A useful summary description of DNA sequence variation in samples from a population is the sample frequency spectrum. Under the infinite-sites model of mutation, every mutation is assumed to occur at a site not previously mutated, in which case the number of polymorphisms equals the number of mutations in the genealogy of the sample. Before presenting our new proof of Eq (1), it is useful to review some elementary properties of the standard coalescent for a sample of size n.
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