Watterson estimator

In population genetics, the Watterson estimator is a method for describing the genetic diversity in a population. It is estimated by counting the number of polymorphic sites. It is a measure of the "population mutation rate" (the product of the effective population size and the neutral mutation rate) from the observed nucleotide diversity of a population. ${\displaystyle \theta =4N_{e}\mu }$, where ${\displaystyle N_{e}}$ is the effective population size and ${\displaystyle \mu }$ is the per-generation mutation rate of the population of interest (Watterson (1975) ). The assumptions made are that there is a sample of ${\displaystyle n}$ haploid individuals from the population of interest, that there are infinitely many sites capable of varying (so that mutations never overlay or reverse one another), and that ${\displaystyle n\ll N_{e}}$. Because the number of segregating sites counted will increase with the number of sequences looked at, the correction factor ${\displaystyle a_{n}}$ is used.

The estimate of ${\displaystyle \theta }$, often denoted as ${\displaystyle {{\hat {\theta }}_{w}}}$, is

${\displaystyle {{\hat {\theta }}_{w}}={K \over a_{n}},}$

where ${\displaystyle K}$ is the number of segregating sites (an example of a segregating site would be a single-nucleotide polymorphism) in the sample and

${\displaystyle a_{n}=\sum _{i=1}^{n-1}{1 \over i}}$

is the ${\displaystyle (n-1)}$th harmonic number.

This estimate is based on coalescent theory. Watterson's estimator is commonly used for its simplicity. When its assumptions are met, the estimator is unbiased and the variance of the estimator decreases with increasing sample size or recombination rate. However, the estimator can be biased by population structure. For example, ${\displaystyle {{\hat {\theta }}_{w}}}$ is downwardly biased in an exponentially growing population. It can also be biased by violation of the infinite-sites mutational model; if multiple mutations can overwrite one another, Watterson's estimator will be biased downward.

Comparing the value of the Watterson's estimator, to nucleotide diversity is the basis of Tajima's D which allows inference of the evolutionary regime of a given locus.