Fu's $F_s$

Fu [3] is based on principles similar to those already described, but instead of comparing estimates of $\theta$ derived in different ways it uses $\theta_\pi$ as an estimate of $\theta$ and generates samples from a neutral coalescent model with that parameter. Using those samples we estimate can calculate the probability that the number of haplotypes is greater than the number we observed, $k_0$, in a sample of size $n$. Fu calls this probability $S'$:

$$\begin{eqnarray*} S' &=& \mbox{P}(k \ge k_0 \vert \theta=\theta_\pi) \\ &=& \... ...k_0} \frac{\vert S_k\vert\theta_\pi^k}{S_n(\theta_\pi)} \quad , \end{eqnarray*}$$

where $S_n(\theta_\pi) = \theta_\pi(\theta_\pi-1)\cdots(\theta_\pi-n+1)$ and $S_k$ is the coefficient of $\theta_\pi^k$ in $S_n(\theta_\pi)$. To determine whether the observed value of $S'$ is unusually large or unusually small, we compare the logit of $S'$

$$F_s = \mbox{log}\left(\frac{S'}{1-S'}\right)$$

with values generated from random samples generated by a coalescent process with $\theta = \theta_\pi$.

A negative value of $F_s$, meaning $S' < S$, is evidence for an excess number of alleles, as would be expected from a recent population expansion or from genetic hitchhiking. A positive value of $F_s$, meaning $S' > S$, is evidence for an deficiency of alleles, as would be expect from a recent population bottleneck or from overdominant selection. Fu's simulations suggest that $F_s$ is a more sensitive indicator of population expansion and genetic hitchhiking than Tajima's $D$. They also suggest that the conventional $P$-value of 0.05 corresponds to a $P$-value from the coalescent simulation of 0.02.