F-statistics

In population genetics, F-statistics (also known as fixation indices) describe the level of heterozygosity in a population; more specifically the degree of (usually) a reduction in homozygosity when compared to Hardy-Weinberg expectation. Such changes can be caused by the Wahlund effect, inbreeding, natural selection or any combination of these.

The concept of F-statistics was developed during the 1920s by the American geneticist Sewall Wright, who was interested in inbreeding in cattle. However, because complete dominance causes the phenotypes of homozygote dominants and heterozygotes to be the same, it was not until the advent of molecular genetics from the 1960s onwards that heterozygosity in populations could be measured.

F-statistics measure the correlation between genes drawn at different levels of a (hierarchically) subdivided population. This correlation is influenced by several evolutionary forces, such as mutation and migration, but it was originally designed to measure how far populations had gone in the process of fixation owing to genetic drift.

Definitions and equations
The measures FIS, Fst, and FIT are related to the amounts of heterozygosity at various levels of population structure. Together, they are called F-statistics, and are derived from F, the inbreeding coefficient. In a simple two-allele system with inbreeding, the genotypic frequencies are:


 * $$ p^2 + Fpq$$ for AA; $$2pq(1-F)$$ for Aa; and $$q^2 + Fpq$$ for aa.

The value for F is found by solving the equation for F using heterozygotes in the above inbred population. This becomes one minus the observed number of heterozygotes in a population divided by its expected number of heterozygotes at Hardy–Weinberg equilibrium:


 * $$ F = 1- \frac{\operatorname{O}(f(\mathbf{Aa}))} {\operatorname{E}(f(\mathbf{Aa}))} = 1- \frac{\operatorname{ObservedNumber}(\mathbf{Aa})} {n \operatorname{E}(f(\mathbf{Aa}))}, \!$$

where the expected value at Hardy–Weinberg equilibrium is given by


 * $$ \operatorname{E}(f(\mathbf{Aa})) = 2\, p\, q\, \!$$

where p and q are the allele frequencies of A and a, respectively. It is also the probability that at any locus, two alleles from the population are identical by descent.

For example, consider the data from E.B. Ford (1971) on the scarlet tiger moth:

From this, the allele frequencies can be calculated, and the expectation of f(AA) derived:

$$p = {2 \times obs(AA) + obs(Aa) \over 2 \times (obs(AA) + obs (Aa) + obs(aa))} = 0.954$$

$$q = 1 - p = 0.046$$

$$F = 1- \frac{ obs(Aa) } { n*2pq } = 1- {138 \over 1612*2(0.954)(0.046)} = 0.023$$

The different F-statistics look at different levels of population structure. FIT is the inbreeding coefficient of an individual (I) relative to the total (T) population, as above; FIS is the inbreeding coefficient of an individual (I) relative to the subpopulation (S), using the above for subpopulations and averaging them; and FST is the effect of subpopulations (S) compared to the total population (T), and is calculated by solving the equation:
 * $$(1-F_{IS})(1-F_{ST}) = (1-F_{IT}),$$

as shown in the next section.

Partition due to population structure


Consider a population that has a population structure of two levels; one from the individual (I) to the subpopulation (S) and one from the subpopulation to the total (T). Then the total F, known here as FIT, can be partitioned into FIS (or f) and FST (or θ):


 * $$ 1 - F_{IT} = (1 - F_{IS})\,(1 - F_{ST}). \!$$

This may be further partitioned for population substructure, and it expands according to the rules of binomial expansion, so that for I partitions:


 * $$ 1 - F = \prod_{i=0}^{i=I} (1 - F_{i,i+1}) \!$$

Fst
A reformulation of the definition of F would be the ratio of the average number of differences between pairs of chromosomes sampled within diploid individuals with the average number obtained when sampling chromosomes randomly from the population (excluding the grouping per individual). One can modify this definition and consider a grouping per sub-population instead of per individual. Population geneticists have used that idea to measure the degree of structure in a population.

Unfortunately, there is a large number of definitions for Fst, causing some confusion in the scientific literature. A common definition is the following:

$$ F_{ST} = \frac{\operatorname{var}(p)}{p\,(1 - p)} \!$$

where the variance of $$ p $$ is computed across sub-populations.

Effective population size
F can be used to define effective population size.