Editing F-statistics (section)

== Definitions and equations ==
The measures F<sub>IS</sub>, [[Fixation index|F<sub>ST</sub>]], and F<sub>IT</sub> 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 [[Coefficient of inbreeding|inbreeding coefficient]].  In a simple two-allele system with inbreeding, the genotypic frequencies are:

:<math> p^2(1-F) + pF\text{ for }\mathbf{AA};\  2pq(1-F)\text{ for }\mathbf{Aa};\text{ and }q^2(1-F) + qF\text{ for }\mathbf{aa}. </math>

The value for <math>F</math> is found by solving the equation for <math>F</math> using heterozygotes in the above inbred population.  This becomes one minus the [[observation|observed]] frequency of heterozygotes in a population divided by the [[expected value|expected]] frequency of heterozygotes at [[Hardy–Weinberg principle|Hardy–Weinberg equilibrium]]:

:<math> F = 1- \frac{\operatorname{O}(f(\mathbf{Aa}))} {\operatorname{E}(f(\mathbf{Aa}))} = 1- \frac{\operatorname{ObservedFrequency}(\mathbf{Aa})} {\operatorname{ExpectedFrequency}(\mathbf{Aa})},  \!</math>

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

:<math> \operatorname{E}(f(\mathbf{Aa})) = 2pq, \!</math>

where <math>p</math> and <math>q</math> are the [[allele frequencies]] of <math>\mathbf{A}</math> and <math>\mathbf{a}</math>, respectively.  It is also the probability that at any [[gene locus|locus]], two alleles from a random individual of the population are [[identity by descent|identical by descent]].

For example, consider the data from [[E.B. Ford]] (1971) on a single population of the [[scarlet tiger moth]]:

{| border=1 cellpadding=5 style="border-collapse:collapse;" align=center
|+ '''Table 1: '''
|-
!Genotype
|White-spotted (<math>\mathbf{AA}</math>)
|Intermediate (<math>\mathbf{Aa}</math>)
|Little spotting (<math>\mathbf{aa}</math>)
!Total
|-
!Number
|1469
|138
|5
|1612
|}

From this, the [[allele frequencies]] can be calculated, and the expectation of <math>f\left(\mathbf{Aa}\right)</math> derived :

: <math>p  = {2 \times \mathrm{obs}(AA) + \mathrm{obs}(Aa) \over 2 \times (\mathrm{obs}(AA) + \mathrm{obs}(Aa) + \mathrm{obs}(aa))} = 0.954</math>

: <math>q  = 1 - p = 0.046\,</math>

: <math>F = 1- \frac{ \mathrm{obs}(Aa) / n } { 2pq } = 1- {138 / 1612 \over 2(0.954)(0.046)} = 0.023</math>

The different F-statistics look at different levels of population structure.  '''F<sub>IT</sub>''' is the inbreeding coefficient of an individual ('''I''') relative to the total ('''T''') population, as above; '''F<sub>IS</sub>''' is the inbreeding coefficient of an individual ('''I''') relative to the subpopulation ('''S'''), using the above for subpopulations and averaging them; and '''F<sub>ST</sub>''' is the effect of subpopulations ('''S''') compared to the total population ('''T'''), and is calculated by solving the equation:

:<math>(1-F_{IS})(1-F_{ST}) = 1-F_{IT}, \, </math>

as shown in the next section.