# F-statistics

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*This article is not about F-statistics as that term is understood in statistical inference, especially analysis of variance and linear regression. See**F*-test and*F*-distribution.

In population genetics, ** F-statistics** (also known as fixation indices) describe the statistically expected level of heterozygosity in a population; more specifically the expected degree of (usually) a reduction in heterozygosity when compared to Hardy–Weinberg expectation.

F-statistics can also be thought of as a measure of the correlation between genes drawn at different levels of a (hierarchically) subdivided population. This correlation is influenced by several evolutionary processes, such as mutation, migration, inbreeding, natural selection, or the Wahlund effect, but it was originally designed to measure the amount of allelic fixation owing to genetic drift.

The concept of *F*-statistics was developed during the 1920s by the American geneticist Sewall Wright,^{[1]}^{[2]} 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* can be used to define effective population size.

## Contents

[show]## Definitions and equations Edit

The measures F_{IS}, F_{st}, and F_{IT} 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:

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:

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

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 a random individuum of the population are identical by descent.

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

Genotype | White-spotted (AA)
| Intermediate (Aa)
| Little spotting (aa)
| Total |
---|---|---|---|---|

Number | 1469 | 138 | 5 | 1612 |

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

The different F-statistics look at different levels of population structure. **F _{IT}** is the inbreeding coefficient of an individual (

**I**) relative to the total (

**T**) population, as above;

**F**is the inbreeding coefficient of an individual (

_{IS}**I**) relative to the subpopulation (

**S**), using the above for subpopulations and averaging them; and

**F**is the effect of subpopulations (

_{ST}**S**) compared to the total population (

**T**), and is calculated by solving the equation:

as shown in the next section.

## Partition due to population structure Edit

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 *F*_{IT}, can be partitioned into *F*_{IS} (or *f*) and *F*_{ST} (or *θ*):

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

## Edit

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:

where the variance of **p** is computed across sub-populations and *p*(1 −*p*) is the expected frequency of heterozygotes.

## in human populations Edit

It is well established that the genetic diversity among human populations is low,^{[3]} although the distribution of the genetic diversity was only roughly estimated. Early studies argued that 85-90% of the genetic variation is found within individuals residing in the same populations within continents (intra-continental populations) and only an additional 10-15% is found between populations of different continents (continental populations).^{[4]}^{[5]}^{[6]}^{[7]}^{[8]} Later studies based on hundreds of thousands single-nucleotide polymorphism (SNPs) suggested that the genetic diversity between continental populations is even smaller and accounts for 3 to 7%^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]} Most of these studies have used the Fst statistics ^{[15]} or closely related statistics.^{[16]}^{[17]} In 2012, Elhaik^{[18]} carried a hierarchical Fst analysis and showed that only a paucity (12%) of the total genetic variation is distributed between continental populations and even a lesser genetic variation (1%) is found between intra-continental populations. The global Fst distribution was shown to closely follow an exponential distribution.

## See alsoEdit

## References Edit

- ↑ (1950). Genetical structure of populations.
*Nature***166**(4215): 247–9. - ↑ (1985). Utilization of emergency toxicology screens.
*The American journal of emergency medicine***3**(6): 573–4. - ↑ (2009). Genetics in geographically structured populations: Defining, estimating and interpreting FST.
*Nature Reviews Genetics***10**(9): 639–50. - ↑ Lewontin (1972). The apportionment of human diversity.
*Evolutionary Biology***6**: 381–98. - ↑ (1991). Drift, admixture, and selection in human evolution: A study with DNA polymorphisms.
*Proceedings of the National Academy of Sciences***88**(3): 839–43. - ↑ (1997). An apportionment of human DNA diversity.
*Proceedings of the National Academy of Sciences of the United States of America***94**(9): 4516–9. - ↑ (2000). The Distribution of Human Genetic Diversity: A Comparison of Mitochondrial, Autosomal, and Y-Chromosome Data.
*The American Journal of Human Genetics***66**(3): 979–88. - ↑ (2004). Genetic variation, classification and 'race'.
*Nature Genetics***36**(11s): S28-33. - ↑ (2006). Similarity of the allele frequency and linkage disequilibrium pattern of single nucleotide polymorphisms in drug-related gene loci between Thai and northern East Asian populations: Implications for tagging SNP selection in Thais.
*Journal of Human Genetics***51**(10): 896–904. - ↑ (2008). Population substructure in Finland and Sweden revealed by the use of spatial coordinates and a small number of unlinked autosomal SNPs.
*BMC Genetics***9**: 54. - ↑ (2008). Correlation between Genetic and Geographic Structure in Europe.
*Current Biology***18**(16): 1241–8. - ↑ (2009). Genome-wide Insights into the Patterns and Determinants of Fine-Scale Population Structure in Humans.
*The American Journal of Human Genetics***84**(5): 641. - ↑ (2009). Genetic Structure of Europeans: A View from the North–East.
*PLoS ONE***4**(5): e5472. - ↑ (2009). Reconstructing Indian population history.
*Nature***461**(7263): 489–94. - ↑ (1965). The Interpretation of Population Structure by F-Statistics with Special Regard to Systems of Mating.
*Evolution***19**(3): 395–420. - ↑ (1991). Long‐term goose breeding for egg production and crammed liver weight.
*British Poultry Science***32**(4): 703–9. - ↑ (1992). Analysis of molecular variance inferred from metric distances among DNA haplotypes: Application to human mitochondrial DNA restriction data.
*Genetics***131**(2): 479–91. - ↑ (2012). Empirical Distributions of FST from Large-Scale Human Polymorphism Data.
*PLoS ONE***7**(11): e49837.

## External links Edit

- Shane's Simple Guide to F-Statistics
- Analyzing the genetic structure of populations
- Wahlund effect, Wright's F-statistics
- Worked example of calculating F-statistics from genotypic data
- IAM based F-statistics
- F-statistics for Population Genetics Eco-Tool
- Population Structure (slides)

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