In population genetics, linkage disequilibrium is the non-random association of alleles at two or more loci, not necessarily on the same chromosome. It is not the same as linkage, which describes the association of two or more loci on a chromosome with limited recombination between them. Linkage disequilibrium describes a situation in which some combinations of alleles or genetic markers occur more or less frequently in a population than would be expected from a random formation of haplotypes from alleles based on their frequencies. Non-random associations between polymorphisms at different loci are measured by the degree of linkage disequilibrium (LD).

Linkage disequilibrium is generally caused by genetic linkage and the rate of recombination; rate of mutation; random drift or non-random mating; and population structure. For example, some organisms (such as bacteria) may show linkage disequilibrium because they reproduce asexually and there is no recombination to break down the linkage disequilibrium.

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Formally, if we define pairwise LD, we consider indicator variables on alleles at two loci, say $I_1, I_2$. We define the LD parameter $\delta$ (delta) as:

$\delta := \operatorname{cov}(I_1, I_2) = p_1 p_2 - h_{12} = h_{11}h_{22}-h_{12}h_{21}$

Here $p_1, p_2$ denote the marginal allele frequencies at the two loci and $h_{12}$ denotes the haplotype frequency in the joint distribution of both alleles. Various derivatives of this parameter have been developed. In the genetic literature the wording "two alleles are in LD" usually means to imply $\delta \ne 0$. Contrariwise, linkage equilibrium, denotes the case $\delta = 0$.

If inspecting the two loci A and B with two alleles each—a two-locus, two-allele model—the following table denotes the frequencies of each combination:

 Haplotype Frequency $A_1B_1$ $x_{11}$ $A_1B_2$ $x_{12}$ $A_2B_1$ $x_{21}$ $A_2B_2$ $x_{22}$

Note that these are relative frequencies. One can use the above frequencies to determine the frequency of each of the alleles:

 Allele Frequency $A_1$ $p_{1}=x_{11}+x_{12}$ $A_2$ $p_{2}=x_{21}+x_{22}$ $B_1$ $q_{1}=x_{11}+x_{21}$ $B_2$ $q_{2}=x_{12}+x_{22}$

if the two loci and the alleles are independent from each other, then one can express the observation A1B1 as "A1 must be found and B1 must be found". The table above lists the frequencies for $A_1, p_1$, and $B_1, q_1$, hence the frequency of $A_1B_1$, $x_{11}$, equals according to the rules of elementary statistics $x_{11} = p_{1} * q_{1}$.

A deviation of the observed frequencies from the expected is referred to as the linkage disequilibrium parameter[1], and is commonly denoted by a capital D [2] as defined by:

 $D = x_{11} - p_1q_1$

The following table illustrates the relationship between the haplotype and allele frequencies and D.

 $A_1$ $A_2$ Total $B_1$ $x_{11}=p_1q_1+D$ $x_{21}=p_2q_1-D$ $q_1$ $B_2$ $x_{12}=p_1q_2-D$ $x_{22}=p_2q_2+D$ $q_2$ Total $p_1$ $p_2$ $1$

When extending these formula for diploid cells rather than investigating the gametes/haplotypes directly, the laid out principle prevails, the recombination rate between the two loci $A$ and $B$ must be taken into account, though, which is commonly denoted by the letter $c$.

$D$ is nice to calculate with but has the disadvantage of depending on the frequency of the alleles inspected. This is evident since frequencies are between 0 and 1. There can be no $D$ observed if any locus has an allele frequency 0 or 1 and is maximal when frequencies are at 0.5. Lewontin (1964) suggested normalising D by dividing it with the theoretical maximum for the observed allele frequencies. Thus $D'=\frac{D}{D_\max}$ when $D \ge 0$ When $D < 0$, $D'=\frac{D}{D_\min}$.

$D_\max$ is given by the smaller of $p_1 q_2$ and $p_2 q_1$. $D_\min$ is given by the larger of $-p_1 q_1$ and $-p_2 q_2$

Another measure of LD commonly reported with D' is the correlation coefficient between pairs of loci, denoted as $r^2=\frac{D^2}{p_1p_2q_1q_2}$. This however is not adjusted to the loci having different allele frequencies. If it was, $r$, the square root of $r^2$ if given the sign of $D$ would be equivalent to $D'$ [3]

## Resources

A comparison of different measures of LD is provided by Devlin & Risch [4]

The International HapMap Project enables the study of LD in human populations online. The Ensembl project integrates HapMap data and such from dbSNP in general with other genetic information.

## References

1. Robbins, R.B. (1918). "Some applications of mathematics to breeding problems III". Genetics 3: 375-389.
2. R.C. Lewontin and K. Kojima (1960). "The evolutionary dynamics of complex polymorphisms.". Evolution 14: 458-472.
3. P.W. Hedrick and S. Kumar (2001). "Mutation and linkage disequilibrium in human mtDNA". Eur. J. Hum. Genet. 9: 969-972.
4. Devlin B., Risch N. (1995). "A Comparison of Linkage Disequilibrium Measures for Fine-Scale Mapping". Genomics 29: 311-322.
5. Hao K., Di X., Cawley S. (2007). "LdCompare: rapid computation of single- and multiple-marker r2 and genetic coverage". Bioinformatics 23: 252-254.