In population genetics, linkage disequilibrium is the nonrandom 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. Nonrandom associations between polymorphisms at different loci are measured by the degree of linkage disequilibrium (LD).
An example is the prevalence of two rare diseases in Finland: there, compared to elsewhere in Europe, cystic fibrosis is less prevalent but congenital chloride diarrhea is more prevalent (see Finnish disease heritage). Both diseases are due to mutations on chromosome 7, in adjacent genes.^{[1]}
The level of linkage disequilibrium is influenced by a number of factors including genetic linkage, selection, the rate of recombination, the rate of mutation, genetic drift, nonrandom 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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If we look at haplotypes for two loci A and B with two alleles each—a twolocus, twoallele model—the following table denotes the frequencies of each combination:
Haplotype  Frequency 
A_{1}B_{1}  x_{11} 
A_{1}B_{2}  x_{12} 
A_{2}B_{1}  x_{21} 
A_{2}B_{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 A_{1}B_{1} as "A_{1} is found and B_{1} is found". The table above lists the frequencies for A_{1}, p_{1}, and for B_{1}, q_{1}, hence the frequency of A_{1}B_{1} is x_{11}, and according to the rules of elementary statistics x_{11} = p_{1}q_{1}.
The deviation of the observed frequency of a haplotype from the expected is a quantity^{[2]} called the linkage disequilibrium^{[3]} and is commonly denoted by a capital D:
D = x_{11} − p_{1}q_{1} 
In the genetic literature the phrase "two alleles are in LD" usually means that . Contrariwise, "linkage equilibrium" denotes the case D = 0.
The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.
A_{1}  A_{2}  Total  
B_{1}  x_{11} = p_{1}q_{1} + D  x_{21} = p_{2}q_{1} − D  q_{1} 
B_{2}  x_{12} = p_{1}q_{2} − D  x_{22} = p_{2}q_{2} + D  q_{2} 
Total  p_{1}  p_{2}  1 
D is easy to calculate with, but has the disadvantage of depending on the frequency of the alleles. 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 when When D < 0, .
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 which is an alternative to D' is the correlation coefficient between pairs of loci, denoted as . This is also adjusted to the loci having different allele frequencies. There is some relationship between r and D'. ^{[4]}
In summary, linkage disequilibrium reflects the difference between the expected haplotype frequencies under the assumption of independence, and observed haplotype frequencies. A value of 0 for D' indicates that the examined loci are in fact independent of one another, while a value of 1 demonstrates complete dependency.
In the absence of evolutionary forces other than random mating and Mendelian segregation, the linkage disequilibrium measure D converges to zero along the time axis at a rate depending on the magnitude of the recombination rate c between the two loci.
Using the notation above, D = x_{11} − p_{1}q_{1}, we can demonstrate this convergence to zero as follows. In the next generation, x_{11}', the frequency of the haplotype A_{1}B_{1}, becomes
This follows because a fraction (1 − c) of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction x_{11} of those are A_{1}B_{1}. A fraction c have recombined these two loci. If the parents result from random mating, the probability of the copy at locus A having allele A_{1} is p_{1} and the probability of the copy at locus B having allele B_{1} is q_{1}, and as these copies are initially on different haplotypes, these are independent events so that the probabilities can be multiplied.
This formula can be rewritten as
so that
where D at the nth generation is designated as D_{n}. Thus we have
. 
If , then so that D_{n} converges to zero.
If at some time we observe linkage disequilibrium, it will disappear in future due to recombination. However, the smaller the distance between the two loci, the smaller will be the rate of convergence of D to zero.
HLA constitutes a group of cell surface antigens as MHC of humans. Because HLA genes are located at adjacent loci on the particular region of a chromosome and presumed to exhibit epistasis with each other or with other genes, a sizable fraction of alleles are in linkage disequilibrium.
An example of such linkage disequilibrium is between HLAA1 and B8 alleles in unrelated Danes^{[5]} referred to by Vogel and Motulsky (1997).^{[6]}
No. of individuals  

Antigen j  Total  
+  −  
B8 ^{+}  B8 ^{−}  
Antigen i  +  A1 ^{+}  a = 376  b = 237  C 
−  A1 ^{−}  c = 91  d = 1265  D  
Total  A  B  N 
Because HLA is codominant and HLA expression is only tested locus by locus in surveys, LD measure is to be estimated from such a 2x2 table to the right.^{[6]}^{[7]}^{[8]}^{[9]}
and
Denoting the '―' alleles at antigen i to be 'x,' and at antigen j to be 'y,' the observed frequency of haplotype xy is
and the estimated frequency of haplotype xy is
Then LD measure Δ_{ij} is expressed as
Standard errors SEs are obtained as follows:
Then, if
exceeds 2 in its absolute value, the magnitude of Δ_{ij} is large statistically significantly. For data in Table 1 it is 20.9, thus existence of statistically significant LD between A1 and B8 in the population is admitted.
HLAA alleles i  HLAB alleles j  Δ_{ij}  t 

A1  B8  0.065  16.0 
A3  B7  0.039  10.3 
A2  Bw40  0.013  4.4 
A2  Bw15  0.01  3.4 
A1  Bw17  0.014  5.4 
A2  B18  0.006  2.2 
A2  Bw35  0.009  2.3 
A29  B12  0.013  6.0 
A10  Bw16  0.013  5.9 
Table 2 shows some of the combinations of HLAA and B alleles where significant LD was observed among Caucasians.^{[9]}
Vogel and Motulsky (1997)^{[6]} argued how long would it take that linkage disequilibrium between loci of HLAA and B disappeared. Recombination between loci of HLAA and B was considered to be of the order of magnitude 0.008. We will argue similarly to Vogel and Motulsky below. In case LD measure was observed to be 0.003 in Caucasians in the list of Mittal^{[9]} it is mostly nonsignificant. If Δ_{0} had reduced from 0.07 to 0.003 under recombination effect as shown by Δ_{n} = (1 − c)^{n}Δ_{0}, then . Suppose a generation took 25 years, this means 10,000 years. The time span seems rather short in the history of humans. Thus observed linkage disequilibrium between HLAA and B loci might indicate some sort of interactive selection.^{[6]}
Presence of linkage disequilibrium between an HLA locus and a presumed major gene of disease susceptibility corresponds to any of the following phenomena:
Ankylosing spondylitis  Total  

Patients  Healthy controls  
HLA alleles  B27 ^{+}  a = 96  b = 77  C 
B27 ^{−}  c = 22  d = 701  D  
Total  A  B  N 
(1) Relative risk
Relative risk of an HLA allele for a disease is approximated by the odds ratio in the 2x2 association table of the allele with the disease. Table 3 shows association of HLAB27 with ankylosing spondylitis among a Dutch population.^{[12]} Relative risk x of this allele is approximated by
Woolf's method^{[13]} is applied to see if there is statistical significance. Let
and
Then
follows the chisquare distribution with df = 1. In the data of Table 3, the significant association exists at the 0.1% level. Haldane's^{[14]} modification applies to the case when either of is zero, where replace x and 1 / w with
and
respectively.
Disease  HLA allele  Relative risk (%)  FAD (%)  FAP (%)  δ 

Ankylosing spondylitis  B27  90  90  8  0.89 
Reiter's syndrome  B27  40  70  8  0.67 
Spondylitis in inflammatory bowel disease  B27  10  50  8  0.46 
Rheumatoid arthritis  DR4  6  70  30  0.57 
Systemic lupus erythematosus  DR3  3  45  20  0.31 
Multiple sclerosis  DR2  4  60  20  0.5 
Diabetes mellitus type 1  DR4  6  75  30  0.64 
In Table 4, some examples of association between HLA alleles and diseases are presented.^{[10]}
(1a) Allele frequency excess among patients over controls
Even high relative risks between HLA alleles and the diseases were observed, only the magnitude of relative risk would not be able to determine the strength of association.^{[11]} δ value is expressed by
where FAD and FAP are HLA allele frequencies among patients and healthy populations, respectively.^{[11]} In Table 4, δ column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high δ values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk = 6.
(2) Discrepancies from expected values from marginal frequencies in 2x2 association table of HLA alleles and disease
This can be confirmed by χ^{2} test calculating
where df = 1. For data with small sample size, such as no marginal total is greater than 15 (and consequently ), one should utilize Yates' correction for continuity or Fisher's exact test.^{[15]}
A comparison of different measures of LD is provided by Devlin & Risch ^{[16]}
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.

