Linkage disequilibrium

Suppose that among the gametes that are formed in a sexually reproducing population, allele A occurs with frequency ${\displaystyle p_{A}}$ at one locus (i.e. ${\displaystyle p_{A}}$ is the proportion of gametes with A at that locus), while at a different locus allele B occurs with frequency ${\displaystyle p_{B}}$. Similarly, let ${\displaystyle p_{AB}}$ be the frequency with which both A and B occur together in the same gamete (i.e. ${\displaystyle p_{AB}}$ is the frequency of the AB haplotype).

The association between the alleles A and B can be regarded as completely random—which is known in statistics as independence—when the occurrence of one does not affect the occurrence of the other, in which case the probability that both A and B occur together is given by the product ${\displaystyle p_{A}p_{B}}$ of the probabilities. There is said to be a linkage disequilibrium between the two alleles whenever ${\displaystyle p_{AB}}$ differs from ${\displaystyle p_{A}p_{B}}$ for any reason.

The level of linkage disequilibrium between A and B can be quantified by the coefficient of linkage disequilibrium ${\displaystyle D_{AB}}$, which is defined as

${\displaystyle D_{AB}=p_{AB}-p_{A}p_{B},}$

provided that both ${\displaystyle p_{A}}$ and ${\displaystyle p_{B}}$ are greater than zero. Linkage disequilibrium corresponds to ${\displaystyle D_{AB}\neq 0}$. In the case ${\displaystyle D_{AB}=0}$ we have ${\displaystyle p_{AB}=p_{A}p_{B}}$ and the alleles A and B are said to be in linkage equilibrium. The subscript "AB" on ${\displaystyle D_{AB}}$ emphasizes that linkage disequilibrium is a property of the pair {A, B} of alleles and not of their respective loci. Other pairs of alleles at those same two loci may have different coefficients of linkage disequilibrium.

For two biallelic loci, where a and b are the other alleles at these two loci, the restrictions are so strong that only one value of D is sufficient to represent all linkage disequilibrium relationships between these alleles. In this case, ${\displaystyle D_{AB}=-D_{Ab}=-D_{aB}=D_{ab}}$. Their relationships can be characterized as follows.[3]

${\displaystyle D=P_{AB}-P_{A}*P_{B}}$

${\displaystyle -D=P_{Ab}-P_{A}*P_{b}}$

${\displaystyle -D=P_{aB}-P_{a}*P_{B}}$

${\displaystyle D=P_{ab}-P_{a}*P_{b}}$

The sign of D in this case is chosen arbitrarily. The magnitude of D is more important than the sign of D because the magnitude of D is representative of the degree of linkage disequilibrium.[4] However, positive D value means that the gamete is more frequent than expected while negative means that the combination of these two alleles are less frequent than expected.

Linkage disequilibrium in asexual populations can be defined in a similar way in terms of population allele frequencies. Furthermore, it is also possible to define linkage disequilibrium among three or more alleles, however these higher-order associations are not commonly used in practice.[1]

The coefficient of linkage disequilibrium ${\displaystyle D}$ is not always a convenient measure of linkage disequilibrium because its range of possible values depends on the frequencies of the alleles it refers to. This makes it difficult to compare the level of linkage disequilibrium between different pairs of alleles.

Lewontin[5] suggested normalising D by dividing it by the theoretical maximum difference between the observed and expected haplotype frequencies as follows:

${\displaystyle D'={\frac {D}{D_{\max }}}}$

where

${\displaystyle D_{\max }={\begin{cases}\max\{-p_{A}p_{B},\,-(1-p_{A})(1-p_{B})\}&{\text{when }}D<0\\\min\{p_{A}(1-p_{B}),\,(1-p_{A})p_{B}\}&{\text{when }}D>0\end{cases}}}$

An alternative to ${\displaystyle D'}$ is the correlation coefficient between pairs of loci, expressed as

${\displaystyle r={\frac {D}{\sqrt {p_{A}(1-p_{A})p_{B}(1-p_{B})}}}.}$

Consider the haplotypes for two loci A and B with two alleles each—a two-loci, two-allele model. Then the following table defines the frequencies of each combination:

Haplotype	Frequency
${\displaystyle A_{1}B_{1}}$	${\displaystyle x_{11}}$
${\displaystyle A_{1}B_{2}}$	${\displaystyle x_{12}}$
${\displaystyle A_{2}B_{1}}$	${\displaystyle x_{21}}$
${\displaystyle A_{2}B_{2}}$	${\displaystyle x_{22}}$

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

Allele	Frequency
${\displaystyle A_{1}}$	${\displaystyle p_{1}=x_{11}+x_{12}}$
${\displaystyle A_{2}}$	${\displaystyle p_{2}=x_{21}+x_{22}}$
${\displaystyle B_{1}}$	${\displaystyle q_{1}=x_{11}+x_{21}}$
${\displaystyle B_{2}}$	${\displaystyle q_{2}=x_{12}+x_{22}}$

If the two loci and the alleles are independent from each other, then one can express the observation ${\displaystyle A_{1}B_{1}}$ as "${\displaystyle A_{1}}$ is found and ${\displaystyle B_{1}}$ is found". The table above lists the frequencies for ${\displaystyle A_{1}}$, ${\displaystyle p_{1}}$, and for${\displaystyle B_{1}}$, ${\displaystyle q_{1}}$, hence the frequency of ${\displaystyle A_{1}B_{1}}$ is ${\displaystyle x_{11}}$, and according to the rules of elementary statistics ${\displaystyle x_{11}=p_{1}q_{1}}$.

The deviation of the observed frequency of a haplotype from the expected is a quantity[6] called the linkage disequilibrium[7] and is commonly denoted by a capital D:

${\displaystyle D=x_{11}-p_{1}q_{1}}$

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

	${\displaystyle A_{1}}$	${\displaystyle A_{2}}$	Total
${\displaystyle B_{1}}$	${\displaystyle x_{11}=p_{1}q_{1}+D}$	${\displaystyle x_{21}=p_{2}q_{1}-D}$	${\displaystyle q_{1}}$
${\displaystyle B_{2}}$	${\displaystyle x_{12}=p_{1}q_{2}-D}$	${\displaystyle x_{22}=p_{2}q_{2}+D}$	${\displaystyle q_{2}}$
Total	${\displaystyle p_{1}}$	${\displaystyle p_{2}}$	${\displaystyle 1}$

In the absence of evolutionary forces other than random mating, Mendelian segregation, random chromosomal assortment, and chromosomal crossover (i.e. in the absence of natural selection, inbreeding, and genetic drift), the linkage disequilibrium measure ${\displaystyle D}$ converges to zero along the time axis at a rate depending on the magnitude of the recombination rate ${\displaystyle c}$ between the two loci.

Using the notation above, ${\displaystyle D=x_{11}-p_{1}q_{1}}$, we can demonstrate this convergence to zero as follows. In the next generation, ${\displaystyle x_{11}'}$, the frequency of the haplotype ${\displaystyle A_{1}B_{1}}$, becomes

${\displaystyle x_{11}'=(1-c)\,x_{11}+c\,p_{1}q_{1}}$

This follows because a fraction ${\displaystyle (1-c)}$ of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction ${\displaystyle x_{11}}$ of those are ${\displaystyle A_{1}B_{1}}$. A fraction ${\displaystyle c}$ have recombined these two loci. If the parents result from random mating, the probability of the copy at locus ${\displaystyle A}$ having allele ${\displaystyle A_{1}}$ is ${\displaystyle p_{1}}$ and the probability of the copy at locus ${\displaystyle B}$ having allele ${\displaystyle B_{1}}$ is ${\displaystyle q_{1}}$, and as these copies are initially in the two different gametes that formed the diploid genotype, these are independent events so that the probabilities can be multiplied.

This formula can be rewritten as

${\displaystyle x_{11}'-p_{1}q_{1}=(1-c)\,(x_{11}-p_{1}q_{1})}$

so that

${\displaystyle D_{1}=(1-c)\;D_{0}}$

where ${\displaystyle D}$ at the ${\displaystyle n}$-th generation is designated as ${\displaystyle D_{n}}$. Thus we have

${\displaystyle D_{n}=(1-c)^{n}\;D_{0}.}$

If ${\displaystyle n\to \infty }$, then ${\displaystyle (1-c)^{n}\to 0}$ so that ${\displaystyle D_{n}}$ converges to zero.

If at some time we observe linkage disequilibrium, it will disappear in the future due to recombination. However, the smaller the distance between the two loci, the smaller will be the rate of convergence of ${\displaystyle D}$ to zero.

HLA constitutes a group of cell surface antigens also known as the 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 HLA-A1 and B8 alleles in unrelated Danes[8] referred to by Vogel and Motulsky (1997).[9]

Table 1. Association of HLA-A1 and B8 in unrelated Danes[8]

			Antigen j		Total
			${\displaystyle +}$	${\displaystyle -}$
			${\displaystyle B8^{+}}$	${\displaystyle B8^{-}}$
Antigen i	${\displaystyle +}$	${\displaystyle A1^{+}}$	${\displaystyle a=376}$	${\displaystyle b=237}$	${\displaystyle C}$
	${\displaystyle -}$	${\displaystyle A1^{-}}$	${\displaystyle c=91}$	${\displaystyle d=1265}$	${\displaystyle D}$
Total			${\displaystyle A}$	${\displaystyle B}$	${\displaystyle N}$
			No. of individuals

Because HLA is codominant and HLA expression is only tested locus by locus in surveys, LD measure is to be estimated from such a 2×2 table to the right.[9][10][11][12]

expression (${\displaystyle +}$) frequency of antigen ${\displaystyle i}$ :

${\displaystyle pf_{i}={\frac {C}{N}}=0.311;}$

expression (${\displaystyle +}$) frequency of antigen ${\displaystyle j}$ :

${\displaystyle pf_{j}={\frac {A}{N}}=0.237;}$

frequency of gene ${\displaystyle i}$, given that individuals with '+/−', '+/+', and '−/+' genotypes are all positive for antigen ${\displaystyle i}$:

${\displaystyle gf_{i}=1-{\sqrt {1-pf_{i}}}=0.170,}$

and

${\displaystyle hf_{ij}={\text{estimated frequency of haplotype }}ij=gf_{i}\;gf_{j}=0.0215.}$

Denoting the '―' alleles at antigen i to be x, and at antigen j to be y, the observed frequency of haplotype xy is

${\displaystyle o[hf_{xy}]={\sqrt {\frac {d}{N}}}}$

and the estimated frequency of haplotype xy is

${\displaystyle e[hf_{xy}]={\sqrt {{\frac {D}{N}}\cdot {\frac {B}{N}}}}.}$

Then LD measure ${\displaystyle \Delta _{ij}}$ is expressed as

${\displaystyle \Delta _{ij}=o[hf_{xy}]-e[hf_{xy}]={\frac {{\sqrt {Nd}}-{\sqrt {BD}}}{N}}=0.0769.}$

Standard errors ${\displaystyle SEs}$ are obtained as follows:

${\displaystyle SE{\text{ of }}gf_{i}={\frac {\sqrt {C}}{2N)}}=0.00628.}$

${\displaystyle SE{\text{ of }}hf_{ij}={\sqrt {\frac {(1-{\sqrt {d/B}})(1-{\sqrt {d/D}})-hf_{ij}-hf_{ij}^{2}/2}{2N}}}=0.00514}$

${\displaystyle SE{\text{ of }}\Delta _{ij}={\frac {1}{2N}}{\sqrt {a-4N\Delta _{ij}\left({\frac {B+D}{2{\sqrt {BD}}}}-{\frac {\sqrt {BD}}{N}}\right)}}=0.00367.}$

Then, if

${\displaystyle t={\frac {\Delta _{ij}}{SE{\text{ of }}\Delta _{ij}}}}$

exceeds 2 in its absolute value, the magnitude of ${\displaystyle \Delta _{ij}}$ is statistically significantly large. For data in Table 1 it is 20.9, thus existence of statistically significant LD between A1 and B8 in the population is admitted.

Table 2. Linkage disequilibrium among HLA alleles in pan-Europeans[12]

HLA-A alleles i	HLA-B alleles j	${\displaystyle \Delta _{ij}}$	${\displaystyle t}$
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 HLA-A and B alleles where significant LD was observed among pan-Europeans.[12]

Vogel and Motulsky (1997)[9] argued how long would it take that linkage disequilibrium between loci of HLA-A and B disappeared. Recombination between loci of HLA-A 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 pan-Europeans in the list of Mittal[12] it is mostly non-significant. If ${\displaystyle \Delta _{0}}$ had reduced from 0.07 to 0.003 under recombination effect as shown by ${\displaystyle \Delta _{n}=(1-c)^{n}\Delta _{0}}$, then ${\displaystyle n\approx 400}$. 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 HLA-A and B loci might indicate some sort of interactive selection.[9]

The presence of linkage disequilibrium between an HLA locus and a presumed major gene of disease susceptibility corresponds to any of the following phenomena:

• Relative risk for the person having a specific HLA allele to become suffered from a particular disease is greater than 1.[13]
• The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by ${\displaystyle \delta }$ value[14] to exceed 0.

Table 3. Association of ankylosing spondylitis with HLA-B27 allele[15]

		Ankylosing spondylitis		Total
		Patients	Healthy controls
HLA alleles	${\displaystyle B27^{+}}$	${\displaystyle a=96}$	${\displaystyle b=77}$	${\displaystyle C}$
	${\displaystyle B27^{-}}$	${\displaystyle c=22}$	${\displaystyle d=701}$	${\displaystyle D}$
Total		${\displaystyle A}$	${\displaystyle B}$	${\displaystyle N}$

• 2×2 association table of patients and healthy controls with HLA alleles shows a significant deviation from the equilibrium state deduced from the marginal frequencies.

(1) Relative risk

Relative risk of an HLA allele for a disease is approximated by the odds ratio in the 2×2 association table of the allele with the disease. Table 3 shows association of HLA-B27 with ankylosing spondylitis among a Dutch population.[15] Relative risk ${\displaystyle x}$of this allele is approximated by

${\displaystyle x={\frac {a/b}{c/d}}={\frac {ad}{bc}}\;(=39.7,{\text{ in Table 3 }}).}$

Woolf's method[16] is applied to see if there is statistical significance. Let

${\displaystyle y=\ln(x)\;(=3.68)}$

and

${\displaystyle {\frac {1}{w}}={\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}+{\frac {1}{d}}\;(=0.0703).}$

Then

${\displaystyle \chi ^{2}=wy^{2}\;\left[=193>\chi ^{2}(p=0.001,\;df=1)=10.8\right]}$

follows the chi-square distribution with ${\displaystyle df=1}$. In the data of Table 3, a significant association exists at the 0.1% level. Haldane's[17] modification applies to the case when either of ${\displaystyle a,\;b,\;c,{\text{ and }}d}$ is zero, where ${\displaystyle x}$ and ${\displaystyle 1/w}$ are replaced with

${\displaystyle x={\frac {(a+1/2)(d+1/2)}{(b+1/2)(c+1/2)}}}$

and

${\displaystyle {\frac {1}{w}}={\frac {1}{a+1}}+{\frac {1}{b+1}}+{\frac {1}{c+1}}+{\frac {1}{d+1}},}$

respectively.

Table 4. Association of HLA alleles with rheumatic and autoimmune diseases among white populations[13]

Disease	HLA allele	Relative risk (%)	FAD (%)	FAP (%)	${\displaystyle \delta }$
Ankylosing spondylitis	B27	90	90	8	0.89
Reactive arthritis	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.[13]

(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.[14]${\displaystyle \delta }$ value is expressed by

${\displaystyle \delta ={\frac {FAD-FAP}{1-FAP}},\;\;0\leq \delta \leq 1,}$

where ${\displaystyle FAD}$ and ${\displaystyle FAP}$ are HLA allele frequencies among patients and healthy populations, respectively.[14] In Table 4, ${\displaystyle \delta }$ column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high ${\displaystyle \delta }$ values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk${\displaystyle {}=6}$.

(2) Discrepancies from expected values from marginal frequencies in 2×2 association table of HLA alleles and disease

This can be confirmed by ${\displaystyle \chi ^{2}}$ test calculating

${\displaystyle \chi ^{2}={\frac {(ad-bc)^{2}N}{ABCD}}\;(=336,{\text{ for data in Table 3; }}P<0.001).}$

where ${\displaystyle df=1}$. For data with small sample size, such as no marginal total is greater than 15 (and consequently ${\displaystyle N\leq 30}$), one should utilize Yates's correction for continuity or Fisher's exact test.[18]

A comparison of different measures of LD is provided by Devlin & Risch[19]

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

• PLINK – whole genome association analysis toolset, which can calculate LD among other things
• LDHat
• Haploview
• LdCompare[20]— open-source software for calculating LD.
• SNP and Variation Suite – commercial software with interactive LD plot.
• GOLD – Graphical Overview of Linkage Disequilibrium
• TASSEL – software to evaluate linkage disequilibrium, traits associations, and evolutionary patterns
• rAggr – finds proxy markers (SNPs and indels) that are in linkage disequilibrium with a set of queried markers, using the 1000 Genomes Project and HapMap genotype databases.
• SNeP – Fast computation of LD and Ne for large genotype datasets in PLINK format.
• LDlink – A suite of web-based applications to easily and efficiently explore linkage disequilibrium in population subgroups. All population genotype data originates from Phase 3 of the 1000 Genomes Project and variant RS numbers are indexed based on dbSNP build 151.

• Haploid — a C library for population genetic simulation (GPL)

• Haploview
• Hardy–Weinberg principle
• Genetic linkage
• Co-adaptation
• Genealogical DNA test
• Tag SNP
• Association mapping
• Family based QTL mapping

• Hedrick, Philip W. (2005). Genetics of Populations (3rd ed.). Sudbury, Boston, Toronto, London, Singapore: Jones and Bartlett Publishers. ISBN 978-0-7637-4772-5.
• Bibliography: Linkage Disequilibrium Analysis : a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918.