Linkage disequilibrium
In population genetics, linkage disequilibrium is the nonrandom association of alleles at different loci i.e. the presence of statistical associations between alleles at different loci that are different from what would be expected if alleles were independently, randomly sampled based on their individual allele frequencies.^{[1]} If there is no linkage disequilibrium between alleles at different loci they are said to be in linkage equilibrium.
Linkage disequilibrium is influenced by many factors, including selection, the rate of recombination, the rate of mutation, genetic drift, the system of mating, population structure, and genetic linkage. As a result, the pattern of linkage disequilibrium in a genome is a powerful signal of the population genetic forces that are structuring it.
In spite of its name, linkage disequilibrium may exist between alleles at different loci without any genetic linkage between them and independently of whether or not allele frequencies are in equilibrium (not changing with time).^{[1]} Furthermore, linkage disequilibrium is sometimes referred to as gametes.
Contents
 Formal Definition 1
 Measures of linkage disequilibrium derived from D 2
 Example: Twoloci and twoalleles 3
 Role of recombination 4
 Example: Human Leukocyte Antigen (HLA) alleles 5
 Resources 6
 Analysis software 7
 Simulation software 8
 See also 9
 References 10
 Further reading 11
Formal Definition
Suppose that among the gametes that are formed in a sexually reproducing population, allele A occurs with frequency p_A at one locus (i.e. p_A is the proportion of gametes with A at the first locus), while at a different locus allele B occurs with frequency p_B . Similarly, let p_{AB} be the frequency of A and B occurring together in the same gamete (i.e. p_{AB} is the frequency of the AB haplotype).
The association between the alleles A and B can be regarded as completely random when the probability of A and B occurring together in a randomly selected gamete p_{AB} is just the probability that A and B independently occur in the same gamete, which is given by p_{A} p_{B} . If p_{AB} differs from p_{A} p_{B} for any reason, then there is nonrandom association between A and B and we say that there is linkage disequilibrium between these alleles.
The level of linkage disequilibrium between A and B can be quantified by the coefficient of linkage disequilibrium D_{AB}, which is defined as
D_{AB} = p_{AB}  p_{A}p_{B}. 
Linkage disequilibrium corresponds to D_{AB} \neq 0 . In the case D_{AB}=0 we have p_{AB} = p_{A} p_{B} and the alleles A and B are said to be in linkage equilibrium. The subscript AB on D_{AB} emphasizes that it is a property of the alleles A and B. Different alleles at the same loci may have different coefficients of linkage disequilibrium.
Linkage disequilibrium can be defined in a similar way for asexual populations using population allele frequencies. Furthermore, it is also possible to define linkage disequilibrium between three or more alleles, however these higher order associations are not commonly used in practice.^{[1]}
Measures of linkage disequilibrium derived from D
The coefficient of linkage disequilibrium 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^{[3]} suggested normalising D by dividing it by the theoretical maximum for the observed allele frequencies as follows:
D' = D/D_\max 
where
D_\max = \begin{cases} \min\{p_A p_B,\,(1p_A)(1p_B)\} & \text{when } D < 0\\ \min\{p_A (1p_B),\,(1p_A) p_B\} & \text{when } D > 0 \end{cases} 
An alternative to D' is the correlation coefficient between pairs of loci, expressed as
r=\frac{D}{\sqrt{p_A(1p_A)p_B (1p_B)}}.
Example: Twoloci and twoalleles
Consider the haplotypes for two loci A and B with two alleles each—a twolocus, twoallele model. Then the following table defines 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 A_1B_1 as "A_1 is found and B_1 is found". The table above lists the frequencies for A_1, p_1, and forB_1, q_1, hence the frequency of A_1B_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^{[4]} called the linkage disequilibrium^{[5]} and is commonly denoted by a capital D:
D = x_{11}  p_1q_1 
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_1q_1+D  x_{21}=p_2q_1D  q_1 
B_2  x_{12}=p_1q_2D  x_{22}=p_2q_2+D  q_2 
Total  p_1  p_2  1 
Role of recombination
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 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
x_{11}' = (1c)\,x_{11} + c\,p_1 q_1 
This follows because a fraction (1c) 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 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
x_{11}'  p_1 q_1 = (1c)\,(x_{11}  p_1 q_1) 
so that
D_1 = (1c)\;D_0 
where D at the nth generation is designated as D_n. Thus we have
D_n = (1c)^n\; D_0. 
If n \to \infty, then (1c)^n \to 0 so that 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 D to zero.
Example: Human Leukocyte Antigen (HLA) alleles
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^{[6]} referred to by Vogel and Motulsky (1997).^{[7]}
Antigen j  Total  

+    
B8^{+}  B8^{}  
Antigen i  +  A1^{+}  a=376  b=237  C 
  A1^{}  c=91  d=1265  D  
Total  A  B  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 2x2 table to the right.^{[7]}^{[8]}^{[9]}^{[10]}
expression (+) frequency of antigen i :
 pf_i = C/N = 0.311\! ;
expression (+) frequency of antigen j :
 pf_j = A/N = 0.237\! ;
frequency of gene i :
 gf_i = 1  \sqrt{1  pf_i} = 0.170\! ,
and
 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
 o[hf_{xy}]=\sqrt{d/N}
and the estimated frequency of haplotype xy is
 e[hf_{xy}]=\sqrt{(D/N)(B/N)}.
Then LD measure \Delta_{ij} is expressed as
 \Delta_{ij}=o[hf_{xy}]e[hf_{xy}]=\frac{\sqrt{Nd}\sqrt{BD}}{N}=0.0769.
Standard errors SEs are obtained as follows:
 SE\text{ of }gf_i=\sqrt{C}/(2N)=0.00628,
 SE\text{ of }hf_{ij}=\sqrt{\frac{(1\sqrt{d/B})(1\sqrt{d/D})hf_{ij}hf_{ij}^2/2}{2N}}=0.00514
 SE\text{ of }\Delta_{ij}=\frac{1}{2N}\sqrt{a4N\Delta_{ij}\left (\frac{B+D}{2\sqrt{BD}}\frac{\sqrt{BD}}{N}\right )}=0.00367.
Then, if
 t=\Delta_{ij}/(SE\text{ of }\Delta_{ij})
exceeds 2 in its absolute value, the magnitude of \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.
HLAA alleles i  HLAB alleles j  \Delta_{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 paneuropeans.^{[10]}
Vogel and Motulsky (1997)^{[7]} 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 Paneuropeans in the list of Mittal^{[10]} it is mostly nonsignificant. If \Delta_0 had reduced from 0.07 to 0.003 under recombination effect as shown by \Delta_n=(1c)^n \Delta_0, then 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 HLAA and B loci might indicate some sort of interactive selection.^{[7]}
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.^{[11]}
 The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by \delta value^{[12]} to exceed 0.
Ankylosing spondylitis  Total  

Patients  Healthy controls  
HLA alleles  B27^+  a=96  b=77  C 
B27^  c=22  d=701  D  
Total  A  B  N 
 2x2 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 2x2 association table of the allele with the disease. Table 3 shows association of HLAB27 with ankylosing spondylitis among a Dutch population.^{[13]} Relative risk xof this allele is approximated by
 x=\frac{a/b}{c/d}=\frac{ad}{bc}\;(=39.7,\text{ in Table 3 }).
Woolf's method^{[14]} is applied to see if there is statistical significance. Let
 y=\ln (x)\;(=3.68)
and
 \frac{1}{w}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\;(=0.0703).
Then
 \chi^2=wy^2\;\left [=193>\chi^2(p=0.001,\; df=1)=10.8 \right ]
follows the chisquare distribution with df=1. In the data of Table 3, the significant association exists at the 0.1% level. Haldane's^{[15]} modification applies to the case when either ofa,\; b,\;c,\text{ and }d is zero, where replace x and 1/wwith
 x=\frac{(a+1/2)(d+1/2)}{(b+1/2)(c+1/2)}
and
 \frac{1}{w}=\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1},
respectively.
Disease  HLA allele  Relative risk (%)  FAD (%)  FAP (%)  \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.^{[11]}
(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.^{[12]}\delta value is expressed by
 \delta=\frac{FADFAP}{1FAP},\;\;0\le \delta \le 1,
where FAD and FAP are HLA allele frequencies among patients and healthy populations, respectively.^{[12]} In Table 4, \delta column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high \delta 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 \chi^2 test calculating
 \chi^2=\frac{(adbc)^2 N}{ABCD}\;(=336,\text{ for data in Table 3; }P<0.001).
where df=1. For data with small sample size, such as no marginal total is greater than 15 (and consequently N \le 30), one should utilize Yates's correction for continuity or Fisher's exact test.^{[16]}
Resources
A comparison of different measures of LD is provided by Devlin & Risch ^{[17]}
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.
Analysis software
 PLINK  whole genome association analysis toolset, which can calculate LD among other things
 LDHat
 Haploview
 LdCompare^{[18]}— opensource 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
Simulation software
See also
 Haploview
 HardyWeinberg principle
 Genetic linkage
 Coadaptation
 Genealogical DNA test
 Tag SNP
 Association Mapping
 Family based QTL mapping
References
 ^ ^{a} ^{b} ^{c} Slatkin, Montgomery (June 2008). "Linkage disequilibrium — understanding the evolutionary past and mapping the medical future". Nature Reviews Genetics 9 (6): 477–485.
 ^ Falconer, DS; Mackay, TFC (1996). Introduction to Quantitative Genetics (4th ed.). Harlow, Essex, UK: Addison Wesley Longman.
 ^ Lewontin, R. C. (1964). "The interaction of selection and linkage. I. General considerations; heterotic models". Genetics 49 (1): 49–67.
 ^ Robbins, R.B. (1 July 1918). "Some applications of mathematics to breeding problems III". Genetics 3 (4): 375–389.
 ^ R.C. Lewontin and K. Kojima (1960). "The evolutionary dynamics of complex polymorphisms". Evolution 14 (4): 458–472.
 ^ ^{a} ^{b} Svejgaard A, Hauge M, Jersild C, Plaz P, Ryder LP, Staub Nielsen L, Thomsen M (1979). The HLA System: An Introductory Survey, 2nd ed. Basel; London; Chichester: Karger; Distributed by Wiley, ISBN 3805530498(pbk).
 ^ ^{a} ^{b} ^{c} ^{d} Vogel F, Motulsky AG (1997). Human Genetics : Problems and Approaches, 3rd ed.Berlin; London: Springer, ISBN 3540602909.
 ^ Mittal KK, Hasegawa T, Ting A, Mickey MR, Terasaki PI (1973). "Genetic variation in the HLA system between Ainus, Japanese, and Caucasians," In Dausset J, Colombani J, eds. Histocompatibility Testing, 1972, pp. 187195, Copenhagen: Munksgaard, ISBN 8716011015.
 ^ Yasuda N, Tsuji K (1975). "A counting method of maximum likelihood for estimating haplotype frequency in the HLA system." Jinrui Idengaku Zasshi 20(1): 115, PMID 1237691.
 ^ ^{a} ^{b} ^{c} ^{d} Mittal KK (1976). "The HLA polymorphism and susceptibility to disease." Vox Sang 31: 161173, PMID 969389.
 ^ ^{a} ^{b} ^{c} Gregersen PK (2009). "Genetics of rheumatic diseases," InFirestein GS, Budd RC, Harris ED Jr, McInnes IB, Ruddy S, Sergent JS, eds. (2009). Kelley's Textbook of Rheumatology, pp. 305321, Philadelphia, PA: Saunders/Elsevier, ISBN 9781416032854.
 ^ ^{a} ^{b} ^{c} Bengtsson BO, Thomson G (1981). "Measuring the strength of associations between HLA antigens and diseases." Tissue Antigens18(5): 356363, PMID 7344182.
 ^ ^{a} ^{b} Nijenhuis LE (1977). "Genetic considerations on association between HLA and disease." Hum Genet38(2): 175182, PMID 908564.
 ^ Woolf B (1955). "On estimating the relation between blood group and disease." Ann Hum Genet 19(4): 251253, PMID 14388528.
 ^ Haldane JB (1956). "The estimation and significance of the logarithm of a ratio of frequencies." Ann Hum Genet20(4): 309311, PMID 13314400.
 ^ Sokal RR, Rohlf FJ (1981). Biometry: The Principles and Practice of Statistics in Biological Research. Oxford: W.H. Freeman, ISBN 0716712547.
 ^ Devlin B., Risch N. (1995). "A Comparison of Linkage Disequilibrium Measures for FineScale Mapping" (PDF). Genomics 29 (2): 311–322.
 ^ Hao K., Di X., Cawley S. (2007). "LdCompare: rapid computation of single and multiplemarker r2 and genetic coverage". Bioinformatics 23 (2): 252–254.
Further reading
 Hedrick, Philip W. (2005). Genetics of Populations (3rd ed.). Sudbury, Boston, Toronto, London, Singapore: Jones and Bartlett Publishers.
 Bibliography: Linkage Disequilibrium Analysis : a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918.
