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| [[Image:Nrm ld.svg|thumb|border||right|220px||alt="Graph showing evolution of linkage disequilibrium in a haploid population with positive assortative mating (inbreeding); graph shows the trajectories of 1,024 populations with randomly generated initial allele frequencies; trajectories beginning close to 0 slope away from 0 sharply then decline to 0 asymptotically, while trajectories beginning near 0 diminish to 0"|A simulation showing the evolution of linkage disequilibrium in randomly generated haploid populations with positive [[assortative mating]]]]
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| In [[population genetics]], '''linkage disequilibrium''' is the non-random association of [[allele]]s at two or more [[locus (genetics)|loci]], that descend from single, ancestral [[chromosomes]].<ref>{{cite journal |last1=Reich |first1=DE |year=2001 |title=Linkage disequilibrium in the human genome |journal=Nature |volume=411 |pages=199-204 |publisher=Nautre Publishing Group |doi=10.1038/35075590 |url=http://www.nature.com/nature/journal/v411/n6834/full/411199a0.html |accessdate=12 July 2013}}</ref> Linkage disequilibrium is wholly a measurement of proximal genomic space. It is necessary to refer to this as '''[[gametic phase]] disequilibrium'''<ref>{{cite book | first=DS | last=Falconer | first2=TFC | last2=Mackay |title=Introduction to Quantitative Genetics | edition=4th | year=1996 | publisher=Addison Wesley Longman | isbn=0-582-24302-5 | location=Harlow, Essex, UK}}</ref> or simply '''gametic disequilibrium''' because it is described through [[DNA recombination]]. In other words, linkage disequilibrium is the occurrence of some combinations of alleles or genetic markers in a population more often or less often than would be expected from a random formation of [[haplotype]]s from alleles based on their frequencies. It is a second order phenomenon derived from [[Genetic linkage|linkage]], which is the presence of two or more loci on a chromosome with limited [[Homologous recombination|recombination]] between them. The amount of linkage disequilibrium depends on the difference between observed allelic frequencies and those expected from a homogenous, randomly distributed model. Populations where combinations of alleles or genotypes can be found in the expected proportions are said to be in '''linkage equilibrium'''.
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| 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]], [[assortative mating|non-random mating]], and [[population stratification|population structure]]. A limiting example of the effect of rate of recombination may be seen in some organisms (such as [[bacteria]]) that [[asexual reproduction|reproduce asexually]] and hence exhibit no recombination to break down the linkage disequilibrium. An example of the effect of population structure is the phenomenon of [[Finnish disease heritage]], which is attributed to a [[population bottleneck]].
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| ==Definition==
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| Consider the [[haplotypes]] for two loci A and B with two alleles each—a two-locus, two-allele model. Then the following table defines the frequencies of each combination:
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| {| class=wikitable
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| |Haplotype
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| |Frequency
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| |-
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| |<math>A_1B_1</math>||<math>x_{11}</math>
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| |-
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| |<math>A_1B_2</math>||<math>x_{12}</math>
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| |-
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| |<math>A_2B_1</math>||<math>x_{21}</math>
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| |-
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| |<math>A_2B_2</math>||<math>x_{22}</math>
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| |}
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| Note that these are [[Frequency (statistics)|relative frequencies]]. One can use the above frequencies to determine the frequency of each of the alleles:
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| {| class=wikitable
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| |Allele||Frequency
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| |-
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| |<math>A_1</math>||<math>p_{1}=x_{11}+x_{12}</math>
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| |-
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| |<math>A_2</math>||<math>p_{2}=x_{21}+x_{22}</math>
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| |-
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| |<math>B_1</math>||<math>q_{1}=x_{11}+x_{21}</math>
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| |-
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| |<math>B_2</math>||<math>q_{2}=x_{12}+x_{22}</math>
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| |}
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| If the two loci and the alleles are [[independent assortment|independent]] from each other, then one can express the observation <math>A_1B_1</math> as "<math>A_1</math> is found and <math>B_1</math> is found". The table above lists the frequencies for <math>A_1</math>, <math>p_1</math>, and for<math>B_1</math>, <math>q_1</math>, hence the frequency of <math>A_1B_1</math> is <math>x_{11}</math>, and according to the rules of elementary statistics <math>x_{11} = p_{1} q_{1}</math>.
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| The deviation of the observed frequency of a haplotype from the expected is a quantity<ref>{{cite journal|author=Robbins, R.B.|title=Some applications of mathematics to breeding problems III|journal=Genetics|date=1 July 1918|volume=3|pages=375–389|url=http://www.genetics.org/cgi/reprint/3/4/375|issue=4|pmid=17245911|pmc=1200443}}</ref> called the linkage disequilibrium<ref>{{cite journal|author=R.C. Lewontin and K. Kojima|year=1960| title=The evolutionary dynamics of complex polymorphisms| journal=Evolution| volume=14|pages=458–472|doi=10.2307/2405995|issue=4|issn=0014-3820|jstor=2405995}}</ref> and is commonly denoted by a capital D:
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| {|
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| |<math>D = x_{11} - p_1q_1</math>
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| |}
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| In the genetic literature the phrase "two alleles are in LD" usually means that ''D'' ≠ ''0''. Contrariwise, "linkage equilibrium" means ''D'' = ''0''.
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| The following table illustrates the relationship between the haplotype frequencies and allele frequencies and D.
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| {| class=wikitable
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| | ||<math>A_1</math> ||<math>A_2</math> ||Total
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| |-
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| |<math>B_1</math>||<math>x_{11}=p_1q_1+D</math> ||<math>x_{21}=p_2q_1-D</math> ||<math>q_1</math>
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| |-
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| |<math>B_2</math>||<math>x_{12}=p_1q_2-D</math> ||<math>x_{22}=p_2q_2+D</math>||<math>q_2</math>
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| |-
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| |Total ||<math>p_1</math>||<math>p_2</math>||<math>1</math>
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| |}
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| <math>D</math> is easy to calculate with, but has the disadvantage of depending on the frequencies of the alleles. This is evident since frequencies are between 0 and 1. If any locus has an allele frequency 0 or 1 no disequilibrium <math>D</math> can be observed. When the allelic frequencies are 0.5, the disequilibrium <math>D</math> is maximal. Lewontin<ref>{{cite journal | author = Lewontin, R. C. | year = 1964 | title = The interaction of selection and linkage. I. General considerations; heterotic models | journal = Genetics | volume = 49 | pages = 49–67 | pmid = 17248194 | issue = 1 | pmc = 1210557}}</ref> suggested normalising ''D'' by dividing it by the theoretical maximum for the observed allele frequencies.
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| Thus:
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| {|
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| |<math>D'</math> = <math>\tfrac{D}{D_\max}</math>
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| |}
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| where
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| {|
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| |<math>D_\max = \begin{cases}
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| \min(p_1q_1,\,p_2q_2) & \text{when } D < 0\\
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| \min(p_1q_2,\,p_2q_1) & \text{when } D > 0
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| \end{cases} </math>
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| |}
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| Another measure of LD which is an alternative to <math>D'</math> is the [[correlation coefficient]] between pairs of loci, expressed as
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| <math>r=\frac{D}{\sqrt{p_1p_2q_1q_2}}</math>.
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| This is also adjusted to the loci having different allele frequencies.
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| 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 <math>D'</math> indicates that the examined loci are in fact independent of one another, while a value of 1 demonstrates complete dependency.
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| ==Role of recombination==
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| In the absence of evolutionary forces other than [[random mating]], [[Mendelian_inheritance#Law_of_Segregation_.28The_.22First_Law.22.29 | Mendelian segregation]], random [[Particulate inheritance|chromosomal assortment]], and [[chromosomal crossover]] (i.e. in the absence of [[natural selection]], [[inbreeding]], and [[genetic drift]]),
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| the linkage disequilibrium measure <math>D</math> converges to zero along the time axis at a rate
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| depending on the magnitude of the recombination rate <math>c</math> between the two loci.
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| Using the notation above, <math>D= x_{11}-p_1 q_1</math>, we can demonstrate this convergence to zero
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| as follows. In the next generation, <math>x_{11}'</math>, the frequency of the haplotype <math>A_1 B_1</math>, becomes
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| {|
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| |<math>x_{11}' = (1-c)\,x_{11} + c\,p_1 q_1</math>
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| |}
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| This follows because a fraction <math>(1-c)</math> of the haplotypes in the offspring have not
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| recombined, and are thus copies of a random haplotype in their parents. A fraction <math>x_{11}</math> of those are <math>A_1 B_1</math>. A fraction <math>c</math>
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| have recombined these two loci. If the parents result from random mating, the probability of the
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| copy at locus <math>A</math> having allele <math>A_1</math> is <math>p_1</math> and the probability
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| of the copy at locus <math>B</math> having allele <math>B_1</math> is <math>q_1</math>, and as these copies are initially on different loci, these are independent events so that the probabilities can be multiplied.
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| This formula can be rewritten as
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| {|
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| |<math>x_{11}' - p_1 q_1 = (1-c)\,(x_{11} - p_1 q_1)</math>
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| |}
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| so that
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| {|
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| |<math>D_1 = (1-c)\;D_0</math>
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| |}
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| where <math>D</math> at the <math>n</math>-th generation is designated as <math>D_n</math>. Thus we have
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| {|
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| |<math>D_n = (1-c)^n\; D_0</math>.
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| |}
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| If <math>n \to \infty</math>, then <math>(1-c)^n \to 0</math> so that <math>D_n</math> converges to zero.
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| 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 <math>D</math> to zero.
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| ==Example: Human Leukocyte Antigen (HLA) alleles==
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| [[Human leukocyte antigen|HLA]] constitutes a group of cell surface antigens as [[Major histocompatibility complex|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.
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| An example of such linkage disequilibrium is between HLA-A1 and B8 alleles in unrelated Danes<ref name=Svejgaard>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).</ref> referred to by Vogel and Motulsky (1997).<ref name=Vogel>Vogel F, Motulsky AG (1997). ''Human Genetics : Problems and Approaches, 3rd ed.''Berlin; London: Springer, ISBN 3-540-60290-9.</ref>
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| {| style="float:right" class=wikitable
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| |+ Table 1. Association of HLA-A1 and B8 in unrelated Danes<ref name=Svejgaard/>
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| |-
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| ! colspan="3" rowspan="3" |
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| ! colspan="2" | Antigen j
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| ! rowspan="3" | Total
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| |-
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| ! <math>+</math>
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| ! <math>-</math>
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| |-
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| ! <math>B8^{+}</math>
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| ! <math>B8^{-}</math>
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| |-
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| ! rowspan="2" | Antigen i
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| ! <math>+</math>
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| ! <math>A1^{+}</math>
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| | align="center" | <math>a=376</math>
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| | align="center" | <math>b=237</math>
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| | align="center" | <math>C</math>
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| |-
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| ! <math>-</math>
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| ! <math>A1^{-}</math>
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| | align="center" | <math>c=91</math>
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| | align="center" | <math>d=1265</math>
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| | align="center" | <math>D</math>
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| |-
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| ! colspan="3" | Total
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| | align="center" | <math>A</math>
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| | align="center" | <math>B</math>
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| | align="center" | <math>N</math>
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| |-
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| | colspan="3" |
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| ! colspan="3" | No. of individuals
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| |}
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| 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.<ref name=Vogel/><ref name=Mittal1973>Mittal KK, Hasegawa T, Ting A, Mickey MR, Terasaki PI (1973). "Genetic variation in the HL-A system between Ainus, Japanese, and Caucasians," ''In'' Dausset J, Colombani J, eds. ''Histocompatibility Testing, 1972,'' pp. 187-195, Copenhagen: Munksgaard, ISBN 87-16-01101-5.</ref><ref name=Yasuda>Yasuda N, Tsuji K (1975). "A counting method of maximum likelihood for estimating haplotype frequency in the HL-A system." ''Jinrui Idengaku Zasshi'' '''20'''(1): 1-15, PMID 1237691.</ref><ref name=Mittal1976>Mittal KK (1976). "The HLA polymorphism and susceptibility to disease." ''Vox Sang'' '''31''': 161-173, PMID 969389.</ref>
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| expression (<math>+</math>) frequency of antigen <math>i</math> :
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| :<math>pf_i = C/N = 0.311\!</math> ;
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| expression (<math>+</math>) frequency of antigen <math>j</math> :
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| :<math>pf_j = A/N = 0.237\!</math> ;
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| frequency of gene <math>i</math> :
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| :<math>gf_i = 1 - \sqrt{1 - pf_i} = 0.170\!</math> ,
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| and
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| :<math>hf_{ij} = \text{estimated frequency of haplotype } ij = gf_i \; gf_j = 0.0215\!</math> .
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| Denoting the '―' alleles at antigen i to be 'x,' and at antigen j to be 'y,' the observed frequency of haplotype xy is
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| :<math>o[hf_{xy}]=\sqrt{d/N}</math>
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| and the estimated frequency of haplotype xy is
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| :<math>e[hf_{xy}]=\sqrt{(D/N)(B/N)}</math>.
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| Then LD measure <math>\Delta_{ij}</math> is expressed as
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| :<math>\Delta_{ij}=o[hf_{xy}]-e[hf_{xy}]=\frac{\sqrt{Nd}-\sqrt{BD}}{N}=0.0769</math>.
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| Standard errors <math>SEs</math> are obtained as follows:
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| :<math>SE\text{ of }gf_i=\sqrt{C}/(2N)=0.00628</math>,
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| :<math>SE\text{ of }hf_{ij}=\sqrt{\frac{(1-\sqrt{d/B})(1-\sqrt{d/D})-hf_{ij}-hf_{ij}^2/2}{2N}}=0.00514</math>
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| :<math>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</math>.
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| Then, if
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| :<math>t=\Delta_{ij}/(SE\text{ of }\Delta_{ij})</math>
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| exceeds 2 in its absolute value, the magnitude of <math>\Delta_{ij}</math> 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.
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| {| style="float:right" class=wikitable
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| |+ Table 2. Linkage disequilibrium among HLA alleles in [[Pan-europeans]]<ref name=Mittal1976/>
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| ! HLA-A alleles i
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| ! HLA-B alleles j
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| ! <math>\Delta_{ij}</math>
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| ! <math>t</math>
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| |-
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| | align="center" | A1
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| | align="center" | B8
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| | align="center" | 0.065
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| | align="center" | 16.0
| |
| |-
| |
| | align="center" | A3
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| | align="center" | B7
| |
| | align="center" | 0.039
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| | align="center" | 10.3
| |
| |-
| |
| | align="center" | A2
| |
| | align="center" | Bw40
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| | align="center" | 0.013
| |
| | align="center" | 4.4
| |
| |-
| |
| | align="center" | A2
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| | align="center" | Bw15
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| | align="center" | 0.01
| |
| | align="center" | 3.4
| |
| |-
| |
| | align="center" | A1
| |
| | align="center" | Bw17
| |
| | align="center" | 0.014
| |
| | align="center" | 5.4
| |
| |-
| |
| | align="center" | A2
| |
| | align="center" | B18
| |
| | align="center" | 0.006
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| | align="center" | 2.2
| |
| |-
| |
| | align="center" | A2
| |
| | align="center" | Bw35
| |
| | align="center" | -0.009
| |
| | align="center" | -2.3
| |
| |-
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| | align="center" | A29
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| | align="center" | B12
| |
| | align="center" | 0.013
| |
| | align="center" | 6.0
| |
| |-
| |
| | align="center" | A10
| |
| | align="center" | Bw16
| |
| | align="center" | 0.013
| |
| | align="center" | 5.9
| |
| |}
| |
| Table 2 shows some of the combinations of HLA-A and B alleles where significant LD was observed among pan-europeans.<ref name=Mittal1976/>
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| | |
| Vogel and Motulsky (1997)<ref name=Vogel/> 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<ref name=Mittal1976/> it is mostly non-significant. If <math>\Delta_0</math> had reduced from 0.07 to 0.003 under recombination effect as shown by <math>\Delta_n=(1-c)^n \Delta_0</math>, then <math>n\approx 400</math>. 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.<ref name=Vogel/>
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| {{further2|[[HLA A1-B8 haplotype]]}}
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| The presence of linkage disequilibrium between an HLA locus and a presumed major gene of disease susceptibility corresponds to any of the following phenomena:
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| * Relative risk for the person having a specific HLA allele to become suffered from a particular disease is greater than 1.<ref name=Gregersen/>
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| * The HLA antigen frequency among patients exceeds more than that among a healthy population. This is evaluated by <math>\delta</math> value<ref name=Bengtsson>Bengtsson BO, Thomson G (1981). "Measuring the strength of associations between HLA antigens and diseases." ''Tissue Antigens'''''18'''(5): 356-363, PMID 7344182.</ref> to exceed 0.
| |
| {| style="float:right" class=wikitable
| |
| |+ Table 3. Association of ankylosing spondylitis with HLA-B27 allele<ref name=Nijenhuis>Nijenhuis LE (1977). "Genetic considerations on association between HLA and disease." ''Hum Genet'''''38'''(2): 175-182, PMID 908564.</ref>
| |
| | colspan="2" rowspan="2" |
| |
| ! colspan="2" | Ankylosing spondylitis
| |
| ! rowspan="2" | Total
| |
| |-
| |
| ! Patients
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| ! Healthy controls
| |
| |-
| |
| ! rowspan="2" | HLA alleles
| |
| ! <math>B27^+</math>
| |
| | align="center" | <math>a=96</math>
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| | align="center" | <math>b=77</math>
| |
| | align="center" | <math>C</math>
| |
| |-
| |
| ! <math>B27^-</math>
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| | align="center" | <math>c=22</math>
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| | align="center" | <math>d=701</math>
| |
| | align="center" | <math>D</math>
| |
| |-
| |
| ! colspan="2" | Total
| |
| | align="center" | <math>A</math>
| |
| | align="center" | <math>B</math>
| |
| | align="center" | <math>N</math>
| |
| |}
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| *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 HLA-B27 with ankylosing spondylitis among a Dutch population.<ref name=Nijenhuis/> Relative risk <math>x</math>of this allele is approximated by | |
| | |
| :<math>x=\frac{a/b}{c/d}=\frac{ad}{bc}\;(=39.7,\text{ in Table 3 })</math>. | |
| | |
| Woolf's method<ref name=Woolf>Woolf B (1955). "On estimating the relation between blood group and disease." ''Ann Hum Genet'' '''19'''(4): 251-253, PMID 14388528.</ref> is applied to see if there is statistical significance. Let
| |
| :<math>y=\ln (x)\;(=3.68)</math>
| |
| and
| |
| :<math>\frac{1}{w}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\;(=0.0703)</math>.
| |
| Then
| |
| :<math>\chi^2=wy^2\;\left [=193>\chi^2(p=0.001,\; df=1)=10.8 \right ]</math>
| |
| follows the chi-square distribution with <math>df=1</math>. In the data of Table 3, the significant association exists at the 0.1% level. Haldane's<ref name=Haldane>Haldane JB (1956). "The estimation and significance of the logarithm of a ratio of frequencies." ''Ann Hum Genet'''''20'''(4): 309-311, PMID 13314400.</ref> modification applies to the case when either of<math>a,\; b,\;c,\text{ and }d</math> is zero, where replace <math>x</math> and <math>1/w</math>with
| |
| :<math>x=\frac{(a+1/2)(d+1/2)}{(b+1/2)(c+1/2)}</math>
| |
| and
| |
| :<math>\frac{1}{w}=\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}</math>,
| |
| | |
| respectively.
| |
| | |
| {| style="float:right" class=wikitable
| |
| |+ Table 4. Association of HLA alleles with rheumatic and autoimmune diseases among white populations<ref name=Gregersen>Gregersen PK (2009). "Genetics of rheumatic diseases," ''In''Firestein GS, Budd RC, Harris ED Jr, McInnes IB, Ruddy S, Sergent JS, eds. (2009). ''Kelley's Textbook of Rheumatology,'' pp. 305-321, Philadelphia, PA: Saunders/Elsevier, ISBN 978-1-4160-3285-4.</ref>
| |
| ! Disease
| |
| ! HLA allele
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| ! Relative risk (%)
| |
| ! FAD (%)
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| ! FAP (%)
| |
| ! <math>\delta</math>
| |
| |-
| |
| ! [[Ankylosing spondylitis]]
| |
| | align="center" | B27
| |
| | align="center" | 90
| |
| | align="center" | 90
| |
| | align="center" | 8
| |
| | align="center" | 0.89
| |
| |-
| |
| ! [[Reiter's syndrome]]
| |
| | align="center" | B27
| |
| | align="center" | 40
| |
| | align="center" | 70
| |
| | align="center" | 8
| |
| | align="center" | 0.67
| |
| |-
| |
| ! [[Spondylitis]] in inflammatory bowel disease
| |
| | align="center" | B27
| |
| | align="center" | 10
| |
| | align="center" | 50
| |
| | align="center" | 8
| |
| | align="center" | 0.46
| |
| |-
| |
| ! [[Rheumatoid arthritis]]
| |
| | align="center" | DR4
| |
| | align="center" | 6
| |
| | align="center" | 70
| |
| | align="center" | 30
| |
| | align="center" | 0.57
| |
| |-
| |
| ! [[Systemic lupus erythematosus]]
| |
| | align="center" | DR3
| |
| | align="center" | 3
| |
| | align="center" | 45
| |
| | align="center" | 20
| |
| | align="center" | 0.31
| |
| |-
| |
| ! [[Multiple sclerosis]]
| |
| | align="center" | DR2
| |
| | align="center" | 4
| |
| | align="center" | 60
| |
| | align="center" | 20
| |
| | align="center" | 0.5
| |
| |-
| |
| ! [[Diabetes mellitus type 1]]
| |
| | align="center" | DR4
| |
| | align="center" | 6
| |
| | align="center" | 75
| |
| | align="center" | 30
| |
| | align="center" | 0.64
| |
| |}
| |
| In Table 4, some examples of association between HLA alleles and diseases are presented.<ref name=Gregersen/>
| |
| | |
| '''(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.<ref name=Bengtsson/><math>\delta</math> value is expressed by
| |
| :<math>\delta=\frac{FAD-FAP}{1-FAP},\;\;0\le \delta \le 1</math>,
| |
| where <math>FAD</math> and <math>FAP</math> are HLA allele frequencies among patients and healthy populations, respectively.<ref name=Bengtsson/> In Table 4, <math>\delta</math> column was added in this quotation. Putting aside 2 diseases with high relative risks both of which are also with high <math>\delta</math> values, among other diseases, juvenile diabetes mellitus (type 1) has a strong association with DR4 even with a low relative risk<math>=6</math>.
| |
| | |
| '''(2) Discrepancies from expected values from marginal frequencies in 2x2 association table of HLA alleles and disease'''
| |
| | |
| This can be confirmed by <math>\chi^2</math> test calculating
| |
| :<math>\chi^2=\frac{(ad-bc)^2 N}{ABCD}\;(=336,\text{ for data in Table 3; }P<0.001)</math>.
| |
| | |
| where <math>df=1</math>. For data with small sample size, such as no marginal total is greater than 15 (and consequently <math>N \le 30</math>), one should utilize [[Yates's correction for continuity]] or [[Fisher's exact test]].<ref name=Sokal1981>Sokal RR, Rohlf FJ (1981). ''Biometry: The Principles and Practice of Statistics in Biological Research.'' Oxford: W.H. Freeman, ISBN 0-7167-1254-7.</ref>
| |
| | |
| ==Resources==
| |
| | |
| A comparison of different measures of LD is provided by Devlin & Risch <ref>{{cite journal|author=Devlin B., Risch N. |title=A Comparison of Linkage Disequilibrium Measures for Fine-Scale Mapping|journal=Genomics|year=1995|volume=29|pages=311–322|url=http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WG1-45S9156-30-1&_cdi=6809&_user=128590&_orig=browse&_coverDate=09%2F30%2F1995&_sk=999709997&view=c&wchp=dGLbVtb-zSkzk&md5=71c2158ad4c51ae80b12a68c68814f78&ie=/sdarticle.pdf|doi=10.1006/geno.1995.9003|pmid=8666377|issue=2}}</ref>
| |
| | |
| The [[International HapMap Project]] enables the study of LD in human populations [http://www.hapmap.org/cgi-perl/gbrowse/hapmap/ online]. The [[Ensembl]] project integrates HapMap data with other genetic information from [[dbSNP]].
| |
| | |
| ==Analysis software==
| |
| * [http://pngu.mgh.harvard.edu/~purcell/plink/ld.shtml PLINK] - whole genome association analysis toolset, which can calculate LD among other things
| |
| * [http://www.stats.ox.ac.uk/~mcvean/LDhat/ LDHat]
| |
| * [[Haploview]]
| |
| * [http://www.affymetrix.com/support/developer/tools/devnettools.affx LdCompare]<ref>{{cite journal|author=Hao K., Di X., Cawley S.|title=LdCompare: rapid computation of single- and multiple-marker r2 and genetic coverage|journal=Bioinformatics|year=2007|volume=23|pages=252–254|url=http://bioinformatics.oxfordjournals.org/cgi/reprint/23/2/252|doi=10.1093/bioinformatics/btl574|pmid=17148510|issue=2}}</ref>— open-source software for calculating LD.
| |
| * [http://www.goldenhelix.com/SNP_Variation/HelixTree/ld_haplotypes.html SNP and Variation Suite]- commercial software with interactive LD plot.
| |
| * [http://www.sph.umich.edu/csg/abecasis/GOLD/index.html GOLD] - Graphical Overview of Linkage Disequilibrium
| |
| * [http://www.maizegenetics.net/index.php?option=com_content&task=view&id=89&Itemid=119 TASSEL] -software to evaluate linkage disequilibrium, traits associations, and evolutionary patterns
| |
| | |
| ==Simulation software==
| |
| * [http://haploid.nongnu.org Haploid] — a [[C (programming language)|C]] library for population genetic simulation ([[GPL]])
| |
| | |
| ==See also==
| |
| *[[Haploview]]
| |
| *[[Hardy-Weinberg principle]]
| |
| *[[Genetic linkage]]
| |
| *[[Co-adaptation]]
| |
| *[[Genealogical DNA test]]
| |
| *[[Tag SNP]]
| |
| *[[Association Mapping]]
| |
| *[[Family based QTL mapping]]
| |
| | |
| == References ==
| |
| {{Reflist|2}}
| |
| | |
| ==Further reading==
| |
| *{{cite book| first=Philip W.| last=Hedrick| title=Genetics of Populations| year=2005 |edition=3rd| isbn=0-7637-4772-6| location=Sudbury, Boston, Toronto, London, Singapore |publisher=Jones and Bartlett Publishers}}
| |
| * [http://www.nslij-genetics.org/ld/ Bibliography: Linkage Disequilibrium Analysis] : a bibliography of more than one thousand articles on Linkage disequilibrium published since 1918.
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| {{Population genetics}}
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| {{DEFAULTSORT:Linkage Disequilibrium}}
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| [[Category:Population genetics]]
| |