= Linkage disequilibrium
{title2=$D=P(AB)-P(A)P(B)$}
{wiki}
= LD
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{synonym}
<Linkage disequilibrium> means that knowing an <allele> at one <genetic locus> changes the <probability> of an <allele> at another <genetic locus> in population <haplotypes>. It compares their <joint probability distribution> with the product of their <marginal distributions>. This is useful because a <genetic marker> can carry information about another <genetic locus>: <tag SNPs> exploit this redundancy, without establishing a causal <genetic association>.
Consider two <genetic loci> with <alleles> $A,a$ and $B,b$. Sample one <haplotype>, rather than an unphased <diploid> <genotype>. Write its four <haplotype frequencies> as $f_{AB},f_{Ab},f_{aB},f_{ab}$, summing to one, and set the marginal <allele frequencies> to $p=f_{AB}+f_{Ab}$ and $q=f_{AB}+f_{aB}$. The <linkage disequilibrium> coefficient is
$$
D=f_{AB}-pq=f_{AB}f_{ab}-f_{Ab}f_{aB}.
$$
The second expression follows by substituting $f_{ab}=1-f_{AB}-f_{Ab}-f_{aB}$. The entire two-locus <joint probability distribution> is then
$$
\begin{array}{c|cc|c}
&B&b&\text{total}\\\hline
A&pq+D&p(1-q)-D&p\\
a&(1-p)q-D&(1-p)(1-q)+D&1-p\\\hline
\text{total}&q&1-q&1
\end{array}
$$
Every row and column sum is unchanged when $D$ varies. <Linkage equilibrium> means $D=0$: all four <haplotype frequencies> equal products of <allele frequencies>, so the two <allele> indicators are <independent random variables>. For $p>0$, the <conditional probability> $P(B\mid A)=q+D/p$ shows directly how positive $D$ enriches $B$ among copies carrying $A$, while negative $D$ depletes it. Positive $D$ also enriches $ab$; negative $D$ enriches $Ab$ and $aB$. Swapping the labels $A$ and $a$ reverses the sign of $D$, so neither sign is intrinsically favorable.
For example, fix the <allele frequencies> at $p=0.60,q=0.40$. At <linkage equilibrium>, the <haplotype frequencies> in the order $AB,Ab,aB,ab$ are $(0.24,0.36,0.16,0.24)$. Positive <linkage disequilibrium> $D=0.12$ instead gives $(0.36,0.24,0.04,0.36)$, with $P(B\mid A)=0.60$; negative <linkage disequilibrium> $D=-0.12$ gives $(0.12,0.48,0.28,0.12)$, with $P(B\mid A)=0.20$. Counting each <allele> separately would miss the difference.
\Image[/haplotype-joint-frequencies-fixed-marginals.png]
{title=Three <haplotype> <joint probability distributions> with identical marginal <allele frequencies>, showing negative <linkage disequilibrium>, <linkage equilibrium> and positive <linkage disequilibrium>}
Nonnegative <haplotype frequencies> constrain $D$. Define the nonnegative bounds $L=\min(pq,(1-p)(1-q))$ and $U=\min(p(1-q),(1-p)q)$. The four entries above give
$$
-L\le D\le U,\qquad \max(0,p+q-1)\le f_{AB}\le\min(p,q).
$$
Thus $|D|\le1/4$ globally, but a particular pair of <allele frequencies> usually permits a smaller, asymmetric interval. The <standardized linkage disequilibrium> convention used here retains the sign:
$$
D'=\begin{cases}D/U&D>0,\\0&D=0,\\D/L&D<0,\end{cases}
\qquad -1\le D'\le1.
$$
This convention requires $0<p,q<1$; if either <genetic locus> has only one <allele> in the population, $D=0$ but normalized measures are undefined. Some presentations report only $|D'|$. At $D'=1$ or $-1$, at least one of the four <haplotypes> has zero frequency; this alone does not prove that <genetic recombination> has never occurred.
For the <indicator random variables> $X=\mathbf{1}_{A}$ and $Y=\mathbf{1}_{B}$, their <covariance> is $D$, and their <variances> are $p(1-p)$ and $q(1-q)$. The <allelic correlation> measure is their squared <correlation coefficient>,
$$
r_{\mathrm{LD}}^2=\frac{D^2}{p(1-p)q(1-q)},\qquad
0\le r_{\mathrm{LD}}^2\le\frac{\max(L^2,U^2)}{p(1-p)q(1-q)}\le1.
$$
The subscript distinguishes it from the <recombination fraction> $r$ in <linkage-disequilibrium decay under recombination>. Squaring loses the association's sign. In the illustrated positive and negative examples, $r_{\mathrm{LD}}^2=0.25$ in both, whereas $D'=0.75$ and $-0.50$, respectively. For another original example, take $p=0.10,q=0.50,f_{AB}=0.10$: then $D=0.05,D'=1$, yet $r_{\mathrm{LD}}^2=1/9$. Every $A$ carries $B$, but most copies carrying $B$ lack $A$. The frequency constraints and the distinction between these normalizations are analyzed in <Mathematical properties of the r² measure of linkage disequilibrium by Jenna M. VanLiere and others> (sections 2.1 and 2.5, pp. 131–132 and 135–136).
<Genetic linkage> describes transmission through <meiosis>; <linkage disequilibrium> describes a population's <haplotype frequencies>. Nearby <genetic loci> on a <chromosome> can be in <linkage equilibrium>. Conversely, mixing populations can create <linkage disequilibrium> even between <genetic loci> on different <chromosomes>. For example, mix equal numbers of independently sampled <haplotypes> from two populations, each internally at <linkage equilibrium>, with $p=q=0.80$ in one and $p=q=0.20$ in the other. The pooled <allele frequencies> are $p=q=0.50$, but $f_{AB}=(0.64+0.04)/2=0.34$, giving $D=0.09$. This is a consequence of <population stratification>, not a measurement of the <recombination fraction>.
Measured <haplotype frequencies> also depend on the sample. Finite <sample size>, related individuals, <genetic-study ascertainment>, <genotyping error> and uncertain <haplotype estimation> can change an estimate. A missing rare <haplotype> can give empirical $|D'|=1$ without demonstrating its absence from the population. Report the sample definition and <allele frequencies> alongside the association measure; a nonzero sample estimate alone does not establish nonzero population <linkage disequilibrium>.
A concrete dataset is the 2005 <A haplotype map of the human genome by The International HapMap Consortium>. The International HapMap Consortium used <genotyping> platforms including Illumina BeadArray to obtain <SNP> <genotype> calls from 269 <DNA> samples across four collections: 30 parent–offspring trios from Ibadan, 30 trios from Utah, 45 unrelated individuals from Beijing and 44 from Tokyo. Parent–offspring trios and the PHASE algorithm supplied <haplotype estimation>, rather than direct observation of every <chromosome>. In Figure 7, 36 <SNPs> resolve seven <haplotypes> in a <chromosome> 2 region in the Utah panel; seven <tag SNPs> capture the 36 <SNPs> at $r_{\mathrm{LD}}^2\ge0.8$. The readout demonstrates reusable <allelic correlation>, not causal effects.
Controls required over 80% complete calls, at most one discordance among five duplicate <genotypes>, at most one <Mendelian segregation> error per trio panel, and a <Hardy-Weinberg equilibrium> test with $P>0.001$; blind reassessment checked performance. Minor <allele frequencies> of at least 0.05 were targeted, the collections were not population-representative, and inferred <haplotypes> retain phase uncertainty. These limits restrict extrapolation of <linkage disequilibrium>.<A haplotype map of the human genome by The International HapMap Consortium>[pp. 1300–1302, Table 1; Figure 7, p. 1306; Methods, p. 1316]{ref}
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