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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 isThe second expression follows by substituting $f_{ab}=1-f_{AB}-f_{Ab}-f_{aB}$. The entire two-locus joint probability distribution is thenEvery 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.

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 giveThus $|D|\le 1/4$ globally, but a particular pair of allele frequencies usually permits a smaller, asymmetric interval. The standardized linkage disequilibrium convention used here retains the sign: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 $|D^{\prime }|$. At $D^{\prime }=1$ 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 $X=1_A$ and $Y=1_B$, their covariance is $D$, and their variances are $p(1-p)$ and $q(1-q)$. The allelic correlation measure is their squared correlation coefficient,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 $D^{\prime }=0.75$ and $-0.50$, respectively. For another original example, take $p=0.10,q=0.50,f_{AB}=0.10$: then $D=0.05,D^{\prime }=1$, 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 $|D^{\prime }|=1$ 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 $r_{\mathrm{LD}}^2\ge 0.8$. 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.^[ref]

Genetic recombination erodes linkage disequilibrium by reshuffling alleles between haplotypes, while preserving marginal allele frequencies in the idealized model. Let $f_{AB,t}$ denote the haplotype frequency in the gamete pool of generation $t$, let $p,q$ denote its marginal allele frequencies, and let $r\in [0,1/2]$ be the constant recombination fraction between two autosomal genetic loci. Assume random mating within one population, nonoverlapping generations, fair Mendelian segregation, equal reciprocal gamete production, the same $r$ in both sexes, and a deterministic population large enough to neglect genetic drift. Exclude natural selection, mutation and gene flow. ThenHere $p$ and $q$ remain constant: recombining changes which alleles occur together, not their total numbers. The other haplotype frequencies change by $+r{D}_t$ for $Ab,aB$ and $-r{D}_t$ for $ab$. Consequently the joint probability distribution stays within its allowed bounds and approaches the product of its marginal distributions when $r>0$.

For the original positive example $p=0.60,q=0.40,D_0=0.12$, a recombination fraction $r=0.10$ gives $D_1=0.108$ and $D_{10}\approx 0.04184$. The negative example $D_0=-0.12$ has exactly the opposite sequence: genetic recombination reduces the magnitude without reversing the sign. With $r=0$, the initial linkage disequilibrium persists in this model; even independent assortment at $r=1/2$ only halves $D$ each generation. A population association does not vanish after one generation merely because the genetic loci are unlinked.

The geometric sequence gives a generation scale: for $r>0$, the continuous interpolation reaches $|D_0|/2$ at $t_{1/2}=\log (1/2)/\log (1-r)$. Actual generations are integers; the first at or below half is $\lceil t_{1/2}\rceil$. For small $r$, $t_{1/2}\approx (\log 2)/r$. Because the denominator of allelic correlation stays fixed, $r_{\mathrm{LD},t}^2=(1-r{)}^{2t}{r}_{\mathrm{LD},0}^2$; its decay is faster than the magnitude of $D_t$.

A short derivation uses random mating to assign frequency $2f_{AB}{f}_{ab}$ to the double heterozygote $AB/ab$ and $2f_{Ab}{f}_{aB}$ to $Ab/aB$. Relative to $r=0$, each loses or gains an $AB$ gamete fraction $r/2$, respectively. Their net contribution is $r(f_{Ab}{f}_{aB}-f_{AB}{f}_{ab})=-rD$. The other genotypes produce no net change in $f_{AB}$, proving the recurrence and, by iteration, its solution.

This temporal model does not identify elapsed generations from physical separation alone. The recombination fraction depends on genetic recombination, including recombination hotspots, while population stratification, natural selection, gene flow or genetic drift can create or maintain linkage disequilibrium. Repeated sampling across time would also require accounting for sample size and haplotype estimation. A cross-sectional plot of association against distance between genetic loci is therefore a different observation from the idealized trajectory plotted here.

An observed genetic marker tags an unobserved causal genetic variant when linkage disequilibrium correlates their allele indicators. Their squared allelic correlation $r^2$ measures prediction strength. Under standard local-alternative score-test approximations, using the marker rather than the causal allele reduces the noncentrality parameter by roughly $r^2$, suggesting a sample-size multiplier about $1/r^2$. This is an approximation requiring the same disease model and sampling design, not a universal power identity.

Linkage disequilibrium means that alleles at different genetic loci are associated in population haplotypes. For two biallelic loci,The first quantity measures departure from independent allele frequencies; the second is the squared allelic correlation. Standardized linkage disequilibrium $D^{\prime }$ instead compares $D$ to its possible range at the given frequencies. High $|D^{\prime }|$ can coexist with low $r^2$, so it need not make one genetic marker a good predictor of another. Genetic linkage is a different notion: it describes within-family transmission and the recombination fraction.

The chromosome-21 study of Patil and colleagues, Science 2001 found consecutive single-nucleotide polymorphisms organized into regions with comparatively few common haplotypes; typically three haplotypes accounted for over 80% of the sampled chromosomes within a block. Such haplotype blocks can be represented efficiently by tag SNPs. Strong association within blocks and weaker association across boundaries give a useful description, though block boundaries depend on allele frequencies, population history, sampling and the operational definition of a block. They do not divide chromosomes into regions where genetic recombination is respectively impossible and certain.

The ancestry explanation uses an ancestral recombination graph. Going backwards, ancestral lineages coalesce, while a genetic recombination event splits one lineage's ancestral material into contributions from two parents. At a fixed genomic position, retaining only its ancestry gives a phylogenetic tree with the appropriate coalescent law. Nearby positions often retain much of the same tree; mutations on shared branches then produce correlated allele states. Breakpoints can change the local tree, weakening that shared history.

Under a simple random-mating two-locus model, linkage-disequilibrium decay under recombination gives $D_{t+1}=(1-r)D_t$. Actual populations also undergo genetic drift, population bottlenecks, mutation and migration, which can create or preserve association. Recombination rates themselves vary along chromosomes: McVean and colleagues, Science 2004 inferred strong local variation from sequence data. These observations motivate an ancestry model with spatially varying recombination, rather than treating physical distance as the only determinant of linkage disequilibrium.

For fine mapping, a typed genetic marker can show genetic association because it is correlated with a nearby causal genetic variant. Linkage-disequilibrium marker tagging makes a sparse first-stage scan possible, but several variants in one block can produce nearly indistinguishable association patterns. Historical recombination and diverse local genealogies can narrow the interval when denser genotyping is available.

An ancestral recombination graph turns that intuition into a likelihood calculation. One can place a proposed disease-origin event on a local ancestral branch and integrate over possible graphs, mutation histories and unobserved haplotypes. The local phylogenetic tree then relates disease-bearing chromosomes and predicts which markers should remain associated. In a Bayesian inference, uncertainty in ancestry, mutation location and demographic parameters enters the posterior distribution rather than being replaced by one guessed tree. Whole-graph inference is computationally demanding, and approximate or pairwise methods often supply practical estimates.

The useful distinction is between finding an associated region and identifying its causal variant. Strong linkage disequilibrium improves marker coverage but can limit resolution inside the associated region. Population stratification, ascertainment, rare variants and misspecified population history can also mislead an association analysis. Combining fine mapping with independent biological evidence is therefore stronger than interpreting an association peak as proof of causation.

Across a human chromosome, neighboring single-nucleotide polymorphisms often show substantial linkage disequilibrium, while average allelic correlation tends to decline with increasing genetic distance. A new mutation starts on one ancestral haplotype; repeated genetic recombination separates it from progressively more distant markers. In an ideal randomly mating two-locus population without further mutation, migration or selection, gamete formation gives $p_{11}^{\prime }=(1-c)p_{11}+c{p}_A{p}_B$, where $c$ is the recombination fraction. The marginal allele frequencies remain fixed, soHere the prime means the next-generation value, not a separately standardized LD statistic. This simple decay explains the distance trend, rather than asserting an exact deterministic law for real finite populations.

The local pattern is heterogeneous. Many regions have haplotype blocks with strong LD and a small number of common haplotypes, separated by intervals showing more historical recombination, often recombination hotspots. A hotspot can therefore cause a rapid drop in LD between markers that are close in physical distance. Block lengths and boundaries are not uniform, perfectly sharp or independent of the population and the statistical block definition. These empirical features are documented in the HapMap data.

Population history also matters. Population bottlenecks can reduce the number of ancestral haplotypes and increase the extent of LD; population stratification or admixture can create longer-range associations, including between distant loci. Different populations can consequently have different haplotype frequencies and LD patterns at the same genomic interval. Raw $D$ also varies with allele frequency, so comparing LD across markers requires attention to the chosen measure. The typical pattern is locally strong, block-like and highly variable LD, with a broad decline over genetic distance rather than a uniform chromosome-wide decay.

Let $A\in \{0,1\}$ and $B\in \{0,1\}$ encode the alleles at two loci on a randomly chosen population haplotype. Write $p_{ab}=P(A=a,B=b)$, $p_A=p_{10}+p_{11}$ and $p_B=p_{01}+p_{11}$. The linkage disequilibrium coefficient isThe determinant identity follows by substituting the marginal frequencies and $\sum p_{ab}=1$. All four haplotype frequencies can be recovered asThus $D=0$ is precisely independence of the two allelic states. Its sign depends on which alleles are labeled one, and its possible magnitude depends on the marginal frequencies.

One other measure is squared allelic correlation. For segregating loci, $0<p_A,p_B<1$, their indicator-variable covariance is $D$, and their variances are $p_A(1-p_A)$ and $p_B(1-p_B)$. HenceThis is the squared correlation, useful for measuring how well one marker predicts the other in association mapping. It is undefined if a locus is monomorphic. Linkage disequilibrium describes population haplotype frequencies; genetic linkage describes transmission within meioses. Physically linked loci can have $D=0$, and population mixing can produce LD even between unlinked loci.

For positive linkage disequilibrium $D$, take $D_{\max }=\min (p_A(1-p_B),(1-p_A)p_B)$; for negative $D$, take $D_{\max }=\min (p_A{p}_B,(1-p_A)(1-p_B))$ and retain its negative sign in $D^{\prime }$. Thus $|D^{\prime }|\le 1$. Its magnitude measures the fraction of the largest disequilibrium possible at the specified allele frequencies. High $|D^{\prime }|$ need not imply high allelic correlation, so it does not alone guarantee good marker prediction.