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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.

Genetic linkage analysis asks whether a genetic marker and a disease genetic locus cosegregate within families more often than would be expected under independent assortment. It uses pedigrees and Mendelian segregation to estimate the recombination fraction $\theta$, with evidence for linkage arising when the data favor $\theta <1/2$. Even when founders are in linkage equilibrium, linked loci can show familial cosegregation.

Genetic association analysis instead asks whether observed allele or genotype frequencies differ with disease status in a sampled population, often comparing cases with controls. It can detect a causal variant or a correlated genetic marker through linkage disequilibrium; population structure and selection can also create association. Linkage is about familial transmission; association is about population dependence. Neither an association alone nor a low family recombination estimate establishes that the marker itself causes disease.

Using cases as disease haplotypes and controls as control haplotypes in the model requested here, the chromosome counts estimateThese are conditional marker allele frequencies, not the reverse conditional probabilities $P(d\mid A)$. Under independent founder chromosomes, conditional on the father's $Dd,Aa$ genotype, the phase $AD/ad$ has weight proportional to $0.60\cdot 0.90=0.54$, and $Ad/aD$ has weight proportional to $0.10\cdot 0.40=0.04$. Their normalized weights are $27/29$ and $2/29$. This conditions on the model's observed parental genotypes; the omitted disease-allele frequencies cancel in the phase-weight ratio.

Using the two phase-specific pedigree likelihoods already derived,The normalizing factor is independent of $\theta$, so maximize $g(\theta )=0.54\theta -1.04{\theta }^2+0.50{\theta }^3$. Its derivative gives exactlyOnly the smaller root lies in $[0,1/2]$:On this interval $g^{\prime \prime }(\theta )=-2.08+3\theta <0$, so this stationary point is the unique maximum. The maximum LOD score, retaining the same linkage disequilibrium frequencies in numerator and denominator, isThis is only a likelihood ratio of about $1.145$ against $\theta =1/2$, hence very weak evidence. The population frequencies are treated as fixed plug-in estimates, as requested; uncertainty in those estimates and possible disease-model differences between controls and normal-allele chromosomes would matter in a fuller analysis.

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.

Suppose a disease-risk mutation first occurs on an ancestral haplotype. Descendants carrying the risk allele initially share that surrounding haplotype. Historical genetic recombination progressively replaces the flanks while retaining the disease allele, leaving linkage disequilibrium between the causal variant and some nearby genetic markers. A noncausal marker can therefore have different allele frequencies in affected individuals and suitably chosen controls. This is the basis of association mapping.

One first tests dense markers or multi-marker haplotypes for genetic association with the disease phenotype. Strongly correlated single-nucleotide polymorphisms can be represented by tag SNPs to locate an associated region efficiently. Within that region, denser genotyping and examination of disease-associated haplotypes locate historical recombination breakpoints: segments retained across independently recombined disease-bearing chromosomes are stronger candidates for containing the causal allele. Compare affected and unaffected carriers as well, since incomplete penetrance and other risk factors can prevent a perfect haplotype-disease correspondence.

Association mapping can achieve fine resolution because it uses recombination accumulated over many generations, rather than only the few observed meioses in a present-day linkage pedigree. Resolution depends on the local recombination history: a long region of strong LD may identify a disease-associated block while leaving many variants statistically indistinguishable. Additional informative haplotypes or populations with different LD can help separate them. Conversely, very weak LD requires closer marker spacing to avoid missing the causal region.

Population stratification must be controlled by ancestry-matched comparisons, statistical adjustment or an appropriate family-based association design, since ancestry can affect both disease frequency and marker frequency. A marker association alone establishes neither causality nor identity of the disease gene; nearby correlated variants must be distinguished by further genetic and functional evidence. Use LD to detect a disease-associated ancestral segment, then use historical recombinants to narrow the causal interval. The correlation between causal variants and their surrounding haplotypes, together with the value of representative markers, is illustrated in the HapMap study.

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.

Five consequences of the ancestral recombination graph are:

Shared ancestry explains correlation; recombination makes ancestry a mosaic; mutation on the resulting branches converts genealogy into observed variation. These explanations use the neutral ARG, with a variable recombination map where stated. Selection, population subdivision or changing population size can also be studied genealogically, but require extending the event model rather than being consequences of the constant-size panmictic neutral process alone.

The pedigree founder factor $P(G_i)$ depends on population allele and haplotype frequencies, ancestry, mating assumptions and any population linkage disequilibrium. At one autosomal genetic locus, Hardy-Weinberg equilibrium with allele frequencies $p,q$ gives genotype probabilities $p^2,2pq,q^2$. For a specified ordered pair of independent pedigree founder haplotypes $h_1,h_2$, the probability is $p_{h_1}{p}_{h_2}$; an unordered distinct pair has twice that probability. Pedigree founders assumed to be related require a joint prior rather than a product of unrelated-founder factors.

The nonfounder factor depends on the parental genotypes and phases, Mendelian segregation, and the recombination fractions between genetic loci. In a two-locus heterozygote of known phase, the two parental haplotypes have transmission probabilities $(1-\theta )/2$ each, and the two recombinant haplotypes $\theta /2$ each. Multiply the maternal and paternal gamete probabilities and sum any gamete combinations producing the child's genotype. Sex-specific genetic recombination, mutation or segregation distortion can be incorporated by changing these gamete probabilities. At one ordinary autosomal genetic locus, each parental copy is transmitted with probability $1/2$.

Genetic linkage analysis follows co-segregation of genetic marker and disease alleles within pedigrees. It estimates or tests a recombination fraction, with independent assortment at $\theta =1/2$ as the ordinary null. It can locate a disease genetic locus even when different families have different genetic marker alleles on their disease chromosomes.

Genetic association analysis tests whether allele or genotype frequencies differ with disease status. It can compare unrelated cases and controls, or compare transmitted and untransmitted parental alleles. A noncausal genetic marker can be associated through population linkage disequilibrium with a causal genetic locus. Association therefore concerns population-frequency relationships, whereas linkage concerns inheritance; population stratification can confound the former without producing genuine within-family linkage.

An unphased multilocus genotype can have several pairs of haplotypes. Its pedigree likelihood must average over these possible phases with prior weights ${\pi }_s$, rather than maximize over phases after observing the offspring. Under linkage disequilibrium, the weights need not be equal. They can be obtained by conditioning population haplotype frequencies on the observed parental genotype; different choices of what is conditioned on change only parameter-independent factors when those frequencies are held fixed.

A population bottleneck is a temporary large reduction in population size. It changes ancestral coalescence rates and can leave fewer ancestral haplotypes and increased linkage disequilibrium after the population expands.

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A recombination hotspot is a short region with an elevated local rate of genetic recombination. Repeated ancestral breakpoints there can separate neighboring haplotype blocks of strong linkage disequilibrium.

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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.

A tag SNP represents other nearby single-nucleotide polymorphisms whose alleles are strongly correlated through linkage disequilibrium. Tagging reduces the number of markers needed for association mapping; it does not establish that the tag itself causes the phenotype.

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