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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.

Without genotyping error, the observed genetic marker genotype is a deterministic function $u(g)$ of the underlying phase-known genotype. In particular $u$ can forget chromosome phase and parental origin. ThereforeSeveral different phased haplotype pairs can receive factor 1 when the observed multilocus genotype is unphased. With laboratory error, missing alleles or uncertain calls, replace the indicator by an error or observation model. A wholly unobserved genetic marker contributes factor 1. Thus a 0-or-1 factor reflects perfect observation of a deterministic genetic marker state, not a general property of every genetic measurement.

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 $1/2$.

Family comparisons condition on the parental genotypes, so cases are compared with the alleles they could have inherited. This protects a transmission test from ordinary population stratification, avoids having to recruit unrelated controls matched for ancestry, and can resolve haplotypes and identify Mendelian inconsistencies. The same pedigree may also provide linkage information.

The costs are recruiting and genotyping relatives, missing or unavailable parents, and fewer informative comparisons because homozygous parents contribute no transmission contrast. These problems are especially important for late-onset disease. Related cases or multiple siblings also require an analysis accounting for dependence, and genotyping error or transmission distortion can mimic overtransmission. Population-based case/control studies are usually easier to assemble and can offer greater power per genotyped person, but need careful control of ancestry and other sources of confounding. Family sampling does not remove every form of ascertainment in a genetic study or observation bias.

Write $D$ for the disease observations and $M$ for the genetic marker observations. The LOD score for complete linkage isThe disease marginal cancels because the single-locus disease segregation law does not depend on the genetic marker–disease recombination fraction. The ten genetic markers form one nonrecombining haplotype; they do not supply ten independent copies of the pedigree's segregation evidence.

There is a distinction between identity by state and identity by descent. The reported observation that affected subjects all have $h/h$ establishes the former. An exact genetic marker likelihood requires pedigree founder haplotype frequencies, or the actual pedigree founder genotypes if conditioning on them. Here is an explicit expression using independent pedigree founder haplotypes, with frequency $p$ for the observed haplotype $h$, and linkage equilibrium between pedigree founder genetic marker and disease states. There are eight genetic marker copies in the four pedigree founders: the original couple and the two generation-2 outside spouses. For genetic marker inheritance configuration $v$, let $K(v)$ be the number of distinct pedigree founder copies ancestral to the sixteen chromosome copies in the eight selected subjects. Thenis the unlinked probability that all those copies have state $h$. At complete linkage the affected subjects all inherit the disease pedigree founder copy twice, and the probability that this copy has genetic marker state $h$ is $p$. Therefore

The denominator can be evaluated without an unspecified inheritance sum. Let $B_a=\left(\genfrac{}{}{0ex}{}{2}{a}\right)p^a(1-p{)}^{2-a}$ for $a=0,1,2$. If parental genetic marker genotypes contain $a,b$ copies of $h$, the child count distribution isFor a generation-2 couple defineFor a particular generation-3 parent with $j$ genetic marker copies, the chance it transmits $h$ to all its $r$ selected offspring is $(j/2{)}^r$, which explains each factor. Conditional on the original founding couple, its two generation-2 siblings and their independent outside spouses give independent left and right contributions. ThusExpanding givesIt satisfies $R(1)=1$, as it must for a monomorphic haplotype. This expression supplies a numerical LOD once $p$ is specified; the problem does not specify that frequency.

In the idealized limit where a matching rare haplotype identifies one ancestral copy, the leading coefficient is particularly simple. Dropping restrictions on the eight unaffected subjects, the probability that all selected subjects are autozygous for one common copy isAccordingly $R(p)/p\to 2^{-22}$ as $p\to 0$, and the affected-only IBD calculation gives $Z(0)=22{\log }_{10}2\simeq 6.623$.

If the genetic marker observations additionally establish the exact pattern $E$ of part (d), including that the eight unaffected subjects are not autozygous, its idealized IBD likelihood ratio is insteadThis latter value requires the additional negative genetic marker observations. It cannot be inferred merely from homozygosity in the affected subjects, and neither simplified IBD score is an exact identity-by-state likelihood for an unspecified common haplotype.

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.

At several genetic loci, a phase-known genotype specifies which alleles occur together on each chromosome, hence its two haplotypes. An unphased multilocus genotype may be compatible with several haplotype pairs. For two heterozygous loci, $AB/ab$ and $Ab/aB$ have identical unphased genotypes but different transmission probabilities when genetic linkage is present.

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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For unaffected genetic carriers of a disease with complete penetrance and recessive inheritance, suppose parental marker-disease haplotype phases are equally likely and the marker is fully informative. Write $u=\theta (1-\theta )$ and $v=1/2-u$, where $\theta$ is the recombination fraction. Conditional on sharing a parental marker copy, the probability that both siblings inherit that parent's disease allele is $[(1-\theta {)}^2+{\theta }^2]/2=v$; conditional on inheriting different marker copies it is $\theta (1-\theta )=u$. Independent paternal and maternal transmissions therefore give $L_0=u^2,L_1=uv,L_2=v^2$. Conditional instead on both siblings being affected, the IBD probabilities are $s^2,2s(1-s),(1-s{)}^2$, where $s=2u\in [0,1/2]$.

// Target: biology.bigb

The recombination fraction is the probability that a transmitted gamete carries a recombinant haplotype between two specified genetic loci, relative to the parent's known haplotype phase. Write it as $r$ here; the symbol $\theta$ is also common in genetic linkage analysis. In the ordinary symmetric two-locus model, $0\le r\le 1/2$, with $r=0$ for complete genetic linkage and $r=1/2$ for independent assortment.

For phase-known genotype $AB/ab$, fair Mendelian segregation and equal reciprocal gamete probabilities giveFor example, 1,000 independent informative transmissions with observed counts $400,400,100,100$ in this order have empirical recombination fraction $(100+100)/1000=0.2$. Those illustrative counts match the expected values in the toy model, but actual counts fluctuate. The genetic linkage diagram shows both the parental and recombinant classes, so their separate probabilities are not confused with the total $r$.

The recombination fraction measures the probability of an odd number of origin switches along the transmitted chromatid. It is neither the expected value of the exchange count nor the probability of at least one exchange: even positive counts can be invisible at the two endpoints. Converting an observed recombination fraction to a genetic map distance therefore needs an explicit model such as the Haldane mapping function. The distinction between transmitted-chromatid counts and four-chromatid exchange counts is discussed in Weeks, Tang and Kwon (2009), Casares' map function: No need for a 'corrected' Haldane's map function, Genetica 135, 305–307.