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An affected-sib-pair analysis examines marker identity by descent between affected full siblings. Without genetic linkage to the disease locus, the probabilities of sharing zero, one and two copies are $1/4,1/2,1/4$. Excess sharing supplies evidence for linkage under disease models that preferentially transmit a common risk allele to both siblings.

// Target: biology.bigb

Full siblings share both parents. With unrelated, noninbred founders in a pedigree, the two parental contributions give independent opportunities for identity by descent.

// Target: biology.bigb

Homozygosity means that the two allele states at a diploid genetic locus are the same; an individual with that property is a homozygote. Observed homozygosity does not by itself establish identity by descent.

A homozygote carries two identical alleles at a locus; a heterozygote carries different alleles. Homozygosity of an observed haplotype is identity by state, and need not be identity by descent.

Two allele copies are identical by state when their observed states agree, regardless of whether they derive from one recent ancestral copy. Equal marker haplotypes can therefore reflect either identity by descent or matching founder haplotypes.

For two individuals carrying labelled copies $(a,b)$ and $(c,d)$, condensed identity coefficients give the probabilities of nine identity by descent patterns, identifying states that differ only by exchanging the two homologues within either individual. They retain within-individual autozygosity, which three ordinary outbred sharing coefficients omit. One convention isEach block consists of copies identical by descent, and different blocks have distinct ancestral origins. For siblings whose parents are otherwise outbred first cousins, these coefficients are $(1,0,2,1,2,1,15,30,12)/64$. This follows by conditioning on whether the parents share one ancestral copy and enumerating the four independent Mendelian segregation choices to their two children. The kinship coefficient is ${\Delta }_1+({\Delta }_3+{\Delta }_5+{\Delta }_7)/2+{\Delta }_8/4$, here $9/32$. Weir's discussion of identity states uses this convention.

For independent pairs of affected full siblings with unrelated, noninbred parents and a fully informative genetic marker, the no-linkage null gives shared identity by descent count $J\sim \mathrm{Bin}(2,1/2)$. Therefore $EJ=1$ and $\mathrm{Var}J=1/2$. An excess-sharing test uses the displayed approximately standard normal statistic and an upper-tail critical value. Pairs sharing family members are dependent, so their variance needs separate treatment. This targeted test differs from an omnibus test of all three sharing probabilities.

Nonparametric linkage analysis tests excess identity by descent among relatives with a shared phenotype, without specifying its complete penetrance model or disease-allele frequency. Its null still uses Mendelian segregation and the specified pedigree.

// Target: biology.bigb

For calculations of identity by descent, an outbred pedigree has unrelated, noninbred founders in a pedigree, and no additional ancestral relationships beyond those explicitly specified. Founder gene copies are treated as distinct ancestral copies even if they have identity by state.

// Target: biology.bigb