#Kinship

Two people share alleles for two reasons: coincidence, and common ancestry. A kinship likelihood ratio separates them.

#IBD coefficients

For a pair of non-inbred people, an entire relationship reduces to three numbers - the probability that at a given locus they share 0, 1 or 2 alleles identical by descent:

A parent and child always share exactly one allele by descent - hence (0, 1, 0). Full siblings inherit each parental allele independently, giving the quarter/half/quarter split.

Other relationships:

def dfc as kinship.Relationship init kinship.relationship(
    "double first cousins", 0.5625, 0.375, 0.0625);

The three must be in [0, 1] and sum to 1.

#Three relationships that look identical

HALF_SIBLINGS, GRANDPARENT and AVUNCULAR have the same coefficients, and therefore produce exactly the same likelihood ratio. Autosomal STRs cannot tell a half-brother from an uncle from a grandfather. That is a property of the markers, not a shortcut in this code - the three constants exist so the hypothesis can be named honestly in a report, not because the numbers differ.

Separating them needs lineage markers, a denser panel, or the extra structure a pedigree supplies.

#The joint probability

At one locus, for genotypes Ga and Gb:

P(Ga, Gb) = k0 · P(Ga) P(Gb)
          + k1 · T1(Ga, Gb)
          + k2 · [Ga == Gb] · P(Ga)

The first term is independence - no shared allele. The third is identity - both shared, so the genotypes must be equal. The middle term T1 is the interesting one: one of B's alleles is a copy of one of A's, and the other is drawn from the population.

kinship.jointGenotypeProbability($db, $ga, $gb, kinship.FULL_SIBLINGS);

T1 is symmetric in its arguments, so the LR does not depend on which person you call A - and the test suite asserts that rather than assuming it.

#Likelihood ratios

def lr as float init kinship.kinshipLR($a, $b, kinship.FULL_SIBLINGS, $db);

tests the relationship against being unrelated, multiplying over every locus the two profiles and the database share. Above 1 favours the relationship; below 1 favours unrelatedness.

To test one relationship against another rather than against unrelatedness:

def lr as float init kinship.kinshipLRAgainst($a, $b,
    kinship.FULL_SIBLINGS, kinship.HALF_SIBLINGS, $db);

That is the form most real questions take. "Are they full siblings or half siblings" is a genuine question; "are they full siblings or complete strangers" usually is not.

Per locus:

kinship.kinshipLRAt($db, $ga, $gb, $rel, $alt);

Which loci contributed:

kinship.pairLoci($db, $a, $b);

Unlike randomMatchProbability, kinship uses the intersection without complaint - two people's panels genuinely differ, and there is no other sensible reading. Print pairLoci if the count matters.

#Numbers worth recognising

With allele frequency p, at one locus:

At p = 0.1, two people both homozygous 10/10 give a sibling LR of (1 + 10)²/4 = 30.25. The test suite checks exactly these closed forms.

Note the last row. Parent and child must share an allele at every locus, so a locus where they share none drives the LR to zero - a single-locus exclusion. Full siblings need not, so the same observation only counts against them.

#Theta is not applied

Kinship uses the plain product rule, θ = 0. The sub-population correction does not carry into the T1 term as a simple substitution, and applying it to some terms and not others would produce a number that answers no question. See Theta and Limitations.

#Inbreeding

The three IBD coefficients describe non-inbred pairs. An inbred pedigree needs the nine condensed Jacquard coefficients, which this pairwise form does not carry. A consanguineous pedigree is also rejected by pedigree.j rather than mis-summed.

#When pairwise is not enough

As soon as a third person is involved - a mother in a paternity trio, a second sibling, an untyped link between two typed people - the pairwise form throws away information. Those cases go to Paternity and General pedigrees, both of which condition on everyone at once.

#Related

• Paternity - the trio special case
• General pedigrees - three or more relatives
• kinship reference