#Kinship

A kinship likelihood ratio weighs allele sharing by descent against sharing by chance, for a pair of people.

#IBD coefficients

A non-inbred pair's relationship is (k0, k1, k2): the probabilities of sharing 0, 1 or 2 alleles identical by descent at a locus.

Any other relationship:

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

Each coefficient must be in [0, 1] and the three must sum to 1 (within 1e-9).

HALF_SIBLINGS, GRANDPARENT and AVUNCULAR have the same coefficients and give exactly the same LR: autosomal STRs cannot separate them. The three names exist so a report can state the hypothesis. Separating them needs lineage markers, a denser panel, or more relatives in a pedigree.

#Joint probability

At one locus:

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

T1 is the probability with one of B's alleles a copy of one of A's and the other drawn from the population; its definition is in Formulae and sources. P(G) is the Hardy-Weinberg frequency at θ = 0.

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

T1 is symmetric, so the LR does not depend on which person is A; the test suite asserts this.

#Likelihood ratios

def lr as float init kinship.kinshipLR($a, $b, kinship.FULL_SIBLINGS, $db);
def lr2 as float init kinship.kinshipLRAgainst($a, $b,
    kinship.FULL_SIBLINGS, kinship.HALF_SIBLINGS, $db);
kinship.kinshipLRAt($db, $ga, $gb, $rel, $alt);   # one locus
kinship.pairLoci($db, $a, $b);                    # the loci used

kinshipLR tests against UNRELATED; kinshipLRAgainst against any alternative, which is the form most real questions take ("full or half siblings?"). Both multiply over the loci both profiles and the table share, and skip the rest without complaint, unlike randomMatchProbability. No shared locus throws. kinshipLRAt throws if the alternative gives the observed pair probability zero.

#Closed forms

At one locus, with allele frequency p:

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

A parent and child must share an allele at every locus, so one locus with none shared makes the parent/child LR zero. Full siblings need not share, so the same locus only counts against them.

#Limits

• θ = 0 throughout; see Theta.
• Non-inbred pairs only. An inbred pedigree needs the nine condensed Jacquard coefficients.
• A third person (the mother in a trio, a second sibling, an untyped link) adds information the pairwise form discards. Use Paternity or General pedigrees.