Page 104 - Applied Probability

5. Genetic Identity Coefficients
88
1
=
Ψ 8
∆ 8
4
1
∆ 9 .
=
Ψ 9
4
It is obvious that the matrix of coefficients appearing on the right of
(5.2) is upper triangular. This allows us to backsolve for the ∆’s in terms
of the Ψ’s, beginning with ∆ 9 and working upward toward ∆ 1 . The result
is
1 1 1 1
∆ 1 =Ψ 1 − Ψ 3 − Ψ 5 + Ψ 7 + Ψ 8
2 2 2 4
1 1 1 3
∆ 2 =Ψ 2 − Ψ 3 − Ψ 4 − Ψ 5 − Ψ 6 + Ψ 7 + Ψ 8 +Ψ 9
2 2 2 4
∆ 3 =2Ψ 3 − 2Ψ 7 − Ψ 8
∆ 4 =2Ψ 4 − Ψ 8 − 2Ψ 9
(5.3)
∆ 5 =2Ψ 5 − 2Ψ 7 − Ψ 8
∆ 6 =2Ψ 6 − Ψ 8 − 2Ψ 9
∆ 7 =4Ψ 7
∆ 8 =4Ψ 8
∆ 9 =4Ψ 9.
It follows that one can compute all of the condensed identity coefficients
∆ 1 ,..., ∆ 9 by computing the coefficients Ψ 1 ,... , Ψ 9 . The algorithm de-
veloped in the next section for calculating generalized kinship coefficients
immediately specializes to calculation of the Ψ’s.
5.6 Calculation of Generalized Kinship Coefficients
Generalized kinship coefficients (kinship coefficients for short) can be com-
puted recursively by a straightforward algorithm having two phases [7, 9,
14, 17]. In the recursive phase of the algorithm, a currently required kinship
coefficient is replaced by a linear combination of subsequently required kin-
ship coefficients. This replacement is effected by moving upward through a
pedigree and substituting randomly sampled parental genes for randomly
sampled offspring genes. In the static phase of the algorithm, boundary
kinship coefficients involving only randomly sampled genes from founders
are evaluated. Not surprisingly, the algorithm is reminiscent of our earlier
algorithm for computing ordinary kinship coefficients. We again assume the
members of a pedigree are numbered so that parents precede their offspring.
Boundary Conditions
Boundary Condition 1 If a founder, or indeed any person, is involved in
three or more blocks, then Φ = 0. This condition is obvious because
a person has exactly two genes at a given autosomal locus.