Page 105 - Applied Probability

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5. Genetic Identity Coefficients
Boundary Condition 2 If two founders occur in the same block, then
again Φ = 0. This condition follows because founders are by definition
unrelated.
Boundary Condition 3 If only founders contribute sampled genes and
neither boundary condition 1 nor boundary condition 2 pertains, then
Φ= ( ) , where m 1 is the total number of sampled founder
1 m 1 −m 2
2
genes over all blocks, and m 2 is the total number of founders sampled.
To verify this condition, imagine choosing one initial gene for each
founder involved in Φ. Since a founder cannot be inbred, a subsequent
gene chosen for him or her must coincide with the initial choice if
the two genes contribute to the same block. If they contribute to
two different blocks, the subsequent gene must differ from the initial
gene. In either case, the correct choice is made with probability 1
2
independently of other choices.
Recurrence Rules
Suppose i is a nonfounder involved in a kinship coefficient Φ. The three
recurrence rules operate by substituting genes sampled from i’s parents
j and k for genes sampled from i. It is required that no person involved
in Φ be a descendant of i. According to our numbering convention, this
requirement can be met by taking i to be the highest-numbered person in
Φ. The form of the recurrence rules depends on whether i belongs to one
or two blocks. In the former case, suppose without loss of generality that
i occupies the first part of the first block. In the latter case, suppose that
i occupies the first parts of the first two blocks. It is noteworthy that all
three recurrence rules preserve or diminish the number of sampled genes
involved in the replacement kinship coefficients relative to the number of
sampled genes involved in the current kinship coefficient. In stating the
rules, we let G i , G j , and G k denote randomly sampled genes from i, j, and
k, respectively.

Recurrence Rule 1 Assume that only one gene G i is sampled from i.
Then
1
Φ[{G i ,...}{} ...{}]= Φ[{G j ,...}{} ...{}]
2
1
+ Φ[{G k ,...}{} ...{}].
2
This rule follows because the gene drawn at random from i is equally
likely to be a gene drawn at random from either j or k.

1
s
Recurrence Rule 2 Assume that the genes G ,... ,G are sampled from
i
i
i for s> 1. If these genes occur in one block, then
1
s
Φ[{G ,...,G ,...}{} ...{}]
i
i