Page 101 - Applied Probability

no genes are i.b.d. Several of the partitions are equivalent if maternally
derived genes and paternally derived genes are interchanged in one or both
of the two people i and j. If the maternal and paternal origins of the two
pairs of genes are ignored, then the 15 detailed identity states collapse
to 9 condensed identity states [6]. Figure 5.3 depicts these nine states
S 1 ,... ,S 9 . Note that 5. Genetic Identity Coefficients 85
= S ∪ S ∗
∗
S 3
2 3
= S ∪ S ∗
∗
S 5
4 5
= S ∪ S ∗
∗
S 7
9 12
= S ∗ ∪ S ∗ ∪ S ∗ ∪ S .
∗
S 8
10 11 13 14
i’s genes

j’s genes

S 1 S 2 S 3 S 4 S 5

S 6 S 7 S 8 S 9

FIGURE 5.3. The Nine Condensed Identity States
Suppose ∆ k denotes the probability of condensed state S k . Although
it is not immediately obvious how to compute the condensed identity
coefficient ∆ k , some general patterns can easily be discerned. For example,
∆ 1 ,∆ 2 ,∆ 3 , and ∆ 4 are all 0 when i is not inbred. Likewise, ∆ 1 ,∆ 2 ,∆ 5 ,
and ∆ 6 are 0 when j is not inbred. The relation

1 1
Φ ij =∆ 1 + (∆ 3 +∆ 5 +∆ 7 )+ ∆ 8
2 4
is also easy to verify and provides an alternative method of computing the
kinship coefficient Φ ij .
By ad hoc reasoning, one can compute the ∆ k in simple cases. For ex-
ample, Table 5.1 gives the nonzero ∆ k for some common pairs of relatives.