Page 114 - Applied Probability

6. Applications of Identity Coefficients
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given j’s relationship to i and i’s genotype, we condition on the various con-
densed identity states that i and j can jointly occupy. (Figure 5.3 depicts
the nine possible states.) This conditioning yields
9
Pr(j = a m /a n | i = a k /a l )= Pr(j = a m /a n | S r ,i = a k /a l )
r=1
× Pr(S r | i = a k /a l).
If i has heterozygous genotype a k /a l and inbreeding coefficient f i [7], then
states S 1 ,...,S 4 are impossible, and
Pr(S r ,i = a k /a l )
Pr(S r | i = a k /a l ) =
Pr(i = a k /a l )
,
0 for r ≤ 4
= ∆ r 2p k p l
for r> 4
(1−f i )2p k p l
0 for r ≤ 4
,
=
∆ r
for r> 4.
1−f i
When i is a homozygote a k /a k , states S 1 ,... ,S 4 come into play. In this
case [2],
Pr(S r ,i = a k /a k )
Pr(S r | i = a k /a k )=
Pr(i = a k /a k )

∆ r p k
f i p k +(1−f i )p
 2 for r ≤ 4
= 2 k
∆ r p
 k 2 for r> 4
f i p k +(1−f i )p
k
&
∆ r
for r ≤ 4
= f i +(1−f i )p k
for r> 4.
∆ r p k
f i +(1−f i )p k
Note that Pr(S r | i = a k /a l ) = Pr(S r | i = a k /a k )=∆ r when f i =0.
The conditional probabilities Pr(j = a m /a n | S r ,i = a k /a l ) can be
computed as follows [7]: In states S 1 and S 7 , j has the same genotype as i.
In states S 2 , S 4 , S 6 , and S 9 , j’s genotype is independent of i’s genotype. In
states S 2 and S 6 , j is also an obligate homozygote and has the homozygous
genotype a m /a m with probability p m . In states S 4 and S 9 , j’s genotype
follows the Hardy-Weinberg law. In states S 3 and S 8 , j shares one gene in
common with i; the shared gene is equally likely to be either of i’s two genes.
The other gene of j is drawn at random from the surrounding population.
Thus, if i is a heterozygote a k /a l in state S 8 , then j has genotypes a k /a r
(a r = a l ), a l /a r (a r = a k ), and a k /a l with probabilities p r /2, p r /2, and
p k /2+p l /2, respectively. If i is a homozygote a k /a k in states S 3 or S 8 , then
j has genotype a k /a r with probability p r . Finally in state S 5 , j is again
an obligate homozygote. If i is a k /a l , then j is equally likely to be either
a k /a k or a l /a l .