Page 61 - Applied Probability

3. Newton’s Method and Scoring
44
1
(3.6)
=
.
θ(1 − θ)
The efficiencies of mating designs can be compared based on their ex-
pected information numbers J(θ) [17]. The phase-known, double-inter-
cross mating A 1 B 1 /A 2 B 2 ×A 1 B 1 /A 2 B 2 offers an alternative to the double-
backcross mating. Table 3.3 shows nine phenotypic categories and their as-
sociated probabilities (column 3) for offspring of this mating. Since some
of these probabilities are identical, the corresponding categories can be col-
lapsed. Thus, categories 1 and 9 can be combined into a single category
2
with probability (1 − θ) /2; categories 2, 4, 6, and 8 can be combined into
a single category with probability 2θ(1 − θ); and categories 5 and 7 can
2
be combined into a single category with probability θ /2. Category 3 has
a unique probability. Based on these four redefined categories and formula
(3.5), the expected information per offspring is
2(1 − 2θ) 2 2(1 − 2θ) 2
J(θ) = 4 + + . (3.7)
2
θ(1 − θ) θ +(1 − θ) 2
TABLE 3.3. Offspring Probabilities for a Double-Intercross Mating
Category i Phenotype c × cp i c × rp i
1 A 1 /A 1 ,B 1 /B 1 (1−θ) 2 θ(1−θ)
4 4
2
θ(1−θ) θ +(1−θ) 2
2 A 1 /A 1 ,B 1 /B 2
2 4
2
3 A 1 /A 2 ,B 1 /B 2 θ +(1−θ) 2 θ(1 − θ)
2
2
θ(1−θ) θ +(1−θ) 2
4 A 1 /A 2 ,B 1 /B 1
2 4
5 A 1 /A 1 ,B 2 /B 2 θ 2 θ(1−θ)
4 4
2
θ(1−θ) θ +(1−θ) 2
6 A 1 /A 2 ,B 2 /B 2
2 4
7 A 2 /A 2 ,B 1 /B 1 θ 2 θ(1−θ)
4 4
2
8 A 2 /A 2 ,B 1 /B 2 θ(1−θ) θ +(1−θ) 2
2 4
(1−θ) 2 θ(1−θ)
9 A 2 /A 2 ,B 2 /B 2
4 4
Besides comparing the double-backcross mating to the coupling × cou-
pling, double-intercross mating, we can compare both to the phase-known,
coupling×repulsion, double-intercross mating A 1 B 1 /A 2 B 2 ×A 1 B 2 /A 2 B 1 .
Column 4 of Table 3.3 now provides the correct probabilities for the nine
phenotypic categories. The odd-numbered categories of Table 3.3 collapse