Page 150 - Applied Probability

7. Computation of Mendelian Likelihoods
134
1 2

3 4

5 6 7
2/2 1/2 2/3

FIGURE 7.6. A Genotype Elimination Counterexample

4. Consider the array product
E(G 1 ,G 2 ,G 3 ,G 4 ,G 5 )
= A(G 1 ,G 2 ,G 3 )B(G 1 )C(G 2 ,G 3 ,G 4 )D(G 5 ),
where the range set S i for the index G i has 4, 2, or 3 elements ac-
cording as i =1, i ∈{2, 3, 4},or i = 5. Show that the greedy tactic
of assembling the product array from the pairwise products of the
multiplicand arrays first multiplies B times D, then A times C, and
finally the product BD times the product AC. Demonstrate that the
alternative of multiplying A times B, then C times D, and finally the
product AB times the product CD requires fewer total multiplica-
tions [18].
5. Verify the numerical entries in Table 7.3.

6. Do by hand the risk prediction calculation for myotonic dystrophy,
showing the various steps of the computations in detail. Neglect the
extremely rare Dm + /Dm+ genotype at the myotonic dystrophy lo-
cus. Using two-locus genotypes, evaluate the two required likelihoods
as seven-fold iterated sums.
7. Figure 7.7 gives a pedigree for an autosomal recessive disease and
a linked marker. The four marker genes a, b, c, and d are assumed
distinct. If the recombination fraction between the two loci is θ, then
show that the risk of the fetus 5 being affected is
2 4
4 2
5
(1 − θ) θ +(1 − θ) θ +(1 − θ) θ +(1 − θ)θ 5
.
2 2
4
(1 − θ) + 2(1 − θ) θ + θ 4