Page 152 - Applied Probability

7. Computation of Mendelian Likelihoods
136
new mutation and passes the disease allele to the grandson 7, or the
grandson 7 is a new mutation. The presence of unaffected uncles 5
and 6 and an unaffected brother 8 modifies the probabilities of these
contingencies. Because µ is very small, you may approximate the
probability of a carrier female passing either the normal or disease
allele as 1/2. You may also approximate the prior probability of a
normal female or normal male as 1.)
10. Consider a nuclear family in which one parent is affected by an auto-
somal dominant disease [11]. If the affected parent is heterozygous at
a codominant marker locus, the normal parent is homozygous at the
marker locus, and the number of children n ≥ 2, then the family is in-
formative for linkage. Because of the phase ambiguity in the affected
parent, we can split the children of the family into two disjoint sets of
size k and n − k, the first set consisting of recombinant children and
the second set consisting of nonrecombinant children, or vice versa.
Show that the likelihood of the family is
1 k n−k 1 n−k k
L(θ)= θ (1 − θ) + θ (1 − θ) ,
2 2
where θ is the recombination fraction between the disease and marker
loci. A harder problem is to characterize the maximum of L(θ) on the
1 n
interval [0, ]. Without loss of generality, take k ≤ . Then demon-
2 2
strate that the likelihood curve is unimodal with maximum at θ =0
2
1
when k =0,at θ = 1 when (n − 2k) ≤ n, and at θ ∈ (0, ) oth-
2 2
erwise. (Hints: The case k = 0 can be resolved straightforwardly by

inspecting the derivative L (θ). For the remaining two cases, write

L (θ)= θ k−1 (1 − θ) n−k g(τ), where g(τ) is a polynomial in τ = θ .
1−θ
From this representation check that θ = 0 is a local minimum of L(θ)
and that θ = 1 is a stationary point of L(θ). The maximum of L(θ)
2
must therefore occur at θ = 1 or some other positive root of g(τ).
2
Use Descartes’ rule of signs [7] and symmetry to limit the number of
1
1
positive roots of g(τ)on τ ∈ (0, 1], that is, θ ∈ (0, ]. Compute L ( )
2 2
1
to determine the nature of the stationary point θ = .)
2
11. Consider the revision

L(β) = ··· Pen(X i | G i )β 1 {G i =g}
i
G 1 G n

× Prior(G j ) Tran(G m | G k ,G l )
j {k,l,m}
of the likelihood expression (7.1), where β is an artificial parame-
ter and g is a fixed genotype. Prove that d ln L(1) is the expected
dβ
number of people in the pedigree with genotype g conditional on the
observed phenotypes in the pedigree. Note that this device is easy to