Page 200 - Applied Probability

If we can sample from marker descent graphs G given the marker types
0
M, then we can employ standard pedigree likelihood programs such as
MENDEL [25] to estimate Pr d (T | M). The basis for this computation is
the obvious decomposition
9. Descent Graph Methods 185
Pr d (T | M)= Pr d (T | G)Pr(G | M), (9.12)
0
0
0 G
which relies implicitly on the assumption of linkage equilibrium between
the trait and marker loci. To evaluate (9.12), we run a Metropolis-coupled
Markov chain on marker descent graphs G. This chain has equilibrium dis-
0
tribution matching the conditional distribution Pr(G | M). If a sequence
0
of descent graphs G 0 ,... , G n−1 is generated by running the chain, then
0
0
the sample average 1 n−1 Pr d (T | G i ) will approximate Pr d (T | M) ac-
0
n i=0
curately for n sufficiently large.
Deterministic computation of Pr d (T ∩ G i ) can be done by MENDEL if
0
it is alerted to recognize a mixture of the marker descent graph G i and the
0
trait phenotypes T as legitimate input [34]. Division of the joint likelihood
Pr d(T ∩ G i ) output by MENDEL by the marginal likelihood Pr(G i ) then
0
0
gives the requisite conditional likelihood Pr d(T | G i ) used in computing the
0
sample average approximation. Since the trait locus is usually biallelic, and
since sampling from the Markov chain fills in all of the missing information
on marker gene flow, the deterministic part of a location score calculation
is generally quick.
9.8 Finding a Legal Descent Graph
The MCMC method of location scores must start with a legal descent
graph. Finding such a descent graph is harder than it first appears, but
fortunately the problem yields to a randomized version of genotype elim-
ination. The successful strategy proceeds locus by locus and constructs a
legal vector of ordered genotypes for a pedigree. From this vector a descent
state and corresponding descent graph are then assembled. Based on the
genotype elimination method of Chapter 7, the following algorithm applies
[34]:

1. Perform step (A) of genotype elimination on the pedigree.
2. Perform steps (B) and (C) of genotype elimination.

3. Consider each individual’s genotype list:

(a) If all people possess exactly one ordered genotype, then use these
genotypes to construct a descent state, assigning sources in the