Applied Probability by Kenneth Lange

Author:Kenneth Lange [Lange, Kenneth]
Language: eng
Format: epub, pdf
Published: 2011-02-20T03:05:49+00:00

=

Pr(ai) i=1

m

=

Pr(aij ). (9.10)

i=1 j

By construction, the founder genes assigned to different components do not impinge on one another. In other words, the set of founder genes consistent with G and M is drawn from the Cartesian product of the sets S1, . . . , Sm of legal allele vectors for the components C1, . . . , Cm, respectively. Applying the distributive rule to equation (9.10) consequently yields m

Prior(G)

=

Pr(Ci), (9.11)

G → G∩M

i=1

where

Pr(Ci) =

Pr(aij ).

ai∈Si j

As mentioned earlier, an allele vector set Si contains either all allele vectors or just two, one, or none. In the first case, Pr(Ci) =

a

Pr(ai) = 1,

i ∈Si

and in the remaining three cases, Pr(Ci) =

a

Pr(ai) contains only

i ∈Si

two, one, or no terms. Hence, calculation of Prior(G) reduces

G → G∩M

to easy component-by-component calculations.

Although likelihood calculation with non-codominant markers or incompletely penetrant traits can be handled similarly, two complications intrude.

First, we need a systematic method of generating the set Si of allele vectors for component Ci. Second, we must include penetrance values in the likelihood calculation, assuming that each person’s phenotypes at the various loci are independent conditional on his or her genotypes at the loci. Regarding the second complication, note that each component Ci carries with it a set Qi of phenotyped people through whom the founder genes pass.

Specifying an allele vector ai ∈ Si determines the genotype of each person k ∈ Qi. In computing Pr(Ci), we must multiply the product Pr(a

j

ij ) by

the penetrance of each k ∈ Qi at the current locus.

The allele vectors ai ∈ Si can be generated efficiently by a backtracking scheme [29]. This entails growing a compatible allele vector from partial vectors that are compatible. The idea can be illustrated by reference to component C2 = {A, C, E} of Figure 9.2 We start with the assignment (aA) = (1), which is consistent with the phenotypes in the pedigree, grow it to (aA, aC) = (1, 1), which is inconsistent, discard all vectors beginning with (aA, aC) = (1, 1), move on to (aA, aC) = (1, 2), which is consistent, grow this to (aA, aC, aE) = (1, 2, 1), which is consistent, discard each 9. Descent Graph Methods

181

of the next three vectors (aA, aC, aE) = (1, 2, 2), (aA, aC, aE) = (1, 2, 3), and (aA, aC, aE) = (1, 2, 4) as inconsistent, backtrack to the partial vector (aA, aC, ) = (1, 3), which is inconsistent, and so forth, until ultimately we identify the second compatible vector (aA, aC, aE) = (2, 1, 2) and reject all other allele vectors. The virtue of backtracking is that it eliminates large numbers of incompatible vectors without actually visiting each of them.

If penetrances are quantitative, so that every genotype is compatible with every phenotype, then Si expands to a Cartesian product having n|Ci| elements, where |Ci| is the number of founder genes in Ci and n is the number of alleles at the current locus. In this case, backtracking will successfully construct every allele vector

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