Page 139 - Applied Probability

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7. Computation of Mendelian Likelihoods
Weinberg and linkage equilibrium, the prior Prior(G i ) factors as
m

(7.4)
Prior(G ij ).
Prior(G i )=
j=1
If i’s phenotype X i also decomposes into separate observations X ij at each
locus, then most penetrance functions exhibit the factorization
m

Pen(X i | G i ) = Pen(X ij | G ij ). (7.5)
j=1
Failures of assumption (7.5) are rare in linkage studies and represent epis-
tasis among loci. Both equations (7.4) and (7.5) are forms of probabilistic
independence.
Factorization of transmission arrays is more subtle. According to Hal-
dane’s model, a gamete transmission probability Tran(H k | G i ) factors into
terms encompassing blocks of loci, with each block delimited by two het-
erozygous loci in the parent i. For example, suppose r and s,1 <r <s<m,
are the only heterozygous loci in the parental genotype G i . Then the trans-
mission probability for the haplotype H k factors as
Tran(H k | G i )=Tran[(H k1 ,...,H kr ) | (G i1 ,...,G ir )]
× Tran[(H k,r+1 ,...,H ks ) | (G ir ,...,G is ),H kr ]
× Tran[(H k,s+1 ,... ,H km ) | (G is ,...,G im ),H ks ],
where the block (r, ...,s) spans the only interval on which recombination
can be counted. Traversing the haplotype from locus 1 to locus m, a factor
of 1 accounts for which parental allele is encountered at the first heterozy-
2
gous locus r. Thus,
1
Tran[(H k1 ,...,H kr ) | (G i1 ,...,G ir )] = .
2
Recombination or nonrecombination between loci r and s is summarized
by

Tran[(H k,r+1 ,...,H ks ) | (G ir ,...,G is ),H kr ]
,
for recombination on interval [r, s]
θ rs
=
1 − θ rs for nonrecombination on interval [r, s],
where θ rs is the recombination fraction between loci r and s. Finally, be-
cause recombination cannot be scored between loci s and m,
Tran[(H k,s+1 ,... ,H km ) | (G is ,...,G im ),H ks ]=1.

For transmission array factorization to be useful, it must take the same
form for all possible multilocus genotypes G i . Clearly, a transmission array