Page 132 - Applied Probability

7. Computation of Mendelian Likelihoods
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given an unobserved genotype G. We denote a penetrance by Pen(X | G).
Penetrances apply to all people in a pedigree, founders and nonfounders
alike. Implicit in the notion of penetrance is the assumption that the phe-
notypes of two or more people are independent given their genotypes. This
restriction rules out complex models in which common environment in-
fluences phenotypes. It is easier to incorporate environmental effects in
polygenic models. In Mendelian models, likelihood evaluation involves com-
binatorics; in polygenic models, it involves linear algebra.
In general, Pen(X | G) can represent a conditional likelihood as well as a
conditional probability. This would be the case, for instance, with a quan-
titative trait X following a different Gaussian density for each genotype.
For many genetic traits, Pen(X | G) is either 0 or 1; in other words, each
genotype leads to one and only one phenotype. When a phenotype is un-
observed, it is natural to assume that the penetrance function is identically
1.
The third and last component probability of a likelihood summarizes the
genetic transmission of the trait or traits observed. Let Tran(G k | G i ,G j )
denote the probability that a mother i with genotype G i and a father j with
genotype G j produce a child k with genotype G k . For ordered genotypes,
the child’s genotype G k can be visualized as an ordered pair of gametes
(U k ,V k ), U k being maternal in origin and V k being paternal in origin. If
all participating loci reside on the same chromosome, then U k and V k are
haplotypes. Because any two parents create gametes independently, the
transmission probability
Tran(G k | G i ,G j ) = Tran(U k | G i ) Tran(V k | G j )
factors into two gamete transmission probabilities. Unordered geno-
types do not obey this gamete factorization rule.
Specification of gamete transmission probabilities is straightforward for
single-locus models. For a single autosomal locus, Tran(H | G) is either
1, 1 , or 0, depending on whether the single allele H is identical in state
2
to both, one, or neither of the two alleles of the parental genotype G,
respectively. For multiple linked loci, Haldane’s model [10] permits easy
computation of gamete transmission probabilities, provided one is willing
to neglect the phenomenon of chiasma interference. For the sake of compu-
tational simplicity, we now adopt Haldane’s model, which postulates that
recombination occurs independently on disjoint intervals.
To apply Haldane’s model, one begins by discarding all homozygous loci
in the parent. This entails no loss of information because recombination
events can never be inferred between such loci. Between each remaining
adjacent pair of heterozygous loci, gametes can be scored as recombinant
or nonrecombinant. Once adjacent intervals have been consolidated to the
point where all interval endpoints are marked by heterozygous loci, cal-
culation of gamete transmission probabilities is straightforward. Invoking
independence, the probability of a gamete is now 1 times the product over
2