Page 142 - Applied Probability

7. Computation of Mendelian Likelihoods
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Example 7.6.2 Gamete Competition Model
When a new locus is investigated, one of the first statistical tasks is
to check whether its proposed alleles and genotypes conform to Mendelian
segregation ratios. The typical way of doing this is to subdivide all available
nuclear families into different mating types. For any given pair of parental
genotypes, there are from one to four possible offspring genotypes. The
numbers of offspring observed in the various genotypic categories are used
2
2
to compute an approximate χ statistic. These approximate χ statistics
2
are then added over the various mating types to give a grand χ . This
2
classical procedure suffers from the fact that the component χ statistics
often lack adequate numbers for large sample theory to apply. If the alleles
at the proposed locus involve dominance, then defining mating types is also
problematic.
An alternative procedure is to assign each allele A i a segregation pa-
rameter τ i . These parameters can be estimated by maximum likelihood
from the available pedigree data. The parameters enter the likelihood cal-
culations at the level of gamete transmission probabilities. For two dif-
ferent alleles A i and A j , take Pr(A i | A i /A j )= τ i /(τ i + τ j ) as suggested
in the Bradley-Terry model for ranking sports teams in the same league.
[3, 12, 14, 20, 29, 30]. For a homozygous parental genotype A i /A i , take
Pr(A i | A i /A i ) = 1. Under the hypothesis of Mendelian segregation, all
the τ’s are equal. Because multiplying the τ’s by the same constant pre-
serves segregation ratios, one should impose a constraint such as τ i = 1 for
one i. Using a likelihood ratio statistic, one can then test whether all other
τ i =1.
As a simple numerical example, consider the four alleles 1+, 1−, 2+,
and 2− of the PGM1 marker locus on chromosome 1. The PGM1 data of
Lewis et al. [23] lists 93 people in 5 pedigrees. For these data, the maximum
likelihood estimates are ˆ τ 1+ =1, ˆ τ 1− = .79, ˆ τ 2+ = .84, and ˆ τ 2− =1.26.
2
The likelihood ratio statistic is approximately distributed as a χ with three
degrees of freedom. The observed value of this statistic,

2 × [(−114.70) − (−115.64)] = 1.88,
suggests that the alleles of the PGM1 locus do conform to Mendelian in-
heritance. Note that this analysis safely ignores the issue of ascertainment.
This gamete competition model also forms the basis of a attractive
parameteric generalization of the transmission/disequilibrium test (TDT).
In the absence of severity data, the most natural implementation uses
Mendelian segregation ratios for transmission to unaffected children and
Bradley-Terry segregation ratios for transmission to affected children. This
tactic permits detection of distorted transmission to affecteds. The model
also accommodates quantitative as well as qualitative outcomes, allows for
covariates, and makes effective use of full pedigree data. To use quantita-
tive outcomes and covariates, it is convenient to reparameterize by writing