Page 143 - Applied Probability

7. Computation of Mendelian Likelihoods
[29]. This eliminates the positivity constraint τ i > 0. Disease sever-
τ i = e
ω i
ity can be taken into account by writing ω i = β i x p , where x p is the child’s
severity index adjusted by gender and standardized to have mean 0 and
variance 1. If large values of the severity index are associated with allele i,
then we expect the estimated value of β i to be positive. Because
1 127
Pr(i/j → i)= , (7.7)
1+ e (β j −β i)x p
we take one β i = 0. Under the null hypothesis of Mendelian transmission,
all β i =0.
It is instructive to see how the gamete competition model handles intra-
genic single nucleotide polymorphisms (SNPs). Unfortunately, com-
bining multiple intragenic SNPs into a single super marker complicates
application of the traditional TDT, which requires codominant markers.
For instance, suppose we score each of three linked SNPs as a 1 or 0. An
observed triple heterozygote is consistent with any of the four unordered
haplotype pairs 111/000, 110/001, 101/010, and 011/100. If we admit the
possibility of incomplete typing within a single individual, the ambiguity
is even greater. Circumventing the phase problem by assigning individuals
their most probable haplotypes is arbitrary and bound to lead to subtle bi-
ases. The gamete competition model neatly circumvents the phase problem
by including each possible haplotype pair weighted by its overall contribu-
tion to the likelihood of the pedigree. This is a virtue of a likelihood-based
method compared to a nonparametric method. Of course, applying the
gamete competition model to multilocus SNP data will require good esti-
mates of haplotype frequencies. The safest course is to estimate haplotype
frequencies simultaneously with transmission parameters.
The pedigree data of Keavney et al. [13] on three SNPs within the
angiotensin-1 converting enzyme (ACE) gene illustrate the gamete competi-
tion model in action. We test whether gamete transmission at these markers
correlates with the quantitative trait, gender adjusted serum ACE levels.
The intragenic SNPs involved are the fourth (A-240T), sixth (T1237C) and
ninth (G2350A) polymorphisms. Because the ACE gene spans only 26kb
(kilobases), the recombination fractions between these SNPs are effectively
zero. For the sake of readability, each of the three SNPs incorporates its
two alleles as the first and last letter of its name. Thus, T1237C has alleles
T and C. The pedigree data consist of 83 white British families ranging
in size from 4 to 18 members [13]. Families were selected without regard
to ACE levels, which were determined on 405 family members. Genotypes
were collected on 555 family members.
We now consider the impact on ACE levels of transmission at all three
loci. The most general model based on equation (7.7) has 8 possible haplo-
types. The likelihood ratio statistic of 80.26 = 2(−682.63 + 722.76) of the
null hypothesis (all β’s 0) has an asymptotic p-value < 10 −6 on 7 degrees of
freedom. In Table 7.4, the maximum likelihood estimates of the haplotype