Page 122 - Applied Probability

6. Applications of Identity Coefficients
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6.5 An Affecteds-Only Method of Linkage Analysis
Genetic epidemiologists are now actively attempting to map some of the
genes contributing to common diseases. This task is complicated by the
poorly understood inheritance patterns for many of these diseases. While
major genes certainly contribute to some common diseases such as breast
cancer and Alzheimer disease, the classical monogenic patterns of inheri-
tance typically do not fit pedigree and population data. One strategy for
identifying disease predisposing genes is to restrict mapping studies to pedi-
grees showing multiple affecteds with early age of onset. Such pedigrees are
more apt to segregate major genes than pedigrees with isolated affecteds
showing late onset. Even this enrichment strategy does not guarantee a
single Mendelian pattern of inheritance in the ascertained pedigrees.
In the absence of a well-defined disease inheritance model, it is still prof-
itable to pursue linkage analysis by robust methods. Robust linkage meth-
ods are predicated on the observation that a marker allele will track a
closely linked disease allele as both descend from a founder through a pedi-
gree. Only recombination can separate a pair of such alleles present in a
pedigree founder. Thus, marker genes can be used as surrogates for disease
genes. Robust linkage tests seek to assess the amount of marker allele shar-
ing among affecteds. Excess sharing is taken as evidence that the marker
locus is closely linked to a disease predisposing locus. The marker locus
may be a candidate locus for the disease. In this case it is perhaps better
to speak of association between the marker and the disease.
Our immediate goal is to examine one robust linkage statistic and to
compute the mean and variance of this statistic using kinship coefficients
[12]. These computations are valid under the null hypothesis of independent
segregation of the marker locus and the disease. Beyond this independence
assumption, nothing specific is assumed about disease causation.
Consider a pedigree and two affected individuals i and j in that pedigree
who are typed at a given marker locus. We assume that the marker locus
is in Hardy-Weinberg equilibrium and that its alleles are codominant with
the kth allele having population frequency p k . At the heart of our robust
statistic is the pairwise statistic Z ij assessing the marker sharing between
i and j. It is desirable for Z ij to give greater weight to shared rare alleles
than to shared common alleles. This weighting is
accomplished via a weighting function f(p) of the population frequency
√
p of the shared allele. Typical choices for f(p) are f(p)= 1, f(p)=1/ p,
and f(p)= 1/p.Now let M i and M j be the observed marker genotypes of
i and j. Imagine drawing one marker gene G i at random from i and one
marker gene G j at random from j.
The statistic Z ij is defined as the conditional expectation
Z ij =E(1 {G i =G j } f(p G i ) | M i ,M j ), (6.7)
where the indicator function 1 {G i =G j } is 1 when the sampled genes G i and