Page 125 - Applied Probability

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6. Applications of Identity Coefficients
where w m is a positive weight assigned to pedigree m. Under the null
hypothesis of independent segregation of the disease phenotype and the
marker alleles, the grand statistic T has mean 0 and variance 1. For a
moderately large number of pedigrees, T should be approximately normally
distributed as well. In practice, p-values can be computed by simulation,
and normality need not be taken for granted. A one-sided test is appropriate
because excess marker sharing increases the observed value of T.
Choice of the weights is bound to be somewhat arbitrary. With r m typed
affecteds in a pedigree, results of Hodge [6] suggest

r m − 1
w m = . (6.8)
Var(Z m )
This weighting scheme represents a compromise between giving all pedi-

grees equal weight (w m =1/ Var(Z m )) and overweighting large pedigrees
(w m = 1). Overweighting is a potential problem because the number of
affected pairs r m (r m − 1)/2 is a quadratic rather than a linear function of
r m . If we suspect recessive inheritance and want to exploit information on
inbred affecteds, then it is reasonable to replace r m − 1by r m in formula
(6.8).
Applications of the statistic T to pedigree data on Huntington disease,
rheumatoid arthritis,Arthritis, rheumatoid breast cancer, and Alzheimer
disease are discussed in the references [5, 9, 12]. Extension of the statistic
to multiple linked markers is undertaken in [13].

6.6 Problems

1. Let the disease allele at a recessive disease locus have population
frequency q. If a child has inbreeding coefficient f, argue that his
2
or her disease risk is fq +(1 − f)q . What assumptions does this
formula entail? Now suppose that a fraction α of all marriages in the
surrounding population are between first cousins [1]. Show that the
fraction of affecteds due to first-cousin marriages is

α( 1 q + 15 2 α(1 + 15q)
q )
16 16 = ,
¯
¯
¯
¯
fq +(1 − f)q 2 16[f +(1 − f)q]
¯
where f is the average inbreeding coefficient of the population. Com-
¯
pute this fraction for α = .02,f = .002, and for q = .01 and q = .001.
What conclusions do you draw from your results?
2. Consider a disease trait partially determined by an autosomal locus
with two alleles 1 and 2 having frequencies p 1 and p 2 . Let φ k/l be the
probability that a person with genotype k/l manifests the disease.