Page 128 - Applied Probability

6. Applications of Identity Coefficients
112
for any two non-inbred relatives i and j [8]. Prove that the two ma-
trices
2

σ
σ axy
axx
2
2
σ
σ
σ axy
σ dxy
dyy
ayy
2
2
are covariance matrices, where σ 2 σ 2 dxx , σ σ dxy , and σ 2 are the
, σ
axx dxx ayy dyy
additive and dominance genetic variances of the X and Y traits,
respectively. (Hints: For the first part, consider the artificial trait
W = X + Y for a typical person. For the second part, prove that
= 2 Cov(A 1 ,B 1 )
σ axy
=Cov(X − A 1 − A 2 ,Y − B 1 − B 2 ),
σ dxy
where A k =E(X | Z k ) and B k =E(Y | Z k ), Z 1 and Z 2 being the
maternal and paternal alleles at the common locus.)
9. In the two-locus heterogeneity model with X = Y + Z − YZ, carry
through the computations retaining the product term YZ. In partic-
ular, let K m be the prevalence of the mth form of the disease, and
let K mR be the recurrence risk for a relative of type R under the
mth form. If K is the prevalence and K R is the recurrence risk to a
relative of type R under either form of the disease, then show that
K = K 1 + K 2 − K 1K 2
KK R = K 1 K 1R + K 1K 2 − K 1 K 1R K 2 + K 1K 2 + K 2K 2R
− K 1 K 2 K 2R − K 1 K 1R K 2 − K 1 K 2 K 2R + K 1 K 1R K 2 K 2R .
Assuming that K 1 , K 2 , K 1R , and K 2R are relatively small, verify the
approximation
λ R − 1
2 2
K 1 K 2
= (λ 1R − 1) + (λ 2R − 1)
K K
K 1 K 2
+ [2K 1 +2K 2 − K 1 K 2 − 2K 1R − 2K 2R + K 1R K 2R ]
K 2
2 2

K 1 K 2
≈ (λ 1R − 1) + (λ 2R − 1),
K K
where λ mR = K mR /K m and λ R = K R /K.
10. In the pedigree depicted in Figure 6.1, compute the marker-sharing
statistic Z and its expectation E(Z) for the three phenotyped affect-
eds 3, 4, and 6. Assume f(p)=1/p, p a =1/2, p b =1/4, and for a
third unobserved marker allele p c =1/4.