Page 178 - Applied Probability

8. The Polygenic Model
163

0
=
w m+1
1
m
satisfy the two equalities (8.19) in R .
10. In the factor analysis model of Section 8.7, we can exploit the approx-
imate multivariate normality of the estimators to derive a different
approximation to the parameter asymptotic standard errors. Sup-
pose the multivariate normal random vector Z has mean µ =(µ i )
and variance Ω = (ω ij ). Verify by evaluation of the appropriate par-
t
t
t
tial derivatives of the characteristic function E(e iθ Z )= e iθ µ−θ Ωθ/2
at θ = 0 that
Cov(Z i Z j ,Z k Z l )= E(Z i Z j Z k Z l ) − E(Z i Z j )E(Z k Z l )
= µ j µ l ω ik + µ i µ l ω jk + µ j µ k ω il
+ µ i µ k ω jl + ω ik ω jl + ω il ω jk .
This translates into the refined approximate covariance
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
Cov(δ ik δ jk , δ il δ jl ) ≈ δ ik δ il ˆ σ jk,jl + δ ik δ jl ˆ σ jk,il
ˆ ˆ ˆ ˆ
+ δ jk δ il ˆ σ ik,jl + δ jk δ jl ˆ σ ik,il (8.20)
+ˆ σ ik,il ˆ σ jk,jl +ˆ σ ik,jl ˆ σ jk,il ,
which can be substituted in the expansion (8.10) of the asymptotic
variance.
11. Any reasonable model of QTL mapping for an X-linked trait must
take into account the phenomenon of X inactivation in females. As
a first approach, assume that all females are divided into n patches
and that in each patch one of the two X chromosomes is randomly
inactivated. If we suppose that the patches contribute additively, but
not necessarily equally, to a quantitative trait u, then we can write
n
c
u = i=1 i u i . Here the u i are identically distributed random vari-
ables of unit variance, and the c i are scale constants measuring the
functional sizes of the patches. For a monogenic trait, we postulate
that u i = α k when allele k is expressed in patch i. If allele k has
population frequency p k , then show that
n

E(u)= c i α k p k .
i=1 k

This mean also applies to males provided we make the assumption
that a male is also divided into n patches. This is a harmless fiction
because the same maternally derived allele is expressed in each patch
of a male.