Page 127 - Applied Probability

6. Applications of Identity Coefficients
111
5. For a locus with two alleles, show that the additive genetic variance
satisfies
2
2
σ
=2p 1 p 2 (α 1 − α 2 )
a
2
=2p 1 p 2 [p 1 (µ 11 − µ 12 )+ p 2 (µ 12 − µ 22 )] .
(6.10)
2
As a consequence of formula (6.10), σ can be 0 only in the unlikely
a
circumstance that µ 12 lies outside the interval with endpoints µ 11
2
and µ 22 . (Hint: Expand 0 = 2(α 1 p 1 + α 2 p 2 ) and subtract from the
2
expression defining σ .)
a
Show that the dominance genetic variance satisfies
2 2
2
σ d 2 = p p (µ 11 − 2µ 12 + µ 22 ) .
1 2
2
It follows that if either p 1 or p 2 is small, then σ will tend to be small
d
2
2
2
compared to σ . Hint: Let µ = p µ 11 +2p 1p 2 µ 12 +p µ 22 . Since µ =0,
1
a
2
it follows that
δ 11 = µ 11 − 2α 1 + µ
2
= p (µ 11 − 2µ 12 + µ 22 )
2
δ 12 = −p 1 p 2 (µ 11 − 2µ 12 + µ 22 )
2
δ 22 = p (µ 11 − 2µ 12 + µ 22 ).
1
2
2
6. Prove that any pair of nonnegative numbers (σ ,σ ) can be realized
d
a
1
as additive and dominance genetic variances. The special pairs ( , 0)
2
and (0, 1) show that the two matrices Φ = (Φ ij ) and ∆ 7 =(∆ 7ij )
defined for an arbitrary non-inbred pedigree are legitimate covariance
matrices. (Hint: Based on the previous problem,
σ 2 a =2p 1p 2 (p 1 u + p 2 v) 2
2 2
σ 2 d = p p (u − v) 2
1 2
for u = µ 11 − µ 12 and v = µ 12 − µ 22 . Solve for u and v.)
7. Show that the matrices Φ and ∆ 7 of coefficients assigned to a pedigree
do not necessarily commute. It is therefore pointless to attempt a
simultaneous diagonalization of these two matrices. (Hint: Consider
a nuclear family consisting of a mother, father, and two siblings.)
8. Let (X 1 ,... ,X n ) and (Y 1 ,... ,Y n ) be measured values for two dif-
ferent traits on a pedigree of n people. Suppose that both traits are
determined by the same locus. Show that there exist constants σ axy
and σ dxy such that
Cov(X i ,Y j )=2Φ ij σ axy +∆ 7ij σ dxy