Page 180 - Applied Probability

8. The Polygenic Model
13. In the hypergeometric polygenic model, Var(X i )= 2n holds for each
person
i in a non-inbred pedigree. In the presence of inbreeding, give a coun-
terexample to this formula. However, prove that
0 ≤ Cov(X i ,X j ) ≤ (2 + q)n 165
for all pairs i and j from a pedigree with q people. Note that the
special case i = j gives an upper bound on trait variances. (Hint:
Argue by induction using the recurrence formulas for variances and
covariances.)
14. In the hypergeometric polygenic model, suppose that one randomly
samples each of the n polygenes transmitted to a gamete with replace-
ment rather than without replacement. If j = i is not a descendant
of i, and i has parents k and l, then show that this altered model
entails

E(X i )=0
1 1
Cov(X i ,X j )= Cov(X k ,X j )+ Cov(X l ,X j )
2 2
1 1 1 1
Var(X i )=2n + 1 − Var(X k )+ 1 − Var(X l )
4 n 4 n
1
+ Cov(X k ,X l ).
2

15. Continuing Problem 14, let v m be the trait variance of a person m
generations removed from his or her relevant pedigree founders in a
non-inbred pedigree. Verify that v m satisfies the difference equation

1 1
v m =2n + 1 − v m−1
2 n
with solution
m

4n 1 1 4n
= + .
v m 1 1 − v 0 − 1
1+ 2 n 1+
n n
Check that v m steadily increases from v 0 =2n to the limit v ∞ = 4n 1 .
1+
n
8.12 References
[1] Amos CI (1994) Robust variance-components approach for assessing
genetic linkage in pedigrees. Amer J Hum Genet 54:535–543