Page 109 - Applied Probability

of pedigree founders with shared offspring but no unshared offspring.
Without loss of generality, number the members of the set 1,... ,m
and the remaining pedigree members m +1,... ,n. Show that
(a) The kinship matrix Φ can be written as the partitioned matrix
t
t
5. Genetic Identity Coefficients 93

a 1 I m + a 2 11 1b
Φ= t ,
b1 C
where 1 is a column vector consisting of m 1’s, I m is the m × m
identity matrix, a 1 and a 2 are real constants, b is a column vector
with n−m entries, and C is the (n−m)×(n−m) kinship matrix
of the n − m pedigree members not in the designated set.
t
(b) The matrix a 1 I m + a 2 11 has 1 as eigenvector with eigenvalue
a 1 + ma 2 and m − 1 orthogonal eigenvectors
i−1
1
u i = e j − e i ,
i − 1
j=1
i =2,... ,m, with eigenvalue a 1 . Note that each u i is perpen-
dicular to 1.

u i
(c) The m−1 partitioned vectors are orthogonal eigenvectors
0
of Φ with eigenvalue a 1 .
7. We define the X-linked kinship coefficient Φ ij between two relatives
i and j as the probability that a gene drawn randomly from an X-
linked locus of i is i.b.d. to a gene drawn randomly from the same
X-linked locus of j. When i = j, sampling is done with replacement.
When either i or j is male, one necessarily selects the maternal gene.
Show how the algorithm of Section 5.2 can be modified to compute
the X-linked kinship matrix Φ of a pedigree [11].

8. Selfing is a mating system used extensively in plant breeding. As its
name implies, a plant is mated to itself, then one of its offspring is
mated to itself, and so forth. Let f n be the inbreeding coefficient of the
1
relevant plant after n rounds of selfing. Show that f n+1 = (1 + f n)
2
n
n
and therefore that f n =(2 − 1)/2 .
9. Geneticists employ repeated sib mating to produce inbred lines of lab-
oratory animals such as mice. At generation 0, two unrelated animals
are mated to produce generation 1. A brother and sister of generation
1 are then mated to produce generation 2, and so forth. Let φ n be the
kinship coefficient of the brother–sister combination at generation n,
and let f n be their common inbreeding coefficient. Demonstrate that