STAT 550 (DL): Homework 5


For this homework, refer to Chapter 2 part 1; sections 2.1 and 2.2 and Chapter 2 part 2; sections 2.3 and 2.4 of the audio lectures.

1. (Based on Lange Ch 6, #1, and an example of Cavalli-Sforza and Bodmer)
In an idealized infinite population, 10% of people marry their first cousins (kinship coefficient 1/16), 25% marry their second cousins (kinship coefficient 1/64), and the remaining 65% marry unrelated individuals. All marriages have the same family size distribution. Consider a very rare recessive trait, with allele frequency q.
(a) Show that the mean inbreeding coefficient in the population is just over 1%.
(b) Show that the overall probability an individual is affected is q(0.01 + 0.99q)
(c) Show that the probability an affected individual is the child of a first-cousin marriage is 0.00625(1+15q)/(0.01 + 0.99q).
(d) How small must q be in order that the posterior probability an affected individual is the child of a first-cousin marriage is larger than the posterior probability the child is from a second-cousin marriage. (Answer : 0.015 (approx).)

2. Consider a male with inbreeding coefficient f1 who is unrelated to a female individual who has inbreeding coefficient f2. The individuals have two offspring B and C.
(a) Show that the probability that B and C receive the same gene from their father is (1+f1)/2. Show that the probability that B and C receive the same gene from their mother is (1 + f2)/2.
(b) Show that the probabilities ki (i=0,1,2) that B and C have i genes IBD are
k2 = (1 + f1)(1 + f2) /4
k0 = (1 - f1)(1 - f2) /4
k1 = 1 - k0 - k2
and that the coefficient of kinship between B and C is
psi(B,C) = (2 + f1 + f2) /8
(c) Suppose the first child has a recessive trait, the allele for which has frequency q. If nothing is known about the trait in the parents, what is the probability the second child will have this trait also? What is the probability the second child will be an unaffected carrier for the trait?
(d) Suppose now it is known that neither parent has the trait, but again the first child is affected. What is the probability the second child will have this trait also? What is the probability the second child will be an unaffected carrier for the trait?

3. (based on Crow and Kimura, Ch 4, #9)
This makes exact the notion due to Wright that gene identity by descent leads to correlations between relatives.
Consider a particular allele A with allele frequency q, and define indicator random variables I(g), for a gene g, where I(g)=1 if the allelic type of gene g is A, and 0 otherwise.
(a) Show E(I(g)) = q, and var(I(g)) = q(1-q)
(b) Show that for genes g1 and g2 segregating from B and from C, the correlation between I(g1) and I(g2)) is the kinship coefficient between B and C.
(c) If g1 and g2 are the two genes in an individual, shown that the variance of (I(g1)+I(g2)) is 2q(1-q)(1+f), where f is the inbreeding coefficient of the individual.
(d) If g1 and g2 are the genes in a parent B, and g2 and g3 are the genes in his child C (that is, the g2 gene is the one inherited by C from B), show that the correlation between (I(g1)+I(g2)) and (I(g2)+I(g3)) is (1+2fC+fB)/(2((1+fC)(1+fB))1/2) where fB is the inbreeding coefficient of B, and fC is the inbreeding coefficient of C.

4. (based on Crow and Kimura, Ch 4, #15)
An individual B has phenylketonuria, a rare recessive condition, for which the allele frequency q is 0.01. What is the probability that B's relative C has phenylketonuria if
(a) C is a first cousin of B
(b) C is a nephew of B
(c) C is a double first cousin of B
(d) C is a quadruple half first cousin of B
(Note the kinship coefficient between B and C is the same for relationships (b), (c) and (d).)