STAT/BIOSTAT 550, 2014, Homework 7: Due Friday May 30

1. (based on Lange, Chapter 7, number 7)
A couple, each of whom is unaffected, have two kids affected by a recessive disease. At a linked marker, the parents have four distinct alleles. One parent is ab, and the other is cd. Both the affected kids are bd at the marker, and the recombination probability between trait and marker is r. A fetus is typed as bc. Show that the probability the baby will be affected is
r(1-r)((1-r)4 + r(1-r) R + r4)/R2
where R = r2 + (1-r)2.

2. Scientists are trying to map a rare autosomal recessive trait, in a population in which the trait allele frequency is 0.01. The trait does not impact survival or fertility of individuals. In the population, 5% of marriages are between first cousins (g = 1/16) and 20% of marriages are between second cousins (g = 1/64). The remainder are between unrelated individuals.

(a) What is the mean inbreeding coefficient in the population? What is the frequency of affected individuals?

(b) Scientists collect their individuals through a registry of affected individuals maintained by a Support Association (SA). Assuming each affected individual has the same probability of being registered with SA, independently of his/her relatives, what proportion of the individuals of SA's list are the offspring of first cousins, and of second cousins?
(Ans: 0.222 and 0.314)

(c) What is the probability that an affected offspring of a first cousin marriage is ibd at the trait locus? And for the offspring of a second-cousin marriage?
(Ans: 0.869 and 0.613)

3. Continuing from Qu.2, through SA the scientist enroll 40 unrelated affected individuals: 20 of them are the offspring of first-cousin marriages and 20 are the offspring of a second-cousin marriages. By good luck, they have a marker so tightly linked to the trait locus that the recombination frequency r is essentially 0. At this marker there are four alleles, each having frequency 0.25.
(a) What is the probability that an affected offspring of a first-cousin marriage is homozygous at the marker locus? And for the offspring of a second-cousin marriage?
(Ans: 0.902 and 0.710).

(b) What is the probability that all 40 affected individuals are homozygous at this marker locus?

(c) Assuming r=0, show that for the offspring of first cousins, each one homozygous at the marker contributes 0.48 to the base-10 lod score, and each one that is heterozygous contributes -0.86. For the offspring of second cousins, show that each one that is homozygous at the marker contributes 0.43, and each one that is heterozygous contributes -0.41. Hence find the overall Elod (expected lod score) from this sample.

4. On the web are data for homozygosity mapping from part of a real study here at UW. There is a list of affected individuals, and their genotypes at a marker locus ApoB. All the affected individuals are offspring of first-cousin marriages. There are a couple of pairs of sibs which you will have to decide what to do with. The families are Japanese, but initially CEPH frequencies were used in the analysis. Assess the evidence for linkage using the CEPH/Caucasian frequencies. Repeat with the Japanese frequencies. What do you conclude?
You also need the two-locus inbreeding coefficient f2(r) for the offspring of first cousins for this -- these can be got from the MORGAN kin program, but it is (more than) enough to know that for r=0, 0.01, 0.05 0.1, 0.18, 0.3 and 0.5, the values are 0.0625, 0.0588, 0.0461, 0.0339, 0.0206, 0.0101 and 0.0039.
Don't spend time doing a detailed analysis (unless you want to). Make any reasonable approximations to simplify.