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Mid-Term Exam - February 25, 1999

- 1.
- For
and
being positive definite, symmetric,
and full rank, and
being rectangular,
**prove**that

You may want to save this question for last. - 2.
- Below are genotypic frequencies for two loci,
and the genotypic values of each.
Genotype Freq. A-Value B-Value Combined *A*_{1}*A*_{1}*B*_{1}*B*_{1}.0504 1 1 2 *A*_{1}*A*_{1}*B*_{1}*B*_{2}.0972 1 1 2 *A*_{1}*A*_{1}*B*_{2}*B*_{2}.0324 1 -1 0 *A*_{1}*A*_{2}*B*_{1}*B*_{1}.1512 2 1 3 *A*_{1}*A*_{2}*B*_{1}*B*_{2}.2916 2 1 3 *A*_{1}*A*_{2}*B*_{2}*B*_{2}.0972 2 -1 1 *A*_{2}*A*_{2}*B*_{1}*B*_{1}.0784 -1 1 0 *A*_{2}*A*_{2}*B*_{1}*B*_{2}.1512 -1 1 0 *A*_{2}*A*_{2}*B*_{2}*B*_{2}.0504 -1 -1 -2 - (a)
- Calculate the genotypic mean and genotypic variance of for each locus separately and for both loci together.
- (b)
- Calculate the covariance between
*G*_{A}and*G*_{B}. - (c)
- Provide an estimate of disequilibrium, D.

- 3.
- The following numbers were taken from a much larger
gametic relationship matrix.
Individual A A1 A2 Individual C1 C C2 0 - (a)
- Compute the additive relationship between A and C.
- (b)
- Compute the dominance relationship between A and C.
- (c)
- What would be the inbreeding coefficient of an offspring of animals A and C?

- 4.
- Border collies (a dog breed) are used extensively in
Scotland for herding sheep. Competitions have been organized
to test dogs for a dozen or more herding abilities. A series
of simpler tests have been designed for untrained 4 month old
puppies to see if they have good natural herding instincts.
Data were collected on 200 puppies of 50 owners from 63 litters
and complete pedigree information was obtained. Tests were
scored from 0 to 100 by 20 different judges over a one year
period. The equation of the model (Part 1) was determined to
be

*y*_{ijk}=*OTY*_{i}+*L*_{ij}+*a*_{ijk}+*e*_{ijk},

where*OTY*_{i}is an owner-test-year subclass effect,*L*_{ij}is a litter effect within OTY subclasses,*a*_{ijk}is an animal additive genetic effect within litter and OTY effects, and*e*_{ijk}is a residual effect. The observations are scores on a particular herding test.- (a)
- Complete the model, identify the fixed and random factors, and
give Parts 2 and 3 of the model. If the model is not
acceptable to you, then make the changes that you think
bring it closer to an ideal model.
- (b)
- Below are scores of some puppies.
OTY Litter Pup Score 1 1 1 9.6 1 1 2 8.5 1 1 3 7.8 1 2 1 6.4 1 2 2 7.2 2 1 1 9.1 2 2 1 8.3 2 3 1 7.6 2 3 2 7.1 3 1 1 5.6 3 1 2 6.2 - i.
- Set up and , and give the dimensions of and .
- ii.
- What is the rank of ?
- iii.
- Set up OLS equations.

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