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10-637 Animal Models
Mid-Term Exam - February 25, 1999

1.
For ${\bf G}$ and ${\bf R}$ being positive definite, symmetric, and full rank, and ${\bf Z}$ being rectangular, prove that

\begin{displaymath}({\bf ZGZ}'+{\bf R})({\bf R}^{-1} - {\bf R}^{-1}{\bf Z}
( {\...
...{\bf Z}+{\bf G}^{-1})^{-1} {\bf Z}'
{\bf R}^{-1} ) = {\bf I}. \end{displaymath}

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
A1A1B1B1 .0504 1 1 2
A1A1B1B2 .0972 1 1 2
A1A1B2B2 .0324 1 -1 0
A1A2B1B1 .1512 2 1 3
A1A2B1B2 .2916 2 1 3
A1A2B2B2 .0972 2 -1 1
A2A2B1B1 .0784 -1 1 0
A2A2B1B2 .1512 -1 1 0
A2A2B2B2 .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 GA and GB.
(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 $\frac{1}{4}$ $\frac{5}{16}$
C      
  C2 0 $\frac{5}{8}$
       

(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

yijk = OTYi + Lij + aijk + eijk,

where OTYi is an owner-test-year subclass effect, Lij is a litter effect within OTY subclasses, aijk is an animal additive genetic effect within litter and OTY effects, and eijk 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 ${\bf X}$ and ${\bf Z}$, and give the dimensions of ${\bf R}$ and ${\bf G}$.
ii.
What is the rank of ${\bf X}$?
iii.
Set up OLS equations.

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Larry Schaeffer
1999-02-17