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10-637 Animal Models
Final Take-Home Exam
Due: April 22, 1999, Thursday
Please Do Your Own Work

1.
(8 points) To do this problem see Henderson (1985, J. Anim. Sci. 60:111-117). Assume a model with the following genetic factors,

\begin{displaymath}y_{ij} = \mu + a_{i} + d_{i} + (aa)_{i}+ (ad)_{i}
+(dd)_{i} + e_{i}. \end{displaymath}

Let the underlying parameters be

$\sigma^{2}_{10}$ = 289, $ \ \ \ $ $\sigma^{2}_{01}$ = 169,
$\sigma^{2}_{20}$ = 81, $ \ \ \ $ $\sigma^{2}_{02}$ = 49,
$\sigma^{2}_{11}$ = 64, $ \ \mbox{and} \ $ $\sigma^{2}_{e}$ = 484.

(a)
Simulate a single observation per animal on five animals, following the above model. Assume the following additive and dominance relationship matrices among these animals.

\begin{displaymath}{\bf A} = \frac{1}{8} \left( \begin{array}{rrrrr}
8 & 0 & 4 ...
...6 & 2 & 6 & 10 & 6 \\
3 & 5 & 5 & 6 & 9 \end{array} \right), \end{displaymath}

and

\begin{displaymath}{\bf D} = \frac{1}{64} \left( \begin{array}{rrrrr}
64 & 0 & ...
... & 16 & 68 & 10 \\
0 & 8 & 16 & 10 & 65 \end{array} \right). \end{displaymath}

(b)
Analyze the data with the model used to generate the observations.
(c)
Analyze the data using the model,

\begin{displaymath}y_{i} = \mu + g_{i} + e_{i}, \end{displaymath}

where gi=ai+di+(aa)i+(ad)i+(dd)i. Refer to the Henderson paper for help. Compare your results.

2.
(10 points) Apply a simple animal model ( ${\bf y}=\mu{\bf 1}+{\bf a}+{\bf e}$) to the following data.

Animal Sire Dam Trait
1     29
2     32
3 1 2 36
4 1 2 30
5 1   34
6 1   33
7 3   38

(a)
Construct and solve HMME using a residual to additive genetic variance ratio of 1.5.
(b)
Absorb the equations for animals 4, 5, 6, and 7 into the equations for $\mu$ and animals 1, 2, and 3. Solve the resulting equations.

The absorbed equations are equivalent to those under a reduced animal model, i.e. RAM. Reduced animal models are practical in situations where many of the animals are never used as parents. For example, in fish breeding, several thousand fish are raised, but only a few are used as parents of the next generation. Thus, solutions for non-parent animals are not really needed. There is a way of constructing RAM MME without doing the absorption. The idea is to have one model for animals that have progeny (i.e. animals 1, 2, and 3), and a different model for non-parent animals (i.e. 4, 5, 6, and 7) which would have different residual variances depending on number of known parents. Below are the ${\bf X}$ and ${\bf Z}$ matrices for RAM.

\begin{displaymath}{\bf X}=\left( \begin{array}{c}
1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \...
... .5 & 0 & 0 \\ .5 & 0 & 0 \\
0 & 0 & .5 \end{array} \right). \end{displaymath}

The residual variance for the first three animals is $\sigma^{2}_{e}$, for animal 4 is $\sigma^{2}_{e}+.5\sigma^{2}_{a} = 1.3333\sigma^{2}_{e}$, and the residual variances for animals 5, 6, and 7 would be $\sigma^{2}_{e}+.75\sigma^{2}_{a} = 1.5 \sigma^{2}_{e}$.
(c)
Construct the RAM MME equations using the matrices for ${\bf X}$, ${\bf Z}$, and ${\bf R}$ given above. They should be identical to those derived from absorption.

(d)
Estimate the residual variance from the full model and from the RAM. They should be equal if done correctly.

3.
(12 points) Selection bias issues will be studied with these questions. Assume a model equation as

yij = (CG)i + aj + eij.

Also, let $\sigma^{2}_{e} = 400$, and $\sigma^{2}_{a} = 225$. The data for Data Set 1 are given in the table below.
(CG)1 (CG)2
Animal Sire Dam Trait Animal Sire Dam Trait
10 1 2 47 15 7 2 -25
11 1 3 20 16 8 4 5
12 7 4 -22 17 1 9 13
13 7 5 -72 18 1 3 92
14 8 6 -44 19 8 5 -93

(a)
Set up and solve HMME.
(b)
Create Data Set 2 by adding the following two poorly producing animals to contemporary group 1 and analyze Data Set 2. Some herd owners try to make their good animals look better by temporarily bringing in very poor performing animals as contemporaries.

Animal = A Sire = C Dam = D Obs. = -105
Animal = B Sire = C Dam = E Obs. = -80

(c)
Create Data Set 3 by deleting two animals from Data Set 1 that have the lowest trait observations, namely animals 13 and 19. This is known as early culling of animals before their records can be used in genetic evaluations. Set up and solve HMME for Data Set 3.
(d)
Create Data Set 4 by adding +60 to animal 10 and subtracting 60 from animal 19. Set up and solve HMME for Data Set 4. This bias is known as preferential treatment of animals.
(e)
Summarize the results from the four data sets assuming that the results from Data Set 1 are unbiased. Write a short report about the effects of the different selection biases. The report should be neat and readable, preferably typed.

4.
(10 points) Below are observations on 3 traits on progeny of 3 unrelated sires. Assume that the same model can be applied to each trait. That is,

yijk = ci + sj + eijk,

where ci represents fixed contemporary group effect, sj represents a random sire effect, and eijk is a residual effect. Note that not all traits are observed on each animal.

ci sj trait 1 trait 2 trait 3
1 1 61 99 445
1 2 - 102 925
1 3 - 100 -
1 1 65 84 -
1 2 66 - -
2 3 17 152 898
2 3 46 88 706
2 1 44 - 686
2 2 - - 468

(a)
Construct HMME using ${\bf R}$ matrices that allow for the missing observations. Assume that

\begin{displaymath}{\bf R} = \left( \begin{array}{rrr}
36 & -18 & 12 \\ -18 & 130 & 38 \\ 12 & 38 & 549
\end{array} \right), \end{displaymath}

and

\begin{displaymath}{\bf G} = \left( \begin{array}{rrr}
4 & -2 & 1 \\ -2 & 10 & 1 \\ 1 & 1 & 14 \end{array} \right). \end{displaymath}

(b)
Partition the solutions for each sire as shown in class.

(c)
Construct HMME using Dr. Bruce Tier's trick of treating each animal as though all traits are observed and assigning missing observations to their own separate contemporary groups.

(d)
Change all of the covariances in ${\bf R}$ and ${\bf G}$ to zero and re-analyze. Compare results to previous solutions.


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This LaTeX document is available as postscript or asAdobe PDF.

Larry Schaeffer
1999-03-26