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ANSC*6370 - QG and Animal Breeding
Fall 2003 - Assignment 13 - Answers

Data on three traits are available on 7 animals as shown below.

Animal	Sire	Dam	Age Group	Trait 1	Trait 2	Trait 3
6	1	3	3	2.5	-	162
7	1	4	2	8.4	35	175
8	1	5	3	8.2	41	-
9	2	3	1	9.0	27	183
10	2	4	1	7.8	32	156
11	2	5	2	2.8	-	168
12	6	10	3	7.4	67	-

Let the model for trait 3 be

$$y_{3i} = \mu_{3} + a_{3i} + e_{3i}$$

Assume that

$${\bf R} = \left( \begin{array}{rrr} 10 & 5 & 30 \\ 5 & 100 & 71 \\ 30 & 71 & 960 \end{array} \right),$$

and

$${\bf G} = \left( \begin{array}{rrr} 1 & 2 & -6 \\ 2 & 15 & -12 \\ -6 & -12 & 800 \end{array} \right).$$

Set up the multi-trait MME and solve.

The solution is most easily shown by giving the SAS IML program that would solve the problem, using Bruce Tier's trick. The program is shown below.

options ls=80 nocenter;
proc iml;

/* Assignment 13, Multiple Trait Models */
y={2.5,  0, 162,
   8.4, 35, 175,
   8.2, 41,   0,
   9.0, 27, 183,
   7.8, 32, 156,
   2.8,  0, 168,
   7.4, 67,   0};

R={10 5 30, 5 100 71, 30 71 960};

G={1 2 -6, 2 15 -12, -6 -12 800};

R1=inv(R);
Ri=I(7)@R1;
Gi=inv(G);

X={0 0 1  0 0 0 0 0  0 0 0, 0 0 0  0 0 0 1 0  0 0 0, 
     0 0 0  0 0 0 0 0  1 0 0,
   0 1 0  0 0 0 0 0  0 0 0, 0 0 0  0 1 0 0 0  0 0 0,
     0 0 0  0 0 0 0 0  1 0 0,
   0 0 1  0 0 0 0 0  0 0 0, 0 0 0  0 0 1 0 0  0 0 0,
     0 0 0  0 0 0 0 0  0 1 0,
   1 0 0  0 0 0 0 0  0 0 0, 0 0 0  1 0 0 0 0  0 0 0,
     0 0 0  0 0 0 0 0  1 0 0,
   1 0 0  0 0 0 0 0  0 0 0, 0 0 0  1 0 0 0 0  0 0 0,
     0 0 0  0 0 0 0 0  1 0 0,
   0 1 0  0 0 0 0 0  0 0 0, 0 0 0  0 0 0 0 1  0 0 0,
     0 0 0  0 0 0 0 0  1 0 0,
   0 0 1  0 0 0 0 0  0 0 0, 0 0 0  0 0 1 0 0  0 0 0,
     0 0 0  0 0 0 0 0  0 0 1};

Z=j(21,15,0)||I(21);

Ai={5  0  1  1  1 -2 -2 -2  0  0  0  0,
    0  5  1  1  1  0  0  0 -2 -2 -2  0,
    1  1  4  0  0 -2  0  0 -2  0  0  0,
    1  1  0  4  0  0 -2  0  0 -2  0  0,
    1  1  0  0  4  0  0 -2  0  0 -2  0,
   -2  0 -2  0  0  5  0  0  0  1  0 -2,
   -2  0  0 -2  0  0  4  0  0  0  0  0,
   -2  0  0  0 -2  0  0  4  0  0  0  0,
    0 -2 -2  0  0  0  0  0  4  0  0  0,
    0 -2  0 -2  0  1  0  0  0  5  0 -2,
    0 -2  0  0 -2  0  0  0  0  0  4  0,
    0  0  0  0  0 -2  0  0  0 -2  0  4};
Ai=Ai/2;

/* Form the MME */
Hi=Ai@Gi;
W=X||Z;
wrw=W`*Ri*W;
wry=W`*Ri*y;
B=J(11,11,0);
add=block(B,Hi);
MME=wrw+add;

/* Solve and print results */
C=inv(MME);
Bhat=C*wry;
D=vecdiag(C);
print wry Bhat D;
res=y-W*Bhat;
print y res;
end;

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