This LaTeX document is available as postscript or asAdobe PDF.

ANSC*6370 - QG and Animal Breeding
Fall 2003 - Assignment 11 - One Set of Answers
Please note the answers depend on your random number generator. Every student will get a different set of observations.

1.

Generate data according to the model

y_iklm = (AX)_i + a_k + m_l + p_l+ e_iklm,

where y_iklm is a birthweight of a Simmental calf k belonging to age of dam-sex of calf subclass i, with dam l; (AX)_i is a fixed effect of age of dam by sex of calf interaction; a_k is a random additive genetic effect of the calf; m_l is a random maternal genetic effect of the dam of the calf; p_l is a random maternal permanent environmental effect of the dam; and e_iklm is a random residual effect.

Use the following data design to generate 10 calf birthweights. None of the calves nor sires and dams are inbred.

Age of	Sex of	Calf	Sire	Dam
Dam	Calf	ID	ID	ID
4	F	10	1	2
4	M	11	3	4
3	F	12	3	6
3	M	13	5	8
4	F	14	7	2
2	M	15	3	10
2	F	16	5	12
4	M	17	9	6
2	F	18	9	14
3	M	19	11	10

Assume that

$$\begin{eqnarray*}\sigma^{2}_{a} & = & 40, \\ \sigma^{2}_{m} & = & 20, \\ \sigm... ... & -2, \\ \sigma^{2}_{p} & = & 11, \\ \sigma^{2}_{e} & = & 74. \end{eqnarray*}$$

Let the (AX) means be

Age	Sex	Mean
2	M	52
2	F	47
3	M	55
3	F	50
4	M	59
4	F	53

2.

Analyze the simulated data using the same model.

3.

Correlate the direct EBVs to the true BVs and also correlate the maternal EBVs to the true maternal BVs. Compare estimates of fixed effects to true values.

STEPS TO GENERATE RECORDS

1.

Generate true genetic values for direct and maternal effects for animals with unknown parents (i.e. animals 1 to 9).

$${\bf TBV} = \left( \begin{array}{rr} 6.3246 & 0.0000 \\ -0.3... ... \mbox{direct genetic} \\ \mbox{maternal} \end{array} \right).$$

For animal 1, for example, if

$$\left( \begin{array}{r} RND_{1} \\ RND_{2} \end{array} \right... ...\left( \begin{array}{r} -0.3694 \\ 0.8444 \end{array} \right),$$

then

$${\bf TBV} = \left( \begin{array}{rr} 6.3246 & 0.0000 \\ -0.3... ...\left( \begin{array}{l} -2.3362 \\ 3.8839 \end{array} \right).$$

2.

Generate true genetic values for animals with known parents (i.e. animals 10 to 19). In this case, the average of the true genetic values of the parents is added to randomly generated Mendelian sampling effects. In this example all animals are not inbred so that the variance of Mendelian sampling effects is one half the additive genetic variances. For animal 10, with sire 1 and dam 2, then

$$\mbox{Average of Parents} = 0.5 \left( \begin{array}{r} -2.33... ...\left( \begin{array}{r} -2.0596 \\ 0.1087 \end{array} \right),$$

and the Mendelian sampling contribution is

$$(0.5)^{.5} \left( \begin{array}{rr} 6.3246 & 0.0000 \\ -0.31... ...left( \begin{array}{r} RND_{1} \\ RND_{2} \end{array} \right),$$

then combining the parent average and Mendelian sampling gives

$${\bf TBV} = \left( \begin{array}{r} -2.7290 \\ -1.3202 \end{array} \right).$$

3.

Generate maternal permanent environmental effects. This is accomplished by drawing a random normal deviate and multiplying by the standard deviation of maternal permanent environmental effects (i.e. square root of 11).

The following table is one set of possible results for true breeding values and maternal PE effects for 19 animals.

Animal	Sire	Dam	TDirect	TMaternal	MatPE
ID	ID	ID
1	-	-	-2.34	3.88
2	-	-	-1.78	-3.67	2.44
3			-3.37	-9.25
4			-1.07	-1.67	-5.22
5			-8.75	0.11
6			-3.29	1.87	4.45
7			2.91	3.41
8			-6.80	6.45	-1.91
9			4.37	4.85
10	1	2	-2.73	-1.32	9.78
11	3	4	-6.23	-7.75
12	3	6	-9.27	-2.67	6.23
13	5	8	-4.91	1.97
14	7	2	-0.91	-0.12	1.43
15	3	10	3.72	-15.36
16	5	12	-6.49	-0.79
17	9	6	0.81	10.71
18	9	14	-5.70	3.68
19	11	10	-7.88	-1.93

4.

Form the records for animals 10 to 19 following the equation of the model. That is,

y_iklm = (AX)_i + a_k + m_l + p_l+ e_iklm.

A residual effect is created for each record, using a random normal deviate times the standard deviation of residual effects (i.e. square root of 74).

To illustrate, for animal 10, the dam was 4 yrs old and animal 10 is female, therefore, the appropriate (AX)_imean is 53. The dam was animal 2, therefore,

$$\begin{eqnarray*}(AX)_{i} & = & 53 \\ a_{10} & = & -2.73 \\ m_{2} & = & -3.67 \\ p_{2} & = & 2.44 \\ e_{10} & = & RND*8.60 = 9.41 \\ \end{eqnarray*}$$

Adding these numbers together gives the observation for animal 10 as 58.45, which can be rounded off to the nearest whole number, 58.

The important points of this step are that the direct genetic effect comes from the animal itself, and the maternal genetic contribution is that of the dam of the animal.

Below is a table of one possible set of completed observations.

Age of	Sex of	Calf	Sire	Dam	AX	TBV	MatBV	MatPE	Res.	BW
Dam	Calf	ID	ID	ID
4	F	10	1	2	53	-2.73	-3.67	2.44	9.41	58
4	M	11	3	4	59	-6.23	-1.67	-5.22	-5.07	41
3	F	12	3	6	50	-9.27	1.87	4.45	1.82	49
3	M	13	5	8	55	-4.91	6.45	-1.91	13.91	69
4	F	14	7	2	53	-0.91	-3.67	2.44	-12.42	38
2	M	15	3	10	52	3.72	-1.32	9.78	12.65	77
2	F	16	5	12	47	-6.49	-2.67	6.23	-6.04	38
4	M	17	9	6	59	0.81	1.87	4.45	-5.77	60
2	F	18	9	14	47	-5.70	-0.12	1.43	4.78	47
3	M	19	11	10	55	-7.88	-1.32	9.78	3.89	59

Care must be taken to construct the mixed model equations correctly. This is shown in the notes.

The solutions using the generated data came out as follows:

Age	Sex	Actual Mean	Estimated
2	M	52	77.82
2	F	47	42.54
3	M	55	64.04
3	F	50	48.38
4	M	59	51.03
4	F	53	47.97

T=True values from simulation, E=estimated values from simulated data.

Animal	Sire	Dam	TDirect	TMaternal	EDirect	EMaternal
ID	ID	ID
1	-	-	-2.34	3.88	1.69	-0.46
2	-	-	-1.78	-3.67	0.01	0.01
3			-3.37	-9.25	-1.99	-0.28
4			-1.07	-1.67	-1.52	-1.19
5			-8.75	0.11	-0.01	0.00
6			-3.29	1.87	0.81	0.85
7			2.91	3.41	-1.67	0.47
8			-6.80	6.45	0.67	0.71
9			4.37	4.85	2.04	-0.10
10	1	2	-2.73	-1.32	2.53	-0.68
11	3	4	-6.23	-7.75	-3.41	-0.65
12	3	6	-9.27	-2.67	-0.93	-0.07
13	5	8	-4.91	1.97	1.08	0.32
14	7	2	-0.91	-0.12	-2.51	0.71
15	3	10	3.72	-15.36	0.27	-0.48
16	5	12	-6.49	-0.79	-1.23	0.00
17	9	6	0.81	10.71	2.70	0.31
18	9	14	-5.70	3.68	0.53	0.27
19	11	10	-7.88	-1.93	-1.18	-0.63

The maternal permanent environmental effects, T=true and E=estimated were

Animal	Sire	Dam	TMatPE	EMatPE
ID	ID	ID
2	-	-	2.44	0.00
4			-5.22	-0.70
6			4.45	0.70
8			-1.91	0.41
10	1	2	9.78	-0.41
12	3	6	6.23	-0.42
14	7	2	1.43	0.42

The above estimated values are not very close to the true value because there were so few observations and animals, and too many age of dam by sex of calf classes. The reader can calculate the correlations between estimates and true values.

This LaTeX document is available as postscript or asAdobe PDF.