Page 115 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

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identity matrix. The transformation matrix T is obtained by carrying out a Cholesky
decomposition of R, the residual covariance matrix for the traits, such that:
R = TT′

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where T is a lower triangular matrix. The transformation matrix T is the inverse of T.
The formula for calculating T is given in Appendix E, Section E.3.
The vector of observations y for the ith animal is transformed as:
ki
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*
y = T y
ki ki
*
where k is the number of traits recorded and y is the transformed vector.
ki
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If traits are missing in y , then the corresponding rows of T are set to zero when
ki
transforming the vector of observation. Thus if y is a vector of observations of n
ki
traits for the ith animal, the transformation of y can be illustrated as:
y = t y
11
*
11 11
22
21
y = t y + t y
*
21 11 21
…
…
…
*
y = t y + t y + t y
n1
nn
n2
n1 11 21 n1
where the t above are the elements of T .
ij
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Given that the variance of y is:
ki
var(y) = G + R
and the variance of the transformed variables becomes:
*
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*
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var(y ) = T G(T )′ + I = G + I = M + I (6.5)
*
where G is the covariance matrix for additive genetic effects and G is the transformed
*
additive genetic covariance matrix. Note that G is not diagonal. Vectors of solutions
*
*
(b and a ) are transformed back to the original scale (b and a ) as:
i i i i
b = Tb * (6.6)
i i
a = Ta * (6.7)
i i
6.3.2 An illustration
Example 6.2
The methodology is illustrated using the growth data on beef calves in Section 5.4.1.
The residual and additive genetic covariance matrices were:
⎡ 40 11⎤ ⎡20 18 ⎤
R = ⎢ ⎥ and G = ⎢ ⎥
⎣ 11 30 ⎦ ⎣ 18 40 ⎦
Now carry out a Cholesky decomposition of R such that R = TT′. For the R above:
⎡ 6.324555 0.000 ⎤ ⎡ 0.1581139 00.000 ⎤
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T = ⎢ ⎥ with T = ⎢ ⎥
⎣ 1.739253 5.193746 ⎦ ⎣ − 0.052948 0.1925393 ⎦
Methods to Reduce the Dimension of Multivariate Models 99