Page 275 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

¶L/¶s = (0.5){8.8753 − (−12.5000) − 18.0733} = 1.6510
2
e
−1
4
2
¶L/¶s = (0.5){a ′ A a/s − q/s + trace[C A ]s /s }
4
−1
2
2
22
a a a e a
2
¶L/¶s = (0.5){6.3461 − 40.0000 + 36.1466} = 1.2464
a
and:
2
A(¶L /¶s ) = −(0.5){(y − Xb − Za) ′ P(y − Xb − Za)/s }
4
4
e e
2
4
A(¶L /¶s ) = −(0.5)16.5346 = −8.2673
e
A(¶L /¶s ) = −(0.5){a ′ Z ′ PZa}/s 4
4
2
a a
4
2
A(¶L /¶s ) = −(0.5)9.1163 = −4.5582
a
2
4
2
2
2
A(¶L /¶s ¶s ) = −(0.5){a ′ Z ′ P(y − Xb − Za)}/(s s )
a e e a
2
2
2
A(¶L /¶s ¶s ) = −(0.5)11.3070 = −5.6535
a e
and:
⎡8.2673 5.6535 ⎤ ⎡ 0.7967 − 0.9882⎤
Ainf = ⎢ ⎥ so Ainf −1 = ⎢ ⎥
⎣ 5.6535 4.5582 ⎦ ⎣ −00.9882 1.4450 ⎦
Using Eqn 15.4 and replacing Einf by Ainf:
⎤ ⎡
q = q + Ainf −1 ( L∂ / ∂ q = ) ⎡0.4 ⎤ ⎥ + ⎢ ⎡ 0.7967 −0.9882 1.6510⎤ ⎥ = ⎡ ⎢ 0.4⎤ ⎥ + ⎡ ⎢ 0.0838⎤ ⎥
⎢
⎥ ⎢
n
0
⎦ ⎣
⎣ 0.2 ⎦ ⎣ −0.9882 1.4450 1.2464 ⎦ ⎣ 0.2 ⎦ ⎣ 0.1695 ⎦
2
2
2
2
so that new estimates of s and s are 0.4838 (kg ) and 0.3695 (kg ), respectively.
e a
Table 15.3 gives six successive iterates and log-likelihood for this data.
In the last iteration:
⎡ 2 4436 −3 2532. ⎤
.
Ainf = ⎢ ⎢ ⎥ ⎥
−1
.
⎢ ⎣ −3 2532. 5 3481 ⎥ ⎦
so that the estimate of s is 0.4835 with standard error 2.4436 = 1.563 and the esti-
2
e
2
mate of s is 0.5514 with standard error 5.3481 = 2.313.
a
2
2
By contrast, if estimates of s = 0.4 and s = 0.2 are used in conjunction with Eqns 15.5
e a
2
2
2
2
and 15.6 then: (n − p)s = (y – Xb − Za) ′ (y) so 3s = 1.9277 so s = 0.6426 (kg )
e e e
2
and qs = a ′ A a + trace[C A ]s so 8s = 0.2538 + 1.4458 so s = 0.2125 (kg ) with
2
2
−1
22
−1
2
2
a e a a
L = −2.3852. After 1000 iterations, the algorithm gives s = 0.4842 (kg ) and s =
2
2
2
e a
2
0.5504 (kg ) with L = −2.1817, showing that this algorithm is slower to converge.
Table 15.3. Estimates of s and s and L.
2
2
e a
2
2
2
2
Iterate s (kg ) s (kg ) L
e a
1 0.4000 0.2000 −2.3852
2 0.4838 0.3695 −2.2021
3 0.4910 0.5126 −2.1821
4 0.4839 0.5500 −2.1817
5 0.4835 0.5514 −2.1817
6 0.4835 0.5514 −2.1817
Estimation of Genetic Parameters 259