Page 272 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

There are three terms in L: the first is a weighted sum of squares of residuals; the sec-
ond, a term that depends on the variance matrix; and a third that depends on the vari-
ance matrix of the fixed effects and can be thought of as a penalty because fixed effects
are estimated. MME (Chapter 3) play an important part in the analysis process.
For the particular model these can be written as (Eqn 3.4):
¢
é XX X Z ¢ ù é ù ˆ b é Xy ¢ ù
ê - ú ê ú = ê ú
1
¢
¢+
ë ZX ZZ A a û a ë û ë Z y ¢ û
ˆ
2
2
2
2
with a = s /s or (1 - h )/h .
e a
Extensive use is made of the prediction error matrix of a. In this case the predic-
22
2
22
tion error matrix is PEV = var(a - aˆ) = C s (Eqn 3.14), where C is associated with
e
the coefficient matrix of the MME.
2
Estimates of s and s are chosen to maximize L. It is useful to express relevant
2
a e
terms in this estimation process in terms of the projection matrix P:
−1
−1
−1
P = V - X(X ′ V X) X ′ V −1
Then:
−1
1
L a( ){-y ′ Py - logdet(V) - logdet(X ′ V X)} (15.1)
2
Estimation of a variance parameter q (q = s , q = s ) involves setting to zero
2
2
i 1 e 2 a
the first derivatives:
¶L/¶q = ( ){y ′ P(¶V/¶q )Py - trace[P(¶V/¶q )]}
1
i 2 i i
These equations could be thought of as equating a function of data (the first term in
the expression) to its expectation.
Normally, finding a maximum requires an iterative scheme. One suggested by
Patterson and Thompson (1971) was based on using the expected value of the second
differential matrix. In this case these are:
1
2
E(¶L /¶q ¶q ) =-( )trace[P(¶V/¶q )P(¶V/¶q )]
i j 2 i j
Using the first and expected second differentials one can update q using terms
that depend on the solution of the MME and PEVs. For the particular animal model
that is being considered, then:
2 1 4 2
e 2 e e
¶L/¶s = ( ){(y - Xb - Za) ′ (y - Xb - Za)/s - (n - p - q)/s
2
22
- trace[C A ]/s } (15.2)
-1
a
−1
4
4
2
1
22
2
-1
2
¶L/¶s = ( ){a ′ A a/s + q/s - trace[C A ]s /s } (15.3)
a 2 a a e a
and:
4
−1 2
1
4
4
2
22
E(¶L /¶s ) = −( ){(n − p − q)/s + trace[(C A ) ]/s }
e 2 e a
2
2
2
4
−1
4
2
22
E(¶L /¶s ) = −( ){trace[{I − C A (s /s )} ]/s }
1
a 2 e a a
2
2
22
−1
4
E(¶L /¶s ¶s ) = −( ){trace[{I − C A (s /s )}{C A }]s }
22
2
1
−1
2
2
a e 2 e a a
Thinking of the variance parameters and the first differentials as vectors q and ¶L/¶q
with ith (i = 1, 2) element q and ¶L/¶q , respectively, and Einf, the expected informa-
i i
tion matrix, a matrix with i,jth element −E(¶L /¶q ¶q ), suggests an iterative scheme
2
i j
with the new estimate q satisfying:
n
256 Chapter 15