Download Multivariate Selection: Methods and Genetic Variance-Covariance Matrix and more Study notes Botany and Agronomy in PDF only on Docsity!
Chapter 18: Multivariate Selection
The economical value of an animal or plant normally depends on several traits. Selection must be considered on these traits simultaneously.
Common methods for multi-trait selection
- Tandem selection
Select in turn for each character singly in successive generations.
- Independent culling levels
Select for all the characters at the same time but independently. Reject all individuals that fail to come up to a certain standard for each character, regardless of their values for any other characters.
The advantage of independent culling level is that if traits are expressed in different stages, this method will allow breeders to select in several stages, referred to as multi- stage selection. Multistage selection is very practical in large animals and trees.
- Index selection
Select an index which is a linear combination of the phenotypic values of all characters.
- Multistage index selection
A combination of independent culling level with index selection.
Variance-covariance matrix
Define (^) X = [ X (^) 1 X (^) 2 " XK ]T as a vector of phenotypic values for K traits
expressed in the same individuals, A = [ A 1 (^) A 2 (^) " AK ]T a vector of breeding values of the
K traits , and E = [ E 1 (^) E (^) 2 " EK ]T a vector of environmental effects.
The multivariate model is X = A + E.
Because X are already expressed as deviations from the population means, E X ( ) = 0 and
Var X Var A Var E
P G E
P G
(^ )^ (^ )^ (^ )
E
E
where
P Var X
Var X Cov X X Cov X X Cov X X Var X Cov X X
Cov X X Cov X X Var X
K K
K K
L
N
M
M
M
O
Q
P
P
P
1 1 2 1 1 2 2 2
1 2
" K
K
K
is the phenotypic variance-covariance matrix of the K traits,
G Var A
Var A Cov A A Cov A A Cov A A Var A Cov A A
Cov A A Cov A A Var A
K K
K K
L
N
M
M
M
O
Q
P
P
P
1 1 2 1 1 2 2 2
1 2
is the genetic variance-covariance matrix for the K characters, and
E Var E
Var E Cov E E Cov E E Cov E E Var E Cov E E
Cov E E Cov E E Var E
K K
K K
L
N
M
M
M
O
Q
P
P
P
1 1 2 1 1 2 2 2
1 2
is the environmental variance-covariance matrix. It should be that
Cov ( X , A ) = Cov ( A + E , A ) = Cov ( A , A ) + Cov ( E , A ) = Var ( A )= G 0
Response to index selection
- Selection response of the aggregate breeding value
The selection response of the aggregate breeding value is denoted by Δ H , which can be predicted using the usual expression:
Δ H = i (^) I rIH σ H
where
σ H^2 = Var H ( ) = Var w A ( T^ ) = w Var A w T^ ( ) = w Gw T
and
r
Cov I H Cov b X w A b Cov X A w b Gw b Pb IH H I H I H I H I H I
I H I
I H
σ σ σ σ σ σ σ σ σ σ
σ σ σ
σ σ
T T T T T 2
Therefore,
Δ H i (^) I I i H
= H I
/ =^ I
σ σ
σ σ
where σ I^2^ = Var b X ( T^ )= b Pb T^.
- Selection responses of individual traits (components of H )
The aggregate breeding value is decomposed as Δ H = w 1 (^) Δ A 1 (^) + w 2 (^) Δ A 2 (^) +" + w (^) K Δ AK
where Δ Ai is the genetic change (response) of the i-th character. Define
Δ A = Δ A 1 (^) Δ A 1 (^) " Δ A 1 T which can be predicted by
Δ Δ Δ
A b I I b I
Cov A I Var I
I
Gb I Gb
i AI s AI I I
I I
( ) σ σ σ
Note that
i (^) I = Δ I / σ I
and
Cov A I ( , ) = Cov A b X ( , T^ ) = Cov A X b ( , ) = Gb.
The detailed expression of Δ A is
A
A
A
A
i
Var A Cov A A Cov A A Cov A A Var A Cov A A
Cov A A Cov A A Var A
b b
b
i Cov A A b
i Cov A A b
i Cov A A b
K
I I
K K
K K K K
I I j j^ j
K
I I j j^ j
K
I I j K^ j^ j
K
L
N
M
M
M
O
Q
P
P
P
L
N
M
M
M
O
Q
P
P
P
L
N
M
M
M
O
Q
P
P
P
L
N
M
M
=
=
=
1 2
1 1 2 1 1 2 2 2
1 2
1 2
1 1
1 2
1
σ
σ
σ
σ
M
M
M
M
M
O
Q
P P P P P P P
