Upload
donna-elliott
View
215
Download
1
Embed Size (px)
Citation preview
Bayesian integration of external information into the single step
approach for genomically enhanced prediction of breeding values
J. Vandenplas, I. Misztal, P. Faux, N. Gengler
1
• Unbiased EBV if genomic, pedigree and phenotypic information considered simultaneously
• Problem – Only records related to selected animals available– Bias due to genomic pre-selection
• Single step genomic evaluation (ssGBLUP)– Simultaneous combination of genomic, pedigree and
phenotypic information (=internal information) – No integration of external information (e.g. MACE-EBV)
Introduction
2
• Integration of a priori known external information into ssGBLUP– By a Bayesian approach– To avoid multi-step methods– By considering
• simplifications of computational burden,• a correct propagation of external information,• and no multiple considerations of contributions due to
relationships.
Objective
3
Methods
• Bayesian approach (Dempfle, 1977; Legarra et al., 2007)
• 2 groups of animals
1) animals I = internal animals with only records in Ia: non genotyped animals
Ib: genotyped animals
2) animals E = external animals with records in and possible records in
Ea: non genotyped animals
Eb: genotyped animals4
Iy
EyIy
Methods
• An internal evaluation – All animals Ia, Ib, Ea, Eb– Only – instead of
where
5
)G,MVN(μ)yup( *EEI ˆ
E1*
I1
I
I1
I
I
I1*
I1
II1
I
I1
II1
I
μGyRZ'
yRX'
u
β
GZRZ'XRZ'
ZRX'XRX'
ˆ
ˆ
G)MVN(0,)up( I ˆIy
'uuuu IEIIbIIaI ˆˆˆˆ
Methods
• An unknown external ssGBLUP – All animals Ia, Ib, E– Genomic information included in – Only ( precorrected for fixed effects)
6
*Ey
IbIbG
E1*
*Eb
1EEb
*Ea
1EEa
EEb
EEa
EIb
EIa
EbEbEEb
1EEb
EbEaEEa
1EEb
EbIbEbIa
EaEbEEb
1EEa
EaEaEEa
1EEa
EaIbEaIa
Ib2bIbEaIbIbIbIa
Ia2bIaEaIaIbIaIa
μG
yRZ'
yRZ'
0
0
u
u
u
u
GZRZ'GZRZ'GG
GZRZ'GZRZ'GG
GGGG
GGGG
ˆ
ˆ
ˆ
ˆ
Methods
• Problem– Unknown external ssGBLUP
• Available– External genetic evaluation of animals Ea and Eb
• without animals Ia and Ib• without genomic information
7
*EE
1EE
*E
1EE
*EE
1*EEEE
1EE uDyRZ'uGZRZ' ˆˆ
EbEb*EbEbEbEa*EbEa
*EaEbEaEb*EaEaEaEa1*
EE1EEEE
1EE GDGD
GDGDGDZRZ'
1*EEG
*EEb
EbEb*EEa
EbEa
*EEb
EaEb*EEa
EaEaE1*
EbEbEbEb*EbEbEbEa*EbEaEbEaEbIbEbIa
EaEbEaEb*EaEbEaEa*EaEaEaEaEaIbEaIa
Ib2bIbEaIbIbIbIa
Ia2bIaEaIaIbIaIa
1*
uDuD
uDuD
0
0
μG
GGDGDGGG
GGDGDGGG
GGGG
GGGG
G
ˆˆ
ˆˆ
Methods
• Substitution in the unknown external ssGBLUP– and
8
*E
1EE yRZ'
EE1
EE ZRZ'
Methods
• Finally, internal evaluation = ssGBLUP integrating external information
9
E1*
I1
I
I1
I
I
I1*
I1
II1
I
I1
II1
I
μGyRZ'
yRX'
u
β
GZRZ'XRZ'
ZRX'XRX'
ˆ
ˆ
EbEbEbEb*EbEbEbEa*EbEaEbEaEbIbEbIa
EaEbEaEb*EaEbEaEa*EaEaEaEaEaIbEaIa
Ib2bIbEaIbIbIbIa
Ia2bIaEaIaIbIaIa
GGDGDGGG
GGDGDGGG
GGGG
GGGG
*EEb
EbEb*EEa
EbEa
*EEb
EaEb*EEa
EaEa
uDuD
uDuD
0
0
ˆˆ
ˆˆ
Methods
• Approximations and simplifications of computational burden
10
E1
I1
I
I1
I
I
I1*11
I1
II1
I
I1
II1
I
μDyRZ'
yRX'
u
β
GDGZRZ'XRZ'
ZRX'XRX'
ˆˆ
ˆ
1*EE*E
1*EE
*EE
*EEI )(G,uGGMNV)uup( ˆˆˆ
'*'E
'EIE uuμ ˆˆˆ
RHS: add a product between a matrix and a vector
Methods
• Approximations and simplifications of computational burden
11
E1
I1
I
I1
I
I
I1*11
I1
II1
I
I1
II1
I
μDyRZ'
yRX'
u
β
GDGZRZ'XRZ'
ZRX'XRX'
ˆˆ
ˆ
traits,...,1 ; ))REL1(REL(
,...,1 ; )(
ijij
*
tjdiag
animalsniblockdiag
i
i10i
11
Δ
ΔGΔΛ
ΛGD
LHS: add a block diagonal matrix
Simulation
•
12