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

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*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