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Raphael Mrode

Training in quantitative genetics and genomics

30th May to 10th June 2016

ILRI , Nairobi

BLUP: Repeatability, Common environment , Multivariate and maternal models

2

Repeatability model

• Repeatability model used when multiple measurements on the same trait are recorded on an individual. For instance: – litter size in successive pregnancies or milk yield in successive

lactations.

• In this situation covariance between individuals can be attributed to environmental factors affecting the successive records of an individual (permanent environmental effects).

• Such environmental effects are usually accounted for in the model to ensure accurate prediction of breeding value

3

Repeatability model

• Some assumptions:

- Genetic correlations of unity

- Equal variance for all records

- Equal environmental correlations

• In real data - all assumptions may hold.

• More recently a random regression model has been used for such data

4

Repeatability model

• The phenotypic structure for 3 observations of an individual under this model is:

• Repeatability (re)= (σ2g + σ2

pe ) / σ2y

σ+σ+σσ+σσ+σ

σ+σσ+σ+σσ+σ

σ+σσ+σσ+σ+σ

y

y

y

2g

2pe

2t3

2g

2pe

2g

2pe

2g

2pe

2g

2pe

2t2

2g

2pe

2g

2pe

2g

2pe

2g

2pe

2t1

3

2

1

= var

5

Repeatability model

• The model then is:

• y = Xb + Za + Wpe + e

y = vector of observations

b = vector of fixed effects

a = vector of random animal effects

pe = vector of random pe effects

genetic effects

e = vector of random residual effect.

6

Repeatability model

• The MME are:

YW'

YZ'

YX'

ep

a

b

γIWW'ZZ'XW'

WZ'αAZZ'XZ'

WX'ZX'XX'1

ˆ

ˆ

22

2 1

h

re

a

e

222

2

2

2 1

hr

r

hr e

e

Pe

e

e

e

s

7

Example : Repeatability model

• Month 1 10 9

• Month 2 12 10

• Month 3 15 12

• Given data above, the genetic merit of all animals are predicted using a

repeatability model. It is assumed that re = 0.6,h2 = 0.4 and

• For this example P

2 10

Sire Dam

Cow 1 Cow 2 Cow 3

1h

r1

σ

σα

2e

2a

2e

2

hr

r1

σ

σγ

2e

e2e

2e

s

8

Example: Repeatability model

• A-1 for the above pedigree is

• Solutions are:

13

1

2

1

2

1

2

1

2

1

3

211

2

1

2

11

2

1

2

1011

3

20

3

400

1-1-020

1-1-002

D

S

C

C

C

D S C C C

3

2

1

321

00000

00000

00300

00010

00002

ZZ'

00110

00101

00101

ZX'

200

020

002

XX'

Month Animal (Genetic) Animal (spec. env.) Sire Dam

1 2 3 1 2 3 1 2 3

9.81 11.31 13.58 0.28 0.50 -0.58 0.08 0.31 -0.39 0.39 0.00

9

Repeatability model

EBV = n1PA + n2YD + n3 PC YD = yield corrected for fixed and pe effects

• Cow 3

• YD = ( 9 + 10 + 12 – 9.81 -11.31 -13.58 – 3(-0.39))/3 = -0.8433

• Only dam is known, numerator of n1 = 4/3α

• Cow has got 3 records : numerator of n2 = 3

• Numerator of n1 + n2 = 4/3 + 3 = 4.333 , since α = 1

• n1 = 1.333/4.333 = 0.307 and n2 = 3/4.333=0.692

• PA = 0 , therefore EBV = 0.692*0.8433 -0.58

10

Models with common environment

• When members of a family are reared together, such litters of pigs, they share a common environment

• This contributes to the similarity between members of the family.

• Thus there is an additional covariance between members of a family due to the common environment they share

• Environmental variance may be partitioned into – between family or group component (σ2

c), usually termed the common environment which causes resemblance between members of a family

– and the within family or group variance (σ2e )

11

Models with common environment

• Phenotypic variance may be partitioned into

• a) additive genetic effects resulting from additive genes from parents

• b) common environmental effects affecting full-sibs or all offspring of the same dam. In the case of full-sibs, it may be confounded with dominance effects peculiar to offspring of same parents

• c) and random environmental effects.

12

The Model

• In matrix notation, the model which is

• y = Xb + Za + Wc + e

• where all terms are as defined for the repeatability model except c which is the vector of common environmental effects

• W now relates records to common environmental effects.

• It is assumed that common environmental and residual effects are independently distributed with means of zero and variance σ2

c and σ2e respectively.

• Thus var(c) = Iσ2c, var(e) = Iσ2

e and var(a) = Aσ2a.

•

13

Mixed model equations

• The MME are:

YW'

YZ'

YX'

c

a

b

IWW'ZZ'XW'

WZ'αAZZ'XZ'

WX'ZX'XX'1

ˆ

ˆ

ˆ

1

2

2

a

e

2

2

1

c

e

14

Example data

• Given the data below on the weaning weight of piglets which are progeny of three sows mated to two boars:

• Piglet sire dam gender Weaning Weight (kg) • 6 1 2 1 90 • 7 1 2 2 70 • 8 1 2 2 65 • 9 3 4 2 98 • 10 3 4 1 106 • 11 3 4 2 60 • 12 3 4 2 80 • 13 1 5 1 100 • 14 1 5 2 85 • 15 1 5 1 68

15

Example data

• X matrix (transpose) is:

• Thus Z will be diagonal matrix for animals with records and W will be of this form

1110000000

0001111000

0000000111

W =

0101101110

1010010001 = X

16

Maternal Models

• The phenotypic expression of some traits in the progeny is influenced by the ability of the dam to provide a suitable environment such as better nourishment.

• Thus the dam contributes to the performance of the progeny in two ways: – The direct genetic effects passed to the progeny and

– Her ability to provide a suitable environment, for instance in producing milk.

• Examples: birth and weaning weights are termed maternally influenced traits.

• The ability of the dam to provide a suitable environment for the expression of such traits in her progeny is partly genetic and partly environmental.

• The environmental part may be partitioned into permanent and temporary environmental components.

• Maternal additive genetic component of the dam is passed on to her offspring but it is expressed only when the female offspring have progeny of their own.

17

Maternal models

Phenotypic variance is partitioned into 1) Additive genetic variance effects – direct effects 2) Additive genetic ability of the dam to nurture – indirect

or maternal genetic effect 3) Pe effects – permanent environmental influences 4) Random residual effects

18

Maternal trait model

y = Xb + Zu + Wm + Spe + e

– where

– y = vector of observations

– b = vector of fixed effects

– u = vector of random animal effects

– m = vector of random indirect genetic effects

– pe = vector of permanent environmental

– e = vector of random residual effects

– X, Z, W and S are incidence matrices

19

Maternal trait Model

σI000

0σI00

00AgAg

00AgAg

e

pe

m

u

2e

2pe

2221

1211

= var

20

Maternal trait model: MME

yS

yW

yZ

yX

I+SSWSZSXS

SWA+WWA+ZWXW

SZA+WZA+ZZXZ

SXWXZXXX

ep

m

u

b

-1-1

-1-1

1

4

32

21

α

αα

αα

ˆ

ˆ

ˆ

ˆ

gg

gg = and

gg

gg = G ;

gg

gg =G

2221

1211

2e

32

21

2221

1211

-

2221

1211 1

21

Maternal trait model

Incidence matrices:

Z = diagonal with for animals with records

S and W both relate records to the dam

For dams with records S = W

W is then augmented with columns and rows equal to number of animals in data

22

Maternal trait model: Example

Calf Sire Dam Herds Sex Birth weight

5 1 2 1 1 35.0

6 3 2 1 2 20.0

7 4 6 1 2 25.0

8 3 5 1 1 40.0

9 1 6 2 1 42.0

10 3 2 2 2 22.0

11 3 7 2 2 35.0

12 8 7 3 2 34.0

13 9 2 3 1 20.0

14 3 6 3 2 40.0

23

Example: Maternal trait model

1011100110

0100011001

1110000000

0001110000

0000001111

= X

00000000100000

00000000000010

00000001000000

00000001000000

00000000000010

00000000100000

00000000010000

00000000100000

00000000000010

00000000000010

W

14

13

12

11

10

9

8

7

6

5

=

0011000000

1000010100

0000001000

0100100011

S

7

6

5

2

=

24

Example: maternal trait model

• Effects Solutions

• Herd

• 1 0.000

• 2 3.386

• 3 1.434

• Sex of calf

• 1 34.540

• 2 27.691

• Animal and Maternal

• 1 0.564 0.262

• 2 -1.244 -1.583

• 3 1.165 0.736

• 4 -0.484 0.586

• 5 0.630 -0.507

• 6 -0.859 0.841

• 7 -1.156 1.299

• 8 1.917 -0.158

• 9 -0.553 0.660

• 10 -1.055 -0.153

• 11 0.385 0.916

• 12 0.863 0.442

• 13 -2.980 0.093

• 14 1.751 0.362

• Permanent environment

• 2 -1.701

• 5 0.415

• 6 0.825

• 7 0.461

• ---------------------------------------

25

Maternal models: Sire and maternal grandsire model

• In some cases, due to inadequate data, a sire and maternal

grand sire model is fitted, especially for calving traits such as calving ease or still birth.

• The calving event is regarded as a direct effect of the service sire ( direct effect). This predicts how easily his progeny are born and is computed by fitting the service sire.

• The maternal effect which predicts how easily the bull daughters calve is computed by fitting the maternal grandsire (MGS) , hence the name, sire-maternal grandsire (S-MGS) model.

26

Maternal models: Sire and maternal grandsire model

• Genetic parameters are calculated from such a model as follows: • • Direct genetic variance component = 4*(service sire variance) • • Covariance between direct and maternal component = • 4*(covariance between service sire and MGS) – 0.5*(direct genetic

variance) • • Maternal genetic variance component = 4*(MGS variance) –

0.25*(direct genetic variance) • – (covariance between direct and maternal component) • • If there are repeated records included in the analysis, maternal

repeatability should also be included in the model

27

Multivariate Models

• Selection usually based on a combination of several traits of economic importance that may be phenotypically and genetically related.

• Such traits may be combined into an index on which animals are ranked.

• A multiple trait evaluation is the optimum methodology to evaluate animals on these traits since it accounts for the relationship between them.

• Advantages: – Increases the accuracy of evaluations.

– More importantly, for trait subjected to culling, a multi-trait analysis is necessary to account for this selection. For instance, animals are measured for yearling weight only they have not been culled due to poor weaning weight. Therefore evaluation of yearling weight on its own will be biased

28

Multivariate model

yi = vector of observations for the ith trait

bi = vector of fixed effects for the ith trait

ai = vector of random animal effects for the ith trait

ei = vector of random residual effects for the ith trait

e

e +

a

a

Z0

0Z +

b

b

X0

0X =

y

y

2

1

2

1

2

1

2

1

2

1

2

1

29

Multivariate model

rr00

rr00

00AgAg

00AgAg

=

e

e

a

a

var

2221

1211

2221

1211

2

1

2

1

yRZ

yRX =

a

b

GA+ZRZXRZ

ZRXXRX

1-

1-

1-1-1-1-

1-1-

ˆ

ˆ

y

y = and

a

a = ,

b

b = , = , =

2

1

2

1

2

1y ab

Z0

0ZZ

X0

0XX

2

1

2

1

ˆ

ˆˆ

ˆ

ˆˆ

30

Multivariate model: MME

2

2

ˆ

ˆ

ˆ

ˆ

yR22

Z'2+y1R21

Z'2

yR12

Z'1+y1R11

Z'1

y2R22

X'2+y1R21

X'2

y2R12

X'2+y1R11

X'1

g22

A1

+Z2R22

Z'2g21

A1

+Z1R21

Z'2X2R22

Z'2X1R21

Z'2

g121

A+Z2R12

Z'1g11

A1

+Z1R11

Z'1X2R12

Z'1X1R11

Z'1

Z2R22

X'Z1R21

X'2X2R22

X'2X1R12

X'2

Z2R12

X'1Z1R11

X'1X2R12

X'1X1R11

X'1

-1

=

a2

a1

b2

b1

2

31

Setting the MME

• Involves setting up X1 for trait 1 and X2 for trait 2 and undertaking the supplication shown below

• Note that X1’R11X1, can be set up by forming initially

X1’X1 and multiplying each element by R11, that is, the inverse element of R-1 corresponding to trait one.

•

=

222

2121

2

212

1111

11-

XRXXRX

XRXXRXXRX

''

'''

32

Example - 2 traits

• Calves gender sire dam WWG PWG

• -------------------------------------------------------------

• 4 male 1 - 4.5 6.8

• 5 female 3 2 2.9 5.0

• 6 female 1 2 3.9 6.8

• 7 male 4 5 3.5 6.0

• 8 male 3 6 5.0 7.5

• --------------------------------------------------------------

and4018

1820

PWG

WWG

RG

3011

1140

PWG

WWG

33

Design Matrices

Inverses are:

Design matrices:

0.0370.010-

0.010-0.028and

0.0420.038-

0.038-0.084 11- = = RG

00110

110011 =X

00110

110012 =X

056.00

0084.0028.0

20

0311

1RXX1

34

• Putting everything together

074.00

0111.0037.0

20

0322

22 RXX

02.00

003.001.0

20

0312

2

21

12 RXXRXX 1

0.0740.000.02-0.00

0.000.1110.000.03-

0.02-0.000.0560.00

0.000.03-0.0000.084

XRXXRX

XRXXRXXRX = =

222

2121

2

212

1111

11-

''

'''

35

Design matrices

• Using the same arguments:

•

• Z’1Z1R11 = diag ( 0. 0. 0. 0.028 0.028 0.028 0.028 0.028)

•

• Z’2Z2R11 = diag ( 0. 0. 0. 0.037 0.037 0.037 0.037 0.037)

• Z’2Z1R21 = Z’1Z2R12 = diag ( 0. 0. 0. -0.01 -0.01 -0.01 -0.01 -0.01)

•

• Putting all together

36

Design matrices

• Z’R-1Z is

• 0

• 0 0

• 0 0 0

• 0 0 0 0.028

• 0 0 0 0.0 0.028

• 0 0 0 0.0 0.0 0.028

• 0 0 0 0.0 0.0 0.0 0.028

• 0 0 0 0.0 0.0 0.0 0.0 0.028

• 0 0 0 0.0 0.0 0.0 0.0 0.0 0

• 0 0 0 0.0 0.0 0.0 0.0 0.0 0 0

• 0 0 0 0.0 0.0 0.0 0.0 0.0 0 0 0

• 0 0 0 -0.01 0.0 0.0 0.0 0.0 0 0 0 0.037

• 0 0 0 0.0 -0.01 0.0 0.0 0.0 0 0 0 0.0 0.037

• 0 0 0 0.0 0.0 -0.01 0.0 0.0 0 0 0 0.0 0.0 0.037

• 0 0 0 0.0 0.0 0.0 -0.01 0.0 0 0 0 0.0 0.0 0.0 0.037

• 0 0 0 0.0 0.0 0.0 0.0 -0.01 0 0 0 0.0 0.0 0.0 0.0 0.037

37

Solutions

• Multivariate analysis Univariate analysis

• Effects Traits Traits

• Sex WWG PWG WWG PWG

•

• 1 4.361 6.800 4.358 6.798

• 2 3.397 5.880 3.404 5.879

• Animals

•

• 1 0.151 0.280 0.098 0.277

• 2 -0.015 -0.008 -0.019 -0.005

• 3 -0.078 -0.170 -0.041 -0.171

• 4 -0.010 -0.013 -0.009 -0.013

• 5 -0.270 -0.478 -0.186 -0.471

• 6 0.276 0.517 0.177 0.514

• 7 -0.316 -0.479 -0.249 -0.464

• 8 0.244 0.392 0.183 0.384

• ---------------------------------------------------------

• In general EBVs for PWG from the multivariate analysis above are similar to those from the univariate

• Much o gain from the multivariate analysis is in WWG and this is due to its lower heritability

38

PREDICTED GENETIC MERIT FOR UNMEASURED TRAITS

When breeding values are predicted for several traits, an animal's

predicted genetic merit can be written as

nn2211 uauauau'a

Not all traits can be measured, but predicted breeding values for correlated traits are

12A

A2 uu

1

21

39

If traits 1, 2 … n are measured and have breeding values

, but breeding values for a DIFFERENT set of traits i, j …k are required

n

2

1

1

2An2An1An

n2A2

2A21A

n1A12A2

1A

Akn2Ak1Ak

Ajn2Aj1Aj

Ain2Ai1Ai

k

j

i

u

u

u

u

u

u

Overall genetic merit is

kkjjii uauauau'a

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