Prediction of breeding values with additive animal models for crosses from 2 populations

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Prediction of breeding values with additive animal

models for crosses from 2 populations

Rjc Cantet, Rl Fernando

To cite this version:

Original

article

Prediction of

_breeding

values

with

additive animal models for

crosses

from

2

_populations

RJC Cantet

RL

Fernando

2

1

Universidad de Buenos

_{Aires, Departamento}

de

Zootecnia,

Facultad de

_Agronomia,

Av San

Martin, 4453,

(1417)

Buenos

_{Aires, Argentina;}

2

University

of Illinois, Department of

Animal

Sciences, 1207,

West

_{Gregory Drive,}

Urbana,

IL _61801, USA

(Received

5 October

1994; accepted

24

_April

₁₉₉₅₎

Summary - Recent

developments

in the

theory

of the covariance between relatives in

crosses from 2

populations,

under additive

inheritance,

are used to

_{predict breeding}

values

(BV)

by

best linear unbiased

_prediction

_(BLUP)

_using

animal models. The consequences

of

_{incorrectly specifying}

the covariance matrix of BV is discussed. The

_theory

of the covariance between relatives in crosses from 2

_populations

is extended for

_predicting

BV in models with

_multiple

traits. A numerical

_example

illustrates the

_{prediction procedures.}

cross

_/

_{heterogeneous} additive variance

_/

segregation variance

_/

genetic group

/

BLUP-animal model

Résumé - Prédiction des valeurs génétiques avec des modèles individuels additifs pour des croisements à partir de 2 _populations. De récentes avancées de la théorie de la

covariance entre

_apparentés

dans les croisements entre 2

_populations,

en hérédité

additive,

sont utilisées pour

prédire

les valeurs

génétiques

(VG)

par le BL UP-modèle animal. Les

conséquences

résultant d’une

_définition

incorrecte de la matrice de covariance des VG

sont discutées. La théorie de la covariance entre

_apparentés

en croisement à _partir de

2

_populations

est étendue à la

_prédiction

de VG pour

plusieurs

caractères. Un

exemple

numérique

illustre les

_procédures

de

_prédiction.

croisement

_/

BLUP-modèle

_{animal / variance}

génétique additive hétérogène

/

INTRODUCTION

A common method used to

_genetically

_improve

a local

_population

is

_by

_planned

migration

of _genes from a

_superior

one. For

_example,

in

developing

_countries,

US Holstein sires are mated to local cows in order to

_genetically

_improve

the local Holstein

population.

Genetic evaluation in such

_populations

must take into

consideration the

_genetic

differences between the local and the

_superior

_populations.

Best linear unbiased

_prediction

_(BLUP)

is

_widely

used for

_genetic

evaluation

(Henderson,

1984).

BLUP

_methodology

_{requires modelling}

_genotypic

means and

covariances. Genetic _groups are used to model differences in

_genetic

means between

populations

(Quaas, 1988).

However, populations

can also have different

_genetic

variances. Under additive

inheritance,

Elzo

₍₁₉₉₀₎

provided

a

theory

to

_incorporate

heterogeneous genetic

variances in

genetic

evaluation

_by

BLUP. His

_procedure

is based on

_computing

the additive variance for a crossbred animal as a

weighted

mean of the additive variances of the

_{parental populations}

_plus

one half the

covariance between

_parents.

Lo et al

₍₁₉₉₃₎

showed that Elzo’s

_theory

did not account for additive variation created

_by

_segregation

of alleles between

_populations

with different _gene

_frequencies.

For

_example,

even

_though

the additive variance

for an F2 individual should be

higher

than for an

Fl,

due to

_segregation

_(Lande,

1981),

Elzo’s formulation

_gives

the same variance for both. Lo et al

₍₁₉₉₃₎

_provided

a

_theory

to

_{incorporate segregation}

variance in

_computing

covariances between crossbred

relatives,

and to invert the

_genetic

covariance matrix

_efficiently.

The

_objectives

of this paper are:

1)

to demonstrate how the

_theory

of Lo et al

(1993)

can be used for

_{genetic evaluation,}

ie to

_predict

_breeding

values

_(BV),

_by

BLUP;

2)

to

_study

the consequences of

using

an incorrect

_genetic

covariance matrix on

_prediction

of

_BV;

and

3)

to extend the

_theory

of Lo et al

₍₁₉₉₃₎

to accommodate

multiple

traits. A numerical

example

is used to illustrate the

principles

introduced here.

MODEL

Even

_though

the

_theory

_presented

_by

Lo et al

₍₁₉₉₃₎

allowed for several breeds

or strains within a

breed,

we focus on the case of 2. A

typical

situation in beef or

_dairy

cattle is when a ’local’

_(L)

strain or breed is crossed with an

_{’imported’}

(I)

one.

Usually

the _{program starts}

by

_mating

genetically

superior

L females with

I males to

_produce

Fl _progeny. Then

_superior

Fl females are mated to I sires to

produce

backcross _{progeny. The program is} continued

_{by repeatedly}

_{mating superior}

backcross females to I sires. It should be noted that

_L,

Fl and backcross sires are also used to

_produce

_progeny.

_Thus,

the crosses

_generated

_by

such a _{program may}

includeFl=IxL,F2=FlxFI,BI=IxFI,BL=FlxL,BII=BIxF

l

,

5/81

= BI x

_Fl,

_3/81

= BI x

_L,

etc. It is shown below how

_genetic

evaluations for

such a mixture of crossbred animals can be obtained

by

BLUP

_using

Henderson’s

(1984)

mixed-model

_equations

_(MME).

Genetic evaluations are based on a vector of

_phenotypic

records

_(y),

which can

where (3

is a vector of

_non-genetic

fixed

effects,

a is a vector of additive

genetic

values or BV and e is a vector of random

_{residuals, independent}

of a, with null

mean and covariance matrix R.

_Although

R can be _any

_general

_{symmetric matrix,}

in

_general

it is taken to be

_diagonal,

and this

_simplifies

_computing

solutions of

(3

and

_predictions

of a. The incidence matrices X and Z relate

₍₃

and _a,

_{respectively,}

to y. The mean and the covariance matrix of the vector of BV

_(a)

for crossbred individuals are modelled as:

and

, , L

where g is a vector of

_genetic

group effects for individuals in the I and L

populations,

Q

is a matrix

relating

a with the

_genetic

_{groups. If} there is

_only

1 group on each

breed,

Q specifies

the breed

_composition

for each individual. The matrix G contains the variances and covariance among BV as defined

_by

Lo et al

_(1993).

In

_modelling

the mean of _a,

_genetic

_groups are

_{only assigned}

to

’phantom’

parents

of known animals

_following

the method

_{proposed by}

Westell et al

_(1988).

_Quaas

(1988)

showed that

Q

can be

expressed

as:

where P relates _progeny to

_parents,

_P

₆

_progeny to

_phantom

_parents,

and

_Q

_b

is

an incidence matrix that relates

_phantom

parents

to

_genetic

_groups. Elements in

each row of

[P

b

:P]

are all _zero,

_except

for two

_1/2’s

in the columns

_pertaining

to

the

_parents

of the animals in a. It should be stressed that the above model for a assumes additive inheritance

_(Thompson,

1979; Quaas, 1988;

Lo et

_al,

_1993).

In the

_{genetic grouping}

_theory

of

_Quaas

_(1988),

all the groups are assumed to

have the same additive variance. In this

_{model, however,}

we allow the I and L

populations

to have different additive

variances,

and the variances and covariances

of crossbred animals are

_{computed following}

the

_theory

of Lo et al

_(1993).

_They

showed that the covariance between crossbred relatives can be

computed

_{using the}

tabular method for

_purebreds

_(Emik

and

Terril, 1949; Henderson,

1976),

provided

that the variance of a crossbred individual i is

_computed

as:

where j

and k are the

_parents

of i,

and

_f

_jl

_,

for

example,

is the breed I

composition

of

dam

_j,

_QAL

is the additive variance of

_{population L,}

_U2

_is

the additive variance for

population I,

and

_QALI

is the

_{segregation variance,}

which results from differences in gene

frequencies

between the L and I

populations.

The term

segregation

variance

where

G,

is a

_diagonal

matrix with the ith

_diagonal

element defined as:

Note that these elements are linear functions of

a

2

2

and

!ALI!

PREDICTION OF BREEDING VALUES

Following Quaas

(1988),

MME for a model with

_genetic

_groups can be written as:

where

and

The matrix H can be constructed

_efficiently

_using

_{algorithms already}

available

(eg,

Groeneveld and

Kovac,

1990).

Quaas

(1988)

gave rules to construct E

effi-ciently

for a model with

homogeneous

additive variances across

_genetic

_groups. To

construct E

_efficiently

for a model with

_{heterogeneous}

additive

_variances,

_replace

x(=

4/[number

of unknown

parents

+

_2!)

in the rules of

_Quaas

₍₁₉₈₈₎

with

_1/Gg

_i

_.

CONSEQUENCES

OF USING AN INCORRECT G

Henderson

_(1975a)

showed that

_using

an incorrect G leads to

_predictions

that are

unbiased but do not have minimum variance. His results are

employed

here to

examine the _consequences of

_using

the same additive variance

(

QA

*

)

for

_L,

I and

crossbred animals.

Let Caa be the submatrix of a

_g-inverse

of the

_{right-hand-side}

of the MME

corresponding

to _a, but calculated with G* =

Aa2 A

.

.

Then,

_as in Henderson

(1975a)

and Van Vleck

_(1993),

the

_prediction

error variance

_(PEV)

of a is not

_equal

to

_C

_aa

_,

but is:

where G is the correct covariance matrix of a.

_Now,

let D be a

_diagonal

matrix with the ith

_diagonal

element

_{being equal}

to

_{0.5[1 - 0.5(Fs}

_i

+

_FDi)!,

if the father

Also,

D

ii

=

0.25(3 -

Fs

i

),

if

_only

the sire of i is

_known,

and

_D

_ii

=

0.25(3 —

F

Di

)

if

_only

the dam of i is known.

_Finally,

if both

_parents

of i are unknown

_D

_ii

= 1.

With this definition of D and after some

_algebra,

_[11]

becomes

equal

to:

Therefore,

PEV of a obtained from Caa will be

_incorrectly

estimated

_by

the second term on the

_right

of

_[11]

or

_[12].

As this term

_depends

on the structure of

_G,

and

_D,

no

_general

result can be

_{given. However,}

if

caa(I-p’)D-1(Go -D)D-1(I-p)caa

is

_{positive definite,}

it adds up to C&dquo; and true PEV is underestimated. This

_happens

if both

_{(G, - D)}

and

_C

_aa

_{(I -}

_P’)D-

₁

are

_positive

definite

_(see,

for

_example,

theorem A.9 _{in page} 183 of

_Toutenburg,

1982).

Now,

the fixed effects are

_{reparameterized}

so that

[X:ZQ]

is a full rank

matrix.

_Then,

_{!Caa(I -}

_{P’)D-’]-’}

=

D(I -

1

)-

aa

(C

1

P’)-

and

C

aa

(I -

P’)D-

1

is

positive

definite.

_Finally,

if

_{(Go - D)}

is

_positive

definite its

_diagonal

elements are

positive

(Seber,

1977,

page

388),

which in turn

happens

when the

_diagonal

elements of

_Gi

are

_strictly

_greater

than

_{corresponding}

elements of D. For

_example,

this _may

happen

whenever

_u

₂

_A

_LI

contributes to the variance of crossbred individuals

_(such

as

F2 or

5/8I),

and this variance

_parameter

is

_ignored.

Under these conditions PEV will be

underestimated,

and the amount of underestimation will

depend

on the

magnitude

of

_{or A}

_LI

_{* 2}

It has been shown that if all data

_employed

to make selection decisions are

available,

then the BLUP of a can be

_{computed ignoring}

selection

_(Henderson,

1975b; GofF!net,

1983;

Fernando and

_Gianola,

_1990).

This result

_only

holds when the

correct covariance matrix of a is used to

_compute

BLUP.

_Thus,

in the

_improvement

of a local breed

_by

_{mating superior}

I sires to selected L

females,

the use of the same additive variance for the I and L

populations

will

_give

biased results if the

2

_populations

are known to have different additive variances and the _process of

selection and

_mating

to

_superior

males is

_repeated.

MULTIPLE TRAITS

The

_{theory presented}

_by

Lo et al

₍₁₉₉₃₎

can be extended to obtain BLUP with

multiple

traits. Consider the extension for 2 traits: X and Y. To obtain BLUP for X and Y the additive covariance matrices for traits X and

_Y,

and between traits X and Y are needed. The covariance matrix for traits X and Y can be

_computed

as described

by

Lo et al

(1993).

It is shown below how to

_compute

the additive covariance matrix between traits X and Y.

Following

the

reasoning employed by

Lo et al

₍₁₉₉₃₎

to derive the additive variance for a crossbred

individual,

it can be shown that the additive covariance

where

_a

_A

₍

_XY

₎

_L

and

_I

_>

_<

_xY

_A

_a

are the additive covariances for traits X and Y

in

_populations

L and

_I,

_{respectively,}

and

_a

_A

₍

_XY

₎

_LI

is the additive

segregation

covariance for

_populations

L and I.

Provided that the covariance between traits X and Y for a crossbred individual is

_computed

_{using equation}

_!9!,

the covariance between X and Y between crossbred

individuals

i and i’

can be

_computed

as:

provided

that i’ is not a direct descendant of i.

Now let

_a(2q

x

1)

be the vector

_containing

the BV

_of

_{q animals}

for the 2

_traits,

ordered

by

trait within animal.

Then,

G is the

2q

x

_2q

covariance matrix between traits and individuals.

_Following

Elzo

₍₁₉₉₀₎

the inverse of

_{G, required}

to

_setup

Henderson’s

_MME,

can be written as:

where for

individual l,

the

_diagonal

elements of

GE!

can be calculated

_{by equation}

(7!,

and the

_{off-diagonals}

elements can be

_computed

as:

All covariances in the above

_equation

are functions of

_,

_L

₎

_XY

₍

_UA

_a

_A

₍

_XYL

₎

_I

and

o-A

(

XY

)

L

i

and can be

computed using equations

[13]

and

!14!.

Equation

[15]

gives

rise to

_simple

rules to

_setup

MME for 2 or more traits.

By appropriately redefining

all vector and matrices to include 2 or more

_traits,

equations

[8],

[9]

and

_[10]

are valid for the

_multiple

trait situation.

_Again,

matrix H can be constructed

efficiently

_{by commonly}

used

_algorithms

(Groeneveld

and

Kovac,

1990).

Cantet et al

₍₁₉₉₂₎

_gave rules to construct E

_efficiently

for a model

with

_homogeneous

additive

_{(co)variances}

across

_genetic

_groups. Their

_algorithm

can be modified to construct E

_efficiently

for a model with

_{heterogeneous}

additive

(co)variances

as follows. Let:

i

r

=

_equation

number of individual i for the rth trait.

j

r

=

equation

number of the sire of i or its sire

group

(if

base

sire)

for trait r.

k

r

=

_equation

number of the dam of i or its dam

group

(if

base

dam)

for trait r.

Now let t be the number of

_traits,

for m = 1 to t and _n = 1 _to

_t,

and add to E

the

_following

9 contributions:

For

_example,

if 2 traits are considered

(t

=

2),

there _are

9(2

2

)

= 36 contributions. If

E is

_half-stored,

there are

[9t(t -1)]/2 + 6t

contributions. For t =

_2,

each individual makes 21 contributions to the upper

triangular

part

of half-stored matrix E.

To obtain BLUP under a maternal effects model

(Willham, 1963),

the additive

covariance matrices for the direct

_effect,

the maternal

_effects,

and between the direct and maternal effects are needed. These matrices can be

_computed

_using

the

_theory

used to

_compute

the covariance matrices for traits X and Y as described above.

NUMERICAL EXAMPLE

Consider a

_single

trait situation where a sire from breed I

(animal 1)

serves 2 dams:

animal

_2,

from breed

_I,

and animal 3 from breed L. Individuals 1 and 2 are the

parents

of 4

_(purebred

_I),

and 1 and 3 are

_parents

of 5

(an

Fl

male).

Finally,

the

F2 animal 7 is the

_offspring

of 5 and

_6,

the latter

being

an Fl dam with unknown

parents.

Individuals

_1,

4 and 5 are males and the rest are females.

Age

at measure

and observed data for animals 2-7 are 100

_(age),

100

_(data);

_{110, 103; 95, 160; 98,}

175; 106, 105;

and

_{100, 114;}

_{respectively.}

There are 2

_genetic

_groups for breed I and 1 for L. The model of evaluation includes fixed effects of age

(as

a

covariate),

sex

and

_genetic

groups

(Al,

A2 and

B),

and random BV for animals 1

through

7. In order for

_[X:ZQ]

to have full rank we

imposed

the restriction: sex 1 + sex

_{2 = 0,}

or sex 1 = -sex 2.

_{Hence, f3}

contains

_only

2

_parameters:

₁₎

the age

covariate;

and

2)

the sex 1 effect

(or

-sex

2).

Matrices _y, X and

Q

are then

_equal

to:

Variance

_components

are

(TÃL

=

80,

QAI

= 120 and

U

p

LI

= 50.

Using

[7],

the

diagonal

elements of matrix

_G

_e

_(the

variances of Mendelian

_residuals)

are

_80,

80, 120,

40, 50,

100 and 100 for animals 1-7

_{respectively.}

Residual variance is R =

1

6

(400).

Matrix G is:

Solutions are

_{-2.903201, -28.95692, 402.73785, 431.59906, 452.07844, 402.68086,}

417.28243, 452.16393, 409.6967, 427.70734,

441.69628 and 434.41686. The

_large

absolute values of the solutions are due to

_{multicollinearity}

associated with

_genetic

groups in the model for the small

example

worked. This is evidenced

by

a small

eigenvalue

(4.36

x

10-

7

)

in the coefficient matrix. Van Vleck

₍₁₉₉₀₎

obtained a

similar result in an

_example

with

_genetic

_groups for direct and maternal effects.

If _groups are left out of the

_model,

solutions are 1.3540226 for the _age

effect,

- 36.19053 for the sex 1

effect,

and

0.602659,

-0.247433, -1.030572, -0.276959,

1.1075823, 0.6457529,

3.6589865 for the BV of animals

_1-7,

_{respectively.}

The smallest

_eigenvalue

is

_0.0046413,

almost 10 000 times

_larger

than the situation where

genetic

groups are in the model.

The consequences of

assuming

an incorrect G can be seen

by taking

G*

-A(100).

The value of 100 for

_QA*

is chosen because it is the average between

ol2 A

L

= 80 and

ai

l

= 120. To alleviate the

problem

associated with

multicollinearity,

the

_system

is solved

using regular

MME

_{(Henderson, 1984)}

rather than the

QP-transformed

_system

_!8!.

Therefore,

PEV are estimated for BV deviated from their means. Incorrect PEV for animals 1-7 are

_{99.91, 99.66, 99.91, 98.30, 98.66,}

99.66

and

_99.91,

respectively.

Whereas true PEV for the same animals

_{computed by}

means of

[11]

or

[12] (or

direct inversion of the

MME)

are:

79.94, 79.79, 119.88,

78.98, 98.52,

99.66 and 149.59 for animals

1-7,

respectively.

DISCUSSION

A model has been

presented

to

predict

BV of different crosses between 2

_populations

under an additive

_type

of inheritance. It allows for different additive means and

variances.

_Computations

are as

_simple

as when there is

_only

1

a A 2

_and,

as

usual,

R is a

_diagonal

matrix. A

practical

application

is the

analysis

of data from crosses

between

_{’foreign’}

and ’local’ strains of a

breed,

as in

_dairy

or beef

breeding. Also,

records from

_registered

vs

grade

animals,

or ’selected vs

_{unselected’,}

_etc,

can be

analyzed

in this fashion.

_Although

the

_developments

presented

were in terms of 2

_populations,

inclusion of more than 2 can be done as indicated

by

Lo et al

(1993).

With _p

_being

the number of

populations,

the number of

_parameters

in G is

_{!p(p +}

_{1)] /2,}

so that for _p = 4 there are 10 variances to consider. Some of these estimates _may be

_highly

correlated

depending

on the

_type

and distribution of the crosses involved.

The

approach

taken in the

_present

_paper differs from Elzo

₍₁₉₉₀₎

in the inclusion

of the

_segregation

variance

_{(o, ALI). 2}

The

_magnitude

of this

parameter

depends

on

differences in gene

frequencies

between the 2

_populations

_(Lo

et

al,

1993).

The

change

in gene

frequency

due to selection is

_inversely

related to the number of loci because

_change

in gene

frequency

at a locus due to selection is

_proportional

to the

magnitude

of the average effect of gene substitution at that locus

(Pirchner,

1969,

page

145),

and the

magnitude

of average effects across loci tend to be

_inversely

number of loci.

_Now,

the

_greater

the value of

_QALI

_,

the

_larger

the difference between the

predictors

calculated

_following

the

_approach

of Elzo

₍₁₉₉₀₎

and the one used in

the

_present

work. This is due to

_QALI

not

_only

_entering

into the

_diagonal

elements

of

_G,

but also into

_{off-diagonals}

which are functions of the

_diagonal

elements

(Lo

et

_al,

_1993).

For

_example,

consider the additive covariance between

_paternal

half sibs

_(cov(PHS))

i and

_i’,

from common sire s and unrelated dams.

_By

_repeated

use

of

_expression

_[10]

in Lo et al

_{(1993), cov(PHS)}

is

_equal

to:

Expression

[17]

shows that

_cov(PHS)

is a function of the additive variance of the

sire,

and is not a function of the additive variance

_present

in progeny

_genotypes.

For

_{I, Fl,}

BI or F2

_sires,

cov(PHS)

are

_equal

to:

Note that

_{or2 A}

_LI

enters into the covariance of individuals whose sire

_belongs

to

later

_generations

than the Fl

_(eg,

BI or

F2).

It must be

pointed

out that

although

predictions

are still

_{unbiased, ignoring}

o,

2

A

LI

would result in a

larger

shrinkage

of

predictions

in

_[8]

than otherwise. As there are no estimates of

_a

₂

_A

_LI

so

_far,

_nothing

can be said about the

magnitude

of the

_parameter

on

_genetic

variation of economic traits in livestock.

If dominance is not

_null,

the model should be

_properly

modified to take into account this non-additive

_genetic

effect.

_{Proper specification}

of the variance-covariance matrix for additive and dominance effects in crosses of 2

_populations

can involve as _many as 25

_parameters

for a

single

trait

(Lo

et

al,

1995).

Therefore,

predictors

of BV and dominance deviations may be difficult to

_compute

for a

general

situation, involving

animals from several crossbred

_genotypes.

Lo

₍₁₉₉₃₎

_presented

an efficient

algorithm

for

computing

BLUP in the case of 2- and 3-breed terminal

crossbreeding

systems

under additive and dominance inheritance.

Up

to this

_point

the animal model has been

_{employed. However,}

the ’reduced animal model’

_(Quaas

and

_Pollak,

₁₉₈₀₎

can

_{alternatively}

be used

_by

_properly

writing

matrix Z with

_1/2’s,

whenever a BV of a

’non-parent’

(an

animal that has no _progeny in the data

set)

is

expressed

as a function of its

_parental

BV. Residual

genetic

variances are obtained

by

means of

_expression

!7!.

In order to solve

_equations

_[8]

the

_parameters

_{o,2 2}

₂

and the residual

components,

have to be known.

_Usually

variance

_components

are unknown and

CONCLUSIONS

For 1 or several traits

_{governed by}

additive

effects,

predictions

of BV from crosses

between 2

_populations

can be obtained

_by

means of animal models that allow

for different additive means and

heterogeneous

additive

genetic

(co)variances.

Calculations

required

are similar to those with

_homogeneous

additive

_{(co)variances.}

ACKNOWLEDGMENT

This research was

_supported

with an UBACYT _grant from Secretaria de Ciencia _y

_Técnica,

Universidad de Buenos

_{Aires, Argentina.}

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