Estimation of genetic parameters of egg production traits of laying hens by restricted maximum likelihood applied to a multiple-trait reduced animal model

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Estimation of genetic parameters of egg production

traits of laying hens by restricted maximum likelihood

applied to a multiple-trait reduced animal model

B Besbes, V Ducrocq, Jl Foulley, M Protais, A Tavernier, M Tixier-Boichard,

C Beaumont

To cite this version:

Original

article

Estimation of

_genetic

_parameters

of

_egg

_production

traits of

_laying

hens

by

restricted maximum likelihood

_applied

to

a

multiple-trait

reduced

animal model

B

Besbes

4

V

_Ducrocq

JL

_Foulley

M

Protais

4

A Tavernier

M

Tixier-Boichard

C

Beaumont

3

1

INRA,

Station de

_{Génétique Quantitative}

et

_Appliquée,

Centre de Recherche

de

_{Jouy-en-Josas,}

78352

_{Jouy-en-Josas Cedex;}

2

INRA,

Laboratoire de

_{Génétique Factorielle,}

78352

_{Jouy-en-Josas Cedex;}

3

INRA,

Station de Recherches

Avicoles,

37380

Monnaie ;

4

Institut de Sélection

_Animale,

Établissements

de Le

_Foeil,

22800

_Quintin,

France

(Received

18 December 1991;

accepted

30 June

₁₉₉₂₎

Summary - Variance components for egg

production

traits

_(No

_{of eggs}

produced

between 19 and 26, 26 and 38 and 26 and 54 wk of

_age),

egg characteristics

(average

egg

weight

at 2 different ages and egg

density)

and

body weight

of hens at _{40 wk of age} were

estimated in two strains of a

_breeding

company

by

univariate and multivariate Restricted Maximum Likelihood

_(REML)

_applied

to a Reduced Animal Model

(RAM).

To allow

tridiagonalization

of the coefficient matrix when RAM is

_considered,

the

approach

of

Thompson

and

_Meyer

₍₁₉₉₀₎

_using

an

_imaginary

random effect with

_negative

variance was

_implemented.

Canonical transformation was also

_employed.

REML

_estimation,

carried

out on 15 random

_samples

of m 7000 recorded hens drawn from each of 2 data files

corresponding

to 2 different

_strains,

showed a rather small

_{genetic antagonism}

between

the group of egg

production

traits _{and egg}

_density

on one hand and that of egg

weights

on the other hand.

_Weight

of hens behaved

_differently

in the 2 strains. It showed also that

traits within these groups were

_positively

correlated. Heritabilities obtained

by

univariate and multivariate

analyses

were _very similar and were lower for egg

production

traits

(from

0.09 to

_0.27)

than for _egg characteristics or

_weight

of hens

_(from

0.34 to

_0.48).

egg production

/

genetic parameter

/

restricted maximum likelihood

_/

animal model

*

Résumé - Estimation des paramètres génétiques des caractères de ponte chez la

poule par maximum de vraisemblance restreinte appliqué à un modèle animal réduit multicaractères. La

_{production d’ceufs}

_(nombre

_{d’œufs produits}

entre 19 et 26, 26 et 38 et 26 et

₅₄

semaines

_d’âge),

les

caractéristiques

de

_{l’œuf}

_(le

_poids

moyen des

œufs

à 2

âges différents

et densité de

_l’a=uf)

ainsi que le

poids

des

poules

à

40

semaines

d’âge

ont été étudié dans deux souches d’une

_firme

de sélection. Les composantes de la variance de

ces caractères ont été estimées à l’aide du maximum de vraisemblance restreinte

_(REML)

uni et multicaractères

_appliqué

à un modèle animal réduit

(RAM).

Pour

_{tridiagonaliser}

la

matrice des

coefficients

et

réduire,

par

conséquent,

les calculs liés à son inversion à

chaque

itération de

_{l’algorithme}

EM

_{(Espérance-Maximisation),}

nous avons utilisé

_{l’algorithme}

proposé

par

Thompson

et

Meyer

(1990).

Ce dernier introduit dans le modèle un

_effet

aléatoire

_{imaginaire supplémentaire,}

ayant une variance

négative.

Ces calculs ont été

également

réduits

_grâce

à une

décomposition canonique.

Les estimées du

REML,

obtenues

à

_partir

de 15 échantillons d’environ 7000 observations

chacun,

ont montré un

_léger

antagonisme

entre, d’une

_part,

le groupe des caractères de

pente

et la densité de

_{l’œuf}

et d’autre part celui du

_poids

des

_œu/s.

Les corrélations entre les caractères

intragroupe

étaient

_positives.

Il est apparu

également

que les caractères de

production d’ceufs

étaient moins héritables

_(de

_0,09

à

_0,27)

que ceux liés auz

caractéristiques

de

1’oeuf ou

de la

poule

(de

0,3¢

à

_0,48),

dans les souches considérées.

_Enfin,

nous

_signalons

_que la

comparaison

entre les héritabilités obtenues avec les

analyses

uni- et multicaractères n’a _pas montré de

différences

nettes.

caractère de _ponte

_/

_{paramètre génétique}

_/

maximum de vraisemblance restreinte

_/

modèle animal

INTRODUCTION

Best Linear Unbiased Prediction

_(BLUP)

_applied

to an Animal Model has been

recognized

as the method of choice for

_estimating

the

_genetic

merit of candidates for selection. Under

_normality

and in the absence of

_prior

_knowledge

about means and

variances,

breeding

values should be

_{predicted using}

BLUP

_methodology,

with the unknown variances

_{replaced by}

their

_{corresponding}

Restricted Maximum Likelihood

(REML;

Patterson and

_Thompson,

₁₉₇₁₎

estimates

_(Gianola

et

_al,

_1986).

REML is

_preferred

to other variance

_components

estimation methods because

of its

_ability

to account for selection bias. These methods

_(BLUP

and

_REML)

are

nowadays

utilized _{in many countries} all over the world and for various domestic

species.

Surprisingly, they

are almost

_{completely ignored}

in

_laying

hens evaluation

systems

even

though

_strong

selection has been carried out on this

_species

for _many

generations.

The _purposes of this

_study

were:

1)

to estimate

_genetic

_parameters

of 7 correlated

egg

production

traits

by

REML

applied

to a

_{multiple-trait}

reduced animal

model;

MATERIALS AND METHODS

Data and traits

_description

Data, including

records of

_165,

748 and

_47,

115 survivor

_laying

hens for strains A

and B

_{respectively,}

were

_{supplied by}

_{the &dquo;Institut de Selection Animale-ISA&dquo;. For} both

_strains,

these records

_represented

6

_generations

of hens.

Traits considered in this

_analysis

were related to _egg

_production

_(number

of _eggs

produced

between 19-26 _{wk of age}

_(P

_l

_),

26-38 wk

_(P

_Z

₎

and 26-54 wk

_(P

₃

₎₎

_{and egg}

characteristics

_(average

_egg

_weight

at 2 different _ages

_(EW

_I

_{, EW}

₂

₎

_{and egg}

_density

(ED)).

Weight

of hens at 40 wk of _age

₎

_(W

_4o

was also included in the

_analysis.

P

I

can be considered as

_being

a combination of sexual

_maturity

and

_early

_egg

production.

ED was a measure of the shell

_strength

determined

_by

_specific

_gravity.

P

i

,

P

2

and

_P

₃

variables exhibited

_markedly

skewed distributions.

_They

were

transformed into new

_variables,

_satisfying

the classical

_hypotheses

for

_describing

traits with

_polygenic

inheritance via a linear model with normal _error,

_using

a power transformation

(Box

and

Cox,

1964).

This transformation relies on a

single

parameter

t and has the

_following

form

_(Ibe

and

_Hill,

_1988):

were _y is the

_geometric

mean of the

_original

observations.

The

_parameter

t was

_empirically

chosen in such a _way that several

_normality

criteria,

such as the low residual sum of _squares of the

_genetic

model used to describe the

_data,

the

_linearity

of half-sib on individual

_regression,

the coefficient of

_symmetry

and the

_{Kolmogorov-Smirnov}

test for

_normality

of the residuals were satisfied as

_closely

as

_possible

and

_{simultaneously,}

as

_proposed

_by

Ibe and Hill

(1988)

and detailed in Besbes et al

_(1992).

Model

_{of analysis}

The model

_describing

the records is the

_following

_{multiple-trait}

animal model:

where Y is the vector

_(n

x

_t)

of observations for the

_{t egg}

_production

traits

considered in the

_{multiple-trait analysis}

_(t

=

7),

b is the vector

_{( f}

x

_t)

of fixed

_contemporary

_groups

_{( f}

was

_equal

to 107 and

70,

for strains A and B

_{respectively),}

a is the vector

_(n

x

t)

of random additive

_genetic

effects associated with the

animal’s

_traits,

e is the vector

_(n

x

_t)

of

_residuals,

X and Z are known incidence matrices associated with b and a and

₍

_*

₎

It was assumed that

A is the

_relationship

matrix between animals. G and R are unknown

_genetic

and

residual

_(co)variance

matrices between the t traits considered.

Computing

strategy

for

_genetic

_parameter

estimation

The

_{multiple-trait}

animal model

_[2]

had one random effect and

equal

design

matrices

for all traits which were recorded for all animals

(no

_missing

records).

Canonical

decomposition

was then

_applied

to

_yield

new uncorrelated variables without loss of

any information contained in the

original

variables. This transformation was first

suggested

for animal

_breeding

_{problems by Thompson}

_(1976,

cited

_by

Jensen and

Mao,

1988).

The transformation matrix

Q,

is chosen such that

_(Quaas

et

al,

1984):

where

G

c

is a

_diagonal

matrix and

_I

_nt

is the

_identity

matrix. After transformation

to the canonical

scale,

model

_[2]

becomes:

The

subscript

c refers to the canonical scale. This transformation reduced the

multivariate

_analysis

to a series of univariate

_analyses

and

_{consequently,}

_drastically

decreased

_{computational}

costs.

These

computational

costs were also lowered

_by

reducing

the number of

equa-tions. Since

_parents

_{represented only}

9%

and

8%

of the total number of

_animals,

for

strains A and B

_{respectively,}

the reduced animal model

_(RAM)

of

_(auaas

and Pol-lak

₍₁₉₈₀₎

was used. With

RAM,

all the

_equations

corresponding

to animals which were

_nonparents

were absorbed into the

_remaining

_equations.

As a consequence, the size of the

_system

was

_brought

down to the number of

_parents.

Considering

model

_[3]

for the

_pth

_trait,

with

RAM,

the vector

_Y

_c

_,

was divided into

_parents

_(denoted

_{by subscript}

_p)

and

_nonparents

_(denoted

_{by subscript}

_n)

as

with

_e:t

_c

₌

_{e!!-!!nc(‘!nc}

_being

the Mendelian

_samplign

on the canonical

scale)

and

P!

is a matrix of 0 and 1

relating

_nonparents

to their

_parents.

Random variables

in

_[4]

have the

_following

_{(co)-variance}

structure:

C7!

is

the

_{pth diagonal}

element of

G

c

. A

n

is a

_diagonal

matrix whose elements are

equal

to

_1/2

if both

parents

are known and

_3/4

if one

_parent

if known. It was assumed that there is

_equal

_parental

information for all

_nonparents

and that

_parents

are not inbred. If this

_assumption

is not

_satisfied,

_nonparents

with unknown

_parents

can have

_dummy

parents

(Thompson

and

_Meyer,

_1990).

Since REML is an iterative

_procedure,

the

_major

cost was the

_need,

at each

iteration,

of direct inversion of a matrix of size

_equal

to the number of

_parents.

This burden was reduced

_by

_{tridiagonalizing}

the coefficient matrix

through

a series

of

_orthogonal

transformations,

as

_{proposed by}

Smith and Graser

(1986).

This

transformation, however,

was not

_{directly applicable}

_because,

as shown in

!5!,

RAM

generates

heterogeneous

residual variances between

parents

and

_nonparents.

To overcome this

problem, Thompson

and

Meyer

(1990)

reparameterized

[4]

adding

an

_imaginary

_effect,

_e, with

_{negative variance;}

e

j =

iwa

d

,

(with

i

2

=

-1),

was chosen such that ep_{c -} ej and

e!c

had the same

variance,

hence

_var(e

_j

_{) =}

_-w2(T!Ip

_(with

w

2

_being

an element of

An).

Hence,

the mixed model

_equations

_{corresponding}

to model

_[6]

can be written as:

where a is the

pth

trait’s variance ratio

cr!/o-!,

with

CT!

= ₁

+(.¡)2CT!

_an element of

For

_estimating

the

_(co)variance

_components,

_equations

for fixed effects are

absorbed. The

_resulting

_system

has

_equations

of the form

_(Thompson

and

_Meyer,

1990):

Further

_{simplification}

of

_[8]

was achieved

by eliminating

A!,1.

This involved the

Cholesky

decomposition

of the numerator

_relationship

matrix

_(Ap

=

LL’)

and

pre-and

_{post-multiplication}

of the left hand side

_by

L’ and L and

multiplication

of the

right

hand side

_by

L’

_(Smith

and

_Graser,

_1986).

The solutions for _ape and ad are of the form:

As shown in

_!9!,

this method led to the

_{tridiagonalization}

of a

_complex

matrix of

size twice the number of

parents.

After the

_tridiagonal

matrix T

_(for

T such that

T =

PHP’,

where P is an

orthogonal

matrix)

was found and

_using

algebra

similar to that of Smith and Graser

₍₁₉₈₆₎

system

[9]

becomes:

Then,

_we

_iteratively

calculate the canonical REML estimates of the

_(co)variance

matrices, G

c

and

_R

_e

_,

between all traits

_using

an

_{Expectation-Maximization}

(EM)

type

algorithm

(Dempster

et

al, 1977; Harville,

1977).

and those in

_R

_c

are:

where N is the total number of

_{observations,}

_f

is the number of levels of the fixed

effect,

p’

is twice the number of

_parents

and

_Y§’Y§

is the

_quadratic

form of the

canonical observations after

absorption

of the fixed effects. It has the

_following

expression:

Once the variance

_components

in

_[11]

_through

_[14]

are

obtained,

we _go back to

the

_original

scale

_by

_performing

back-transformation as follows:

then,

the new estimates of G and R are used to obtain a new

Q

transformation matrix and new

_G

_e

and

R

e

,

so the _process from

[10]

to

_[15]

is iterated until

convergence is reached.

The

_single

trait

_analysis

_corresponds

to the

_special

case where

Q =

I

(Hence

G and R are

diagonal),

and the EM-REML

_equations

are those in

[11]

and

!13!.

Sampling

procedure

Despite

these

_{cost-reducing techniques}

_(canonical

_{decomposition,}

use of

RAM,

tridiagonalization)

the

_{time-consuming}

_{tridiagonalization}

and above all the amount

of

_computer

_memory

_{required, prohibited}

the

_application

of REML estimation

to the whole

_population.

Therefore,

15

_{samples, reflecting}

as well as

_possible

the

population’s

structure and

_selection,

were drawn in each strain from the data file

in the

_following

manner:

1. Choose S sire

_(S

=

5)

_at random among the

_{youngest parents.}

2. Include the sire and dam of each selected sire.

3. For each of the selected sires

_S

_i

in

₍₁₎

and

_(2),

choose 3 sires

_S

_j

at random

among those whose

offspring

are

_contemporary

to those of

S

i

.

4. For each of the sire selected in

₍₁₎

to

_(3),

choose 3 females at _{random among} its mates.

5.

_Repeat

_step

₍₂₎

to

₍₄₎

until the

_generation

which is assumed to be the base

population

has been reached.

As mentioned

above,

the

_system

to solve is of size twice the number of

parents.

This,

depending

on the

_computer

capacity,

limits the

sample’s

size. The values 5

and 3 in

steps

1,3

and 4 were chosen

_by

trial and error in order to obtain a maximum

of 800

parents,

corresponding

to

_sampling

rates of m 5 and

17%

_respectively

for

strains A and

B,

which is as _many as can be handled.

This

_{sampling procedure}

has the

_following

characteristics:

_only

the

_pedigrees

of males are

_{complete. Step}

3 leads to the inclusion of hens

contemporary

to those

hens whose records were used to select sires.

_Hence,

selection on the male side is

(approximately)

accounted for. This is not the case on the female sire.

However,

the

_change

from one

_generation

to the next in the

expected

value

_E(a)

= _g of

the

_genetic

merit

of

hens,

due to

_selection,

is accounted for since the levels of the

fixed effect

_{(contemporary groups)}

are defined within

_generation.

In other

words,

this

_expected

value,

considered as

_fixed,

_using

the

_approach

of Westell

(1984)

or

Quaas

(1988)

is

_completely

confounded with the

_contemporary

group effect.

Only

the effect of selection on the female side on the

genetic

variance is not accounted for as

usually

indicated when

using

animal models.

As

_{already mentioned, only}

the

_pedigrees

of males are

_{complete. Hence,}

neither Henderson’s rules

₍₁₉₇₆₎

for

_computing

the numerator

_relationship

matrix A or its inverse

_{A -}

_I

_,

nor

_Quaas’s

_algorithm

(1989)

to eliminate

A-

1

from mixed model

equations

used in REML could be utilized. A and its

_Cholesky

factor

_L(A

=

LL’)

were therefore calculated

directly considering

the

pedigree

available for the entire

population

(Tier, 1990).

RESULTS AND DISCUSSION

The _{average numbers} of

_parents

_per

_sample

were 710 and 600 for strains A and B

_{respectively,}

which

_represented

samples

of 7000 and 6500 recorded hens

partitioned

into 107 and 70

_contemporary

groups. The

tridiagonalization

of the coefficient matrix of size twice the number of

_parents

_required

_respectively

113 and 60 min of CPU time on an IBM 3090-17T. This

_represented

98%

of the total

computational

time needed for

_genetic

_parameter

estimation for such

_samples.

As a _consequence of a smaller number of sires _per

generation

for strain

B,

the

_{corresponding samples overlapped}

in a

_{higher proportion.}

On _average,

38%

of

_parents

were common to _any 2

_samples

vs

17%

for strain A. This

overlap

was

neglected

when

_computing

means and standard deviations of the estimates obtained from the 15

_samples.

As a _{consequence of}

_working

with

_imaginary

_terms,

the

algorithm proposed

by Thompson

and

_{Meyer produced}

a

_tridiagonal

matrix T with some

_negative

eigenvalues.

When

_{by chance, during}

the EM

_iteration,

the variance ration a was very close to one of those

_{negative eigenvalues,}

the matrix of

_system

[10]

was

singular.

This led to an infinite

trace,

preventing

the EM-REML from

_converging.

When this

occurred,

it was _{necessary to}

_{&dquo;jump&dquo;}

over the value a to avoid numerical

problems.

Single

trait

_genetic

_parameter

estimates

Heritability

values

_{(table I)}

exhibited very small differences between strains A and

B. The

_highest

difference was observed for the number of eggs

produced

between

26 and 38 wk of _age: 0.09 for strain A vs 0.13 for strain B.

However,

it should be

noted that for similar

_{heritabilities,}

strain B

_presented,

in

_general,

_larger

additive

genetic

and residual variances.

These estimates also showed that _egg

_production

traits

_(P

_l

_{, P}

₂

and

_P

₃

₎

are much less heritable _{than egg characteristics} or

body weight.

Heritabilities of egg

production

traits were lower than

_{usually reported}

in the

literature, especially

those for

P

2

and

_P

₃

_.

_{Regarding heritability}

of

_P

_i

_,

a combined trait of both sexual

_maturity

and

_early

egg

production,

it was

_roughly

within

the range of

Liljedahl

and

_Weyde

₍₁₉₈₀₎

estimates but closer to that of _egg

production.

We

_{should, however,}

be careful with such

comparisons

since most

reported

heritabilities and variance

_components

were obtained on the

_original

scale

without

_performing

_any transformation but

_using

methods

_{assuming normality}

of

the data distribution.

_{Strictly speaking,}

such results should be

_interpreted

with caution.

The _purpose of the

Box-Cox

_{transformation, applied}

for

_P

_l

_,

P

2

and

P

3

is then

to

_change

the scale of measurements in order to make the

_analysis

more valid.

Besbes et al

₍₁₉₉₂₎

showed that this transformation resulted in an increase of all heritabilities without

_{drastically modifying}

the

_genetic

and residual correlations between these traits.

Heritability

values _{of egg and hen}

_{weight, though}

rather

small,

remained within the range of estimates

reported by

King

and Henderson

_(1954,

cited

_by

_Kolstad,

1980),

Kolstad

₍₁₉₈₀₎

and Sorensen et al

_(1980).

The same trend was observed for the estimates of egg

density

(specific gravity).

Multiple-trait genetic

parameter

estimates

As shown in table

_II,

heritabilities obtained

_by

multivariate EM-REML were very

close to those obtained

_{by single}

trait

_analysis.

This result was in

_agreement

with that of Colleau et al

_(1989).

The

_comparison

of

_genetic

correlations of both strains revealed

_quite

similar

global

trends

_concerning

the

_sign

of these correlations. But those of strain A _were,

in

_general,

smaller in absolute values.

These correlations showed a rather small

antagonism

between the group of _egg

production

traits

_(P

_I

_{, P}

₂

and

_P

₃

₎

and egg

density

(ED)

on one hand and that of average egg

weights

on the other hand. The

_largest

_antagonism

was observed between egg

weight

(EW

l

)

and egg

density

(-0.17

and -0.23

_respectively

for strain

A and

_B)

but remained rather small.

In the

_literature,

there is a

_large

variation in the

_reported

scale of the

_genetic

correlation between number of eggs

produced

and _{average egg}

_weight.

For Sorensen

et al

_(1980),

this correlation was -0.32 and -0.17

_depending

on the

population.

Liljedahl

and

_Weyde

₍₁₉₈₀₎

_reported

an evolution of this correlation from

_slightly

which showed some evidence for a decline in the

genetic

correlation from -0.18

in the base

_population

to -0.11 in the selected lines.

_Nearly

all these correlations

and heritabilities were estimated

_{by analyzing}

the resemblance between

_paternal

half-sibs and

_separately

for each

_generation

and trait.

Hence,

they

were biased

_by

selection. This was

_{theoretically}

not the case with our REML estimation.

_However,

it must be said that our &dquo;base&dquo;

_population

was

_already

a selected one

_{and, also,}

the

_sampling

_procedure

could not take all the selection process into account.

From table

_II,

it can be seen that correlations between _egg number and

_body

weight

of hens varied

_considerably

_{among strains}

_(the

correlation between

_P

₃

and W40 was 0.25 for strain A and -0.12 for strain

B)

which was also found

_by

Kolstad

(1980).

This author

_reported

that an intermediate

_{body weight}

is

optimum

for

reproduction,

of _{which egg}

_production

is a

_component,

and hence one should

expect

these correlations to be

_positive

or

_{negative depending}

on whether the

_weight

is

below or above the

_optimum

of the

_{population. Therefore,}

it

_might

be

_thought

that

strain A was closer to an

_optimum

for

body

size than strain B. Great variations of these correlations were also observed _among the 3

_{recording periods}

_(they

varied,

in these case of strain

A,

from -0.09 for

P

l

to 0.25 for

_P

₃

while correlations between

T ble II also shows that traits within each group of traits were

_positively

correlated. The

_highest

correlations were between

P

2

and

P

3

for the first _group of

traits,

respectively

0.66 and 0.78 and

EW

I

and

EW

z

for the second group, 0.92 and 0.94. To some

_degree,

ELI’

1

and

EW

z

were measurements of the same

characteristic.

_T!e

correlation

between P

2

and

P

3

is lower than the literature _average values

(0.79

and

0.85,

respectively

for selected and unselected White

_Leghorn

_breed)

obtained

by

Fairfull and Gowe

_(1990).

This is

_probably

due to a low correlation

between

P

2

and _{the residual egg}

_production

_(38-54

wk of

_age).

Hence,

P

2

and

_P

₃

cannot

be considered as

_measuring

the same trait even if

_{they overlap.}

The

positive

correlations between

_P

_I

and the other egg

production

traits

(P

z

and

P

3

)

_indicate

that this trait has to be considered as an _egg

_production

trait rather

thanl a sexual

_maturity

trait

_(sexual

_maturity

is

_{usually negatively}

correlated with the

number

of

_eggs).

The

_{nonsignificant}

correlation between these traits and shell

quality

was also

reported by

Kolstad

(1980).

However Fairfull and Gowe

(1990)

obtained,

for selected White

_Leghorn,

a

_negative

correlation of m -0.2 between survivor _egg

_production

from

_housing

to 40 or 55 wk of age and

specific gravity.

CONCLUSION

The

_present

_study

showed a rather small

_antagonism

between egg

production

traits and egg

weights

and a

_positive

correlation between traits within the same

_category.

Comparison

of the heritabilities obtained with univariate and multivariate

RE1V$L

estimation showed no

significant

differences.

By

employing

a series of transformations in the variance

_components

estimation

process,

especially

the

_{tridiagonalization}

of the

_system

_{proposed by}

_Thompson

and

_,Meyer

_(1990),

the

_problem

of slow _convergence of the EM

_algorithm

was

largely

alleviated since the

_computations

necessary at each round of iteration were

_performed

in linear time. This allowed us to

_{study large samples}

with an

acceptable computational

time.

However,

this

_algorithm

encountered numerical

problems, depending

on the matrix

_structure,

that

potential

users should be aware

of.

_;

Tue

animal model used involved additive

_genetic

effects

_only.

The

_complexity

of

calculations

precluded

the use of a more

sophisticated

model

including

_eg

non-additive,

maternal and common environmental effects.

Comparison

of sire and

dam components

estimated elsewhere

_{(Besbes, 1992)}

revealed very small differences

between

these

_2,

which support

the

choice of model

_!2).

ACKNOWLEDGMENTS

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