Use of relationship matrix in the evaluation of natural service Limousin bulls

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Use of relationship matrix in the evaluation of natural

service Limousin bulls

D Laloë, G Renand, J Sapa, F Ménissier

To cite this version:

Original

article

Use of

_relationship

matrix

in the evaluation of natural service

Limousin bulls

D

Laloë

G

_Renand,

_{J Sapa,}

F

Ménissier

Institut National de la Recherche _Agronomique, Station de _Génétique Quantitative et _Appliquée, Centre de Recherches de _{Jouy-en-Josas,}

78352 _{Jouy-en-Josas Cedex,} France

(Received

8 _{April 1991 ; accepted} 12 December

₁₉₉₁₎

Summary - In _France, natural service bulls of beef breeds are _progeny tested _using reference sires and _{planned matings.} The evaluation is made with a _single-trait best linear unbiased

_prediction

_(BLUP)

_{procedure applied} to a sire model on _pre-weaning calf

performance, where sires are considered as unrelated. The _purpose of this _{investigation} was to examine the impact of

_using

_{relationships} between sires on _breeding value _estimates, their accuracies, and the connectedness between sires. Two mixed models, with or without

sire interrelations were used. _{Weaning weights} of 5 207 Limousin calves from 219 sires were

analysed. A total of 53 sires were related _by a Mal6cot relationship coefficient > 0.25. The correlation between _breeding value estimates obtained with the 2 models was 0.969. The gain in accuracy was characterized by the reduction of prediction error variances and expressed by the ratio of these variances obtained with the 2 models. This _{gain averaged}

4%,

reached 26% and was related to the sire _relationship coefficient means

(correlation

of

0.63).

Overall connectedness indexes were _computed and their _{interpretation} discussed. The practical interest of _using the _relationship matrix in these evaluation programmes

was examined in the _light of French beef cattle _breeding.

relationship matrix _/ connectedness _/ accuracy / sire model / beef cattle

Résumé - Prise en _compte des _parentés pour l’évaluation des taureaux Limousins de

monte naturelle. En _France, les taureaux de races à viande de monte naturelle sont évalués sur descendance en utilisant des taureaux de référence et des _{accouplements planifiés.} La méthode d’évaluation est un BL UP unicaractère _appliqué à un modèle _père sur les

est de _0,969. Le _gain de _précision, camctérisé par la réduction des variances d’erreur de

prédiction, est _{exprimé par} le rapport de ces variances obtenues dans les 2 modèles. Le _gain se situe en _moyenne à

4%

et _peut atteindre 26%. Il est lié à la moyenne des coefficients

de _parenté entre _pères

_{(corrélation}

de

_0,63).

Divers indicateurs du

_degré

de connexion

sont calculés et leur _{interprétation} est discutée. L’intérêt _pratique de considérer la _parenté

entre taureaux dans ces _programmes d’évaluation est examiné à la lumière de la situation française.

matrice de _{parenté /} connexion _/ modèle _{«père» /} bovins viande _/ précision

INTRODUCTION

An efficient evaluation of natural service bulls in the French beef breeds is difficult

to obtain

_using

field

_{performance, mainly}

because the number of _contemporary sires

within the

_herd,

and the number of sires used in more than one herd is limited due

to the small

_proportion

of artificial insemination.

With such a data _structure, disconnectedness results.

_{Consequently,}

bulls used

in different herds or _{years cannot} be

compared

to each

other,

or

_comparison

will be

carried out

_imprecisely.

Therefore,

a _system for

evaluating

the

breeding

values of natural service beef

sires was _{set up} in 1980 in order to reduce this disconnectedness

_(Foulley

and

_Sapa,

1982;

Sapa

and

_M6nissier,

_{1987; M6nissier,}

_1988).

It is based on

planned

_matings

and artificial insemination with references

_sires,

_using

cows of different ages and

morphological

types

within the same intra-herd

_period

in order to obtain a random

choice of mated cows. The evaluation concerns

weight

at birth and at 210

_days

as

well as muscle and skeleton

_development

scores at

_weaning.

The method used is a

single-trait

BLUP

_applied

to a sire model

including

the

following

effects:

herd-year,

sire, dam

_parity

and

_morphology,

sex-feed

_{supplementation level,}

calf birth season.

The sires are considered as unrelated. A selection index has been established and a connectedness index between all the sires is

_computed

_(Foulley

et

_al,

₁₉₈₄₎

to

obtain an automatic

_ranking

of all sires

_exhibiting

connectedness indexes above a

given

level

_(Foulley

et

_al,

_1984,

_1990).

The

_increasing

use of the best dams and sires of the herd or the breed leads to

relationships

among these sires. The use of

relationships

_among sires is an

alter-native method to

_improve

the beef sire evaluation

_(Henderson,

1975b;

Slanger

and

Lewis,

1986).

It was therefore decided to use these

_{relationships}

in the evaluation. The purpose of this

investigation

was to examine the

_impact

of

_{relationships}

on

the

_ranking

of sire

_breeding

_values,

their accuracies and connectedness between the

sires in the French _system.

MATERIALS AND METHODS

Methods

unbalancedness can result in

_{disconnectedness, ie,}

contrasts between levels of

one factor are no

longer

estimable. For random

factors,

the contrasts are

always

estimable,

but the variance of the contrasts will increase with the unbalancedness of the

_design,

which we called here

&dquo;degree

of disconnectedness&dquo; .

Consequently,

comparison

of the

_degree

of disconnectedness between BLUP

es-timates obtained with different models consists of

_comparing

variances of contrasts

between estimates: the _greater these variances are, the _greater the

_degree

of discon-nectedness from one model to another. The variances are

quadratic

forms associated

with the variance-covariance matrices of

_prediction

errors, and we will therefore

ex-amine

_quadratic

form ratios. Two models were used

_involving

the same

_effects,

but

differing

in terms of the variance-covariance matrix of

_genetic

effects

_{(sire effects),}

ie,

model a

_including

between-sire

_{relationships}

and model i not

_including

these

rela-tionships.

In order to

_study

the

_impact

of the

_relationship

_matrix,

_A,

on the

_degree

of

_{disconnectedness,}

we used a third

model,

model

a’,

where all the

off-diagonal

coefficients of A were

_0.25,

and the

_diagonal

coefficients were

_1,

_{corresponding}

to

a

population

where the animals are

non-inbred, and,

for _instance, half sibs.

Models of

_analysis

The model is:

where y is the vector of calf

_{performance ;}

b is the vector of fixed effects

_(herd-year,

sex-level of feed

_{supplementation,}

calf birth _season, dam

_parity)

and of covariables

_(withers

_height

and dam

muscle

_development

score, age of calves at

_{scoring) ;}

u is the vector of direct additive random effects of sires

_(variance

_U,2,);

e is the vector of random residuals

_(variance

_a2)

X and Z are incidence matrices

relating

effects to observations.

The first 2 moments of the random variables are:

According

to whether sire

_{relationships}

are taken into account or not, D will be

_equal

to the

_relationship

matrix A

_(model

_a, model

_a’)

or the

identity

matrix I

(model

i).

We then assume that e is the same in the 3 models.

Computation

of A and A-1

A was

computed according

to the recursive

_procedure

described

_by

Henderson

(1976).

A-’

was obtained

_by

direct inversion of A.

For each

_sire,

2

_relationship

criteria have been

computed,

the mean

relationship

Mixed model

_equations

After

_absorption

of the fixed

_effects,

the mixed model

_equations

are:

Computation

of variance-covariance matrices of

_prediction

errors

Suppose

that the true between-sires variance-covariance matrix is

_A(1!,

the

variance-covariance matrix of

_prediction

errors

(V)

is obtained

(Henderson, 1975a)

according

to the model

_by:

*

model a, model a’:

* _{model i:}

Note that model a

_supplies

BLUP estimates of sire

_effects,

while model i does

not.

_{Hence, they}

are of minimum variance in the class of unbiased linear estimators.

Consequently,

the value of _any

_quadratic

form associated with

_C

_a

will be less than

or

equal

to the same

_quadratic

form associated with the variance-covariance matrix

of

_prediction

errors

_resulting

from another unbiased linear estimator.

In

_particular,

this is true for the estimator

_{provided by solving}

the mixed model

equations corresponding

to model i.

_{Thus, L’C}

_a

_{L <_}

_L’C

_I

_L

for _{any vector} L.

These matrices

_being

both

_positive

_definite,

the ratio

_L’C

_i

_L/L’C.L

will be > 1. Evaluation of

_gain

_{in accuracy of estimates}

The variance of

_prediction

error of the

_breeding

value estimate of sire t is:

and the

_gain

in _accuracy

_{provided by considering relationships}

can be

computed

for sire t

_by

the ratio:

- I

The overall

_gain

in _accuracy was the ratio between the traces of both matrices:

Evaluation of the

_gain

in connectedness between sires

The connectedness

_improvement

and hence the

_gain

in between-sire

_{comparability}

was estimated

_by

the ratio between

_quadratic

forms:

where

L

t

is the contrast between the

_breeding

value of sire t and that of all the other sires. This ratio varies from 1 to

_infinity

and increases with the

_improvement

of the connection between sire t and the other sires. The correlation between the

connection

_improvement

of 1 sire and its 2

_relationship

criteria

_(r&dquo;

₁

and

_r

_M

₎

can

thus be

_computed

as well as its

_gain

_{in accuracy.}

Overall

_gain

in connectedness

Furthermore,

because

_C

₈

and

_C;

are

_positive

definite

matrices,

the ratios of the

quadratic

forms associated with these 2 matrices

_ranged

between

_min(k)

and

max(k),

k

_being

solutions to the

equation:

Thus,

Foulley

et al

₍₁₉₉₀₎

_suggested

2 indicators of the

_degree

of connectedness between sires

_depending

on the

following

solutions:

where n is the number of sires

(bulls),

_{’/’1 is} the arithmetic mean of solutions

k,

and

,y

2 represents the

_product

of solutions k.

Instead of !2; we _suggest the

_geometric

mean of solutions k as an indicator of

the

_degree

of _connection, ie:

noting

that _y3 is < _yl, the difference between the 2 values

_reflecting

the

_dispersion

of solutions k. The further these indexes are from

_1,

the

larger

the mean

_gain

in

connectedness

_resulting

from the consideration of sire

_{relationships.}

Animals

The data used in this

_study

were those included in the 1990 evaluation program

involving

on-farm _progeny

_testing

of Limousin natural service sires. The entire

data file was used. The evaluation concerned 219

bulls,

ie: 8 reference sires and 211

natural service sires which had been progeny tested

during

the last 5 years. The

number

_being

1.8. A total of 5 207 _progeny were tested

_involving

an _average 19 calves _per natural service sire and 159 calves per reference sire. In the last 5 years,

227

_herd-year

effects were

considered,

each

_involving

an _average of 6 progeny from

reference sires and 17 from natural service sires.

Because of the

_objective

of this

_{study, only}

calf

_weight

at 210

_days

was

_analysed.

The

_heritability

value used for this trait was

_0.20,

which

corresponds

to

_previous

estimates

_(Lalo6

et

_al,

_1988).

The

_pedigrees

of the bulls were established from the

Limousin breed

file,

provided by

&dquo;UPRA-France-Limousin-Selection&dquo;.

RESULTS AND DISCUSSION

Relationship

matrix structure

The

_relationship

matrix between the 219 sires was

computed

from their known

ancestry over 5

_generations,

_ie, a total of 2 144 animals. The structure of

_A,

the

matrix of

_relationship

between these 219 sires is

_given

in table I. The coefficients

in A are twice the Mal6cot coefficients.

Almost

_2/3

of the

_relationship

coefficients were null and

_only

3.6% of the

pairs

of sires exhibited a

relationship

coefficient >

(1/2)

4

.

The 33

relationship

coef&cients >_ 0.5

_belonged

to 53 sires

_{(sire-son pairs).}

In 1986

_(H

_Roy,

_personal

communication),

a similar

_relationship

structure was

_already

observed between

Limousin sires of this evaluation programme.

Breeding

values

Elementary

statistics

_concerning

the

_breeding

values are

_given

in table II. Their

distribution did not

_markedly

differ whether sire

_{relationships}

were used or not.

The correlation coefficients between the

_breeding

values of the 2 evaluations

were 0.969

(Pearson’s coefficient)

and 0.961

(Spearman’s correlation).

The

coe!cient >_

0.5 were 0.920

(Pearson’s correlation)

and 0.907

_{(Spearman’s}

correla-tion). Although

these coefficients indicate a close

_relationship

between the 2

evalua-tions,

the individual

_breeding

values nevertheless exhibited substantial

differences,

as shown in table III. Means and maxima of the absolute value of the differences

both in terms of sire

_breeding

values and

_rankings

increased with the mean

relation-ship

level

_(mean

of sire

_relationship

_{coefficients).}

The

_ranking

of sires with extreme

breeding

values remained similar.

Gain _{in accuracy}

The _average

_gain

_{in accuracy per} sire was

_1.038,

_ranging

from 1 to 1.255

_{(table III).}

This

_gain

_{in accuracy} was correlated with the mean level of

_relationship

(r

=

0.63)

and with

_{the maximum}

_relationship

_(r

=

0.81)

of the sire. As

expected,

the

_gain

in accuracy increased with

_increasing

mean

relationship

between sires.

_Comparing

model i and model

a’,

the _{average gain} in _accuracy was

_{1.101, ranging}

from 1.082

Gain in connectedness

The

_gain

in connectedness per sire reached a mean value of 1.040 and

_ranged

from

1 to 1.253

_(table

_III).

The

_gain

increased with

_{relationship coefficients,}

as shown

by

the correlation between this

_gain

and

_r,&dquo;(r

=

0.60),

and

r

M

(r

=

0.81).

There was a _very

high

correlation

(r

=

0.998)

between

_gain

in connectedness

and

_gain

in _accuracy.

_However,

_comparing

model i and model

_a’,

the

_gain

in

connectedness _per sire reached a mean value of 1.019 and

ranged

from 1 to 1.023.

The correlation between

_gain

in connectedness and

_gain

in _accuracy was

only

-0.22.

The values of indicators _’)’1 and _y3 of the overall

_degree

of connectedness were

1.038 and _1.035,

_{respectively.}

_Comparing

model i and model

_a’,

the values of

indicators yl and 73 of the overall

degree

of connectedness were the same: 1.020.

CONCLUSION

Use of the

_relationship

matrix in the evaluation of natural service sires leads to

marked estimation

_changes

and also to an

_improvement

(+4%)

in the _accuracy

and connectedness of estimates. An

_increasing

_relationship

matrix increased the

precision

of the

_evaluations,

but not the

_degree

of connectedness. More studies

are

_required

to

_study

the

_impact

of

_{including relationships}

on the

degree

of

connectedness on

_genetic

evaluations.

ACKNOWLEDGMENTS

We are _{very grateful} to UPRA-F)rance-Limousin-S61ection who _{supplied pedigree} data and to Mrs K R6rat

_{(INRA-UCD, Jouy-en-Josas)}

for the translation of the manuscript into

English.

REFERENCES

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JL,

Sapa

J

₍₁₉₈₂₎

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