Restricted maximum likelihood estimation of genetic parameters for the first three lactations in the Montbéliarde dairy cattle breed

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Restricted maximum likelihood estimation of genetic

parameters for the first three lactations in the

Montbéliarde dairy cattle breed

C. Beaumont

To cite this version:

Original

article

Restricted

maximum

likelihood

estimation

of

_genetic

_parameters

for

the

first

three lactations

in

the Montbéliarde

_dairy

cattle breed

C.

Beaumont

Institut National de _la Recherche

_Agronomique,

Station de Recherches _{Avicoles, Nouzilly,} 37380 _Monnaie, France

(received

12 _{January 1989, accepted} 24

_{August 1989)}

Summary - Genetic _parameters for the first three lactations have been estimated for the main _dairy traits

_(milk,

_{fat, protein} and useful _{yields adjusted} for lactation _length, fat and _protein

_contents).

Two data sets were _analysed,

_including

records on 30 751 cows

born from 128 young sires and 52 proven sires. Daughters’ performances from the most

widely used proven sires were _incorporated in order to _improve the

_degree

of connectedness among herds. The model fitted young sires as random and proven sires, herd-year,

season-year of

calving,

age at first

calving

and

length

of the previous lactation as fixed effects.

Relationships among bulls were included.

Analysis

was _by restricted maximum likelihood

using an EM-related _algorithm and a

_Cholesky

transformation. All _genetic correlations were _larger than 0.89. Correlations between the first and third lactations were

_slightly

lower than the others. Heritabilities of _{milk, fat, protein} and useful _{yields ranged} from 0.17 to 0.27.

_Phenotypic

correlations between successive lactations were

_higher

than 0.6 and those between lactations 1 and 3 lower than 0.55. Heritabilities of fat and _protein contents were

_higher

than 0.44 with _phenotypic correlations _being stable at about 0.70. The _{"repeatability} model" which considers all lactation records as a _single trait can be considered in _genetic evaluation _procedures for _dairy traits without

_significant

losses in

efficiency.

dairy cattle - milk yield - fat and _protein contents - _genetic parameters - maximum likelihood

Résumé - _Application de la méthode du maximum de vraisemblance restreint

(REML) à l’estimation des _{paramètres génétiques} des trois premières lactations en

race montbéliarde. Ce travail a

pour

=but l’estimation des _{paramètres génétiques} des 3

premières lactations des

_femelles

Montbéliardes et porte sur les _{principales caractéristiques} laitières

_{(productions,}

_ajustées pour la durée de lactation, de lait et de matières utiles,

grasses et protéiques,, tnux _butyreux et

_protéique).

Deux

_fichiers

sont étudiés. Ils

rassem-blent les

_performances

de 30 751 _femelles issues de 128 taureaux de testage et de 52

tau-reaux de service. Ceux-ci sont introduits dans _{l’analyse pour} améliorer les connexions entre

troupeaux. Le modèle _comporte

_l’e,!’et

aléatoire _"père de

_testage"

et les

_{effets fixés}

_"père de

service", "troupeau-année", "âge au _{premier vêlage",} "année-saison de

_vêdage"

et "durée de la lactation _{précédente". L’apparentement} des _{reproducteurs} mâles est considéré. Les données _{transformées} par la décomposition de _Cholesky sont _analysées par le maximum

de vraisemblance restreint avec un _{algorithme apparenté} à l’E.M.

Les corrélations

_génétiques

des 6 _{caractères, toujours}

_supérieures

à _0,89, sont _légèrement

de 0,17 à 0,27. Les corrélations _{phénotypiques} sont _supérieures à _0,60 pour les

lacta-tions successives et

_inférieures

à _{0,55 pour} les lactations 1 et 3. Les taux _présentent une héritabilité _supérieure à 0,44 et des corrélations _{phénotypiques} voisines de 0,7 et

pra-tiquement indépendantes du _couple de lactations considéré. Ces résultats _indiquent que les

différentes lactations _peuvent être traitées comme des _{répétitions} d’un même caractère. Ce

modèle, dit de _{&dquo;répétabilité&dquo; permet d’alléger} les calculs sans diminuer

_{l’efficacité}

de la sélection.

bovins laitiers - _production laitière - _composition du lait - paramètres génétiques

-maximum de vraisemblance

INTRODUCTION

The

_goal

of

_dairy

selection is to

_improve

lifetime

_production

of _cows, which

_implies

taking

into account the different lactations. Until now,

genetic

evaluation of the

animals has in most cases been made under the

_assumption

that these lactations are influenced

_by

the same _{genes. In} some countries

_only

the first lactations are

considered;

in others the so-called

&dquo;repeatability

model&dquo;

_{(Henderson, 1987)}

in which all lactations are treated as

_repetitions

of one trait is fitted. But the lactations are made at _{various ages} and

_{physiological}

status of the animals and may therefore be

determined somewhat

_by

different genes. The accuracy of the

genetic

evaluation

and thus the

_effi!ciency

of

_dairy

selection

_might

be

_{improved by}

_fitting

a

multi-trait model to the lactations. Reliable estimates of the

_genetic

_parameters for the

different lactations are needed to

_appreciate

this

_possible

_gain

in _accuracy. Data

_usually

available for such estimations are selected as breeders cull about one _quarter of the animals

_by

the end of each lactation. Their decision is

_mostly

based on

_{dairy performance.}

Useful methods of estimation of these _parameters have been available

_{only recently.}

Henderson’s methods

₍₁₉₅₃₎

assume animals

are measured for all

lactations,

thus

_leading

to results biased

_by

the selection.

However,

the maximum likelihood

_{(ML) (Hartley}

and

_Rao,

₁₉₆₇₎

and restricted maximum likelihood

_{(REML) (Patterson}

and

_Thompson,

₁₉₇₁₎

estimators can

take into account this selection

_(Im

et

_al.,

_1987),

a _{necessary condition}

_being

that the selection process is based

only

on the observed data or on observed data

and

_independant

variables. REML was

prefered

to ML as it accounts for the loss

of

_degrees

of freedom in simultaneous estimation of the fixed effects.

_Moreover,

theoretical studies have shown that the

optimum

statistical

_{procedure maximising}

the

_genetic

merit of selected animals consists of

_estimating

variance and covariance

components

by

REML and thereafter

_applying

these estimates in the mixed model

equations

(Gianola

et

_al.,

_1986).

MATERIALS AND METHODS

Data

Records for the first 3 lactations of Montb6liarde cows whose first

_calving

occurred between

_1/09/1979

and

_30/08/1982

were extracted from the National Milk

Record-ing

files. The conditions of

_editing

are

_presented

in Table 1. Records made after cows

within herds and the

_absorption

matrix of the herd effects was block

_diagonal

for herds.

Two

_populations

of females were considered. The first was made of

_daughters

of test bulls. It was used to estimate sire _components of variance and covariance. The second consisted of

_daughters

of the most

_widely

used _proven sires. As these bulls had been

_selected,

_they

were treated as fixed effects and were not considered for the

estimation of sire _components. The

performances

of their

_daughters

were introduced

in the

_analysis

in order to

_improve

the accuracy of the estimation

through

additional

information,

increased herd size and

_degree

of connectedness between herds.

A total of 180 bulls of which 128 were random test bulls was considered. To

simplify

computation,

records were

_split

into 2 data sets, as did

_Meyer

_(1984,

1985a)

and Swalve and Van Vleck

_(1987).

The first data set consisted of the

daughters

of

_sampling

bulls born in 1975 and the second of the

_daughters

of

proven sires were determined and their

daughters

added. Table II summarizes their

main characteristics.

The main

_dairy

variables were considered:

milk,

fat,

protein

and useful

yields,

and fat and

_protein

contents. Useful

_yield

_(UY)

is defined as:

Yield traits were corrected

_{multiplicatively}

for lactation

_length

_prior

to

_analysis,

according

to Poutous and

_Mocquot

₍₁₉₇₅₎

as:

Corrected

_yield

=

(total

_yield

x

_{385)/(lactation length}

+

₈₀₎

Data were scaled to reduce

_rounding

errors.

Model

The

_following

model was used for each of the 6 variables:

where

y is the vector of the

_{observations;}

h is the vector of fixed herd effects

_(the

number of levels of which is shown in Table

_II);

b is the vector of fixed year-season of

calving

(15

levels for each

lactation),

age

at 1st

_{calving (10}

levels for each

_lactation)

and

_length

of the

_preceding

lactation effects

₍₈

levels for each

_lactation);

u is the vector of the sire effects

_(this

effect was treated as fixed when the sire was a _proven

_bull,

and as random and

_normally

distributed when the sire was a

young

bull);

and

e is the vector of residual

_effects,

assumed

_normally

distributed;

X,

W and Z are known incidence matrices for the herd

effects,

the other fixed effects and the sire effects.

Expectations

and variances are defined as:

where ui is the subvector of u

corresponding

to the effects of _{the young} bulls and:

where A is the

_relationship

matrix of the _young

_bulls,

T the matrix of the sire

components and * _the

right

direct

_product

_{(Graybill, 1983).}

Let n denote the number of

_animals;

with data ordered

_by

lactations within

_animals,

R is block

diagonal having n

blocks

_R,!

_(k

=

_{1, ... n).}

If the kth _cow has made the first 3

lactations, R,!

= E where E is the matrix of residual _components; if it has been

culled

_before,

the rows and columns

_{corresponding}

in E to the

_missing

records are deleted.

Method

Data were

_Cholesky

transformed

_(Schaeffer,

1986)

_but,

because the incidence matrix

not be transformed and the mixed model

_equations

for this _parameter remained

unchanged.

A combined

_REML/ML

_procedure

_(Meyer,

_{1983a, 1984,}

_1985a)

was used. From a

_{Bayesian viewpoint, only}

herd effects were

integrated

in the

posterior density.

From a classical

_viewpoint,

the inference was based on the likelihood of

(n — r(X ))

error contrasts

_{K’y (where}

K is an n x

(n — r(X))

matrix such that K’X = 0

(Harville, 1977).

An

_algorithm

&dquo;related to the E.M.&dquo;

_{(Henderson, 1985)}

was used. First derivatives of the restricted likelihood function are set to zero

(eqn. (7)

of

Meyer

(1986)).

For the

computation

of residual components, use is made of _eqn.

(8)

of

_Meyer

_(1986).

For

_{computations,}

the iterations were

stopped

when the relative difference between the estimates of the _components from one round to the

_following

fell below 1%.

The

_asymptotic

standard errors of the estimates were calculated. This

required

the

_computation

of the information

_matrix,

which is very extensive.

Therefore,

in

this _part of the

_analysis,

the fixed effects of the _year-season of

_calving,

_{of the age}

at 1st

_calving

and of the

_length

of the

_preceding

lactation were

_ignored,

so that the

Cholesky

transformation was

fully

efficient. The

relationships

among bulls were also

ignored

in order to reduce the

_{computational requirements.}

The information matrix

I

e

of the transformed data was first calculated

_and,

after back

transformation,

the information matrix I of the

_original

data was obtained as it can be showed that:

where D is the

₍₆

x

₆₎

matrix whose element

_(i,j)

is

(where

0

_{0c )}

is the vector of

_{(transformed)}

sire and residual variance and covariance

components).

Computations

of the

_1,

matrix was made

_using

Meyer’s algorithm

(1983a)

and

_{taking advantage}

of the

_{simplifications}

the

_Cholesky

transformation made

_possible.

Because of the

_{computational}

_costs, the two data sets were

analyzed separately

and the mean of the estimates calculated

_although

the two data sets were not

totally independent. Similarly,

the

_asymptotic

standard errors of the means of the estimates were obtained as if the

_asymptotic

standard errors of the estimates in the

2 data sets were

independent.

RESULTS

The estimates from the 2 data sets differed

_by

less than 1 standard _{error, except}

for the variance of the

_protein

content in the third lactation which differed

_by

a little less than 2 standard errors. The

_asymptotic

standard errors in the two data sets differed

_by

less than 0.01

_{(Table III).}

The estimates of the

_genetic

parameters for the

_yields

were similar. The

phe-notypic

variances increased with lactation number. The

_change

of the

_phenotypic

standard deviations was

proportional

to the increase in the

_{corresponding}

means and may be considered to

_be,

at least

_partly,

due to a scale effect.

By

contrast, the

(except

for

_protein

_yield

where it differed

_by

26%,

but this difference was not

sig-nificant).

Except

for fat

_yield,

the

_genetic

_component of the third lactation was the

highest

but the difference was not

_significant.

The heritabilities for the 3 lactations were therefore

_slightly

but not

_{significantly}

different

_{(Table IV):}

the heritabilities for the first and third lactations were similar

and

_higher

than for the second lactation. The

_only

_exception

was

_protein

_yield,

where the heritabilities for the first and second lactations were

_equal

to 0.18 and

slightly

smaller than for the third lactation

_(0.22).

All

_genetic

correlations were

higher

than 0.89. The correlation between the first and third lactations was smaller than the

_others,

which _except for useful

_yield

were _very similar. The same trend was observed for the

_phenotypic

correlations: the correlations between

_adjacent

lactations

_(first

and second lactations or second and third

_lactations)

was

_higher

than between first and third lactations. All

_phenotypic

correlations were between 0.53 and 0.65.

Genetic _parameters for the contents measured showed different trends. First the means for the different lactations were _very similar

_{(Table II)}

so there was no

scale effect. The

_genetic

_components decreased with lactation

number,

whereas the

phenotypic

components remained constant. Thus the heritabilities decreased with lactation

_number,

but the differences were not

_significant.

The

_genetic

correlations showed the same trend as for

_yields

and were between 0.90 and 0.96.

_By

_contrast, the

_phenotypic

correlations were

_higher

than for

_yields

_(between

0.67 and

_0.71)

and

did not _vary much.

DISCUSSION

Choice of the method

Different iterative

_algorithms

_may be used for REML estimation.

They

differ

in _convergence

_speed

and

_{computational}

_requirements

_per round of iteration.

Primarily,

3

_algorithms

have been advocated for

_analysis

on selected data: Fisher’s

method

_{(Meyer (1983a)),}

_algorithms

related to the E.M.

_algorithm

of

_Dempster

et al.

₍₁₉₇₇₎

and

_{Meyer’s algorithm}

_{(Meyer, 1986). Although}

Fisher’s method has the

_highest

_convergence

_speed,

it

_appears

to be the most

_expensive

_{(Meyer, 1986)}

and was therefore

_disregarded.

_Algorithms

called &dquo;related to the E.M.

_{algorithm&dquo;}

_by

Henderson

₍₁₉₈₅₎

_{converge very}

_slowly

but have the _property

_{of forcing}

the estimate

within the _parameter space.

Meyer’s

&dquo;short cut&dquo;

_algorithm

estimates the residual

components via an

_algorithm

related to the E.M. and the

_genetic

_components via

Fisher’s method. Thus the convergence

speed

is

quicker

than for

algorithms

related

to the

_E.M.,

but the estimates may lie outside the parameter space. The first

data set was

_{analysed using Meyer’s algorithm,}

and the estimate of the

_genetic

correlation between first and third lactations was

_equal

to 1.05 and between second

and third to 1.09. Such estimates cannot be considered as maximum likelihood

estimators

_(Harville,

_1977).

_They

cannot be used in the mixed model

_equations

without

_using

a transformation of the results such as the

_{&dquo;bending&dquo;}

of

_Hayes

and Hill

_(1981).

In contrast, the E.M. _type

_algorithm

gave estimates that were

achieved

_(instead

of 6 with

_Meyer’s

_algorithm),

as each iteration took 25% more

time with

_{Meyer’s algorithm}

than with the E.M. type

algorithm.

The

_Cholesky

transformation makes the

_absorption

of the herd effects

_quicker.

In

reference to the time necessary for the

absorption

of untransformed

_data,

the same

process took after

Cholesky

transformation 43% more time on the first round and

48% less time on the

_following.

This transformation also spares a lot of

_computations

for the estimation of the

_asymptotic

standard errors. Results of different methods

Maximum likelihood estimators can

_only

take into account selection if it is based on

observed data or on observed data and other

_independent

variables. In this

_analysis,

selection occurred both between

_generations

_(for

the choice of the

_parents)

and

within a

_generation

(by

the end of each

_lactation).

As the

_performance

of the parents

of the animals could not be

_analysed

because of the

_{computational}

_{requirements,}

only

the later selection was considered. This is the case for most REML estimates of

genetic

parameters for lactations. The results are in accordance with studies

_using

maximum likelihood related estimators

_(Tables

V and

_VI).

_Except

for Rothschild and Henderson

_(1979),

the authors used restricted maximum likelihood estimates

but the

_algorithms

varied: Fisher’s method for

_Meyer

_(1983a)

and

_Hagger

et al.

(1982);

&dquo;short cut&dquo; for

_Meyer

_(1985a);

E.M. _type for Colaco et al.

_(1987),

Rothschild

and Henderson

_(1979),

Simianer

_(1986b),

Swalve and Van Vleck

₍₁₉₈₇₎

and

_Tong

et al.

_(1979).

All considered a sire model _except for Swalve and Van Vleck

_(1987),

who used an animal model which _may better take into account

_selection,

since all

relationships

are included in the model

(Sorensen

and

_Kennedy,

₁₉₈₄₎

and thus did not observe _any decrease in

_heritability

for the second lactation. This decrease that

most authors observe

_might

be due to a selection bias. However Swalve and Van

Vleck

₍₁₉₈₇₎

_{neglected relationships}

_among herds and thus

_ignored

selection across herds.

Henderson’s methods lead to different results as

they

are affected

by

selection of

the data

_(Rothschild

et

_{al., 1979; Meyer}

and

Thompson,

1984).

Because of selection

at the end of first

_{lactation, they}

underestimate heritabilities for later lactations.

For milk

_yield,

the

_weighted

means of the estimates in the literature are 0.26 for first

_lactation,

0.20 for second lactation and 0.17 for third lactation

_(Maijala

and

_Hannah,

_1974).

In the first data set, Henderson’s method III estimates for

useful

_yield

were

_respectively

_0.21,

0.08 and 0.19. The first lactation estimate is in

accordance with REML estimates because selection has not _yet occurred. The third lactation estimate does not differ much either. As the criteria of selection

_depend

less on milk

_production

at the end of the second lactation than of first

_lactation,

the

selection bias _may be less

_important.

This result _may also

_be,

at least

_partly,

due to

sampling

errors.

_Similarly,

the decrease in the heritabilities for the later lactations for content measures _may

_partly

be due to the fact that the selection bias is not

well removed for these

_variables,

because the selection criteria are

_usually

based more on milk

_yield

than on content.

a sire model. The determinism of the second lactation _may also be

_slightly

different:

this

_{performance depends}

both on the

_dairy

value of the animal and on its

_ability

to recover from both

_growth

and first lactation. The

_genetic

correlations are _very

high

but the correlation between first and third lactations is

_{significantly}

different from 1. The older the cow

_is,

the more disease it has had to resist and the more

its

_ability

to resist is

_important

in the determinism of its lactations.

_However,

the

differences in the _parameters of the lactations are _{very small.}

It does not seem to be necessary to

_modify

the current French

_genetic

evaluation

procedure

which fits a

_{repeatability}

model to the different lactations. All available

lactations are taken into account because of the small mean herd size

(34.5

cows

per

herd).

Accuracy

is increased

using

all records instead of first records

_only.

This

gain

is due to both extra

_genetic

information and increased

_degree

of connectedness

among herds

(Meyer, 1983b).

Ufford et al.

(1979)

reported

such an increase even for

young bulls whose

daughters

had

only

first lactations.

Fitting

a multi-trait model would

_imply

a very

large

increase in

computational

requirements,

as time needed for an iterative inversion of the coefficient matrix of the mixed model

_equations

is

proportional

to the square of its size. But

only

a _very small

_gain

in accuracy could be

_expected.

Simianer

(1986a)

and

_{Schulte-Coerne (1983)}

estimate this increase to

be less than 1% when 3 lactations are considered and all the

_genetic

correlations

are 0.80. The difference is

_expected

to be even smaller in our case because the

correlations are

higher.

However,

they

restricted their

_analysis

to

_complete

data

(i.e.

all animals were

_supposed

to have made 3

_lactations).

In

_reality,

some selection occurs. The selection bias can be

_totally

removed

_only

when the true

_genetic

parameters are

used,

i.e. with the multi-trait model. But the difference between the 2 models is still

_expected

to be small.

ACKNOWLEDGMENTS

This research was conducted at INRA

_D6partement

de

_G6n6tique

Animale at the Station de

_{G6n6tique Quantitative}

et

_Appliqu6e.

The author wishes to thank F.

Grosclaude and L. Ollivier for their _support, B. Bonaiti and 2 referees for their

useful comments on the

_manuscript

and J.J. Colleau and J.L.

_Foulley

for their

helpful

discussions and

_suggestions.

REFERENCES

Colaco

_J.A.,

Fernando R.L. & Gianola D.

₍₁₉₈₇₎

_Variability

in milk

_production

among

sires,

herds and cows in

_Portuguese

_dairy

cattle. In: 38th Annual

_Meeting

_of

the

_European

Association

_for

Animad

_{Production, Lisbon,}

_{Portugal, Sept.}

27-Oct.

1,

1987 (Abstr.)

Dempster A.P.,

Laird N.M. & Rubin D.B.

₍₁₉₇₇₎

Maximum likelihood from

incomplete

data via the E.M.

_algorithm.

J.R. Stat. Soc. B.

_39,

1-22

Gianola

_{D., Foulley J.L.,}

Fernando C.L.

₍₁₉₈₆₎

Prediction of

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