Genetic improvement of litter size in sheep. A comparison of selection methods

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Genetic improvement of litter size in sheep. A

comparison of selection methods

M Pérez-Enciso, Jl Foulley, L Bodin, Jm Elsen, Jp Poivey

To cite this version:

Original

article

Genetic

_improvement

of

litter size

in

_sheep.

A

_comparison

of selection

methods

M

Pérez-Enciso

1

JL

_Foulley

L

Bodin

2

JM Elsen

JP

_Poivey

₂

1

Institut national de la recherche

_agronomique,

station de

_génétique

quantitative

et

_{appliqu6e, Jouy-en-Josas}

78352

_cedex;

2

Institut national de la recherche

_agronomique,

station d’amelioration

_génétique

des animaux, BP

_27,

3i826 Castanet-Tolosan

_cedex,

France

(Received

22

_{February 1994; accepted}

1

_August

₁₉₉₄₎

Summary - The

_objectives

of this work were to examine the usefulness of

_measuring

ovulation rate

_(OR)

in order to

_{improve genetic}

progress of litter size

_(LS)

in

_sheep

and to

study

different selection criteria

_combining

OR and

_prenatal

survival

_(ES)

_performance.

Responses

to selection for 5

_generations

within a

population

of 20 male and 600 female

parents were

_{compared using}

Monte-Carlo simulation

techniques

with 50

replicates

_per

selection method. Two breeds with low

_(Merino)

and medium

_(Lacaune)

_prolificacy

were

considered. Records were

_{generated according}

to a bivariate threshold model for OR and ES. Heritabilities of OR and ES in the

_underlying

scale were assumed constant over breeds and

_equal

to 0.35 and 0.11,

respectively,

with a

_genetic

correlation of -0.40 between these

traits. Four methods of

_genetic

evaluation were

_compared:

univariate best linear unbiased

prediction

(BLUP)

using

LS records

_only

_(b-LS);

univariate BLUP on OR records

(b-OR);

bivariate BLUP

_using

OR and LS records

_(b-ORLS);

and a maximum a _posteriori

predictor

of a

generalised

linear model

whereby

OR was

_analysed

as a continuous trait

and ES as a

_binary

threshold trait

(t-ORES).

_Response

in LS was very similar with

b-LS,

b-ORLS and

_t-ORES,

whereas it was

significantly

lower with b-OR.

_Response

in OR was

maximum with b-OR and minimum with b-LS. In contrast, response in ES was maximum

with b-LS. This

study

raised the

_question

as to

_why

selection based on indices

_combining

information from both OR and ES did not

_perform

better than selection

_using

LS

_only.

litter size

_/

ovulation rate

_/

prenatal

survival / sheep /

threshold model

Correspondence

and

_{reprints: UdL-IRTA,}

Area of Animal

_Production,

Rovira

Roure, 177,

Résumé - Amélioration génétique de la taille de portée chez les ovins. Comparaison

de méthodes de sélection. Cet article discute l’intérêt du taux d’ovulation

_(OR)

pour

accroître le

_{progrès génétique}

sur la taille de

_portée

_(LS)

et étudie à cet

_effet

divers critères de sélection combinant OR et le taux de survie

_embryonnaire

_(ES).

On a examiné

par simulation les

réponses

à la sélection en 5

_{générations}

dans une

_population

de 20 et

600

reproducteurs

mâles et

_femelles

avec 50

réplications

par méthode. On a considéré

2 races, de

_prolificité

_faible

_(Mérinos)

_{et moyenne}

_(Lacaune).

Les

_performances

ont

été

_générées

à

_partir

d’un modèle bicaractère à seuils. Les héritabilitiés d’OR et ES

ont été

_supposées

constantes sur l’échelle

_sous-jacente

dans les 2 races et

_prises

_égales

respectivement

à

_0,35

et

_0,11

avec une corrélation

_génétique

entre ces 2 caractères de

- 0,40.

Quatre

méthodes d’évaluation

_génétiques

ont été

_{comparées :}

_i)

_Blup

unicaractère basé sur LS

(b-LS) ;

_ii)

_Blup

unicaractère basé sur OR

(b-OR) ; iii)

_Blup

bicaractère basé sur OR et LS

_{(b-ORLS) ; iv)}

Prédicteur du maximum a

_posteriori

bicaractère d’un modèle linéaire

_{généralisé}

où OR est traité comme un caractère continu et ES comme un caractère

à seuils

_(t-ORES).

Les

réponses

observées étaient très voisines avec

_b-LS,

b-ORLS et

t-ORES,

alors que b-OR donne une

_{réponse significativement inférieure.}

La

réponse

sur

OR était maximum avec b-OR et minimum avec

_b-LS,

tandis que la

réponse

sur ES était

maximum avec b-LS. Cette étude _pose la

_question

de savoir

_pourquoi

la sélection basée sur

des indices combinant OR et ES ne donne _pas de résultats

_{significativement supérieurs}

à

la sélection sur LS.

modèle à seuils

_/

ovins

_/

prolificité

/

survie prénatale

/

taux d’ovulation

INTRODUCTION

Several studies

_support

the conclusion that increased

_{reproductive performance}

will

improve

economic

_efficiency

of

_sheep

_breeding

schemes

_(Nitter,

_1987).

Litter size

(LS)

is the trait

_receiving

_highest

relative economic value in the

_Norwegian

scheme

(Olesen

et

_al,

_submitted);

the British Meat and Livestock Commission

₍₁₉₈₇₎

includes ewe

_{reproduction performance}

in the selection indices in all

_except

terminal sire

_breeds;

selection schemes to

_improve

LS are

_implemented

in most breeds in France. Recommended economic indices used in the Australian Merino should result in substantial

_gain

in number of lambs

_weaned,

_according

to theoretical studies of Ponzoni

_(1986).

Litter size in

sheep

has been increased

_by

direct selection

_(Hanrahan,

1990;

Schoenian and

_Burfening,

₁₉₉₀₎

but the

_gains

have not been very

large

because of the low

_heritability

of LS. The _average

_{figure reported}

in the literature is 0.10

(Bradford,

1985).

The

_categorical

nature of this trait

_together

with a

possible

physical

upper limit

(uterine

capacity)

may also have hindered

genetic

progress. Ovulation rate

_(OR)

is considered to be the

principal

factor

_limiting

litter size in

sheep

(Hanrahan,

1982;

Bradford,

1985).

Heritabilities of OR are

typically larger

than those of LS in most

species,

including sheep.

Further,

correlation between OR and LS in

_high

and there is a

_nearly

linear

_relationship

with LS at ovulation rates _up

to 4

_(ie

Dodds et

_al,

_1991).

These results led Hanrahan

₍₁₉₈₀₎

to _propose OR as an indirect criterion to select for LS. Before routine evaluation of OR is

_implemented,

however,

its

_advantage

as selection criterion has to be assessed

_{experimentally.}

1992)

and in Romanov

_(Lajous

et

al, quoted

in Bodin et

_al,

₁₉₉₂₎

_{but, despite}

theoretical

_{expectations,}

most of _{response in} OR did not result in an increase of LS. The same

phenomenon

has been observed in mice

(Bradford, 1969).

In

pigs,

OR was increased

_by

selection but correlated response in LS was smaller than

expected

(Cunningham

et

_al,

_1979).

A second

_possible

criterion of selection is an index that combines OR and

prenatal

survival

_(ES).

Johnson et al

₍₁₉₈₄₎

derived a linear index of OR and ES and

_{they predicted}

that _response

_using

the index would be about

50%

larger

than

with conventional direct selection on LS in

pigs.

Similar

predictions

are

_given

by

Bodin et al

₍₁₉₉₂₎

in

_sheep.

_However,

selection

_experiments

have not confirmed the

expected advantage

of an index for LS

_components,

in mice

(Gion

et

al,

1990;

Kirby

and

_Nielsen,

₁₉₉₃₎

or in

_pigs

_(Neal

et

_al,

_1989),

whereas there is no

_experimental

evidence in

_sheep

_yet.

In all

species,

the

_apparent

reason

_why

OR or an index was no better criterion than LS was a correlated decrease in ES.

Cited

_predictions

of response are

_implicity

based on an infinitesimal model with a continuous

normally

distributed trait. This model can be

justified

for OR in

pigs

or mice but

certainly

not for

ES,

which is a dichotomous trait. P6rez-Enciso et al

_(1994a)

examined the

_implications

of

_generating

OR

and ES records

_according

to a bivariate threshold model. In this

model,

2

_underlying

_(unobserved)

normal

variates which are

negatively

correlated and a set of fixed thresholds are assumed. The main

_implications

of a bivariate threshold model for litter size

_components

are:

(i)

the existence of a non-linear

_{antagonistic relationship}

between OR and

_ES,

ie

correlation between LS and OR decreases as OR

_augments;

(ii)

as a _consequence, a linear index

combining

OR and

ES,

which

gives

a constant

weight

to ES over all the _range of

_OR,

is not the

_optimum

selection criterion to increase LS in all

generations;

and

_(iii)

litter size behaves as a natural index close to the

_optimum

selection criterion

combining

OR and

ES,

at least in the situation

analysed

(mass

selection and

_equal

information on

candidates).

Points

(ii)

and

(iii)

are

_especially

relevant because the

_theory

based on a linearisation of the model

predicted

an

advantage

of the index over

LS,

which was not

fully

achieved in the simulation. This is

_precisely

the situation encountered in selection

_experiments,

where a linear index of OR and ES has not

proved

to be

_{significantly}

better than direct selection on LS

(see

review of Blasco et

al,

1993)

regardless

of

_optimistic

_predictions.

The

_objectives

of this work were:

(i)

to examine in a more realistic situation than in a

_previous

_report

(P6rez-Enciso

et

al,

1994a)

the usefulness of

measuring

OR in

order to

_{improve genetic}

progress of LS in

sheep using

overlapping generations

and all

_family

information;

and

_(ii)

to

_study

different selection criteria

_combining

OR and ES

_{performances.}

The influence of

_genetic

correlation between OR and ES has also been considered. Work was carried out

_using

stochastic

_computer

simulation.

Records were

generated according

to a bivariate threshold model. Two breeds with

MATERIALS AND METHODS

Selection scheme

A

_population

of 600 dams and 20 sires was simulated. After each

breeding

_season, when new records from OR and LS were

_available,

old and newborn animals were evaluated

_according

to 1 of several methods described below. The worst 120 dams

(20%)

and the worst 10 sires

_(50%)

were discarded and

_{replaced by}

the best 120 newborn females and the best 10 newborn males.

_Only

1 female and 1 male

offspring

per dam per

breeding

season were allowed and the maximum number of male

_offspring

to be selected from each sire was set to 3. A control line was simulated where sires and dams were chosen at random. Results were

expressed

as deviations from the control

_line,

in order to correct for the effect

_assigned

to each

_breeding

season. Five

_cycles

of selection were simulated and 50

_replicates

for each selection

method were run.

Two

_populations

were

_considered,

a low

_prolific

breed

_(Merino)

and a more

prolific

breed

_(Lacaune).

_Phenotypic

means and variances are shown in table I for

_nulliparous

and

_{non-nulliparous}

ewes of both breeds. These

_figures

are based on

performances

in INRA

experimental

herds for Merino and on-farm

recording

for Lacaune. Ovulation rate and LS increased with

_parity

order even if

_prenatal

survival was lower.

_Phenotypic

correlations between OR and ES were -0.56 and -0.38 in Merino and

_Lacaune,

_{respectively.}

Note that

_populations

with

_higher

means had

higher

variances but that coefficient of variation remained

_{approximately}

constant,

as

_commonly

observed

_(Nitter,

_1987).

- .. - -.... - - . - .... , ..

Generation of records

Records of

OR,

ES and LS were

generated

as described in detail in P6rez-Enciso et al

_(1994a).

In

_short,

both OR and ES were

_categorical

variates assumed to be determined

_by

a threshold

_liability

process with

normally

distributed

underlying

from a Bernoulli distribution with

_appropriate

_parameters.

Litter size was the number of

_{embryos surviving.}

Thresholds were set to match observed

frequencies

in each

_category

of OR and ES. Heritabilities of OR and ES in the

_underlying

scale were 0.35 and

_0.11,

_{respectively,}

in both breeds.

_{Repeatabilities}

of OR and

ES were 0.70 and

0.22,

respectively.

Genetic correlation between OR and ES was - 0.40 in both breeds. Environmental correlations were -0.32 and -0.22 in Merino and

_{Lacaune, respectively.}

Given that there exists

_uncertainty

about the

_genetic

parameters,

especially

for

_genetic

correlation between OR and

_ES,

other correlations were considered in the Lacaune breed. The model used to simulate records included

animal

plus

common environment as random effects. Fixed effects were

_parity,

with 2

_levels,

first and

_following

_parities,

and year, with 5 levels. Values for the effect of

_parity

in the

_underlying

scale were chosen as to match

_figures

in table I. The

effect of year was simulated such that maximum differences between the ’best’ and ’worst’ _years were about

10%

in LS.

Genetic evaluation

Four methods of

_genetic

evaluation were

_compared:

1)

univariate BLUP

_using

LS records

_only

_(b-LS);

₂₎

univariate BLUP on OR records

(b-OR); 3)

bivariate BLUP

_using

OR and LS records

_(b-ORLS);

and

₄₎

a bivariate non-linear model

whereby

OR was

_analysed

as a continuous trait and ES as a

_binary

threshold trait

(t-ORES;

Foulley

et

_al,

_1983).

Here

equations

derived

_by

Janss and

_Foulley

₍₁₉₉₃₎

were

adapted

to take into account that for each record of the ’continuous’ trait

_(OR),

there were as many observations of the

binary

trait

_(ES)

as number of ova shed

(see

A

P

pendix).

The

_original

_program

_(LLG

_Janss,

_personal

_{communication)}

was

optimised

to solve the

_system

of

_equations

_by

_sparse matrix methods

_using

FSPAK

(Perez-Enciso

et

_al,

_1994b).

In

_agreement

with the

_simulation,

the statistical model in all methods included

_parity

and _year as fixed

effects,

and animal and

_permanent

environmental effect as random effects. Criterion b-LS is that

_{currently implemented}

where

_sheep

are evaluated for their

reproductive performance

(Bolet

and

Bodin,

1992;

Olesen et

al,

submitted),

whereas b-OR

_responds

to Hanrahan’s

₍₁₉₈₀₎

suggestion

of

_using

OR as indirect criterion to select for LS.

_Finally,

b-ORLS and

t-ORES are different ways of

combining

OR and ES

performance.

In

b-ORLS,

a direct estimation of the

_breeding

values for LS is

_obtained,

whereas in t-ORES the

estimated

_breeding

values of OR and ES have to be combined in an index for LS. The index chosen was that

_{suggested by}

Wilton et al

_(1968),

ie for the ith

_animal,

where

_!OR,

the

_predicted

ovulation

_performance,

is

where

h

ORI

and

_poR,

are estimations

_(maximum

a

_posteriori,

MAP)

of first _year

and first

_parity

obtained

_{by solving}

the t-ORES

_equations

_and,

_similarly,,a

_OR

_,

and

effect, respectively.

In

_[1],

the

_predicted

ES

_probability

is

In

_equation

_!3!,

h

ES

&dquo;

_{PES&dquo; aesi}

and

_F

_E

_si

are MAPs obtained from

_solving

the t-ORES

_equations.

Note that because

_[1]

is a nonlinear index

_genetic

merit

depends

on levels of fixed effects. The index was chosen to maximise _response in first

_parity.

Alternatively,

a

weighted

_{average of all}

_parities

could also have been

_applied

_(Foulley

and

_Manfredi,

_1991).

Methods were evaluated in terms of elicited response to selection but

goodness

of

fit,

as

_{suggested by}

P6rez-Enciso et al

_(1993),

was also studied. Correlations between observed and fitted records were

_computed.

For b-LS and b-ORLS

_methods,

fitted LS records of ith animal in the

_jth

_year and kth

_parity

were obtained from

where

_!LS,

_{PLSk’ âLsi’}

and

_F

_L

_si

are best linear unbiased estimate

_(BLUE)

and BLUP solutions to fixed and random effects obtained for the LS location param-eters. In the case of

_t-ORES,

fitted LS records were

computed

from an expres-sion similar to

_[1]

except

that

_{corresponding}

_year and

_parity

solutions were used. Ovulation records were fitted from

_expressions

similar to

_[2]

and ES records from

expressions

similar to

_!3!.

Note that OR was treated in all cases as a continuous variate even

_though

it was simulated

_following

a threshold model. There is

_evidence,

_nonetheless,

that the

advantage

of a threshold model over a linear model for

_genetic

evaluation diminishes very

quickly

for more than 1 threshold

_(Meijering

and

_Gianola,

_1985).

Genetic variances used for

_genetic

evaluation were those in the observed

_scale,

except

for ES in t-ORES.

They

were obtained

_by

simulation from the definition of

_breeding

value in the observed

_scale,

ie mean

_phenotypic

value conditional on

genotype.

Heritabilities of LS were 0.14 and

_0.15,

and 0.23 and 0.30 for OR in Merino and

_{Lacaune, respectively.}

The

_heritability

of ES was 0.06 in both breeds.

Genetic correlation between LS and OR was 0.83 and between LS and

_ES,

0.17.

RESULTS

Selection responses for LS in the first

_generation

are shown in tables II and III

for Merino and

_Lacaune,

_{respectively.}

An increase in LS of about

5-6%

of the mean was achieved in both breeds.

_Changes

were

_relatively

more

_important

in first than successive

_{parities. Response}

in LS was _very similar whether selection was

directly

on LS or

_using

OR as indirect selection criterion.

_Considering

information

on both OR and LS

_(or

_equivalently

OR and

_ES)

_{produced only}

a small increase in response with

_respect

to direct selection on LS. Performance of b-ORLS and

t-ORES was almost identical. Even if index

[1]

was derived to maximise _response in first

_parity,

correlated _response in successive

_parities

was as

_high

as with linear

methods,

ie

_b-ORLS,

in which

_weights

do not

_depend

on location

_parameters.

This

Subindices refer to first

₍₁₎

and

_{following parities}

_(>

_1).

Maximum

empirical

standard

errors were 0.02 for ALS and AOR and 0.004 for AES.

Subindices refer to first

₍₁₎

and

_{following parities}

_(>

_1).

Maximum

_empirical

standard

errors were 0.02 for ALS and AOR and 0.004 for AES.

and Manfredi

_(1991,

_equation

_[62]),

_whereby

_predicted

_performance

is

_weighted

according

to

_frequencies

of the different

_{subclasses, might}

be robust to different

weights.

Changes

in LS were

relatively

similar across selection methods but

_they

were

not for the

_components,

OR and ES

_(tables

II and

_III).

Ovulation rate increased twice as much when selection was on OR than on LS.

_However,

_only

about half of

that increase

_corresponded

to an increase in LS with b-OR

_{(!LS/ !OR ::::i}

_0.50),

whereas the ratio

_ALS/AOR

was

_{always larger}

than 0.80 with b-LS. Methods

b-ORLS and t-ORES induced

_changes

in OR similar to b-OR. Correlated

_changes

in ES were also different

_depending

on the method of selection. Selection

_using

b-LS was

_accompanied

_by

an increase in

_prenatal

survival of about

2%.

All other

methods,

especially b-OR,

resulted in lower ES.

Response

of LS in the

_following

_generations

is

_plotted

in

_figure

l. Unlike results in tables II and

_III,

it is evident that indirect selection on OR was the

_poorest

method in the

_long

term,

especially

in the more

_profilic

_breed,

Lacaune. Direct selection on LS was

_only

_slightly

worse than selection based on either b-ORLS or t-ORES.

Responses

were not

_{significantly}

_{different among methods in any}

_generation

with the sole

_exception

of b-OR.

_Figure

2 shows correlated

_changes

in OR

_phenotype.

As

expected,

b-OR caused the

_largest

increase in OR and

_b-LS,

the minimum. Methods

in-creased more in Lacaune than in Merino. Direct selection for LS in Merino induced no correlated

_{phenotypic change}

in ES

_(except

in the first

_generation),

whereas

survival increased

_regularly

in Lacaune. In

Merino,

phenotypic

trends were

_negative

with b-ORLS and t-ORES in the first 2

_generations

but ES remained constant or increased thereafter. This

_highlights

that b-ORLS and t-ORES are nonlinear selection criteria with

_respect

to

_underlying

_genotypes.

Selection on OR induced a

negative

response in ES because of the

negative genetic

correlation between both traits.

Correlations between observed and fitted records are shown in table IV for the traits studied. All methods were _very close to each other

_regarding

the

_degree

of fit within trait.

Previous results assumed a

_genetic

correlation between OR and

ES(pg

oR

,

Es

)

of -0.4.

_Consequences

of

_using

different values of the

_genetic

correlation were examined in Lacaune. Table V shows how different

parameters

are affected

by

a

change

in

_p

_9oR

_,

_ES

_.

In all cases

_phenotypic

correlation and heritabilities of OR and ES were

_constant,

thus environmental correlation decreased as

_genetic

correlation

increased. Genetic correlations and heritabilities were obtained

_by

simulation. Genetic correlation between LS and its

components

increased with

_pg

_oR

_,

_ES

_,

but the increase was much more noticeable between LS and ES than between LS and OR. In the extreme case of

_pg

_oR

_,

_ES

=

-!.9, genetic

correlation between LS and ES

was

less than zero. This

_negative

_genetic

correlation was confirmed

by

a decrease in mean ES

_genotype

when selection was

_performed

on b-LS

(results

not

_shown).

However,

the ES

_phenotype

did not

_change

_{(table VI)}

_perhaps

because of the small decrease in the

_breeding

value of ES.

_Heritability

of LS decreased as

_pg

_oR

_,

_ES

did.

Thus a small

_heritability

of LS could be due either to a low

_heritability

of its

components

or to a

_highly

_{negative genetic}

correlation between them

_{(equation !11!}

in P6rez-Enciso et

_al,

_1994a).

The effect of

_genetic

correlation on _{response to} selection is shown in table VI. Given the similar results between b-ORLS and

_{t-ORES, only}

b-ORLS was

_studied,

in addition to b-LS and b-OR. Method b-ORLS was chosen because of its lower

pg,

genetic correlation;

pe, environmental

correlation; OR,

ovulation rate;

ES, prenatal

survival; LS,

litter

_size;

_h

₂

_,

_heritability

in the observed

_{scale; figures}

refer to Lacaune.

Figures

refer to Lacaune.

genetic

correlation increased. Methods b-LS and b-ORLS were not

_{significantly}

different for moderate

_{genetic correlations,}

whereas b-OR was

_consistently

less

efficient. The behaviour of the ratio

_ALS/AOR

_{depended strongly}

on

_genetic

correlation. An increase in number of ova shed was followed

_{closely by larger}

litter size for moderate correlations.

However,

as

_pg

_oR

_,

_ES

decreased so did

ALS/AOR.

DISCUSSION

The

results

_presented

here are in

_agreement

with simulation results

_reported

previously

(P6rez-Enciso

et

_al,

_1994a)

where selection on OR or on an index

combining

OR and ES did not

produce

a

significantly larger

_response than direct selection on LS. The

_{slight advantage}

of b-ORLS and t-ORES over b-LS in the first

breeding

season did not

_persist

over

_generations,

due to a

_larger

decrease in ES when

information from OR was included in the selection

_criterion,

Methods

b-LS,

b-ORLS and t-ORES behaved _very

_similarly

with

_respect

to LS

_{(fig 1),}

but

_differently

with

respect

to OR and ES

_(fig

2 and

_3).

_{Oddly enough, including}

information from OR

_(a

trait of moderate

_heritability

and

_highly

correlated with

_LS)

did not result in a much

_higher

_response for LS but rather in a redistribution of

weights

_given

to OR and ES. Selection pressure on ES decreased in bivariate methods

(b-ORLS

and

_t-ORES)

because correlation between OR and LS is much

_higher

than that between ES and LS

_{(table V).}

_Phenotypic

differences in ES among lines were small

(about

3%

between b-LS and b-ORLS in

_Lacaune)

but it sufficed to

_compensate

for different ovulation

_rates,

a difference of

_{approximately}

0.2 ova between lines selected with b-LS and b-ORLS. In

_fact,

one of the

_arguments

adduced in favour of OR as indirect selection criterion is that

_phenotypic

differences between control and selected lines were much

_larger

in OR than in ES

_{(Hanrahan, 1982).}

However,

prenatal

survival cannot be

_neglected

even if its contribution to total variation of

LS in the base

_population

is

_small,

in

_particular

when

_genetic

correlation between

OR and ES is

_negative.

Using b-LS,

increase in litter size

_{corresponded exactly}

to the increase in OR for

P90R,ES !

-0.4

(ALS/AOR

= 1 in table

VI).

This _can be

interpreted

_as if

response to selection for LS were

_{completely explained}

_by

a

_change

in OR. In

_contrast,

selection

_using

OR

_produced

a smaller _{response in} LS with a much

_larger

increase in OR

_(ALS/AOR

=

0.48).

This

phenomenon

_was described

by

Bradford

₍₁₉₈₅₎

as &dquo;a

_{striking example}

of

_asymmetrical

correlated

_{response&dquo;}

_(underlining

is

_ours).

It is evident from results in table

VI, however,

that this

_{apparent asymmetry}

is due to a different

_emphasis

on ES in the two criteria. It is current

_opinion

_among

_sheep

breeders that selection _pressure on ES should become more

_important

as mean OR increases. The results in table V

_highlight

that this pressure also

depends

on the value of

_{/0gon !g -}

As this correlation becomes more

_negative,

selection _pressure

on OR relative to ES increases

_{dramatically,}

_especially

if information on OR is used.

_Then,

if

_{prenatal mortality}

increases too

_much,

as a result of a correlated

change

with

_OR,

decline in ES will offset the increase in OR. For

instance,

for

P

g

OR

,

ES

=

-!!9 response in LS was

slightly larger

with b-ORLS than with b-LS in the first

_generation

_(results

not

_shown)

but the reverse was true in the fifth

generation

(table VI).

Matos

₍₁₉₉₃₎

reported

correlations between fitted and observed LS records about 0.65 in Rambouillet and

_{Finnsheep using}

linear models. These

_figures

are

_relatively

close to those

_reported

here if we consider that in Matos’

(1993)

_study

variances had to be estimated from the same data set. In a similar

_study,

Olesen et al

₍₁₉₉₄₎

reported

lower correlations between fitted and observed LS records of

_Norwegian

terms of

_goodness

of fit but the differences between methods were _very

small,

in

agreement

with results in table IV.

The

_question

whether LS can be increased more

rapidly by

using

information on its

_components

rather than

by

direct selection remains open. All

experiments

have failed in this

_respect

_(Blasco

et

_al,

_1993).

A

_{likely explanation}

is that indices

combining

OR and ES

_(or

OR and

_LS)

have not been

_{optimum. Only}

linear indices with a constant

weight

to ES over all the _range of OR have been tested

_{experimentally,}

and these do not take into account the nonlinear

_phenotypic

relationship

among LS and its

components.

P6rez-Enciso et al

_(1994a)

have shown

that a

_separate

index for each OR should be used. Nonlinear indices

might

overcome some of these

handicaps.

In this

work,

a

simple

nonlinear index

(equation

[1])

was examined. The exact

_equation

is an

_integral

that

implies marginalization

with

respect

to a

_large

number of variables. The

_analytical

solution to this

_integral

is

unknown.

_Equation

_[1]

is a first-order

approximation

which will

only

be close to the

optimal

criterion if the amount of information on each individual is

large

(Gianola

and

Fernando,

1986).

Results

_{showed, however,}

that nonlinear indices

_(t-ORES)

did not elicit a

_larger

_{response in} LS than linear indices

(b-ORLS),

perhaps

because of the lack of information. Moreover

genetic

parameters

in the observed scale are assumed to be constant but

_{they depend}

on the

population

_mean, which

changes

with selection. It should be

recalled, nonetheless,

that OR could be used as an

early

predictor

for

_LS,

_given

its

_high

correlation with LS and its

_{high repeatability.}

Measurement of OR would then allow us to decrease

_generation

interval and increase the _accuracy of

_genetic

evaluation of young animals.

Certainly,

the results

presented

here

depend

crucially

on how

_likely

a bivariate threshold model

_is,

ie on the existence of 2

underlying

continuous normal variates and a set of fixed threshold

_points.

Because a statistical model is

necessarily

an

oversimplification

of

_reality,

these conditions will never be met with strict

_rigour.

There

_{is, nonetheless,}

considerable literature on the

_plausibility

and

biological

justification

of the threshold

_model,

see reviews

by

Curnow and Smith

(1975)

and

Foulley

and Manfredi

_(1991).

With

_respect

to

_reproductive

_traits,

there is a

complex

interaction between continuous

_variates,

_eg, hormone

_levels,

and discrete

_variates,

ie

number of ova and survival

(Haresign, 1985).

A threshold-like mechanism has been advocated to

_explain

_{embryo mortality}

as a function of the

_degree

of

asynchrony

between uterus and

_embryo

_(Wilmut

et

_al,

_1985).

In

_addition,

covariation _among LS

_components

and selection

_experiment

results can be used to check the

validity

of the threshold model.

First,

under the bivariate threshold model considered

_here,

the

_{phenotypic probability}

of

_embryonic

survival at a

_given

ovulation level is

_given

_by

(P6rez-Enciso

et

_al,

_1994a)

_{where p.}

is the

underlying

mean, x is the

underlying

variable,

b is the

_regression

of xES on _Rxo and _p is the correlation between xES and x

oR

.

Equation

[4]

allows us to describe a

_decreasing

correlation between OR and LS as OR

increases,

as

_commonly

observed in real data

(Hanrahan,

1982;

Dodds et al

1990).

Further,

[4]

can be written as

.1,O

)]

R

f

R -

o

<I>[a+,B(x

_{with /3}

=

b/(1-p

2

)

1

/

2

.

Now

on number of ova

(P6rez-Enciso

et

_al,

_1994a),

or from

_phenotypic

covariances and variances

_{(table I),}

_assuming

that

_[4]

holds as well in the observed scale. With the

_parameters

used

here,

values for

_!3

from the

_{probit regression}

were -0.34 and - 0.25

_and,

from

_phenotypic

_covariances,

-0.37 and -0.18 for Merino and

_Lacaune,

respectively. Agreement

seems

_reasonable,

_especially

for the less

_prolific

breed.

Secondly,

table V

_emphasises

that

_genetic

_parameters

for

_OR,

ES and LS are interrelated. In

_{particular, heritability}

of LS can be

_{expressed, approximately,}

as

(P6rez-Enciso

et

_al,

_1994a),

_{where py} is the

_phenotypic

_mean;

_Varg(Covg)

the

genetic

variance

_(covariance)

in the observed

_scale;

and

_Vary

the

_phenotypic

variance. It follows that for a

_given

_variability

of OR and

_{ES, heritability}

of

LS is

_inversely

related to

_genetic

correlation between OR and ES. Thus

_given

the

_phenotypic

_parameters

in table I and

_provided

heritabilities are moderate for OR and low for LS and

_ES,

_equation

_[5]

allows us to

_predict

a

_negative

genetic

correlation between LS

_components.

Table VII shows how well

_[5]

_predicted

heritabilities of LS in different studies

_reporting

multivariate variance estimates of

OR,

LS and ES in

_pigs.

_Agreement

between

_expected

and estimated heritabilities was excellent in

_reported

estimates in

_pigs.

Note that

_Haley

and Lee

₍₁₉₉₂₎

found no additive variation for ES and that in this case the smaller

_heritability

of LS occurs due to the additional noise from

_{embryonic mortality.}

A

_{negative genetic}

correlation between OR and ES has also been evidenced

_{by selecting}

on

_OR,

which has been

_accompanied

_by

a lower ES

_(Bradford,

_1969;

_Cunningham

et

_{al, 1979;}

Hanrahan,

1992).

Most

_reported

estimates of

_genetic

covariances between OR and ES in

_pigs

and rabbits are also

_negative

_(Blasco

et

_al,

_1993).

_Furthermore,

the

magnitude

of this correlation

_greatly

influences the ratio of _response in LS relative to OR

_(R

=

ALS/AOR).

The _more

negative

the

_{genetic correlation,}

the

_bigger

the difference in R between direct selection on LS and indirect selection on OR

(table VI).

There is

_experimental

evidence

_supporting

that the ratio R is

_larger

for

direct selection than indirect selection on OR as

_expected

from results in table VI. For

_instance,

R was 0.61

_using

direct selection in the

_Galway

breed

_(Hanrahan,

1990),

and 0.67 in Rambouillet

_(Schoenian

and

_Burfening,

_1990),

whereas R was

only

0.26 when

_Finnsheep

were selected for OR

(Hanrahan, 1992).

n, number of

records;

hL

S

,

expected heritability

of litter size

_(from

_equation

_[5]);

other abbreviations as in table V.

Finally,

the most critical

_implication

of this model refers to the relative

_advantage

Using

mass

selection,

P6rez-Enciso et al

(1994a)

found in a simulation

study

that response in LS with an index

combining

OR and ES was similar to direct selection.

Both methods were better than indirect selection on OR. In this

_study,

where

_family

information was

used,

the same conclusion

_applies.

These results are

compatible

with

_experimental

evidence in

_pigs

and mice

_(Bradford,

_1969;

_Cunningham

et

al,

1979;

Neal et

_{al, 1989;}

Gion et

_{al, 1990;}

_Kirby

and

Nielsen,

1993)

where OR or an index

_combining

OR and ES has not

_proved

to be

_{significantly}

better than direct

selection on LS.

Several

_problems

with the infinitesimal threshold model remain nonetheless.

First, major

genes

affecting

OR in

sheep

are

known,

the gene Fee B of the Booroola Merino

_being

the best documented case. The _presence of a

_major

_gene per se does not invalidate a threshold

model,

since it can be considered as a fixed effect that shifts the

_underlying

mean, but it does

change

the

dynamics

of the

_population

under selection. Thus

_predicted

or simulated responses under an

infinitesimal threshold model will be

quite

inaccurate. Genes with a

_significant

effect on OR are also

being

identified in other

species

such as mice

(Spearow

et

al,

1991)

but evidence is

_conflicting

in

_pigs

_(Mandonnet

et

_al,

_1992;

_Rathje

et

al,

1993).

With

respect

to

_{embryonic survival,}

a number of recessive _genes that

cause

_{embryo lethality}

in mice have been identified

(Rossant

and

Joyner,

1989).

As information on effects and

_frequencies

of

(aTLs affecting

litter size

accumulates,

the

_implications

of an infinitesimal threshold model should be reconsidered. The existence of

_major

chromosomal abnormalities as a cause of

_embryonic

_mortality

would also pose

problems

for the threshold model. In a recent

review,

Blasco et al

₍₁₉₉₃₎

_quoted

that lethal chromosome abnormalities are found in

5-10%

of

early

pig

and rabbit

_embryos,

a

_{non-negligible}

_proportion.

Deleterious

reciprocal

translocations and

_aneuploidies

have been

_reported

in

_sheep

but the

_proportion

of

embryo

deaths due to these abnormalities is uncertain

_(Bolet,

1986;

Wilmut et

_al,

1986).

CONCLUSIONS

The main conclusions can be summarised as follows:

(i)

A bivariate threshold model can be

justified

based on statistical and

experimental

evidence.

(ii)

Ovulation rate was an effective criterion of selection for litter size in

_{sheep only}

in the _very short term. The

_advantage

of

_using

OR as an

_early

predictor

in order

to decrease

generation

interval should be

_{investigated.}

(iii)

The ratio of response in litter size relative to ovulation rate

(ALS/AOR)

depends

strongly

on

_{genetic correlation;}

the more

_negative

the

_{genetic correlation,}

the

_larger

the difference in this ratio between direct selection for litter size and indirect selection on ovulation rate. Direct selection on litter size maximised the ratio

_ALS/AOR.

(iv)

Using

information from both ovulation rate and

_embryonic

survival was not

significantly

better than selection

using

litter size records

_exclusively.

Selection with

b-LS

_puts

more _pressure on survival than methods

combining

ovulation rate and

Several unresolved

_problems

can be

_quoted:

(i)

The extent to which these conclusions

_apply

to other situations and

_species

such as

_mice,

rabbits and

_pigs.

(ii)

The

_{interpretation}

of

_breeding

value for litter size and how to combine information from ovulation rate and

_prenatal

survival in an

_optimal

_way.

(iii)

An

_{analytical approach}

to

_predict

_{response to} selection with this model

_along

the lines of

_Foulley

₍₁₉₉₂₎

would be

_highly

desirable.

(iv)

The

implications

of more realistic

_{genetic models,}

ie the influence of different distributions of gene effects and

frequencies

in a finite loci model.

ACKNOWLEDGMENTS

We thank LLG Janss for

_{kindly providing}

his program, and G Bolet and R Ponzoni for

useful comments. The senior author

_{gratefully acknowledges}

a _grant from INRA.

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APPENDIX

Genetic evaluation for 1 continuous and 1

_binary

trait when there are several observations of the

_binary

variable per record of the continuous trait

Suppose

that the number of ova shed is nl

+no,

with nl

_embryos

that survive

(litter

size =

n

l

)

and _no

embryos

that do _not. Ovulation rate is considered as a continuous trait and

_prenatal

survival as a dichotomous trait. There are nl + no observations of the

_binary

trait for each ovulation record.

_Breeding

values for ovulation rate and

embryo

survival can be estimated

_{by solving}

until convergence a

_system

of

equations

identical to

_equation

_[18]

in Janss and

_Foulley

₍₁₉₉₃₎

where the

_weighting

vector W is

_{replaced by}

if both ovulation rate and litter size are observed or

if

_only

litter size is recorded. Above

and

where c’ is the conditional residual variance of

_embryo

survival

_(Janss

and

_Foulley,

1993,

p

185)

and p is the conditional

expectation

of

_embryo

survival

_given

the