Prediction of the response to a selection for canalisation of a continuous trait in animal breeding

HAL Id: hal-00894220

https://hal.archives-ouvertes.fr/hal-00894220

Submitted on 1 Jan 1998

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Prediction of the response to a selection for canalisation

of a continuous trait in animal breeding

Magali San Cristobal, Jean-Michel Elsen, Loys Bodin, Claude Chevalet

To cite this version:

Magali San Cristobal, Jean-Michel Elsen, Loys Bodin, Claude Chevalet. Prediction of the response

to a selection for canalisation of a continuous trait in animal breeding. Genetics Selection Evolution,

BioMed Central, 1998, 30 (5), pp.423-451. �hal-00894220�

Original

article

Prediction of the

_{response to}

a

selection

for canalisation of

a

continuous trait

in

animal

_breeding

Magali

SanCristobal-Gaudy

Jean-Michel Elsen

b

Loys

Bodin

Claude Chevalet

a

a

Laboratoire de

_{génétique cellulaire,}

Institut national

de la recherche

_{agronomique, BP27,}

31326 Castanet-Tolosan

cedex,

France

b

Station

d’amélioration

_génétique

des animaux, Institut national

de la recherche

_{agronomique, BP27,}

31326 Castanet-Tolosan

_cedex,

France

(Received

7 November

1997; accepted

31

_August

₁₉₉₈₎

Abstract -

Canalising

selection is handled

_by

a heteroscedastic model

_involving

a

genotypic

value for the mean and a

_genotypic

value for the

_{log variance,}

associated with a

_{single phenotypic}

value. A selection

_objective

is

_proposed

as the

expected

squared

deviation of the

phenotype

from the

_optimum,

of a _progeny of _any candidate

for selection. Indices and

_{approximate expressions}

of

_{parent-offspring regression}

are derived. Simulations are

_performed

to check the accuracy of the

analytical

approximation.

Examples

of fat to protein ratio in _goat milk

_yield

and muscle

_pH

data in

_{pig breeding}

are

_provided

in order to

_investigate

the

ability

of these

_populations

to be canalised towards an economic _optimum.

_©

Inra/Elsevier,

Paris

canalising selection

_/

_{heteroscedasticity}

_/

selection index

*

Correspondence

and

_reprints

E-mail: msc@toulouse.inra.fr

Résumé - Prédiction de la _réponse à une sélection canalisante d’un caractère continu en _génétique animale. Le

problème

de la sélection canalisante est traité

_grâce

à un modèle

_{hétéroscédastique}

mettant en

_jeu

une valeur

_génétique

_pour la _moyenne et une valeur

_génétique

_pour le

_logarithme

de la

_variance,

toutes deux associées à une

seule valeur

_{phénotypique.}

Pour un

_objectif

de sélection visant à minimiser

_{l’espérance}

des carrés des différences entre le

_phénotype

et

_l’optimum,

_pour un descendant

d’un candidat à la

sélection,

des index sont estimés et des

_{expressions approchées}

de la

régression parent-descendant

sont calculées. La

_précision

de ces

_expressions

ces

_populations

à être canalisées vers un

_{optimum économique,}

des

exemples

sont

donnés : le rapport entre matière grasse et matière

protéique

du lait de

chèvre,

et le

pH

d’un muscle chez le porc.

©

Inra/Elsevier,

Paris

sélection canalisante

_/

hétéroscédasticité

_/

index de sélection

1. INTRODUCTION

Production

_homogeneity

is an

_important

factor of economic

_efficiency

in

animal

breeding.

For

_instance,

optimal weights

and ages at

_slaughtering

exist for

broilers,

lambs and

_pigs,

and the breeder’s

_{profit depends}

on his

_ability

to send

_large

_homogeneous

_groups to the

_{abattoir; optimal}

characteristics of

meat such as its

_pH

24 h after

_slaughtering

exist but

_depend

on the

_type

of

_{transformation;}

ewes

_lambing

twins have the maximum

_{profitability}

while

single

litters are not

_sufficiently

_productive

and

_triplets

or

_larger

litters are

too difficult to

_raise;

with extensive conditions where food is determined

_by

climatic

_{situations, genotypes}

able to maintain the level of

_production

would be of interest.

Hohenboken

_[22]

listed different

_types

of

_matings

_(inbreeding,

_outbreeding,

top

crossing

and assortative

_matings)

and selection

_{(normalising,}

directional and

_canalising)

which can lead to a reduction in trait

_variability.

Stabilisation of

_phenotypes

towards a dominant

_expression

has been known

for a

_long

time as a

_major

determinant of

_species

_evolution,

_similarly

to

muta-tions and

_genetic

drift

_(e.g.

_[4]

for a

review).

Different

_{hypotheses explaining}

these natural

_stabilising

selection forces have been

_proposed

_(2,

_{3, 8, 15,}

_{16, 19,}

27, 38,

45-47,

49,

52!.

A number of models assume that trait stabilisation is

controlled

_by

fitness genes

(e.g.

[9]

for a

review),

which

_keeps

the mean

phe-notype

at a fixed

_{’optimal’ level,}

without a _necessary reduction of the trait

variability.

Alternative

_hypotheses

were

proposed

for

canalisation;

for instance

Rendel et al.

_{[32, 33]}

assumed that the

_development

of a

_given

_organ is under

the control of a set of _genes, while a

_major

_gene controls the effects of these

genes within bounds to

_keep

the

_{phenotype roughly}

constant.

Whatever its

_origin

stabilisation is to be related to the

_{environment(s)}

in

which it is

_observed,

which makes it essential

_[48]

to

_distinguish

stabilisation of a trait in a

_precise

environment

(normalising selection)

from the

_aptitude

to maintain a constant

_phenotype

in

_fluctuating

environments

_(canalising

selection).

Various artificial

stabilising

selection

_experiments

have been carried out with

laboratory

animals:

_drosophila

_[17,

_{23, 29, 30, 34, 40, 41, 44,}

_48],

tribolium

[5,

6, 24,

43]

and mice

_[32].

Most

often,

selection was of a

normalising

_type

with a

_culling

of extreme

_individuals,

this selection

_{being applied}

_globally

_[5,

29, 30,

41,

43,

44],

within

_family

_[24]

or between

_family

[6, 34].

_Canalising

selection was

_{experimentally}

_applied

_by

_Waddington

_[48]

and

by

Sheiner and

Lyman

!40!,

their rule

_being

the selection of individuals less sensitive to

_breeding

temperature

and

_by

Gibson and

Bradley

[17]

who

applied

a

culling

of extremes

in a

_population

bred in unstable environment

_(fluctuating

_{temperature).}

Some

_general

conclusions from these

_experiments

_may be

_proposed:

₁₎

_very

phenotypic

variance;

2)

heritability

estimations

_during

and at the end of the selection

_experiments

often showed that the selected trait

_genetic

variance

decreased,

this conclusion not

_being

_general;

₃₎

in many cases it was

_possible

to _prove that the environmental

_variance,

or the

_sensitivity

of individuals to

environmental

_fluctuation,

was reduced

_by

selection.

In this _paper we

_investigate

mathematical tools for the evaluation of the

pos-sibility

and

_efficiency

of

_organising

_canalising

selection in animal

_populations.

Existence of a

_genetic

_component

in variance

_{heterogeneity}

between groups is

a

_prerequisite

for such a selection

_goal

to be feasible. Statistical

_modelling

and

estimation

_procedures

have been

_developed

to take account of variance

hetero-geneity

(e.g. [10,

11, 35,

36!),

in

_{particular using}

a

logarithmic

link between

variances and

_predictive

parameters

[12,

13,

39!.

In the

_following,

we extend such models

_by

_introducing

a

_genetic

value

among these

parameters,

consider the

_possibility

of

_estimating

this new

_genetic

value,

then discuss the

_efficiency

of selection based on this model.

Although

our

objective

is to

_apply

such

_{methodologies}

to continuous and discrete

_traits,

we

first concentrate here on continuous traits.

_Applications

to artificial

_canalising

selection towards an economic

_optimum

in

_goat

and

_pig

_breeding

are

_given.

2. GENETIC MODEL

2.1.

_Building

of a model

Our

_approach

was motivated

_by

the extensive literature mentioned in the

Introduction,

and in

_particular

the _paper of Rendel and Sheldon

_[34]

shows that artificial

_canalising

selection does

_work,

in the sense that the

_population

mean reaches the

_{optimum and,}

more

_importantly,

the environmental variance is reduced. Some individuals are less

_susceptible

to environment than

_others,

this

_{particularity}

_{being genetically}

_controlled,

since it

_responds

to selection. Some genes are now known to control

_variability,

_e.g. the

_{Apolipoprotein}

E

locus

_[31]

in

_humans,

the Ubx locus in

_Drosophila

_[18],

the dwarfism locus in chickens

_{(Tixier-Boichard,}

_pers.

_comm.),

and some

_(aTLs

with effects on

variance are

_already

_suspected

!1!.

Like

_Wagner

et al.

_[50]

in their

_equation

_7,

the effect of

_polymorphism

at a

given

locus on the environmental _{variance may} be

_expressed

_by

a

genotype-dependent multiplicative

factor for this variance. The same

_hypotheses

(in

particular

no interactions between

genes)

and

_reasoning

as in the Fisher model allow the

_previous

one-locus model to be extended to a

polygenic

or

infinitesimal

_model,

in which each individual has a

_genetic

value

_governing

a

multiplicative

factor for the environmental variance.

Since the

_analysis

needs the evaluation of

_phenotypic

variances associated with

_{genetic values,}

it must be based on

_experimental

_designs

_allowing

for the

repeated

expression

of the same or of

_closely

related

_genetic

values.

_Although

not

_necessarily

_efficient,

_any

_population

scheme

_might

be

_considered,

but

we focus here on two

_{simple situations, repeated}

measurements on a

single

individual,

and evaluation of one individual from the

performances

of its

2.2. Animal model: basic model

A model

_linking

a

_phenotype

_yjof a

_given

animal

(from

repeated phenotypes

y =

(!1, ...,

yj , ... ,

yn ) )

with two

genetic

values u and v is considered.

According

to the infinitesimal model of

_{quantitative genetics,}

these

_genetic

values u and

v,

possibly correlated,

are assumed to be continuous

_normally

distributed

variables,

and contribute to the mean and to the

_logarithm

of the environmental

variance. The

_simplest

version of the model can be written as:

where p

is the

_population

mean

_and

_the

_{population log}

variance mean, while:

and the _Ejs are

_{independent identically}

distributed

N(0, 1)

Gaussian

variables,

independent

of u and v. Additive

_genetic

variances are denoted

by

afl and a V ’ 2

and r is the correlation coefficient between u and v. The distribution of the

conditional random variable

_Ylu,

v is Gaussian

_./1!(!

+

_{u, exp(!}

+

_v)),

but the unconditional distribution of Y is not. The unconditional mean and variance

(the

phenotypic

variance

_{or y 2}

_of

the random variable Y are

_equal

to

Note that the

_{v genetic}

value and its variance

_{o, are}

_{dimensionless;}

_exp(

₇₇

₎

has the same units as the

_{phenotypic variance,}

and

exp(w/2)

is the _average

(genetic)

scale factor of the environmental variance.

2.3. Animal model: extensions

More

_general

formulations of the model are needed to _cope with real

situations.

_{First, introducing}

_permanent

environmental effects

_(denoted

by

p

and

_t)

common to several

_performances

of the same individual is _{necessary to}

take account of

_{non-genetically}

controlled

correlations,

both on the mean value

- as it is usual to deal with

_{repeatability -}

and on the

_log

variance of the within

performance

environmental effect.

Thus,

the

_jth

_performance

of an individual

is modelled as:

where

_{(u, v),}

_{(p, t)}

and follow

independent

Gaussian distributions: the

bivari-ate normal

_(2),

a similar bivariate distribution with

components

_{o, 2,}

at and

When _q individuals are measured in several

_{environments,}

a more

_general

heteroscedastic model can be stated as:

,-/

where _yg is the

_{jth performance}

of a

_particular

animal in a

particular (animal

x

_environment)

combination i. This full model

₍₆₎

is a

_{generalisation}

of model

₍₁₎

_introducing

environmental and

_genetic

_parameters

to be estimated: location

_parameters

_{({3, u, p)}

and

_dispersion

_parameters

_(6,

_v,

_t)

with incidence

matrices

_i

_,

_(x

zi,

z

i

)

and

, z

),

i

, z

i

i

(q

respectively.

Vectors _{u, p,} v and t have the

same

_length

_q.

_!3

and 6 denote fixed

_effects,

while u, v and _p, t are random

genetic

and random

_permanent

environmental effects attached to

_individuals,

respectively.

The vectors of

_genetic

values u and v have then a

_joint

normal distribution:

where ©

denotes the Kronecker

_product

and A is the

_relationship

matrix

between the animals

_present

in the

analysis.

Permanent environmental effects

p and t are

similarly

distributed as:

where I is the

_identity

_matrix,

_{independently}

of

_{(u, v).}

This

_general

_way of

_setting

_up the model

_needs,

_however,

some caution when

applied

to actual

_data,

to assess which

_parameters

are

_estimable,

_taking

account

of the structure of the

_experimental

_design.

_{Specifically, analysing}

a

possible

genetic

determinism of

_{heteroscedasticity}

needs a sufficient number of

_repeated

measures to be available for the same

(or

related)

_genotypes.

2.4. Sire model

In a _progeny test

_scheme,

the

_phenotypic

values attached to an individual are the

performances

of its

_offspring.

From the

_previous

animal

_model,

the

performance

y2! of the

jth offspring

of sire i can be written as

_follows,

conditional on the

genetic

values ui and vi of the sire and

_assuming

unrelated dams:

It is assumed here that the terms _aZ! and

_{

₃

_ij

include the

_genetic

effects in

offspring

not accounted for

_by

the

_part

transmitted

_by

the sire. Permanent environmental effects in the

_offspring

_(the

p and t variables of model 5 are

This can be rewritten as

with

_E’(

_Etj

₎

=

_0,

Var(e!)

= 1. The distribution of

e! is

only

approximately

normal

_N

_(0,1).

Models

₍₉₎

and

₍₁₀₎

are not

_{strictly equivalent, but,}

since the first two moments of _yj are

_equal

under both

_{models, they}

are

_equivalent

in

the sense of Henderson

[21]

(see

_e.g.

[37]

for an

application

of this

concept).

For

example,

for

_large

numbers of

_offspring

per

sire,

the mean sire’s

_performances

and

sample

within sire variances have

_{asymptotically}

the same structure of

variances and covariances between relatives under both models. The

_{corresponding generalised}

_approximate

sire model is written as

with the

_joint

densities

₍₇₎

for u and v, and

(8)

for p and t.

Methods needed to estimate

_parameters

are outlined in

Appendix

A. In

particular,

they

allow the

_genetic

values of individuals to be

_estimated,

as

the conditional

_expectations

of

_{genetic values, given}

observed

phenotypes

y:

h =

E(u!y)

and v =

E(vly),

if _variance

components

are known. Estimation

of variance

_components

was

_{similarly developed}

to make the method

_possible

to

_apply.

In the

_following

we first focus on

_developments

of the basic

_model,

which is

simple

enough

to derive

approximate analytical predictions

of the _response to

selection and to _compare several selection criteria. In a second

_step

we check the

validity

of the theoretical

_approach

_by

means of simulations and test the

_ability

of the extended models and

corresponding

numerical

procedures

to tackle actual data and evaluate the

_potential

for

_canalising

selection.

3. SELECTION OBJECTIVE AND CRITERION

3.1.

_Objective

and criterion

One

_objective

that summarises the

_{breeding goal}

_(progeny

_performances

close to the

_optimum

and with low

_variability

around

_it)

is the minimisation of the

_{expected squared}

deviation of

_{offspring performances}

from the

_optimum

y o

. This is the one we have chosen. For an individual characterised

by

a

set y of

performances

(on

itself and on its

relatives),

a selection criterion

is defined as the

_expectation

of the

_squared

deviation E

_{!(Yd - yo)2lyJ}

of

offspring performance Y

d

,

conditional on _y, and selection will

_{proceed by}

keeping

individuals with minimal values of this

_index,

such that:

In classical linear

_theory,

it is

_equivalent

to

_giving

an individual a merit with

respect

to the selection

_objective,

defined as the

_expectation

of its

_offspring

performance,

or to consider its

_genetic

value _u, since the former is

_{just equal}

to half the latter.

_Breeding

animals are ranked

_according

to their estimated

genetic

value.

In the

_{present context,}

due to the

_{non-linearity}

of the

_model,

we

define,

for a

candidate to selection with

_{given genetic}

values u and _v, its merit for

_canalising

selection as the

_{expected squared}

deviation of an

_{offspring performance:}

Its conditional

_expectation

_E(M

_*

_ly)

is

_equal

to the index

With

_{complications}

due to the non-linear

_setting

of our

_model,

we derive in

the

_following

the mean and variance of an individual’s

_{phenotype distribution,}

conditional on the

_performances

of a relative.

3.2. Conditional mean and variance

We need the distribution of a

_phenotype

_Y

_d

of a _progeny

_{d, given}

perfor-mances _y of a relative F. Let _{ud, vd} be the

_genetic

values of

_d,

y =

fy

j

1,

j

=

1, ...n,

_u and _v the

phenotypic

and

_genetic

values of animal F.

Perfor-mances of animals F and d follow model

_(1),

with:

where a is the

_relationship

coefficient between animals F and d

_(a

= 0.5 if d is

the _progeny of

_F).

The

_density

_{f (yd!y)}

_describing

the distribution of

_Y

_d

_,

conditional on _y

can-not be

_explicitly

_derived,

but its moments are calculable or can be

approxi-mated. We have:

This is first

_integrated

over _yd,

_owing

to

and

_finally

the distribution of u and v conditional on _{y is}

approximated

as:

where u =

E(u!Y)!

v =

E(v!Y),

C

uu

=

Var(!!Y)!

C

vv

=

Var(vly),

C

w

=

Cov (u, v ly),

are the estimated first and second moments of the

_genetic

values

_(see

_Appendix

A for the estimation

_method).

It follows that

and that

These

_expressions

are

_given

numerical values after estimates of

_genetic

values

and of variance

_components

are available.

General formulae can be derived that take into account all

_performances

of the whole

_pedigree,

not

_{only performances}

of a

_single

relative. The

_explicit

forms of the extensions of

_{equations (18)}

and

₍₁₉₎

are

_given

in

_Appendix

B. The combination of

_equations

₍₁₈₎

and

₍₁₉₎

_gives

the index

_I

_*

_(y)

in

equation

(14),

equal

to the conditional

_{expectation E(M*!y)}

of the

_genetic

merit

_M

_*

_,

as in Goffinet and Elsen

_!20!.

3.3.

_Approximate

criteria

When the conditional variance terms

(C)

can be

_neglected,

for instance when

n is

_large,

I* is

_{approximately}

_equal

to the maximum likelihood estimate of the merit M*:

where hats

denote,

in this case, modes of the

density

of

v,, v!y.

This is to be related to the work of Wilton et al.

_!51!,

who

_developed

a

_quadratic

index for

a

_{quadratic merit, by &dquo;minimising}

the

_expectation

of the

_squared

difference

between total merit and

_index,

both

_expressed

as deviations from their

expec-tations&dquo; . In their

_setting,

_normality

was assumed for the distributions of

_genetic

values and of

_{performances,}

so that this criterion was

_equal

to the maximum

likelihood estimate of the merit.

The

_previous

calculations make it

_{numerically possible}

to set _up a selection

scheme,

but do not allow

analytical predictions

of the

_efficiency

of selection

according

to the values of variance

_components

_{o’!, or2and}

r. Some

_insight

can

In the individual model

_(1),

_assuming

that

_repeated

measures are available

for the candidates for

_selection,

we consider the

_following

selection index I

which is

_equal

to the

_sample

mean _square

_deviation,

_y

_denoting

the

_sample

mean and

_S’y

the

sample

variance of the

_performance

set of an

individual,

1 n

6! ! -

!(!j -

y

)

2

.

Note, however,

that this index measures the value of a

n

j

=1

i

candidate,

not

_directly

the

_expected

value of its future

_offspring.

Truncation selection would be

_accordingly

characterised

_by

a

_step

fitness function _wt

defined as:

Instead,

we consider a continuous fitness function

where s is a selection coefficient which can be

_adjusted

to obtain the same

selection differential as

_equation

(22).

The

_positivity

of

_w(y)

in

_equation

₍₂₃₎

necessitates a small s value. Hence we assume that selection is

_{weak, allowing}

first-order

_{approximation}

of the _response to selection.

For _progeny test selection the model for _y is

_equation

_(10),

but without p

and

t,

and

_yields

a similar selection

_index,

_y values

_being

made _up of the

performances

of the

_offspring

of the candidate for selection. The selection

criterion

₍₂₁₎

is then a true measure of the candidate’s

value,

and can be

considered as an

_{approximation}

of the criterion

₍₁₂₎

for this

_{simple population}

structure.

4. RESPONSE TO CANALISING SELECTION

We seek the _responses to selection for the

_genotypic

values u and v, the

genetic merit,

and the

_performance

_{(Y -}

_YO

₎

₂

_.

We

_quantify

the effects of selection

_by

the

_regression

of

_offspring

on the selected

_parent

(e.g. !9)),

in a

general

way as:

where X _{is any trait} of

_{interest, E!(X)}

its

_expectation

in the selected

_part

in

the candidate

_population,

and

_{Ed (X )}

the

_expectation

of

_phenotypes

_among the

_offspring

of the w-selected

_parents.

The numerator is the _response

_R(w,

_X)

to selection based on the fitness function w in the trait X of

_interest,

measured

in the next

_generation.

The denominator is the selection differential

_{S’(w, X),}

measured _among

_parents.

As a

_rule,

we restrict the

_following

derivations to

4.1.

_Analytical

_{approximations}

4.1.1. Animal model

We first derive the distribution of u and v in the

_parent

population

after

selection

_according

to the fitness function w, then calculate the

corresponding

distribution in the

offspring

population.

Let

_{f (y)}

be the unconditional distribution of

Y,

and

_{f (u, v)}

the

_{joint density}

of u and v. The

density

of Y in the selected

parental

population

is

Following

Gavrilets and

Hastings

!14!,

we introduce the mean fitness of the

genotype

(u, v):

As with

_M

_*

_(u,v)

in

_equation

_(13),

this function

_M(u,v)

=

E(I(Y)!u,v)

can be considered as a

_genetic

merit

referring

to a candidate’s own value and

not as in

_equation

(13)

to that of a future

_offspring.

The mean fitness of the

population

is the

_proportion

of selected individuals:

where

We obtain the distribution of

genetic

values _among selected

_parents:

4.1.1.1.

Genetic _response

Since

_genetic

values are transmitted

linearly

to the

offspring,

the

genetic

responses to

selection,

R(w,u)

and

R(w,v),

are the differences of

expected

genotypic

values u and v,

respectively,

between candidates and selected

indi-viduals

_(assuming

that selection occurs in a

single

_sex,

only

half of this _progress

where wg refers to

_equation

_(26).

The effects of

_{non-linearity}

are seen in the above

_equations.

Note that if

_genotypes

are correlated

(if

r is not

_zero),

the

_efficiency

of selection is reduced if r and

₍

_M

_-

_y

_o

₎

are of

_{opposite signs.}

4.1.1.!.

Parent-offspring regression

The

_efficiency

of individual

_canalising

selection towards _{yo is} evaluated

_by

the

regression

coefficient

₍₂₄₎

calculated for the trait X =

II(Y)

_

(Y -

y

o

)

2

.

The

fact that the

_expectation

of the trait II of interest is

equal

to the

_expectation

of the index I involved in the fitness function w defined in

_equation

₍₂₃₎

makes the

_following

derivations feasible.

_Summarising

the detailed calculations

_given

in

_Appendix

_C,

we state that the numerator of

_equation

₍₂₄₎

is

_equal

to the w-selection _{response in} the

_genetic

merit M:

since M =

E(II!u, v).

The denominator of

_equation

(24)

is the selection differential:

This leads

to,

if

_{r = 0,}

where V stands for

_exp(?

_l

+

_a!j2).

If

_genotypes

are

correlated,

an extra term

2rau

av

V

(p, -

yo +

4 ra

u

av)

is added to the

numerator,

and

_4(l

+ n)r

OUUV

V 2

1

-t -

yo

+ ! 2 rauav )

is added to the denominator. The response to selection can be written as:

i.e. as the

_product

of selection

intensity

(1 = ! ) , of

a realized

heritability,

the B tf7

ratio

_{b(w, II)}

defined in

_equation

_(34),

and of the standard deviation Qn of the

selection index. 4.1.2. Sire model

As for the individual

_model,

the

_genetic

merit for the sire model is defined as:

and the fitness

The

_{expectation E(M)}

=

E[I(Y)]

is the same as

_given

in

_equation

(27).

The response to selection in the trait

_II(Y)

among male

parents

is

and the selection differential is

The

_regression

coefficient b

_giving

the response to

_canalising

selection in a

progeny test scheme is

equal

to the ratio of

(36)

to

(37).

Figure

1

plots

the response

given

in

equation (36)

in units of selection

intensity

and

phenotypic

variance,

from an

equation

similar to

equation (35).

4.1.3. Extensions

The

_previous

exact

_results,

obtained

_using

the fitness function

₍₂₃₎

and

analogous

for the sire

_model,

hold for weak

_selection,

and their

_expressions

as ratios of a covariance to a variance indicate that

they

can also be obtained

from a linear

approximation.

This comment makes it

possible

to extend

easily

the

_{approximate prediction}

of response in cases when different

weights

are

_given

to the variance of

_performances

and to their deviation from the

optimum.

Considering

the animal model with

_repeated

measurements

_(5),

let us denote

II

1

(

Y

) =

(y -

YO

)

2

,

II

2

(

Y

) =

_Sy,

the two

_components

of II =

(II

l

(y),II

2

(

Y

))&dquo;

s =

(

81

,

S2

)’

a vector of selective

_values,

a =

(cr

l

, cr

2

)’

a vector of

_weights.

We

are interested in the _response for the trait

a’II,

when

using

the index s’ll as

selection criterion. The

_{parent-offspring}

regression

is

_equal

to

introducing

the

_following

notations

_h2

=

or2 2 ,

c

2

-

(or2

₊

2 2

A =

_{(y -}

_y

_o

_)lay.

From

_equation

_(38),

_{parent-offspring}

_regressions

for the mean

and for the variance can be written

separately.

With si = 0 and _a1 = 0 for

instance,

b tends to

as n tends to

_infinity

and if

_at’

= 0. This

parent-offspring

regression

is lower

than a

half,

and tends to

1/2

as

afl

tends to zero.

Note that the

_{parent-offspring regression for y}

is

which tends to

_1/2

as n tends to

infinity

and if

Q

p

= 0.

1 n

If the unbiased estimate of variance

_H[

= ! 1

_{1 !(yj -}

_y)

₂

is used in the n - I

!’

j-i

index,

then the variance term

_P!2

=

Var(II2

2

)

is

proportional

to

When

_{o, =}

_0,

the _{response in}

_IIZ

_is

null and the selection differential is

corresponds

to the variance of the trait

_{(Y -}

_y)

₂

_(squared

deviation from the

mean),

up to a

multiplicative

term. When

_afl

₌

_0,

the _{response in}

Y

is null and the selection differential is

_equal

to

_V/n,

More

_generally,

this extension shows that a selection index

_(weights

s =

(s

l

, s

2

)’)

can be

_adjusted

to

_optimise

the _{response in} a

_{given objective}

specified

by weights

a =

(a

l

, a

2

)’.

4.2. Simulations

Simulations were used to check the _accuracy of

_{previous analytical}

expres-sions of _response as

_proposed

in

_equations

₍₃₄₎

and

_(36)/(37),

in more

_general

situations:

- intermediate selection

intensity,

since the

_analysis

assumes

_only

weak

selection;

- behaviour of the

population

parameters

(mean, variance)

during

several

generations

of

_selection;

-

comparison

of the relative efficiencies of different selection

_criteria,

re-placing

in the simulation the theoretical continuous selection scheme

₍₂₃₎

_by

truncation

_selection

_according

to the

_simplified

index

₍₂₂₎

and

_by

the likelihood based index

1 (20).

Simulations were restricted to the case of the sire model with no

_genetic

correlation

_(r

=

0).

4.2.1. Selection scheme

The selection scheme was as follows.

1)

Genetic values of sires and dams of the base

_population

were

ran-domly

drawn from the

_joint

distribution

₍₇₎

with no

_{relationships}

(A

is

the

_identity

_{matrix), giving}

the sets

_{{(ui, vi),}

i =

_{1, ... , ,S} for}

the

_sires,

and

(u

j

,

v

j

), j

=

_{1, ... ,}

D}

for

the dams.

2)

Sires and dams were mated at random.

3)

For each

_couple

_(i,j),

the

_performance

_y2! of a

_daughter

was

_generated

according

to:

where _{Etj, aZ!} and

_{3

_ij

were drawn from the Gaussian distributions

N(0,1),

jV(0,o’!/2)

and

_N(0,

_a!/2),

respectively.

The terms _a2! and

_{3

_ij

represent

Mendelian

sampling.

4)

An index for each sire was

computed

and elite sires were selected.

5)

The elite sires

_produced

S sons with the same female cohort used in

_steps

1-2.

Step

2

_(with

sons of

step

5 and

_daughters

of

step

3)

to

_step

5 were

_repeated

until the 10th

_generation.

The sire selection of

_step

4 was a truncation selection based either on the

of the merit

_I(

_Yi

₎

=

M(Û

i

, V

i

)

=

3!!

+

_exp(!7

_{+ v}

_Z

+

3w

)

+

_{(p, -}

yo +

ii 2

)2,

_ _ 428 8 2

with

_u

_i

and

_v

_i

maximum likelihood estimates of ui and vi

respectively,

according

to model

_(10),

but

_allowing

for no

_permanent

environmental

_effect,

and

_assuming

that variance

_components

were known.

4.2.2. Simulation

_experiments

For a constant

_phenotypic

standard deviation for the base

population,

several values of variance

_components

were tested:

or =

0.033 and

_{0.114, corresponding}

to a ’low’

(hu =

_0.10)

and a

_{’high’ heritability}

(h!

₌

_0.3);

a ’low’

_variability

variance

_Q

_v

₌ 0.03 and a

_’high’

or2

₌ 0.15

(corresponding

to ratios of maximum

to minimum variance

_equal

to 3 and

_10,

_{respectively).}

Three base

_phenotypic

means were considered: pt=o

= 1,

1.8 and

2,

for an

optimum equal

to _yo = 2

(giving

discrepancies

A

t=

o

=

(

ut=

o -

yo)/

QY

,

c=

o

between

population

mean and

optimum,

expressed

in

_phenotypic

standard

_{deviations, equal}

to

_1.75,

0.35 and

_0).

For

_given

values of the set

_{a!, a!, /1}

and _y, of the numbers of sires and dams and of selection

intensity,

100 selection

experiments

were

_performed,

and statistics

_averaged

over the runs. The evolution of

phenotypic

mean

and

_variance,

estimated merit over the ten

_generations

and

_{parent-offspring}

regression

are

_highlighted.

4.2.3. Results

Figure 2 displays

the curves

given by

the

analytical approximation

of the

response, with

point

estimates and confidence intervals obtained with 100 simulated selection

_experiments,

_{showing good}

_agreement

of the

_{approximation}

with truncation selection

on

the

simplified

index I

(not shown),

but also with

the likelihood based index

I,

_except

for intermediate values of A for which the

theory provides

underestimates.

Figure

3

_plots

the evolution of

_phenotypic

means and standard deviations over

_generations

of

_canalising

selection. Several

aspects

appear:

-

with a

_{high heritability}

h!,

the

_population

mean tends in a linear manner

towards the

_optimum

in a _very efficient _way;

- the

convergence of the mean is

_slightly

better if

w

is

_low;

- the decrease in

phenotypic

variance has a linear

tendency, although

more

fluctuating

than the evolution of the mean;

-

this decrease is even more evident as

Q

v

_{is higher}

and

h2

_is

lower.

This

_general

balance was encountered

throughout

the simulation

experi-ments: a

_particular

_aspect

was

_{maximally improved}

when the other

_aspects

were not under selection _pressure. Variances are best reduced when the

popu-lation mean is at the

_optimum.

The

_optimum

is more

rapidly

reached when no

genetic variability

of the variances is

_present.

Figure

4 compares

the

_performances

of the two indices I and

T.

The likeli-hood based index

_gives

more efficient results for the trait mean _pt,

_probably

because

_{heterogeneous}

variances were taken into account in the evaluation of the animal

_genetic

values u,

giving

less biased estimates. On the

contrary,

the

phenotypic

variance

_QY

_is

best reduced with the

_simplified

_index,

presum-ably

due to the lack

of

robustness of v estimation

_by

maximum likelihood. A full

_Bayesian

estimation

_procedure

with

_marginal

_{posterior expectation}

of

parameters

might

be more

_appropriate.

It was nevertheless not

_performed

be-cause of the heaviness of the

_algorithm,

since numerical

_integrations

are then needed. The two indices

_{give, however, equal}

values of the

_global

criterion

(p’t -

yo)

2

+

₀

_{,2 yl}

_t

at _any time t.

The

_phenotypic

variance and

_squared

difference between mean and

_optimum

are lowered more and more as selection

_intensity

is

_increased,

while the

parent-offspring

regression

remains constant in the simulations as in the

_approximate

theory

(not

shown).

5. DISCUSSION

5.1. Model for the variance

The introduction of a

log

linear model is an _{easy way} to handle a

mul-tiplicative

model on the variance. It is known that the distribution of

_InS

₂

_,

the

_logarithm

of the

_sample

variance

_estimator,

is

_{approximately}

normal

_(e.g.

!25!).

Similarly,

Bayesian

considerations on

_{prior/posterior}

densities show that the Gaussian distribution is a

_good

_{approximation}

to a

_log

inverted

_chi-square

(see

[13]).

This led us to focus all

_analytical

derivations on the first two

moments of

_{distributions, assimilating}

when needed _any distribution to the Gaussian distribution

_sharing

these same moments.

_Although

this _may be a

crude

approximation

if it is used for

prediction

of

_genetic

response over several

generations,

it allows first order solutions to be

_derived,

and makes it

_possible

to build statistical evaluation

procedures.

The model allows estimation of the

_importance

of

_genetic

determinism in

the

_{heterogeneity}

of

_variances,

and hence

_prediction

of how the

_population

may

respond

to selection

_against

_variability.

For

_example,

the

_proportion

of the selection _response due to the

_{genetic variability}

in the

_v-component

is

_{given by}

the ratio

where the Gs are

_given

in

_equation

_(40).

It is all the more

_important

as the

population

mean is closer to the

_optimum,

the

_u-genetic

variance is

lower,

and the

_v-genetic

variance is

_larger.

Estimation of

_genetic

parameters

(or u 2,

r,

av 2)

may be somewhat

imprecise,

especially

for

_u2

_and

r. Hence it _may be worth

_considering

the robustness

of

_predictions

with

_respect

to

_badly

known

parameters.

As far as a

_simple

global

criterion is

_used,

the

_question

can be dealt with

easily,

considering

the

expected

responses as functions of

_parameter

values. The situation would be more difficult to handle for selection schemes that would

_rely

on the

_knowledge

of

_parameter

_values,

for

_example

if a balance between selection for the mean or for the variance were

_adjusted

each

_generation.

5.2. Data

The

_generalised

version of the sire model

_(11),

_including

fixed and random

permanent

environmental

_effects,

was

_applied

to actual data in

_goats

_(dairy

production)

and in

_pigs

_(pH

of muscles after

_{slaughtering).}

5.2.1. Milk data

Protein and fat contents were measured on milk from 2 383 first lactation

goats

between 1992 and 1995. The

_goats

were

_daughters

of 54 artificial

insemination

_sires,

with 20 observations at least in the data set. The trait of interest is the ratio of fat to

_{protein contents,}

with a desired

_{optimum equal}

to

1.3. This

_objective

would be

_{complementary}

to

_yield

traits such as milk

yield

or

protein

yield.

The

_phenotypic

mean and variance are

_equal

to 1.1 and

_0.0135,

respectively,

i.e. the

_population

mean is 1.7

_phenotypic

standard deviations

away from the

optimum.

Data are

_normally

distributed. For

_{computational}

ease, data were

_{pre-corrected}

with the additive model

_including

herd,

season,

lactation

_length

and age, on a much

_larger

data set

_including

all lactations of all herds where the 2 383

kept

daughters

had been

_producing.

The variance

components

were

_{estimated, leading}

to a null

_correlation

coefficient

(r - 0)

and zero

variability

variance

(Qv -

0),

and a

heritability

h!

= 0.44 of the _same

order as those for the

_protein

and fat.

A

_canalising

selection

_experiment

is

_expected

to drive the

_population

mean

rapidly

towards the

_optimum,

but without

_change

in environmental variance. For

_example,

_assuming

selection of the best 10

%

of

_sires,

a reduction of 1.5

phenotypic

standard deviations of the

population quadratic

deviation

_(!c-yo)2

2 would be

_expected

in one

_generation.

5.2.2.

_pH

data

pH

values of semi-membranous muscle were measured on 947

_piglets

from 25

_Large

White sires. Data were

normally

distributed. Each sire had at least 20

_piglets.

Data were

_{pre-corrected by}

the usual linear model

_accounting

for

sex,

line,

year and

slaughtering

date effects on the trait _mean, on a much

larger

data

_set,

in order to

_simplify

further

_{computations.}

_Thereafter,

a sire model

for the residuals of the

_previous

model was fitted.

Estimated values of variance

components

under model

₍₁₁₎

with

’perma-nent

environmental effects’

(non-genetic-sire effects)

were

_equal

to

_a2

₌

_0.15,

hu

= 0.26

(with

a

2

y

=

0.037),

r =

_0.79,

QP

=

_0.00045,

_Qt

= 0.046 and p = 0.79.

With an

_optimum

value y_{o =} 5.7 not different from the overall _{mean p} =

5.75,

the estimated variance

components

should allow a

high

_response to

_canalising

selection to be obtained

_through

a

_strong

reduction of the

_genetically

controlled

part

of environmental variance:

_assuming

that selection sorts out the best 10

%

of male

parents,

a reduction of about 12

%

of the initial

phenotypic

variance

It must be stressed that

_predictions

derived from the above

_analysis

of fat

to

_protein

ratio in

_goats

and of

_{pig pH}

muscle data are

only

indicative. For

example,

the effect of a

_wrongly

estimated correlation

value

r remains to be

assessed,

even if - in the

_goat

_{example -}

no

_significant

_genetic

_component

of

variance was found for variances.

_{Also, although}

_precision

of

_{the previous}

_early

estimates was not

_evaluated,

_larger

data sets are

_probably

needed. A _proper

prediction

of

_expected

_{response to} selection cannot be

_proposed

until these

analyses

are carried out.

So far we do not have results from an actual selection

_experiment,

based on our index selection

rules,

which would be _necessary to

_completely

validate the

approach

through

the

_comparison

of observed realised heritabilities with our

predictions.

It is one of the

_perspectives

of the current work to

_organise

such selection

_experiments.

5.3. Selection criteria .

We have considered a

_{single global}

criterion that combines selection for the

mean and selection

_against

the variance of the trait.

Shnol and

_{Kondrashov [42]}

considered the action of selection with fitness

w(y)

on a

_quantitative

_{trait y.}

_They

concluded that truncation selection

min-imises the

_genetic

load and the variance of the trait after selection. Linear

selec-tion

_{(corresponding}

to our continuous fitness with low

_selection)

_gives

minimal variance of the relative fitness and is less efficient than truncation selection.

However,

linear selection _gave us the

_opportunity

for

robust

analytical

approx-imations of realised

_{heritability.}

Calculations were

_impossible

for truncation

selection,

even with

the

simpler

index. Within the limits of the

_present

com-parisons

with

simulations,

the fitness

_{approximation}

_{proved useful,}

even in cases with

_strong

departure

from

linearity,

and with a rather

_strong

selection

intensity

(proportion

of selected individuals

_equal

to 20

_%).

More

_{sophisticated}

selection _{criteria may} be

_defined,

_allowing

selection to be

differentially

directed towards

_changing

the mean value of the trait or

_reducing

the environmental variance. In fact

using

a

_{global genetic}

merit to be maximised in the next

_generation

is a _way to distribute selection

_intensity

between both

parameters.

It is

_possible

that a

_{higher multi-generation}

_response could be

ob-tained if selection were controlled each

_generation

in view of the

_objective.

For

example,

the index

_{(y -}

_YO

₎

₂

_{+ S;}

can be

_generalised

into

81(Y -

YO

)

2

+

_S2’S’!,

allowing

a

_greater

selection _pressure either on the location near the

_optimum,

or on the

_dispersion,

as illustrated in the above theoretical section. The same

remark is available for the index

_{(.E(y!y) —}

_y

_o

₎

_z

+

_Var(Yd!y).

More

_generally,

the selection criterion

_might

be based on the economic worth of

offspring.

The

criterion would then be defined as the

_expected

economic value of

_offspring,

a function

depending

on the distribution of

expected phenotypes

and on the

economic value of

_phenotypic

values. But of course other

_types

of indices and

mating

systems

are

_potentially

_interesting

to

_consider,

for instance a linear

index when

_Q

_v

_is

_small,

mate selection or _group selection.

Managing

the balance between location and scale could be

_interesting

in a

long-term

selection process,

provided

some

_{analytical approximation}

is

avail-able in order to include

_{one-generation expected}

_{response in} a

_dynamic