Optimization of selection response under restricted inbreeding

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Optimization of selection response under restricted

inbreeding

M Toro, M Pérez-Enciso

To cite this version:

Original

article

Optimization

of

selection

_response

under restricted

_inbreeding

M

Toro*

M Pérez-Enciso

Universidad

Complutense,

Facultad de _{Biologia, Departamento} de Genética, 280l0 Madrid, Spain

(Received

4 _{January 1989; accepted} 13 _September

₁₉₈₉₎

Summary - A reasonable _objective for selection _{programs in small populations} is the maximization of response, with a restriction on the increase

_{of inbreeding.}

This restriction will be

_especially

_important when information on relatives is used for evaluation of candidates for selection. To achieve this objective, different strategies have been proposed:

(i)

to reduce the _intensity of _selection;

_(ii)

to lower the _{weight given} to _family information in an index below the _{optimal value;}

_(iii)

to restrict the variation of

_family

size,

(iv)

to make

matings

between the selected _animals, so as to minimize _{the average coancestry}

coefficient;

and

_(v)

to find a

_general

solution _using linear _programing. These _strategies have been illustrated _by

_genetic

simulation of a _{simple example.} The population consisted of 8 males and 8 females selected from 32 animals evaluated in each sex. The candidates were evaluated _by an index

_using

information on the individual and its 7 sibs. Five _generations of selection were _practised. It was concluded that there are several alternative

_strategies

which ensure that

_inbreeding

is below the fixed level

_(5%

per

generation)

without a

significant

loss of response, in comparison with classical

_strategies,

where

_inbreeding

is

not restricted. A substantial reduction of

_inbreeding

was found with the use of _matings

having

minimal _{coancestry. However,} this reduction was due principally to a _delay of

1 _generation in the _{appearance of}

_inbreeding.

Linear

_programing

was also efficient in

achieving

these aims. It _is, in _principle, more flexible than the other

_{strategies, but}

its

heavy

cost of _computation is a

_{disadvantage,}

_and, in practice, comparable results can

probably be obtained _using much _{simpler strategies.}

effective size _/ artificial selection _/ linear _{programing / computer} simulation _/

inbreeding

Résumé - _Optimisation de la réponse à la sélection avec une restriction sur la

consanguinité - Une _proposition raisonnable dans les programmes de sélection en _petits

troupeaux est la maximisation de la _réponse avec une restriction sur _{l’augmentation} de

la

_{consanguinité.}

Cette restriction sera spécialement importante quand l’information sur

les parents est considérée pour l’évaluation des candidats. Pour arriver à cet

_objectif,

différentes

méthodes ont été _proposées:

_(i)

réduire l’intensité de la _sélection;

_(ü)

ramener le

_poids

de

_{l’information}

sur les _parents en dessous de la valeur optimale dans un

indice

_familial;

_(iii)

restreindre la distribution des tailles de

_famille;

_(iv)

réaliser des

accouplements

entre les animaux sélectionnés avec un

_coefficient

de _{parenté minimal;} et

(v)

appliquer

une solution _générale avec l’utilisation de la _{programmation} linéaire. Ces

*

méthodes ont été illustrées par des simulations génétiques sur un exemple. La population

était

_formée

de 8 mâles et 8

_femelles

sélectionnés parmi 32 animaux évalués dans _chaque

sexe. Les candidats ont été évalués selon un indice _comprenant l’information sur l’individu

et ses

_{7 frères.}

_{Cinq générations} de sélection ont été réalisées. On est arrivé à la conclusion

qu’il existe _plusieurs méthodes alternatives _qui assurent une consanguinité en dessous de la valeur _fixée

_(5%

par

génération)

sans _{perte significative} de la _réponse en _comparaison avec

les méthodes classiques, où la consanguinité n’est pas restreinte. On a trouvé une réduction

substantielle de la _{consanguinité} avec des _{accouplements} de _parenté minimale. _Cependant

cette réduction a été due _{principalement} au retard d’une _génération dans _{l’apparition} de la _{consanguinité.} La _{programmation} linéaire a été efficace

également

_pour arriver à ces

fins.

Elle est, en _{principe, plus flexible que} les autres _méthodes, mais son coût _important

de calcul est un _{inconvénient, et,} dans la _pratique, des résultats similaires peuvent être

probablement

obtenus avec des méthodes _beaucoup

_{plus simples.}

effectif _génétique

_/

réponse à la sélection _{/ programmation} linéaire _/ simulation

aléatoire

_/

consanguinité

INTRODUCTION

The total number of individuals under control in an animal

_breeding

program is

usually

constrained

_by

economic factors. The choice of effective

_population

size

depends mainly

on

fertility

and

fecundity

_parameters, as well as on

_predictions

of _{response to} selection. Of all the variables an animal breeder can

manipulate,

population

size is the one that has the widest range of _{consequences.} In the short

term, it influences the selection

_{differential,}

the

_inbreeding

_depression

and the

reduction of

_genetic

variance due to

_genetic

drift. In the

_long

term, it affects the selection limit and the utilization of a new variation

_arising

from mutation

_(see

_Hill,

1986,

for a

_review).

Furthermore,

in a

_population

under artificial

_selection,

the effective

_population

size will be lower than that

_expected

in a

_{random-mating}

control

_population

of

equal

size because _parents do not have an

_equal

chance of

_{contributing offspring}

to the

next

_generation,

even if all

_pairs

of _parents contribute an

_equal

number of progeny to be measured

_{(Robertson, 1961).}

Moreover,

the efficient use

_{of family}

information

by

selection indices or BLUP

_methodology

will lead to more individuals from the best families

_being

selected

_and,

_therefore,

considerable reductions of

_population

size will follow.

Some

_problems

related to the

_optimization

of _{response in} selection _programs in

populations

of finite size have been

_{explored by}

Robertson

_{(1960; 1970).}

_Using

the infinitesimal model for the

_decay

of

_{genetic variability,}

he showed that if individual selection is carried out from a constant number 2M of individuals scored per

generation,

the maximum advance at the limit is achieved when the best half

is selected. He also showed that the

_proportion

selected to

_give

the maximum cumulated

_gain

after t

_generations

is a function of

_t/2M.

_Experimental

checks on the

_theory

have been

_{reported by}

Ruano et al

₍₁₉₇₅₎

and Frankham

_(1977).

Dempfle

(1975)

investigated

the effect of

_{within-family}

selection on selection

limits, showing

that this method is more efficient than individual selection when

the

_heritability

is _very

_high,

because of a

_relatively

lower

_decay

of the additive variance

_during

selection. This

_prediction

was

_{experimentally}

checked

by Gallego

have

_proposed

a

_{simple method,}

called

weighted

_selection,

that also leads to

_higher

selection limits.

A different

_approach

focuses attention on

_{inbreeding depression,}

and several methods have been

_proposed

to minimize the rate of

_inbreeding

in selection

programs;

ie,

by reducing

selection

intensity, by ignoring

some

_family

_information,

or

by imposing

restrictions on

family

_size,

such as

_practising

within-sire selection.

Other

methods,

such as minimum _coancestry

_mating,

may also be advisable

(Toro

et

_al,

_1988a).

The _purpose of this work is to

_analyse

the above

_{methodologies}

and to discuss

some _aspects of

_optimization

of

_genetic

progress when the

acceptable

level of

inbreeding

is fixed a

_p

₇

_iori.

_Although

there are other

_{possible approaches,}

such as

maximizing

cumulated selection

_gain

in a

_{given period}

of

_time,

the

_approach

taken here is

perhaps

a more realistic one as, in

practice,

breeders choose an

empirical

level of

_inbreeding

such that the selected or

_reproductive

traits will not be

_impaired

by

an excess of

_{consanguinity}

_(Smith,

_1969;

_Land,

1985).

These

_{methodologies}

will be illustrated with a

_simple

_computer simulation

example.

Strategies

of

_optimization

in selection with restricted

_inbreeding

A selection _{program consists of 2 main steps:}

_{(1) ranking}

and choice of

_candidates;

(2)

mating

of selected animals. To attain the

_objective

outlined

above,

these 2

aspects can either be considered

separately

or

_jointly.

The first 3

_strategies

analysed

in this paper refer to _step

_1,

the 4th to _step 2

_exclusively,

while in the 5th _strategy,

a

_general

solution

_combining

both _steps is

_sought.

The

_breeding

structure considered here is a closed

_population

with k

_families,

each

_{family contributing}

n males and n females as candidates for

_selection,

and k individuals are selected out of the M = kn

eligible

from each sex. In all

strategies,

selection is based on a

_family

linear index of the form:

where

_P,

F and

P

are the individual’s own

performance,

its

family

mean and the

population

mean,

respectively,

and A is the

weight

given

to

_family

information.

Optimal

selected

_proportion

The choice of an

_adequate

_proportion

of selected individuals is the

_simplest

_way of

diminishing

the level of

_inbreeding

and it will not be discussed further.

_However,

it should be

_pointed

out _{that the range of choice is limited if} a balanced

_family

structure is to be maintained.

Optimal weight given

to

_family

information

The second alternative that can be considered is to

_ignore

some

_family

information or, more

_strictly,

to reduce the relative

_importance

_given

to the

_family

mean below

the value that maximizes the correlation between the index and

_breeding

value. As A

_decreases,

the

_intrafamily

_(intraclass)

correlation of index decreases

_{(see eqn(2)}

Restriction on the distribution of

family

size

The third _strategy that can be utilized to maintain a desired rate of

_inbreeding

is to

impose

some constraints on the number of selected individuals contributed

by

different

families,

such as

_practising

some kind of

within-family

selection with

respect to the index. Given a fixed number of

families,

with

_{within-family}

selection,

the variance of

_family

size is zero and the effective

_population

size is

_maximum,

while with

_family

_selection,

the rate of

_inbreeding

and the variance of

_family

size will be maximum.

_{Nevertheless,}

there is a wide range of intermediate selection methods

which differ in the

_magnitude

of the variance of

_family

size that can be

_{imposed and,}

apparently,

this

_possibility

has

_commonly

been overlooked. All

_possible

distributions of

_family

size are

_equivalent

to all the

_possible

forms of

_arranging

k marbles

_(the

selected

_individuals)

among k boxes

(families),

each of

capacity n

(maximum

family

size).

These _arrangements will follow a

multi-hypergeometric

distribution.

Minimum

_coancestry

_matings

Following

a different

_approach,

Toro et al

_(1988a)

have

_emphasized

the

_utility

of

2 methods to minimize

_inbreeding

in selection _programs. The first is minimum

coancestry

mating

(MC),

where

_matings

are chosen to _{minimize average}

_pairwise

coancestry coefficients between males and females in the selected _group. The second

is a method

_{proposed by}

Toro and Nieto

_(1984),

which is called

_{&dquo;weighted}

selection&dquo; and is

_fully

_explained

in the article. Both methods were evaluated

_(Toro

et

_al,

(1988a)

by

computer simulation and it was concluded that the first is the most

promising

in the short term

_{and, therefore,}

it will be the

_only

one considered here. Mate

selection;

a

general

solution for the maximization

of

_genetic

progress under restricted

inbreeding

It is desirable to have a

_general

solution that could

incorporate

the main features of the methods

_previously

described. Such a solution can be obtained

by

means of

linear

_{programing techniques.}

If k males and k females are to be selected out of M available from each sex, we must choose the best k

_pairs

_among the k!

k

1

r

M

J

[M]

J

l Jk

possible mating

combinations;

&dquo;best&dquo;

_meaning

that we seek to maximize

_genetic

progress while

maintaining

the rate of

_inbreeding

below a certain value.

The

_problem

can be solved

_{using integer}

linear

_programing

_algorithms

which is

reduced to find a X =

[x

ij]

(i, j

=

1, M)

matrix,

where Xij _{represents a} decision

variable

_indicating

whether the

i

th

male and the

_j

_th

female are

(

z

jj

=

1)

or are

not

_(x2!

=

0)

_to be selected and mated. Such a matrix is chosen _to maximize the

expected genetic

progress:

where

_ai

and

_a

_j

are the best available estimates of the

_breeding

values of the

i

th

where

_F,

OF and

_f

_ij

are,

respectively,

the

_population

mean

_inbreeding

coefficient in

generation

t, the maximum rate of increase

_permitted,

and the _coancestry coefficient

between the

it’

male and the

_jt!’

female. Restrictions

_(iii)

and

_(iv)

_simply

indicate that a male or a female will be mated

_only

once at

_{maximum, ie,}

there are no

half-sibs.

METHODS

Prediction of selection _response and

_inbreeding

The value of A that maximizes correlation between value and index _score,

_assuming

an infinite

_population,

is:

where r = 0.50 for full-sib

families,

and

p is the intraclass

phenotypic

correlation.

The selection

_intensity

for finite

_populations

under selection was obtained from

the tables in Hill

_(1976),

_using

the

_appropriate

_intrafamily

_(intraclass)

correlation of the

_index,

_pl,

_{given by}

Expected

responses were obtained from standard methods

(Falconer, 1981),

and

N

e

from Burrow’s

₍₁₉₈₄₎

_formula,

that is

_strictly

valid

_only

for 1

_generation,

and

can also be

_{computed by}

the numerical

_integration

methods described in

_Ducrocq

and Colleau

_(1986).

Expected

F values for

_generation

t were obtained

_using

Crow and Kimura’s

(1970)

formula

In the case of fixed

family

size

(3

d

strategy),

selection response can be

approxi-mately predicted by assuming

that selection has occurred in 2 distinct _steps.

_First,

families are ranked

_according

to their mean index

_values,

the best

family(ies)

are

se-lected

_and,

in a second _step, the best

_{individual(s)}

from each

_family

_is(are)

selected

according

to their

_previously

fixed contribution. Since the between- and

within-family

components are

uncorrelated,

the total response

(R)

can be

split

up into 2 _parts, those due to

_family

_(R

_f

₎

and

_{within-family}

_(R&dquo;,)

_{selection, respectively.}

Thus,

if we denote

_by

_ci, the number of selected individuals of each sex contributed

by

the

i

th

_family

_{(0 <}

_{ci <}

_n)

where or is the standard

_deviation,

as defined in Falconer

(1981),

and

_{subscripts f}

and w refer to

_family

and within

_{family, respectively.}

The intensities of

_selection,

if

and

i

w

,

are

where

S

i

is the

it!’

order statistic from k

_independent

normal variables and

_5&dquo;

is

the

_p

_h

order statistic from an n-dimensional normal distribution with correlation

equal

to

_{-1/(n -}

_1),

obtained from Owen

₍₁₉₆₂₎

and Owen and Steck

_(1962).

The effective

_population

size for a constant distribution of

_family

size was

obtained from Crow and Denniston’s

₍₁₉₈₈₎

_formula,

where

_u¡

_is

the variance of

_family

size.

Simulation methods

In the

_{genetic simulations,}

the trait was assumed to be controlled

_by

100 biallelic additive

_loci,

with

_equal

effects and initial

_{frequencies, spaced}

with recombination

rates of 0.50. The

_genotypic

_(additive)

values _per locus were

_4,

3 and 2 for the

AA,

Aa and aa allelic

_{combinations,}

_{respectively.}

The initial

_frequency

of the A allele

was

_0.50,

_implying

an additive

_genetic

variance

_{or’ A}

= 50.

Phenotypic

values were

values, corresponding

to heritabilities of 0.10

_(a

₅

=

450)

and 0.30

(a

5

=

116.66),

respectively.

Genetic values were

_independent

of environmental effects.

In the

_{example considered,}

the number of

_{families, k,}

was

_8,

with n = 4

individuals of each sex _per

_family.

Five

_generations

of selection were

performed

and the desirable maximum rate of

_{inbreeding imposed}

was 5% _per

_generation.

The

_performance

of the linear

_programing

_strategy

was carried out

_introducing

the MIF

_{integer programing}

subroutines

_(Land

and

Powell,

1973)

in the

_genetic

simulation program. In order to

_simplify

the

_problem,

the best 16 males and the

16 best females

_(out

of the 32

_eligible)

were considered. This was done to facilitate

computing,

but it is

_{intuitively appealing}

_since,

in

_practice,

as

_{suggested by}

Smith

(1969),

it would be better to use unscored

_individuals,

than individuals which are

below average. The number of runs was

_400,

_except in the

_{integer programing}

method in

_which,

in order to save

_computing

_time,

50

_replicates

were run.

RESULTS

Optimal weight given

to

_family

information

Table I _presents the

_{theoretically}

_{predicted genetic}

progress

(R

E

)

and the

inbreed-ing

coefficient

_(F

_E

₎

attained after

_{5.generations}

of

_{selection,.for}

different values of A and the 2 values of

_heritability

considered

_(h

_l

= 0.10 and

0.30).

Notice that

A = 0 means unrestricted within

_family

_{selection; ie,}

that an _{individual is selected}

solely according

to its deviation from

_family

_mean, and

_thus,

each

_family

can con-tribute between 0 and

_{min(n, k)}

individuals

_{(Dempfle, 1988),}

and A = 1

phenotypic

individual selection.

_Optimum

A values obtained from

_eqn(1)

are also included in

the lower row of the Table. It is

_interesting

to notice that the

_relationship

between

R

E

and A follows a law of

_diminishing

_returns;

_ie,

a

change

in A from 0 to

_1,

or

from 1 to

_2,

results in an

_important

increase in _response,

_whereas,

a

_change

from

3 to 4 results in

_practically

no _progress,

_and,

more

_importantly,

further increments in A are even

_expected

to reduce response. This is

because,

as A

_gets

_larger,

the

increasing

correlation between the index and

_breeding

value is

_{overcompensated by}

the reduction in the

_intensity

of selection

_{and, consequently,}

_A

_o

_p

does not

_give

the maximum _{response. In}

_{parallel, expected inbreeding}

coefficients

_(F

_E

_),

_computed

from

_eqn(4),

_steadily

increase with A.

_Considering

_{jointly R}

_E

and

F

E

values,

it can be seen that values of A = 3

(h

2

=

0.10)

and A = 2.5

(h

2

=

0.30)

should be chosen

in order to restrict the increment in

_inbreeding

below 5% per

generation.

The above

_prediction

for

F

E

is

_strictly

valid for

_only

1

_generation

and

_applies

solely

to _{neutral genes} which affect neither fitness nor the trait under

_selection,

and which are not linked to _genes affected

_by

selection. In successive

_generations,

there will be a cumulative effect on the variance of

_family

sizes up to a

_limiting

factor of

4

_{(Robertson, 1961)}

_but,

at the same

_time,

there will be a reduction in the

_genetic

variance and

_changes

in other _parameters such as _pi

_acting

in the

_opposite

direction. In order to check the

_adjustement

of the

_{predictions, genetic}

simulations were

performed.

The

_results,

_R

_o

and

_F

_o

_,

also _appear in Table I. In

_general,

_disagreement

between observed and

_expected

values for both _response and

_inbreeding

becomes

larger

as A increases. This should be taken into account when

_predictions

on

_possible

confirm

_expectations

in the sense that the

_largest

response was obtained with a A

value below the

_optimum

in infinite

_populations

_(eqn(l)).

Since

_inbreeding

was

larger

than

_expected,

and differences between observed responses for A > 1 were

small,

perhaps

in

_practice,

a value of A = 2 should be chosen for both heritabilities.

Restriction on the distribution of

_family

size

In the

_example,

there are as _many as 15 different distributions of

_family

_size,

and

they

are shown in Table II. Case 1

corresponds

to

_family

selection

_(with

respect to

the

_index),

in which the best families for each sex were

selected,

each

contributing

4

individuals. In case

2,

families were ranked

according

to their 4 individual means for

each sex, and the 4 full-sibs

belonging

to the best

_family

were selected.

_Then,

the

remaining

families were ranked

_{again by}

the means of their best 3 individuals and the 3 individuals from the best

_family

were selected.

Finally,

the best individual from a

_remaining

_family

was chosen. The same

_logic

_applies

to the

_following

cases. Case 15 is

_obviously

the well-known

_{within-family}

_selection,

with respect to the index.

For the sake of

_comparison,

the

_optimum

combined selection method is included in

The

_expected

_genetic

progress,

R

E

,

and

_inbreeding

coefficient,

FE are shown in

Tables II and III. As is well

_known,

_{within-family}

selection leads to a poor

genetic

progress

but,

as soon as the worst families are not allowed to

_reproduce,

response

quickly

increases

_(cases

13 and

_14),

_although

none of the fifteen cases

_{gives R}

_E

_,

as

_large

as that

_{for A >}

3 in Table I. This is because selection acts

_{independently}

on within- and

_{between-family}

_genetic

variation in this _strategy, and

_therefore,

selection will be less efficient than with unrestricted

_family

size.

As

_expected,

_F

_E

decreased as the distribution of

_family

size became more

uniform. In case

_8,

_inbreeding

was maintained below the maximum desired

_level,

with a reduction _{in response} of less than

10%,

with _respect to the

_optimum.

The _agreement between observed response

(R

o

),

obtained

by genetic simulation,

and

_expected

results

_(R

_E

₎

was better in those cases in which the variance

of family

size was small. In case

_9,

the desired

_inbreeding

is

_maintained,

with a reduction in response of about

5%,

with respect to the

_optimum

combined selection

_{(lower row).}

Minimum _coancestry

_matings

The observed

_genetic

_{progress attained}

_during

the first 5

_generations

of

_selection,

both with

_{random, R}

_R

_,

and minimum _coancestry

_matings,

_R

_MC

_,

_together

with

the

_{corresponding inbreeding}

_{coefficients,}

_F

_R

and

_F,

_uC

_,

are shown in Table IV

(A

o

p

was

_used).

The selection _response obtained was similar in both cases, as

expected

in a

strictly

additive model.

However,

minimum _coancestry

_matings

dramatically

reduced

_{inbreeding, compared}

with random

_{mating. Nevertheless,}

it

Mates selection

Table IV shows the observed response,

R

MS

,

and the

_{inbreeding coefficient,}

Fms.

It can be seen

that,

while

conforming

with

inbreeding

restrictions,

response was not

smaller than that attained under the

_optimum

unrestricted

scheme, R

R

.

DISCUSSION

It is

_{generally accepted}

_(Hill,

_1985;

₁₉₈₆₎

that the

importance

of

_population

size in a selection _program

_depends

on the time horizon in which the breeder _operates.

If we are

_only

interested in the first few

_generations,

it matters little and selection should be as intense as

_possible.

As the time horizon

_increases,

_{larger population}

sizes will be needed in order to compensate for

_{inbreeding depression}

and the loss of

useful genes

initially

present in the

_{population. Although}

the influence of a limited

population

size on selection _response has been

extensively

studied

(Hill,

1985;

1986),

there are no

_general

_equations

that can be used to

_predict

either the

_inbreeding

coefficient or cumulative response to selection in intermediate

_generations,

and

they

would

_probably

_require

information on the distribution of _gene effects and

frequencies

that is not

_likely

to be available. Recent results derived

_{by Keightley}

and Hill

_(1987),

Chevalet

₍₁₉₈₈₎

and Verrier et al

₍₁₉₈₈₎

could be of

_importance

in

this context.

Small herds are found in many situations and in some cases it has been

increasingly

popular

to advocate the use of

_family

information in their

_genetic

evaluation;

eg, in selection for

prolificacy

in

pigs

(Avalos

and

_Smith,

_1987),

or

MOET schemes in cattle and

_sheep

_{(Smith, 1988a).}

Nevertheless,

the inclusion

of information from relatives has some undesirable consequences, as

_previously

emphasized by

Toro et al

_(1988b)

and

_Dempfle

_(1988).

_{First, discrepancies}

between theoretical and actual rates of responses

steadily

grow with the

complexity

of the

family

index utilized because of the

_increasingly

correlated structure of the index values

_causing

a lower than

_expected

response.

Second,

the rate of

inbreeding

will be

higher

than that obtained with

_{simpler methods,}

because the

_probability

of

_selecting

related individuals is

_{higher. Additionally,}

the reduction of

_genetic

variance due to

generation

of

_linkage

_{disequilibrium,}

the so-called &dquo;Bulmer effect&dquo;

_{(Bulmer, 1971),}

is

_greater

when the accuracy of selection increases.

A reasonable

_proposal

for these small herds is to maximize

_genetic

progress,

while

_imposing

at the same time a restriction on the rate of

_inbreeding.

_Here,

we have

_explored

several

_{possibilities}

of

_attaining

this

_goal

and have illustrated them with a

_{simple example.}

The most obvious one

(ie,

to increase the

_proportion

selected)

is not an easy task if a balanced

_population

structure is to maintained.

A more flexible _strategy is to reduce the

_{weight given}

the

_family

information

_and,

as has been

_shown,

a considerable reduction in

inbreeding

can be attained with little loss in response. In more

complex

situations,

such as with BLUP

evaluation,

a

deliberately

overestimed

heritability

could be used in

evaluating

the

animals,

since the

_higher

the

_{heritability,}

the smaller the

_{weight given}

to information from relatives. In this _case, records should be first corrected for fixed effects

_using

the

appropriate

parameters. The third

_strategy

_{considered, imposes}

a restriction on the

family

size

_(case

₉₎

has been found that meets our

_objective,

while

_maintaining

the

desired level of

inbreeding

and

_attaining

an observed response of

95%,

with

respect

to the

_optimum.

A drawback of this method is

that,

if the number of families is

_large

and the number of selected males and females

differ,

it can become an

extremely

tedious task to find the

_optimum

situation. In

_practice,

another

_shortcoming

could arise since the number of selected

_offspring

left

_by

each

_family

cannot

_always

be

completely

controlled. An additional

_advantage

of these

_{&dquo;within-family&dquo;}

selection methods is the slower decrease of the

_genetic

variance,

as shown

by Dempfle

(1975)

for classical

within-family

selection

_{and, therefore,}

the

_long-term

response will be

expected

to increase.

Another course of action that can be

_taken,

is to

_practice

minimum

_coancestry

matings.

This is

_especially

valuable when

_family

index selection is used. Since

an

_important

fraction of the

inbreeding

reduction arises from the

delay

in the appearance of

consanguineous matings,

its usefulness will be _greater in the short

term,

whereas,

in the

_long

_term, its effectiveness may well be reduced.

Anyway,

a substantial reduction in the final

_inbreeding

was attained in our 5

_generation

example.

Finally,

the use of

_integer

linear

_programing

can

provide

a

_general

solution

for the selection and

_{mating policy,}

_regardless

of

_population

_structure, evaluation

method,

or selection

intensity,

as

previously suggested by

Jansen and Wilton

(1984),

Smith and Allaire

₍₁₉₈₅₎

and

_Kinghorn

_(1987).

In this _context,

_genetic

evaluation should be made

_using

the best available

_{technique, namely,}

the animal

_model,

and individuals and

_matings

should be chosen to conform to the

_inbreeding

restriction.

Unfortunately,

both processes are very

demanding

from the

computation point

of view

and,

in

_pratice,

similar results can

probably

be obtained

_using

some

of the

_simpler

methods outlined. The results that can be obtained with

_integer

programing depend critically

on

_AF,

the restriction on

_inbreeding

_imposed

in a

specific

program.

Thus,

if OF is

large, only

the best animals

(regardless

of their

genetic

relationship)

will be

_chosen,

and the results will not difFer from those obtained with

_{conventional,}

_{random-mating}

_{systems. On} the _contrary, if OF is

_kept

at a _very low

_value,

the less related animals

_(which

are not

_necessarily

the

_best)

would to be selected. This case would be similar to that of minimum coancestry

mating

(among

the

_preselected

best-half

_animals).

The

_advantage

of the

_general

strategy, therefore,

is

_expected

to be

_greater

at intermediate values of OF. In

practice,

breeders

_design

the structure of the

_population

in order to maintain the

rate of

_inbreeding

below the desired level. For

_example,

Smith

_(1988b)

has

_suggested

that the size of a MOET nucleus unit should be such that the level of

_inbreeding

is similar to that attained in a national program

(0.005

per

year).

In some

_situations,

it can be

_argued

that the aim of a selection program should be

to increase the accumulated selection response up to a fixed

_time,

_regardless

of the

inbreeding

coefficient,

because those lines will

_eventually

enter a _program of

_regular

crossing.

The most

_appropriate

course of action in such cases would be to estimate a value of

N

e

,

giving

the maximum response in a fixed number of

generations,

and

ACKNOWLEDGMENTS

We are indebted to C

_{L6pez-Fanjul,}

L

_Sili6,

A

_{Garcia-Dorado,}

A

_Gallego

and A Caballero for _very

_helpful

comments. We are also

_grateful

to the staff of Secci6n

de Proceso de Datos of the INIA for their kind

_cooperation.

This work has been

supported by

a CICYT _grant.

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