Mass selection in livestock using limited testing facilities

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Mass selection in livestock using limited testing facilities

L Ollivier

To cite this version:

Original

article

Mass selection

in livestock

_using

limited

testing

facilities

L Ollivier

Institut National de la Recherche

_Agronomique,

Station de _{G6n6tique Quantitative}

et

_Appliquee,

78350 _{Jouy-en-Josas,} France

(Received

29 May 1989; accepted 20 _September

₁₉₈₉₎

Summary - This _paper considers the _problem of

_maximizing

the _expected annual response

to mass selection when

_testing

facilities are limited and so do not allow _testing of all

potential candidates. In such _situations, there is room for variation both in the proportion

of

_breeding

animals selected on the basis of the test

_{result "nd}

in the allocation of

_testing

places between male and female candidates. When _testing facilities are _very limited

_(case

1), males

have

_priority

in _testing and the maximum _proportion to select based on test

results is 27%. This means that it is then better to use untested males, i.e. taken at

random, than males which are in the lower 73%. This situation holds until the ratio

_(k)

of tested to _potential candidates reaches kl =

1.85/c(4aA + 1),

where _c is the

degree

of

polygyny

(mating ratio),

a the _age at first

_offspring

_(yr)

and _À, the annual

_fecundity

(s.e.

half the dam _progeny

_crop).

As k increases above kl

(case

2),

all _replacement males should be tested and

_testing

_{space should be}

_entirely

devoted to _males, with random choice of females. This situation holds until k reaches a critical _{value, k2,} above which testing space should be equally distributed between the 2 sexes

_{(case 3).}

The value of _k2, obtained _{iteratively for any}

_given

set of _{parameters c,} a and _À, as defined _above, is shown

to increase when c increases and when aA decreases. The _{strategies recommended,} which

imply

contrasting

turn-over rates between selected candidates and candidates chosen at

random, are compared to those aimed at _maximizing selection _intensity for a fixed value of

the _generation interval. Numerical _examples are _{provided, covering} the range of situations prevailing in farm livestock _species.

mass selection _/ selection _{response /} selection _{intensity / generation} interval Résumé - La sélection massale chez les animaux domestiques avec une _capacité

de contrôle limitée - Cet article traite de la maximisation du _gain

_génétique

annuel attendu en sélection massale

quand

la _capacité de contrôle est limitée et ne permet pas de

contrôler tous les candidats _potentiels à la sélection. Dans une telle _situation, on _peut

faire

varier à la

_fois

la _proportion des _{reproducteurs} sélectionnés sur leur résultat de

contrôle et la _répartition des _places de contrôle entre les 2 sexes. _Quand la _capacité de

contrôle est restreinte

_{(cas 1),}

les mâles ont la

_priorité

et le taux de sélection maximal à l’issue des contrôles est de 27%. Il vaut mieux alors utiliser des mâles non _controlés,

c’est-à-dire choisis au hasard, _que des mâles se trouvant dans les 73% inférieurs. Cette situation _prévaut tant _que le _rapport

_(k)

des candidats contrôlés aux candidats potentiels ne dépasse pas kl = _1,

85/c(4aλ+1),

_où c est 1_e degré de

_{polygynie (nombre}

de _{reproducteurs}

femelles/nombre

de _{reproducteurs}

_mâles),

à _l’âge au ler descendant

(an)

_{et ) g la fécondité}

femelle).

Quand k _dépasse ki

(cas 2)

tous les mâles de renouvellement doivent être contrôlés et toutes les _places de contrôle doivent être réservées aux _mâles, les _femelles

étant choisies au hasard. Cette situation _prévaut

_jusqu’à

une valeur _critique k = _k2,

au-dessus de

_laquelle

les

_places

doivent être

_également

_réparties entre les 2 sexes

(cas

3).

On

montre que cette valeur _{k2, qui} est obtenue par itération pour tout ensemble donné des

paramètres c, a et _À,

_définis

_ci-dessus, augmente avec c et diminue _quand aa augmente.

Les

_stratégies

_{recommandées, qui impliquent} des taux de renouvellement très

_différents

entre les candidats sélectionnés et les candidats choisis au hasard, sont comparées à celles

qui visent à maximiser l’intensité de sélection à intervalle de

_génération

_fixé. Des _exemples

sont donnés pour illustrer le cas des diverses _espèces animales

_domestiques.

sélection massale _/ réponse à la sélection _/ intensité de sélection _/ intervalle de

génération

INTRODUCTION

Mass selection is a

_simple

and

_widely

used selection method for farm animals.

Considering

a trait

_expressed

in both _sexes, and

_following

a normal

_{distribution,}

the

_expected

annual response can be shown to be a function of the mean _ages of males and females at

_culling.

_{The maximum response is} obtained

_{by determining}

an

optimal

balance between selection intensities and

_{generation intervals,}

as shown

by

Ollivier

₍₁₉₇₄₎

for the case when all

_potential

candidates are tested. The purpose

of this _paper is to extend the treatment to situations where

_testing

facilities are limited and so do not allow

_testing

of all

_potential

candidates. In such _cases, there is room for variation both in the

_proportion

of

_breeding

animals selected on the

basis of the test

_resultgand

in the allocation of

_{testing places}

between male and female candidates. The effect of such a variation on the overall selection

_intensity

has

_previously

been considered

_by

Smith

_(1969).

The

_general

method

Dickerson and Hazel

₍₁₉₄₄₎

_gave a

_general

formula for the

_expected

annual _response

to

_selection,

R

d

=

(i

1

+

l

2

)/(t

i

₊

t

2

),

as a function of female and male selection intensities

_(i

_l

and

_i

₂

_{,’ respectively)}

and

_generation

intervals

_(t

_l

_,

_t

₂

_),

R

a

being

expressed

in

_genetic

standard deviations for a trait assumed to have a

_heritability

equal

to 1. With selection of

_respective

_{proportions f}

and m of the females and males

_required

for

_breeding,

and

_{corresponding}

proportions

1 —

_f

and 1 — m taken

at

_random,

the

_expected

_{annual response} becomes:

where tll and t12 are the

_generation

intervals for the females selected and the females taken at

_random,

_{respectively,}

and t21 and t22 are

_similarly

defined for males. &dquo;

If selection is

_by

truncation of a normal

_{distribution,}

i

l

l, = znlwhere ni is the

number of female candidates tested _per female selected and zl the ordinate of the

normal curve for a

_proportion

_1/n,

_selected,

and

_i

₂

is

_similarly

defined.

_Moreover,

generation

intervals may be

expressed

as functions of

_demographic

_parameters

pertaining

to any

given

species,

and of the distribution of

_testing

space between

males and females.

_Using

the

_{simple demographic}

model assumed

_by

Ollivier

_(1974};-.

for

_instance,

one can write:

-where a is the

parents’

age

(in years)

at birth of first

_offspring,

assumed

_equal

for

both sexes; c is the

_degree

of

_polygyny,

or

_mating

_ratio;

A is the annual female

fecundity, referring

to the number of candidates of 1 sex

_(sex

ratio assumed to be

1/2)

able to breed

_{successfully;}

_h

and ₁₂ are the

_respective

numbers of female and

male candidates tested

_annually

_per dam.

Expressions

(2)

are based on the definition taken for the

_{generation interval,}

which is assumed to be the arithmetic mean of the

_parents’

_ages at birth of first

(a)

and of last

_offspring.

The latter is determined

by

the time _necessary to

_replace

1

_{breeding animal,}

either selected _among n candidates or taken at random. For

instance,

knowing

that h

female candidates are tested

_annually

_per

_breeding

_female,

ie,

Illf

candidates per female

selected,

and that each selected female is chosen among n1

candidates,

the time

required

is

fnl/l1

years, which leads to

eqn(2a).

On

the other

_hand,

_{(A - 11)}

females are

_untested,

_ie,

_{(À -1¡)/(1- f)}

_per female chosen

at _{random. The time necessary} to obtain 1

_candidate,

if one takes the first

_born,

is

_{(1 - f)/(A - 1}

₁

_),

which leads to

_eqn(2b).

_Equations

_(2c)

and

_(2d)

are

_similarly

obtained.

Now

_h

and

₁

₂

_depend

on the overall

_testing

_capacity,

defined as the

_proportion

k of available candidates which can be tested

_annually,

and of the distribution of

testing places

between females and

_males,

defined

_by

the sex ratio a _among the

tested

_candidates,

so that:

The

_possible

_range of a extends from 0 to 1 as

_long

as k _< 0.5.

Then,

as k exceeds

_0.5,

the _{range is}

_{progressively}

_narrowed,

until a = 0.5 when k = 1.

Case 1:

_only

males are tested

(a

=

1);

_a

proportion

(m

_<

1)

of males

required

for

_breeding

is tested

In this case,

f

= h

= 0 and

2

1

= 2kA.

Expression

(1)

reduces _{to a} function of 2

variables,

m and _n2, such that:

with

The maximum of

_R

_’f

_,)with

_respect to m is obtained for:

With this value of m,

R

a

becomes a

_quantity

_{approximately proportional}

to

z

2

ng.

5

,

which is maximum for n2 - 3.7.

Thus,

the critical value of k for which

m =

or, with n2 -

3.7,

Consequently,

when

_testing

_capacity

is limited to a value k <

_k

_i

_,

a

_proportion

of

untested males should be

_used,

in order to maintain a constant

_proportion

selected

of about 27%

_(1/3.7)

_among those tested. Under these

_conditions,

the

_expected

annual response is

approximately proportional

to

_k

_l

_-’,

as

Case 2:

_only

males are tested

(a

=

1);

all males

required

for

_breeding

are tested

_{(m = 1)}

As k becomes

_equal

to

_k

_l

_,

and then increases above

_,

_k

_l

m = ₁ and

eqn(4a)

reduces to:

which can be maximized

_iteratively

with _respect to !2. But the

question

then arises

as to whether a

_higher

_response can be

_{expected by diverting}

some

_testing

space

for the selection of females. This case will now be considered.

Case 3: all males tested

_(m

=

1)

and a

proportion

of females

(a

<

_1;

_f

>

₀₎

With selection of all males

_{(m = 1),}

and of a

_proportion

( f )

of the females

required

for

_breeding,

_R

_a

becomes:

which is a function of

f,

_{a, ni} and _n2 for _any

_{given testing capacity.}

It can

easily

be shown that the derivative of

R

a

;

with _respect to

_f,

is

_positive

when 0 <

_f

<

_1,

_provided

_2i

₂

> _{. il} As selection should

_generally

be more intense

in males

_(i

₂

>

_i

_l

_),

this condition is

_always

_fulfilled,

and the

_optimum

value of

_f

is

therefore

_1,

_irrespective

of the other _parameters.

Then, assuming f

= 1

(ie,

all females

required

for breeding

are

_tested),

the

question

is how to allocate the

_{testing places}

between 2 sexes, within the limits

previously

indicated for the sex ratio a _among tested candidates. In

fact,

the value of

_R

_a

is rather insensitive to variations of a

_(although

the

_optimal

value of a is

slightly

below

_0.5),

as shown

by

Ollivier

(1988:

see

eqn(6),

p

446).

One can then

take a to be

_0.5,

and the

_optimal

values of n1 and n2 are obtained

by

_maximizing:

where _t, l = a +

_Mi/

_2A:A,

and _t21 = a +

_n

₂

_/2ckA,

as h = 12 = _kA.

For any

given testing capacity,

the maximum

_{of eqn(10)}

can be

compared

to the

maximum of

_eqn(8)

considered in case

_2,

and

(by iteration)

the

_k

₂

value

yielding

equal

responses in the 2 cases is obtained.

_Thus,

when

_{testing capacity}

is below

_k

₂

_,

all

_testing

_space should be devoted to

_males,

and when k >

_k

_2;

it should be

_equally

distributed between the 2 sexes.

Numerical illustration

As an illustration of the above

_results,

Table II

_gives

_k

_i

and

_k

₂

values for 9 sets

of

_demographic

_parameters

_implying

3 values of aA

_(0.5,

1 and

₅₎

valid for

_sheep,

cattle and

_pigs,

_{respectively,}

and 3

_degrees

of

_polygyny,

either

_{corresponding}

to

natural

_mating

_(c

=

10)

_or artificial insemination

(c

= 100 _or

1000).

The Table

also

_gives

the

_expected

response for

k = k

1

and k =

k

2

,

expressed

relative _to the

maximum _response

_expected

with k = 1.

The Table

_clearly

shows

_that,

for a

given degree

of

polygyny,

k

l

and

k

2

both decrease when

_fecundity

increases. For

_species

of

_high

_fecundity,

such as

poultry

and

_rabbits,

_k

_l

becomes

_negligible

and the low value of

_k

₂

is

_likely

to fall below

the actual

_testing

_{capacity, owing}

to the low cost of

_testing.

_{Therefore; case}

3 will

usually apply

to those

species.

On the other

hand, k

l

decreases

when

polygyny

increases,

as it is

inversely proportional

to c,

(from eqn(6))

!where.-t§;,k2

increases with c _up to a

_point

_{where, particulary}

when

_fecundity

is

_low,

a

_{large proportion}

of the _{maximum response} can be

expected

from

testing

males

only.

It is also worth

noting

that when

_fecundity

is low

_(below

a limit which is somewhere between 1 and 5 for

_aa),

the critical

_{testing capacity, k}

₂

_;yis

above 0.5. As this

_corresponds

to

one of the _patterns illustrated in

_Fig

_1,

_according

to whether

_k

₂

< 0.5 or

k

l

> 0.5.

In the latter case, rather

paradoxically,

the extra _space available when all males are tested should not be used for

_testing.

The worst solution would

_actually

be to

use it for

_{testing females,}

as shown

_{by point}

C in

_Fig

la. This is because the extra

selection

_intensity

obtained

_{by testing}

females is more than offset

_by

the increase in their

_generation

interval.

DISCUSSION

A

_parallel

can be draw between the above results and those of Smith

_(1969).

He

considered

_maximizing

selection

_intensity,

or response per

generation,

for a

_given

number of

_{testing places}

_(T)

_{available, assuming}

a fixed

_generation

interval. Here

the

_objective

is to _{maximize annual response,} with variable

_generation

_length,

and the

_{testing capacity}

_(k),which

is defined on a

yearly

basis. If

_generation

interval is

set at a value _t, and T is defined as the number tested _per

_breeding

female over a

period

of time

_equal

to _{the average}

_breeding

life of sires and

_dams,

_{2(t &mdash; a),}

the

relationship

between T and k is:

In case

_1,

the selection _strategy recommended

_{here, may}

be

_compared

to the

rule

_{given by}

Smith

_(1969),

which states that &dquo;if

_testing

facilities are _very

_limited,

it is better to use untested

_males,

than males which are below

_{average&dquo;.}

_Thu!!

1/2

is the maximum

_proportion

to select in order to maximize the

_response

per

generation,

as

_against

1/3.7 ,‘two, )

if the response per year is considered.

The two

_approaches

_{can, for}

instance,

be

_compared

in terms of

_expected

annual

response for a

_{testing capacity equal}

to

k

l

.

Using

Smith’s

_approach,

the critical number of

_{testing places}

below which untested males should be

_used,is

T =

2/c,

i _e, 2 male candidates tested per sire to be

_replaced.

This

_{implies a}

_generation

interval t = 2.1a ₊

1/3.7A,

a value obtained from

solving

eqn(11)

for T =

2/c

and

1

2

=

2k, A.

The

supplementary

gain

expected

from

applying

Table I _strategy when

k =

k

l

,

_using

the value

_R

_a

=

1.2A/(1

₊

4aA)

derived from

eqn(7),

_can then be

shown to _{range from 33} to

_53%,’when

_{aa goes from 0.5} to 5.

For case 2 and

_3,

Smith’s

approach

leads to

_{recommendation}

of a

_gradual

increase

in the

_proportion

_(f)

of females

_selected,

whereas,

here no intermediate

optimum

for

_f

_exists,

but rather an

abrupt change

from- f

= 0 to

_f

=

_1,

between

case 2

and case 3. In Smith’s

_approach,

the

_gradual

increase in

f

should start at T -

_1,

and case 3 is reached when T ! 3.

_Taking

a

_generation

interval t =

2a,

usual

in livestock

_populations,

it can be seen from

eqn(ll)

that the

equivalent testing

capacity

necessary to reach case 3 is k =

3/4aA.

This _means that when the

generation

interval is not acted _upon, case 3 can be reached

_only

if aa > 0.75. The model used in this

_study,)

rests on several

_simplifying

_assumptions,

of which a detailed discussion has een

_given

by

Ollivier

(1974).

The

population

undergoing

selection is

_supposed

to be

_large

and

stationary

in

_size,

and a uniform age distribution of the

breeding

animals is assumed. It

perhaps

should be stressed

that

_testing

space is defined relative to a

_given

population

size. If the

testing

space were defined as an absolute

_value,

the

_population

size

_might

be reduced to match the

_testing

_capacity

in order to increase the immediate response. Loss

of

_genetic

variance

_{would, however,}

be

_incurred,

thus

_{compromising long-term}

response.

Optimal

strategies

for

_maximizing

_long-term

_{response to} selection in such

situations have been

_{explored by}

Robertson

_{(1960; 1970),}

Smith

_(1969;

_1981mand

James

_(1972),

_among

_others,

under the

_assumption

of a fixed

_generation

time. In a situation of restricted

_{yearly testing facilities,}

it would then be advisable to maximize the

_generation

_length

in order to also maximize the number of candidates

per

generation.

The

_assumption

of a uniform _age distribution is not

_generally

met in

_practice

and can

_only

be

_accepted

as an

_{approximation}

in situations of fast

replacement,

or when the increase in

_fecundity

with age can _compensate for the

_gradual

_decay

in the number of

_breeding

individuals. With low

_testing

_capacity,

_however,

the

procedure

recommended

_herp

_r

_,

_-

_-implies

_contrasting

turnover rates between males

(selected)

and females

(takei

r

at

_rand

_Q

_m).

When k <

_k

₂

_,

the female

_generation

interval should be

_{minimized, whereas,}

the male

_generation

interval will

_gradually

increase as k decreases. When k =

k

l

;‘£his

interval exceeds 3 times

the

_{age at} birth

of the first

_offspring,

as shown in Table I.

Obviously,

such a

_strategy

can be

strictly

implemented,,

provided

enough

semen from the selected sires can be stored and if

breeding

can

be

carried

_{on$ artificially}

if necessary.

In

spite

of the limitations discussed

_above,

the results

_presented

may serve as

guidelines

for the

_optimal

use of limited

_testing

facilities.

They

also show that a sizeable

_proportion

of the maximum

_{genetic gain}

can be obtained with very limited

testing

facilities. The conclusions are restricted to mass

_selection,

which

_requires

no

_pedigree

information. Extension to

_family

selection _may be considered. Extra selection

_{intensity could,}

for

instance,

be obtained

_{by applying}

_family

selection to

untested candidates whenever tested relatives are available.

_However,

in a situation of limited

_{testing facilities,}

information on relatives would also be limited and the extra selection

_intensity

would have to be set

_against

the

_resulting

increase in

_generation

interval. Evaluation of tested candidates could also be made more

and Rouvier

₍₁₉₇₁₎

or, more

_generaly,

with best linear unbiased

_prediction

of

breeding

value

_{(Henderson, 1963).}

One would then expect the accuracy of selection to increase with the

_testing

_capacity.

This would mean, for any

given

testing

capacity,

a lower response relative to

_complete

_testing,

than with mass selection. N An

_optimal

_strategy

_{would, therefore,}

be more

complex

to establish for combined

selection,

as it would

_depend

on the

_relationship

between the

_testing

_capacity

and

the selection _accuracy.

REFERENCES

Dickerson

_GE,

Hazel LN

₍₁₉₄₄₎

Effectiveness of selection on progeny

performance

as a

_supplement

to earlier

_culling

in livestock. J

_Agric

Res

_69,

459-476

Henderson CR

₍₁₉₆₃₎

Selection index and

_expected

_genetic

advance. In: Statistical Genetics and Plant

_Breeding

_(Hanson

_WD,

Robinson

_HF,

_eds)

NASNRC Publ.

_982,

141-163

James JW

₍₁₉₇₂₎

_Optimum

selection

_intensity

in

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