Restricted selection and effective population size

HAL Id: hal-00894133

https://hal.archives-ouvertes.fr/hal-00894133

Submitted on 1 Jan 1996

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.

Restricted selection and effective population size

Rp Wei

To cite this version:

Original

article

Restricted

selection and

effective

population

size

RP

Wei*

Department of Forest

Genetics and Plant

_Physiology,

Swedish

_University

of

Agricultural

Sciences,

901 83

_Urrcea,

Sweden

(Received

22

_May

_1995;

_accepted

6 March

₁₉₉₆₎

Summary - A

simple

and flexible selection

method,

’restricted truncation

_{selection’,}

has been

_developed

to screen

_superior

individuals from

populations

with

family

structure. ’Restricted’ means

_placing

limits on the contributions of families to the selected _group and on the number of families allowed to contribute. Selection is made on the basis of

individual

_{performance judged by phenotype}

or

_breeding

value estimate. Formulae have

been derived to

predict

the

_approximate

effective

_population

size in the selected group.

Changes

in the restrictions used

_modify

the distribution of

_family

contributions and thus lead to different effective sizes in the selected

population.

Effective size is influenced

by

sib type,

heritability,

selection

_intensity,

initial

family

number and size. It is decreased

_by

restrictions on the

_family

number but increased

_by

restrictions on

_family

contributions. The

_application

of the

_predictions

of effective sizes to

_planning

a

_{breeding population}

is discussed.

selection

_/

family

/

truncation

_/

effective size

Résumé - Sélection avec restriction et

effectif

génétique. Une méthode de sélection

simple

et

_{flexible, appelée}

«sélection _{par troncature} avec _{restriction»,} a été mise au

point

pour retenir les individus

_supérieurs

dans des

_populations

à structure

_familiale.

La restriction revient à imposer des contraintes sur la contribution des

_familles

au _groupe sélectionné et sur le nombre de

familles

autorisées à contribuer. La sélection est basée

sur la

performance

individuelle

_{phénotypique}

ou sur une estimée de valeur

génétique.

Des

_{formules approximatives}

de calcul de

_{l’effectif génétique}

du groupe sélectionné sont données. Des

changements

dans les contraintes

_{appliquées modifient}

la distribution des contributions

familiales

et conduisent ainsi à des

_{effectifs génétiques différents}

dans les

populations

sélectionnées.

L’effectif génétique dépend

du type de

famille,

de

l’héritabilité,

du nombre initial de

_familles

et de leur taille. Cet

_effectif

diminue si on _impose des contraintes sur le nombre de

_familles

mais _augmente si les contraintes _portent sur

les contributions

_{familiales. L’application}

des

_{prédictions d’effectif génétique}

dans la

planification d’expériences

de sélection est discutée. sélection

_{/ famille /}

troncature

_/

effectif génétique

*

Correspondence

and

_{reprints: Department}

of Renewable Resources,

University

of

INTRODUCTION

In a

population

with uniform

family structure,

selection leads to different families

making

different contributions to the _{selected group. Effective}

_population

_size,

if

concerning

family

number

_{(Robertson, 1961),}

is thus

_always

lower in the selected

population

than the initial

_size,

_except

when all families contribute the same

numbers of individuals. Reduction of effective

_population

size is an inevitable

effect,

for

_example,

of truncation selection based on either

_phenotype

or

_optimal

index

(Lush,

1947; Robertson, 1961; Burrows, 1984;

Falconer,

1989).

Wei

₍₁₉₉₅₎

has

therefore

_proposed

a modified truncation selection method in which restrictions are

placed

on the number of individuals selected from a

family,

and also on the number

of families from which selections are made. It has been shown that truncation

selection with restrictions can be used for the

_manipulation

of both

effective

size and

genetic

gain

in the selected

_population

_{(Wei, 1995).}

This

_study

attempts

to increase

the

_{general applicability}

of the method and to derive

_approximate

formulations for

predicting

the effective size

_following

selection.

ASSUMPTIONS AND SELECTION THEORY

Consider a _group of m unrelated or

_equally

related

families,

each of s members that are

_genetically

related

_by

the coefficient of

_relatedness,

r. The observed

phenotypes

of all individuals are recorded. The

phenotype

of the kth individual of the

jth family

could be

_expressed

as the sum of two

_independent

variables.

where _Xj are

family

means, and

_d!k

are

within-family

deviations. If the

family

means have variance

_o,

and

_{within-family}

deviations have variance

_Q

_w,

the total

phenotypic

variance,

!t ,

is

_af

₌

_Qb

_-f-

_<7!

and the ratio of the

_phenotypic

variance of

_family

mean to the total

_phenotypic

variance is

Selection criteria will be the

phenotypic

value or

_{optimal index}

that best

_predicts

the

_breeding

value of an individual

(Lush,

_1947;

Falconer,

1989).

In the

following

development

the index will be treated in the same _way as

_phenotypic

value but

with a different value for the ratio

_(K

_*

₎

of the variance of

_family

mean to the total

variance

_given

_by

Wei and

_Lindgren

₍₁₉₉₄₎

in the form

A

_proportion

_(P)

of individuals will be selected. Two restrictions are

_imposed:

one on the maximum number

(s

r

)

of individuals that _may be contributed

_by

a

family

and one on the number

(m

r

)

of families that are allowed to contribute.

Thus,

the mr

top-ranking

families with the Sr

top-ranking

individual in each

family

are shortlisted.

_Superior

individuals are

_finally

truncated from the

shortlist,

on the

Let

P

1

=

s

r/

s

and

_P

₂

=

m

r/

m,

the

_proportions

of the restrictions _sT and

m r

to the

_family

size and number

_{respectively.}

The extreme cases of restricted

selection with

_specified

_P

₁

and

_P

₂

values describe conventional selections and

one-step

restricted selections as follows

(Falconer,

_{1989; Wei, 1995;}

Wei and

_Lindgren,

1996).

When both

_P

₁

and

_P

₂

are _one, selection is based on either

phenotypic

or

optimal

index value but is unrestricted.

P

1

= P

(and

thus

P

2

=

1)

describes

within-family

selection,

and

_P

₂

= P

(and

thus

_P

₁

=

1)

corresponds

to

between-family

selection. Selection with

P

1

P

2

= P

_represents

the

permutations

of combined

between-family

and

_{within-family}

truncation.

_One-step

restricted selection means

restrictions

_being

_imposed

on either

_just

_family

number

(P

2

=

1)

or

_{just family}

contributions

_(P

_i

=

1).

EFFECTIVE POPULATION SIZE AND APPROXIMATIONS

Let n! denote the number selected from the

_{jth family}

and total selections n =

En

j

.

Two

_types

of definitions for effective

_population

size are considered:

and

where

_{E(n! )}

is the second moment _{of nj}

_samples

and

_{E[n! (n! -}

_1)]

is the second factorial moment. Both were

developed

for

considering inbreeding

effect in

offspring

(N

R

by

Robertson,

1961;

N

B

by Burrows,

1984)

although

the values _may have

potential

uses in other senses.

_{Considering selfing}

of selected individuals into random

_mating,

_0.5[r/N

_R

+

_{(1 - r)/n]}

is the _average

_inbreeding

coefficient

_(OF)

of _progeny. If

_selfing

is

_excluded,

_0.5/N

_B

is then _{the average}

_pairwise

_coancestry

in the selected group relative to the

_parents

of the

_population,

and therefore is the average

inbreeding

coefficient of _progeny

_{produced by}

random

_mating

_among selected individuals.

For

_{planning breeding}

_programmes,

breeders

often need to

_predict

selection

differentials

_{(genetic gain)}

as well as effective

_population

size

_by

_{using existing}

information

_(eg,

K

_value)

and

_designated

_operation

parameters

(m,

s,

P, P

i

,

P

2

in the

_present

_case).

Here we

_proceed

in

_analogy

with Burrows

₍₁₉₈₄₎

and Wei and

Lindgren

(1996)

to reformulate

_[1]

and

_[2]

to enable the

prediction

of effective size. We assume that

_U!

_represents

the number of individuals in the

_jth

sampled

family

with

_performance

_exceeding

the truncation

_point

relative to

_P,

_P

₁

and

_P

₂

_.

Thus, they

sum to a random variable and are distributed over

_integers

_{0,1, ... ,}

sr.

and

where

_N

_r

_(K,

_{P, P}

₁

_,

_P

_Z

₎

stands for the

_{corresponding}

effective

_population

size under selection from a

_population

of

_{infinitely large family}

number and size. Let

_{f (x)}

denote the

_density

function of the

_family

mean

(x)

in an infinite

population,

and

_p(x)

the

_proportion

of members selected from a

_family.

Then the term

7Vr(-f!,f,fi,-P2)

is

_expressed

in the

_integral

form

_(Wei,

_1995;

Wei and

_Lindgren,

1991)

Clearly

N

r

(K,

P, P

l

,

P

2

)

measures the relative value of effective size

_compared

to

that before selection. As K =

0,

selection is

completely

based on

within-family

deviations,

thus

_N

_r

_(K,

_{P, P}

_l

_,

_);

₂

_P

as K =

l,

selection is

completely

based on

_family

means, thus

!(!,f,fi,f2)

=

P/P

1

.

To obtain

, P

)

Z

N

l

(K, P, P

r

for K between

0 and

_1,

numerical

_computation

is needed

_(for

full details see

_{Burrows, 1984;}

Wei

and

_Lindgren,

_1991;

_Wei,

_1995).

In contrast to

_U

_j

_,

_n!

_represents

a _consequence of

censorship applied

to the

population sample.

The n

j

are constrained to sum to n and are distributed over

integers 0,1, ... , min(s

T

, n).

Thus we do not

directly employ

E(UJ)

and

E(U! (U! -1)!

as the

_respective

approximations

of

E(n! )

and

E[nj(nj -1)].

Instead,

we assume that

3

and

in which W and V will be

obtained

for two cases.

When

_only

K = 0 is considered

When K = 0 there is no difference _among

_families,

and selection is based on

within-family

deviations but is random with

_respect

to

_{pedigree. By using K}

= 0 _{as a}

factor,

formulations for

_predicting

effective

_population

sizes have been

_developed

for unrestricted selection and

one-step

restricted selection

_(Burrows,

_1984;

Wei and

Lindgren,

1996).

Proceeding

in a similar _way, the relevant

hypergeometric

sampling

moments for the

_present

situation

_yield

When K =

0,

[7]

yields

the same result as

(10!.

_{By using}

(K,

r

N

_{P, P}

_i

_,

P

Z

)

=

_P

₂

_,

!4!,

[7]

and

_!10!,

W is solved in the form

Therefore an estimative

approximation

to

[1]

is

As

_[8]

_yields

the same result as

[9]

at K =

0,

we could obtain

M id

When both K = 0 and K = 1 _are considered

Obtaining

the consequences of selection for the

special

case when K = 1 is easy.

Because K = 1 means no variation within

families,

and truncation selection is

completely

based on

family

means with

_family

contributions either

P

1

or zero, we

have

a

-and

Meanwhile,

for infinite

_populations,

_Nr(K,

P, P

i

,

P

2

)

=

P/P

l

.

A linear

relationship

between W and

_N

_r

_(K,

P, P

1

,

),

Z

P

which _passes the two

_limiting

cases

[10]

and

!15!,

produces

Thus,

[1]

could be

_{predicted using}

and

RESULTS AND DISCUSSION

The effective

_population

size

_following

selection illustrates the

_possible

disadvan-tages

(eg,

inbreeding

depression)

of

_using

selection in

_{production populations,}

and the richness of

_genetic

resources achieved for further selection and

_breeding.

Knowl-edge

of effective size

_helps

a breeder assess the benefits and risks associated with a

breeding

operation

such as selection in

_planning

a

_breeding

_programme.

When selection is

_applied

to a

_breeding

_population

or _progeny

_test,

effective

population

sizes can be calculated

_directly

from

_pedigrees

of selections

_using

_[1]

and

(2!.

Before

_testing,

the

_prediction

of effective sizes

_{requires prior}

_knowledge

of K and

_N

_r

_(K,

_{P, P}

_l

_,

_P

₂

_),

_except

in the extreme cases

_P

₁

=

P,

P

2

= P and

P

i

P

2

= P.

To

_plan

an advanced- or

_{improved-generation breeding}

_population,

values of K

derived from measurements of the last

_generation

could be

_directly

_employed.

In

planning

a new

_breeding

_programme, a reliable value of K is often not available.

Any

information about the

_genetics

of the

_species

under consideration can then

be used to

_{synthesize K,}

such as data from similar tests in the same or similar

environments,

or trials at

_{clonal, individual, family}

_(sibs),

_population,

_provenance

or even

_species

levels in

_greenhouse,

_nursery or field conditions. Breeders should use

_{existing knowledge}

to make the best

_possible

estimate of K.

The effective

_{population size,}

_N

_r

_(K,

_{P, P}

₁

_,

_P

₂

_),

from infinite

_populations

is used

to draw

_general

conclusions and to

_{predict N}

_R

and

_N

_B

from finite

_populations.

For unrestricted

_selection,

_{Nr (K,}

_{P, P}

_I

_,

_P

₂

₎

could be

_{computed using}

the same

pro-cedure as Burrows

₍₁₉₈₄₎

and Wei and

Lindgren

(1991).

Truncation

_points

corre-sponding

to P can be obtained or

_interpolated

from

_existing

tables and compu-tational programmes. When restrictions are

imposed,

the

_population

for selection

has truncated distributions of

family

means and

_{within-family}

deviations.

Search-ing

for a truncation

_point

_{corresponding}

to

_{P, P}

₁

_,

and

_P

₂

becomes

_complicated.

A numerical

_procedure

to calculate

_N

_r

_(K,

P, P

l

,

P

2

)

has been documented in

_previous

studies

_(Wei,

_1995;

Wei and

_Lindgren,

_1996).

With

_assumption

that

_family

mean and

_{within-family}

deviations are

normally

distributed,

and the total

_phenotypic

variance is one, an

_example

is

_given

to

show

_{Nr(K, P, P1, P2)}

at P = 0.01 for different

_K,

_P

₁

and

_P

₂

values

(table I).

As K

_increases,

effective size

_rapidly

decreases

_except

where

P

1

=

P,

P

2

= P

and

_P

_l

_P

₂

= P. At

given

K

values,

the restrictions

yield

different

_results,

in

comparison

with unrestricted

selection,

depending

on which

_type

of restriction is

used. A restriction on

family

contributions leads to a more

_dispersed

distribution of selections among

families,

and thus

_higher

effective size. The trend becomes most

evident when K is

_high

and the restriction is

_strong.

In

_contrast,

a restriction on

family

number

_drastically

reduces effective

_size,

_especially

at low K.

Analysis

of

_predicted

effective

_population

sizes for a small

_population

_{(table II)}

suggests

the same effects. The

_optimal

index selection

_is,

for

_instance,

worse than

K. This is consistent with unrestricted selection

_(Robertson,

1961;

Burrows, 1984;

Wei and

_Lindgren,

_1991).

The two

_definitions,

_[1]

and

_[2],

are different in value

but are related in a _way

(Kimura

and

Crow,

1963; Burrows,

1984).

The

inbreeding

coefficient of progeny

produced

by

random

mating

among selections can be

_easily

obtained

_using

either

_[1]

or

(2!.

For both of

them,

two

_types

of

approximations give

very close

results, especially

when a

_strong

restriction on

_family

number is used.

Several other factors may influence effective size

following

selection. For charac-ters with

_{given hereditary ability}

_{(heritability),}

half-sib families have

larger

within-family

variation

_{(lower K)}

than full-sib

families,

so

_they

are a better choice for

conservation of

_high

effective size. Intense selection

_{(low P),}

which is often used for

rapid genetic gain

in selective

_breeding,

often leads to drastic reductions in effective

size

_(Wei

and

_Lindgren,

_1991).

Values of P should be

_deliberately

chosen. Table III

shows the effects of

_family

number and size on the effective size

following

increase with

_family

size is trivial. This

_suggests

that if effective size is of concern,

having

many families in

_{breeding populations}

is more

_important.

Because selection is

_totally

at random when K =

0,

a drift effect due to small

family

size is included in the

approximations

of effective size

_(eqs

_{(12!, !14!, [18]}

and

_(20!).

Small differences between

_N

_R

and

_N

_R2

_(or

_N

_B

and

_N

_B2

₎

_{(table II)}

_may be

_{explained by}

the

_decreasing

influences of K on drift effect as it

_approaches

one

(Wei

and

_Lindgren,

1994,

1996).

The

_{approximation}

_{(eqs [12]}

and

_[14])

_yield

the

exact values when K = 0 and m ! oc

(Burrows,

_{1984; Woolliams,}

1989).

This is

also true for

_[18]

and

_[20].

Moreover

_[18]

and

_[20]

_give

the exact value at K = 1.

It has been found that the

_{approximations}

for unrestricted selection underestimate effective size when K > 0 and

_family

number is small

_{(Woolliams, 1989).}

_Computer

simulation shows

_{(table IV)}

that restriction

_(both

_P

₁

and

_P

₂

₎

_greatly

_improves

the

prediction

of effective size.

_{Increasing family}

number could also better the

_prediction

at a

_given

_family

_size,

_{although large family}

size has a

_negative

effect

_{(table IV).}

The

quantity

N

r

(K,

P, P

i

,

)

2

P

is a

_limiting

result for

_{large family}

number

(m).

The

prediction

of effective size for small m could then be

_{improved by}

the

_adjustment

which is bounded at

P

2

as

N,.(K,P,P

1

,P

2

)

=

P

2

,

especially

N

;

(K,P,P

1

,P

2

) =

ACKNOWLEDGMENTS

I am indebted to D

_Lindgren

for encouragement and

helpful

comments, and J Blackwell for

_revising

the

_English

text. This

_study

has been

_{supported by Skogsindustrins}

Forskn-ingsstiftelse.

REFERENCES

Burrows PP

₍₁₉₈₄₎

_Inbreeding

under selection from unrelated families. Biometrics

_40,

357-366

Falconer DS

₍₁₉₈₉₎

Av, Introduction to

_Quantitative

Genetics

_{(3rd edition),}

Longman

Scientific and

_{Technical, London,}

UK

Kimura

_M,

Crow JF

₍₁₉₆₃₎

The measurement of effective

population

number. Evolution

17,

279-288

Lush JL

₍₁₉₄₇₎

_Family

merit and individual merit as bases for selection. Am Nat 81, 241-261; 262-379

Robertson A

₍₁₉₆₁₎

Inbreeding

in artificial selection programmes. Genet Res 2, 189-194 Wei RP

₍₁₉₉₅₎

Optimal

restricted

phenotypic

selection. Theor

_Appl

Genet 91, 389-394 Wei RP,

Lindgren

D

₍₁₉₉₁₎

Selection effects on

_diversity

and _genetic

gain. Silva

Fenn 25,

229-234

Wei

RP, Lindgren

D

₍₁₉₉₄₎

Gain and effective

population

size

_following

index selection with variable

_weights.

For Genet 1, 147-155

Wei

_{RP, Lindgren}

D

₍₁₉₉₆₎

Effective

_family

number

following

selection with restrictions. Biometrics 52, 198-208

Woolliams JA

₍₁₉₈₉₎

Modifications to MOET nucleus

breeding

schemes to improve rates

APPENDIX

Restricted selection is

_applied

to the

₍₌

_r

_s

_r

_m

_msP

_i

_P

₂

₎

individuals shortlisted

by

considering

both restrictions. Those with the

_{highest performances}

are truncated at

the

_point

XT,

corresponding

to

_P,

_P

₁

and

P

2

.

Assume that the

_family

_{mean, Xj} =

x, is a continuous random variable with a

_density

function

f (x).

The truncation

point

for a

_{particular family}

with mean value x on a standardized scale is

_expressed

as _y =

(x

T

-

z) law.

Let F denote the unit distribution function. In the

jth

family

with _xj = _x, the

probability

that the

performance

of an individual exceeds

XT is

[1

-

F(y)]/P

1

,

and the

_probability

that the

_performance

is less than xT is

!F(y)

+

_P

₁

_{-1)]/ P}

_l

_.

We can

_reasonably

assume that 1-

)

F(

y

= 0 and

F(

y

)

= 1 for

the m - mr

rejected

families. Then the

probability

that in the

jth

family,

U! (=

u)

individuals have a value

_greater

than xT and - rs u have a value less than _xT,

Pu =

pr(U

j

=

u)

is

given by integrating

over the distribution of x

(Burrows,

1984)

in the form

where the

_integral

part

is the

expectation

with

_respect

to the

_family

mean x.

_Thus,

the

_{moment-generating}

function for

_U!

is

The

_expectations

of

_U!

and

_U§

_are

_{given by}

the values of the first and second derivatives of

_M(8)

at B = 0:

and

Similarly

we can obtain the

_{probability generating}

function for

_U

_j

_:

The

_expectation

of

_U

_j

_(U

_j

_-

₁₎

is

_given

_by

the value of the second derivatives of