Estimation of heritability in the base population when only records from later generations are available

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Estimation of heritability in the base population when

only records from later generations are available

L Gomez-Raya, Lr Schaeffer, Eb Burnside

To cite this version:

Estimation of

_heritability

in

the base

population

when

_only

records

from

later

generations

are

available

L

_Gomez-Raya

LR

Schaeffer

EB

Burnside

University of Guelph, Centre _for Genetic _{Improvement of} Livestock Animal and _{Poultry Science , Guelph, Ontario,} Canada NI G 2W1

(Received

8 November _{1990; accepted} 28 November

₁₉₉₁₎

Summary - The genetic variance and

_heritability

of a quantitative trait decrease under

directional selection due to the _generation of _linkage

_{(gametic phase)}

_{disequilibrium.} After

a few _cycles of directional selection in a _population of infinite sire a _{steady-state equilibrium}

is approached. At this _point there is no further reduction in these parameters since the

disequilibrium generated by selection is offset _by free recombination. In many situations records available to estimate genetic parameters come from _populations at the _steady-state

equilibrium. A _simple method to obtain estimates of genetic variance and _heritability in the base population using estimates of these _parameters at the _equilibrium is described. The method makes use of _knowledge of the effect of _{repeated cycles} of selection on _genetic

variance and _heritability to infer the base population parameters.

genetic variance _{/ heritability /} estimation of _{genetic parameters / linkage} disequi-librium

Résumé - Estimation de l’héritabilité dans la _population initiale en utilisant

seule-ment les données des générations subséquentes. Lorsqu’il y a sélection directionnelle sur un caractère _quantitatif, la variance _génétique et l’héritabilité sont réduites à la suite de la formation d’un _{déséquilibre} de liaison

_(phase

_gamétique).

_{Après quelques cycles} de

sélection directionnelle dans une _population de taille _infinie, un _équilibre stable est at-teint.

À

_partir de ce _moment, il n’y a plus aucune réduction de ces paramètres puisque

le _{déséquilibre} créé par la sélection est _compensé par la recombinaison. Dans plusieurs

situations, les données _disponibles pour estimer les paramètres génétiques proviennent de

populations en _équilibre stable. Une méthode _simple d’estimation de la variance _génétique et de l’héritabilité dans la _population initiale est _présentée. Cette méthode tient _compte de _l’effet d’une succession de _cycles de sélection sur la variance génétique et l’héritabilité pour inférer la valeur de ces _paramètres dans la _population initiale.

variance _{génétique /} héritabilité _/ estimation des _{paramètres génétiques / déséquilibre} de liaison

INTRODUCTION

The estimation of

_genetic

variances and heritabilities of

_quantitative

traits in

populations

under artificial or natural selection is a common

objective

in animal

breeding

and

_evolutionary

_biology

of natural

_populations.

Standard methods to

estimate

_genetic

variances and heritabilities when information is available on the parents and

_offspring

are the correlation _among sib and the

_regression

of

_offspring

on _parents

(Falconer, 1989).

_Analysis

of variance of half-sib

_yields

biased estimates

of

_heritability

if the _parents are a selected

_sample

from the

_population

_(Robertson,

1977;

Ponzoni and

James,

1978).

Unbiased estimates of

_heritability

_by

half-sib correlation can be obtained after

_correcting

for the bias induced

by

selection of

sires

_(Gomez-Raya

et

al,

1991).

Regression

of

_offspring

on _parents is not altered

by

selection of animals to be _parents

_{(Pearson, 1903)}

and therefore estimates of

heritability

by

regression

are unbiased

(Robertson,

1977).

In

both,

half-sib and

regression

analyses,

unbiased estimates of

_heritability

are obtained after one

_cycle

of

selection.

_Regression

estimates of

_heritability

are not unbiased for the accumulated reduction in

_genetic

variance after

_{repeated cycles}

of selection

_(Fimland,

_1979).

The

changes

in

_genetic

variance under selection were described

_by

Lush

(1945)

_using

genetical

arguments and a numerical

example.

Bulmer

(1971)

formally

established

the

_theory

to

_explain

the

_changes

in the

_genetic

variance under continued

_cycles

selection. Under the

_assumption

of an infinitesimal _gene effect model the

_genetic

variance and

_heritability

are reduced due to the

_build-up

of

_linkage

_(gametic

phase)

disequilibrium

in a

_population

of infinite size and with discrete

_generations.

After

_only

a few

_cycles

of directional or

_stabilizing

selection a

_limiting

or

steady-state

_equilibrium

value for these _parameters is

_approached.

At this

_point

the new

disequilibrium generated

by

the selection of parents is offset

_by

free recombination. Most animal

_populations

are

probably

in the

steady-state

or close to it since the

equilibrium

is

_approached

_very

_quickly.

The use of standard methods to estimate

genetic

variance and

_heritability

_yields

estimates of these _parameters in the limit

situation.

_However,

_{in many cases,} interest is on the _parameters in the

non-selected base

_population.

Sorensen and

_Kennedy

₍₁₉₈₄₎

have shown that mixed model

_methodology

_may be used to estimate the

_genetic

variance and

_heritability

in the base

_population.

_They

carried out a simulation

_experiment

for several

cycles

of mass selection and then

proceeded

to estimate

_genetic

variances

_using

a minimum variance

_quadratic

unbiased estimator

_(MIVQUE)

under the correct

model.

_They

found close _agreement between observed and simulated parameters.

The

_requirement

of

_using

the correct model

_{implies making}

use of the

_relationship

matrix with

_{complete pedigree}

information back to the base

_population.

Natural

populations

are

currently

under selection and

pedigree

information is not known. In

livestock

_species,

such as

dairy

cattle,

_pedigree

information is

_only

recorded from the

later _years. In

_general,

mixed model

_methodology

_requires

the

_genetic

variance of the base

_generation

as determined

_by

the available data and

_{corresponding pedigree}

information.

_Therefore,

if the available data and

_pedigrees

are

only

for animals at

the

_point

of selection

_equilibrium,

then the

_genetic

variance at selection

_equilibrium

is needed to evaluate animals

_by

mixed model methods.

However,

knowledge

of

genetic

variance in the base

_population

_(prior

to

_starting

_selection)

is _necessary

and/or

accuracy of evaluation differ from those in the current

_breeding

_programme.

Any

changes

in those _parameters alter the amount of

_{disequilibrium}

maintained in

the

population.

After a few

cycles

of selection a new

_equilibrium

will be

_approached

which can be

_predicted

with

_knowledge

of new selection

_intensity,

new _accuracy of

evaluation,

and the

_genetic

variance in the base

_population.

The

objective

of this _{paper is to} describe a method to estimate base

_population

genetic

variance and

_heritability

from data available at the

_{steady-state equilibrium.}

Use is made of effect of

_{repeated cycles}

of selection on

_genetic

variance and

heritability.

Assuming

the

_population

is at the

_equilibrium,

the base

_population

parameters are obtained

_by

_reversing

Bulmer’s _arguments.

THEORY

Consider an additive infinitesimal _gene effect model. The trait under selection

is determined

_by

a _very

large

number of loci with recombination rates of

_1/2.

Assume that selection

_intensity

is

_constant

across discrete

_generations

and that each individual

_belonging

to the same sex is evaluated with the same _accuracy.

Population

size is infinite. Selection is

_by

truncation. Assume that there are no

departures

from

_normality

after selection

_{(Bulmer, 1980).}

The basic

_theory

to

_explain

the

_changes

in

_genetic

variance in

_populations

undergoing

selection was first

_{given by}

Bulmer

(1971).

The

_breeding

value of an

individual i in a

_given

_generation

is:

where a8 and aD are the

_breeding

values of the sire and dam

_respectively

and ei

is the mendelian

_sampling

effect in individual i. ei is distributed

_normally

with variance

_((1/2)

_{0’ A}

_.

_{) 2}

in a

population

of infinite size where

or2A

O

is the

_genetic

variance in the base

_population.

The

_genetic

variance in the selected group of

parents is reduced

_by

kr

2

_(Pearson,

_1903),

where r is the _accuracy of selection and

k =

(Ø(x)/p)((Ø(x)/p) - x)

for selection of the

top

ranking

individuals

_(directional

selection)

and k =

2x(Ø(x)/p)

for selection of the middle

_ranking

individuals

(stabilizing selection);

x = standard normal

deviate;

§(x)

= ordinate _at cutoff

points

for p =

_proportion

selected. The

_genetic

_{variance in} the

offspring

can be

partitioned

into between and within

_family

_components. The

_{within-family}

variance is not affected

_by

selection of _parents and has value

_((1/2)

_QAo

_).

This is true on the

assumptions

of a _very

large

number of loci and infinite

population

_size,

ie no

_change

in the gene

frequencies

of the

_segregating

loci for the trait. The

_{between-family}

variance has a value of

(1-krLl)(1/2)0’!t-l’

_where

QAt

_

1

_is

the

_genotypic

variance in the

_{previous generation.}

_Therefore,

the

_genetic

variance in a

_{given generation,}

t,

assuming

different selection intensities and accuracy of selection in the 2 sexes is:

where r,,_, =

accuracy of selection of sires in

generation

t -

1; rDt-1 =

accuracy

of selection of dams in

_generation

t —

_1;

_k

₉

and

_k

_D

are the values of _parameter k

At the limit there are no further

_changes

in the

_genetic

variance since the new

disequilibrium generated

in that

_generation

is

_compensated

for

_by

free

recombina-tion.

_{Then, genetic}

variance becomes:

After some

_{algebraic manipulation}

this reduces to:

Assuming

constant environmental variance across

_generations

and

_substituting

expression

[1]

in the standard formula of

heritability,

the

_heritability

at the

equilibrium

limit is:

If the

population

is at the

_{steady-state equilibrium}

and records are available

to estimate

_genetic

_variance,

then estimates of base

population

parameters can

be found

_{by solving}

_expressions

_[1]

and

_[2]

for

_ar2A

and

_ho,

_{respectively,}

and

_by

substituting

true values

_by

their estimates.

_Thus,

genetic

variance and

_heritability

in the base

_population

can be obtained

_by:

where &dquo;&dquo;’&dquo;

denotes estimate.

!

If selection criterion is the individual

_phenotype

then

_ri&dquo;,

=

F2 D,

=

!2

and

expressions

[3]

and

_[4]

reduce to:

respectively.

In these

_{expressions k}

=

0.5k

8

₊

0.5k

D

.

The

required

estimates of the

genetic

variance and

_heritability

at

_equilibrium

can be obtained

_by

either

_regression

or maximum likelihood methods. It is

_{generally accepted}

that maximum likelihood

information is included in the

_analysis.

If REML

_(restricted

maximum

_likelihood)

account for

_selection,

_say, in

_generations

0 to _6, then it will also account for selection

in

_generations

6 to 10 when

_only

data from these

_generations

are available. In the

former _case, the _component of variance estimates the

_genetic

variance in the base

population,

and in the latter the

_genetic

variance in

_generation

6 which it is assumed

to be the

_equilibrium

_genetic

variance.

The

_approximate

_sampling

variance of the estimate of

_heritability

in

_the

base

population

can be obtained

_{by differentiating}

_expression

_[6]

with _respect to

!2 L7

Therefore,

the

_sampling

variance of the estimate of

_heritability

in the base

population depends

on a factor

_{f ,}

which is a function of

h)

and k because under

phenotypic

selection

_h

_{depends only}

on

h)

and

_k,

and on the

_sampling

variance of

_h

₁

_.

Values of the

_f

factor for different

£)

are

represented

in

figure

1 for

_varying

selected _percentages

_(p)

_{50%, 20%, 10%,}

and 1%. The value of

_hi

was obtained

_by

solving

expression

[6]

as a function of known

h) :

as described

by

_Gomez-Raya

and Burnside

(1990).

For traits with

_{heritability values}

less than

0.70,

f

is

_larger

than 1 and therefore the

_sampling

variance of

!2

_will

be

_increased

with _respect to the

_sampling

variance of the estimates at the limit

(Var

(hL)).

Selection

_intensity

_{appears to} have small effect on

_f.

In

_practical

animal

_breeding,

the

_performance

of relatives can be used to

max-imize response

by

the use of selection indices. For

example,

consider a

population

where sires are selected on the _average of records of d

_daughters

each with one record

and dams are selected on the average of n records each. Estimates of

heritability

in

the base

_population

can be obtained

by

_substituting

in

_{expression [4]}

the

appropri-ate

_equilibrium

values of accuracy for sires

T

SL

= [dhLf(4+(d-l)hlW/2

and dams

T

v

L

= [nhLf(l + (n-l)rêpL)]1/2,

where

rep

L

_{# [(8 fl! + 8 $! ) / (8 fl! + 8$! + 8$! )] ,}

8$ !

= estimated _permanent environmental variance and

_QT

_E

= estimated

DISCUSSION

In this _paper a method to estimate

_heritability

in the base

_population

from data

at the

_{steady-state equilibrium}

is

_presented.

The method to obtain estimates of

parameters at the

_equilibrium

is assumed to be unbiased

_by

selection of _parents in

that

_particular

_generation.

Estimates of

_{heritability by regression}

of

_offspring

on parents is unbiased

_by

selection in a

_{given generation}

_{(Robertson, 1977).}

Estimation

of

_{heritability by}

half-sib correlation is biased

_by

selection of sires

_(Robertson,

_1977;

Ponzoni and

_James,

_1978),

but estimates can be corrected

(Gomez-Raya

et

_al,

_1991),

and then final estimates free of selection bias can be obtained. Another alternative

is to use the method

_given

_by

Sorensen and

_Kennedy

_(1984).

_{They proposed}

the

estimation of

_genetic

variance in later

_generations

_using

the

_{MIVQUE algorithm}

and

paper

they

carried out a simulation

experiment

to test the

_validity

of this method. In

generation

7 actual

_genetic

variance had decreased from 10 to 8.41. The simulated

environmental variance was

_10,

so

_heritability

at the limit was

_{0.457, assuming}

that environmental variance was known without error. The _percentage selected in

males was 50%

_(k

_s

=

0.637)

in each

generation.

Dams were not selected

_(k

_D

=

0).

Using

expression

[6]

after

_substituting

estimated with true _parameter values and

corresponding

values of

_h

_i

_,

_k

_s

and

_k

_D

the

_heritability

in the base

_population

is

expected

to be

_0.491,

which _{is very} close to the simulated

_heritability

in the base

population

(0.50).

On the other

_hand,

Van der Werf

₍₁₉₉₀₎

carried out 2 different

simulation

_experiments

in which mass selection was

_practised

on males at different selection intensities

_{corresponding}

to _percentage selected _p = 10% and

p = 25%. He

proceeded

to estimate _components of variance

_using

REML

_(restricted

maximum

likelihood)

and the data from

_generations

4 and 5 with

_pedigree

information known

back to

_generation

3.

_Treating

sires as random in the model he obtained biased

estimates

_(8.58

for p = 10% and 8.71 for

p =

25%)

of the base

population genetic

variance

_(10).

If we assume that the

population

is at the

_steady-state

_equilibrium

in

generation

3 then

_genetic

variance in the base

_population

can be estimated

_using

[5]

after

_substituting

_appropriates

values of

_k(k

_s

= 0.830 for

p = 10%

and k,

= 0.759 for _p =

25%;

k

D

=

0)

and

!2

(0.45

for

p = 10% and 0.46 for

p =

25%).

The values of

_!2

_{can be obtained from} the estimates of

_genetic

_(8.58

for p = 10% and 8.71 for p =

25%)

and residual _variances

(10.44

for

p = 10% and 10.17 for

p =

25%)

given

by

Van der Werf

₍₁₉₉₀₎

in table II.

_Proceeding

in this way, estimates of the

genetic

variance in the base

_population

are 10.18

(p

=

10%)

and 10.23

(p

=

25%).

These values are _very close to the simulated

_genetic

variance in the base

_population

_(10).

The

_{slight discrepancy,}

in these

studies,

occurs because the formulae derived in this

paper have not taken into account the effect of

_inbreeding

in the reduction of

_genetic

variance.

_Throughout

this _paper,

_population

size has been assumed

_infinite,

and

therefore, inbreeding

effects on

_genetic

variance were not considered. Both natural

and livestock

_populations

are finite. The reduction in

_genetic

variance due to the

build-up

of

_{linkage disequilibrium}

occurs

_rapidly

in the first

_generations

whereas

inbreeding

effect is small but accumulates

gradually

in later

_generations.

After the

steady-state equilibrium

is

_achieved,

the

_genetic

variance reduces

_gradually

due

to

_inbreeding

and so does the amount of

_{linkage disequilibrium}

maintained in the

population. Thus,

correction for selection at this

_point

would not

_yield

estimates of the

_genetic

variance and

_heritability

in the

_original

base

_{population. Rather,}

the

estimates of these _parameters would be those obtained after

_relaxing

selection for

several

_generations,

in other

_words,

the

_genetic

variance due to _{the gene}

_frequencies

segregating

in the

_population

at the

_generation

in

_question.

In most

situations,

these

are the _parameters of interest because

_{they explain}

how much

_{genetic variability}

could be used in selection _programmes. Prediction of

the

_joint

effects of

_inbreeding

and selection on

_genetic

variance is rather difficult

(Robertson,

_1961;

Verrier et

_al,

1990;

Wray

and

_Thompson,

_1990).

The method described in this paper is able to correct for the bias

_generated

after

_{repeated cycles}

of selection

_assuming

_equal

information on each individual

evaluated and constant selection

_intensity

across

_generations.

In

_practice,

both

assumptions

may not hold. Methods to estimate

_breeding

values such as best linear

of livestock. Each individual

_breeding

value has a different _accuracy in BLUP evaluations. If

_pedigree

information is known back to the base

population

then mixed model

_methodology

could be used

_(Sorensen

and

_Kennedy,

_1984).

The effect of different _{accuracies among} selection candidates on

_genetic

variance is

not known. Further work is needed to

_incorporate

this kind of selection in the method

_presented

in this _paper.

_Changes

in selection

_intensity

across

_generations

result in

_changes

in the

_{disequilibrium}

in the

_population

_parameters over time. In natural

_populations,

selection

_intensity

could oscillate due to

_changes

in the

pattern of _{interaction among}

_species

_and/or

environmental fluctuations. In livestock

populations,

selection

intensity

may fluctuate due to

_changes

in

_production

_system

or in market conditions.

_Therefore,

the

_procedure

described in this paper would

give

only

approximate

values of the base

_{population heritability. However,}

oscillation

in the selection

_intensity

across

_generations

has small effect on the estimation

of

_heritability

in the base

_population

because the _parameter k

_{changes only}

_very

slightly

with selection

_intensity.

For

_example,

if we use a _wrong value of selection

intensity corresponding

to selection of the _{top 1%}

_(k

₉

=

_{0.903; k}

D

=

0)

in the simulation

_experiment

of Sorensen and

_Kennedy

_(1984),

then

_heritability

in the base

_population

after

_{using expression}

_[6]

is 0.504. This value is

_again

_very close

to the simulated

_heritability

_(0.50).

_Therefore,

even

though

selection

_intensity

is

not constant across

_generations

the method described in this _paper could be used

to

estimate,

in a _very

_approximate

_manner, the value of

_heritability

in the base

population.

ACKNOWLEDGMENTS

We _gratefully thank C Smith and B Villanueva for very useful comments. This research

was _{supported by} Instituto Nacional de _{Investigaciones Agrarias}

_(Spain)

and the Ontario

Ministry of

_Agriculture

and Food

_(Canada).

REFERENCES

Bulmer MG

₍₁₉₇₁₎

The effect of selection on

_genetic

_variability.

Am Nat 105,

201-211 ’

Bulmer MG

₍₁₉₈₀₎

The Mathematical

_{Theory of Quantitative}

Genetics. Clarendon

Press,

Oxford

Falconer DS

₍₁₉₈₉₎

Introduction to

_Quantitative

Genetics.

_Longman

Press, Essex,

3rd edn

Fimland E

₍₁₉₇₉₎

The effect of selection on additive

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_parameters. Z Tierz

Zuchtungsbiol96,

120-134

Gomez-Raya

L,

Burnside EB

₍₁₉₉₀₎

The effect of

_{repeated cycles}

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genetic

variance,

heritability,

and _response. Theor

_AppL

Genet

79,

568-574

Gomez-Raya

L,

Schaeffer

_LR,

Burnside EB

₍₁₉₉₁₎

Selection of sires to reduce

sampling

variance in the estimates of

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half-sib correlation. Theor

Appl

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