Changes in genetic correlations by index selection

HAL Id: hal-00893883

https://hal.archives-ouvertes.fr/hal-00893883

Submitted on 1 Jan 1991

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.

Changes in genetic correlations by index selection

Y Itoh

To cite this version:

Original

article

Changes

in

_genetic

correlations

by

index selection

Y Itoh

Kyoto University, Department of

Animal

Science,

Faculty of

Agriculture, Kyoto

606,

Japan

(Received

13

_{August 1990; accepted}

5 June

₁₉₉₁₎

Summary - A formula

_expressing

_changes

in

_genetic

variances and covariances

_by

index selection in one

_generation

is derived. Then

_changes

in

_genetic

correlation are discussed in 2

_simple

cases

_using

that formula. When 2 traits involved in the index have

equal

heritabilities and

_{equal weights,}

the

_change

in the

_genetic

correlation is

_{always negative}

and

_{generally large.}

When selection is on one _trait, the

_genetic

correlation with another trait after selection is inclined toward zero.

selection index

_/

genetic correlation

_/

Bulmer effect

Résumé - Changements des corrélations génétiques dûs à la sélection sur indice. Une

formule

est établie pour

exprimer

les

changements

des variances et covariances

_génétiques

dûs à la sélection sur indice. Les

_changements

des corrélations

génétiques

sont ensuite discutés dans 2 situations

_simples,

à l’aide de cette

_{formule. Quand}

les 2 caractères de l’indice ont des héritabilités et des

_coefficients

de

_{pondération égaux,}

le

_changement

de la corrélation

_génétique

est _toujours

_négatif

et

_{généralement}

_important.

_Quand

la sélection

se

_fait

sur un

_caractère,

les corrélations

génétiques

avec les autres caractères tendent à se

rapprocher

de zéro.

indice de sélection

_/

corrélation _génétique

_/

effet Bulmer

INTRODUCTION

Assuming

a trait influenced

_by

_many

_loci,

Bulmer

(1971)

showed that a substantial

change

in additive

_genetic

variance due to selection is caused

_by

_gamete

_phase

disequilibrium

and derived a formula for the

_{disequilibrium}

_component

of the

genetic

variance. Tallis

₍₁₉₈₇₎

proposed

an alternative

procedure

which obtained

the same result as that of Bulmer

(1971),

and extended it to be

_applicable

to

multiple

traits.

_Furthermore,

Tallis and

_Leppard

₍₁₉₈₈₎

studied the

_joint

effects of index selection and assortative

_mating

on

_multiple

traits. Index selection affects

*

various

_genetic

parameters.

Especially changes

in

_genetic

correlations are

_important

in

_{multiple-trait}

selection. The

objectives

of this note are to show

_explicitly

the formula

_expressing

the

_changes

in

_genetic

variances and covariances due to index selection in one

_generation

based on the results of Tallis

(1987)

and Tallis and

Leppard

(1988),

and to discuss the

_changes

in

_genetic

correlations in some

_simple

cases based on that formula.

A GENERAL FORMULA FOR CHANGES

IN VARIANCES AND COVARIANCES

Let P and G be

_phenotypic

and

_genetic

variance-covariance

_matrices,

_{respectively,}

and let b be a vector of index

_weights.

If the variance of a selection

_index,

denoted

by

a’

=

b’Pb,

is

changed by

selection _to be

QIS

=

(1

₊

k)a¡,

then P becomes :

which can be derived from _eq 6 of Tallis and

_Leppard

_(1988).

This result holds without

_assuming

a normal distribution. If a normal distribution and truncation

selection are

_assumed,

the value of k is determined

only by

a selection rate.

_Then,

from Robertson

₍₁₉₆₆₎

or Bulmer

_(1980,

_p

_{163, Eq}

_9.29),

k is

_expressed

as :

where x and i are

_respectively

the abscissa at the truncation

_point

and the mean of

the selected

_population

in the standard normal

_{distribution,}

that

_is,

i is the selection

intensity.

The formula obtained

_by

_substituting

₍₂₎

into

₍₁₎

becomes identical to

_Eq

10 of Tallis

₍₁₉₆₅₎

which was obtained

_by

_assuming

_normality

completely.

Because

i > 0 and i < x in truncation

_selection,

k < 0. And from

_QIS

>

_0,

k > &mdash;1.

_Thus,

the

_possible

_{range of k} is :

This

_inequality

will be used in the

_succeeding

sections. The value of k for various selection rates can be calculated from a table on the normal distribution

( eg

_Pearson,

1931,

table

_II)

as shown in table I.

Now consider cases where selection intensities and index

_weights

are different in 2

sexes, and let us denote b and k in the

_jth

sex

_{( j}

=

1, 2)

by

b

j

and

k!,

respectively.

When selection

_changes

P to

_{P 5j =}

_(I+K!)P

in the

_jth

_sex, it can be shown that G in the next

_generation

after selection becomes :

Substituting

this into

_(3),

we obtain :

It can be shown that the

_diagonal

elements of the latter term of the

_right

hand side of

₍₅₎

are

_{always negative.}

It follows that the additive

_genetic

_variances,

as

well as

heritabilities, always

decrease

_by

index selection

_irrespective

of the values

of

_genetic

parameters and index

_weights.

When the same index

_weights

are used in

both sexes,

(5)

reduces to :

where

k

=

(k

i

₊

k

2

)/2

and b is a common vector of index

_weights.

This

_equation

will be used in the

_following

sections to derive

_changes

in

_genetic

correlations.

The

_change

in G in

_only

one

_generation

of selection has been described above.

But this

_change

is

_transitory.

If selection is not

_practiced

in the next

_generation,

this

change

is halved and G _goes back toward its

_original

value. When index selection is

_repeated

for _many

_generations,

G and P continue to

_change

until

_equilibrium

is attained. The values of G and P in each

_generation,

as well as in

_equilibrium,

can

be

_{computed iteratively}

if

_Eq

24 of Tallis

₍₁₉₈₇₎

is used with

_K

_j

in

_(4).

Then

_k

_j

in

each

_generation

needs to be known.

_However,

the distribution of a

_population

after

selection is no

_{longer normal,}

_{so k}

_j

cannot be determined

_precisely.

If we assume

normality throughout,

we can

_compute

G and P in

_equilibrium.

But we do not

know whether this

_{approximation}

is

_appropriate

or not.

_Therefore,

we will discuss

mainly

the

_parameter

values after

_only

1

_generation

of selection and show the

the

_changes

of

_genetic

variances and covariances are of course taken into

_account,

but environmental variances and covariances are assumed to remain constant. When

genetic

parameters

are

changed, generally

index

_weights

should be recalculated in

each

generation. However,

in the

_following

_sections,

_{only simple}

cases are discussed

in which index

_weights

can be assumed to be constant.

EXAMPLE 1 : TWO TRAITS WITH

_EQUAL

HERITABILITIES AND

EQUAL

WEIGHTS

First we consider the

_simplest

index selection with 2 traits which have

_equal

heritabilities and

_equal

index

_weights

in both sexes. The traits are assumed to

be standardized to have unit

_phenotypic

variances for

simplicity. Then,

P and G before selection and b are :

where rp and rG are the

_phenotypic

and

_genetic

correlations and

h

2

is the

heritability. Substituting

these matrices and vector into

_(6),

we obtain :

Thus,

the

_genetic

correlation in the next

_generation

after selection becomes :

The

_change

in the

_genetic

correlation is :

It is obvious that the numerator of

₍₈₎

is

_always

_negative.

On the other

_hand,

the

fact that the denominator of

₍₈₎

is

_always

_positive

can be

_proved

in the

_following

way. Because the environmental variance-covariance

_matrix,

P -

_G,

is

_positive

definite,

its characteristic roots should all be

_positive.

It can be shown that the

characteristic roots of P - G in this

_example

are :

From

_À

_l

>

_0,

we obtain :

Thus,

it has been

_proved

that

_{always Arc}

<

_0,

that

_is,

rGs < rG.

Therefore,

the

_genetic

correlation after selection is inclined toward -1 as

_compared

with the

genetic

correlation before selection. This effect of the selection is undesirable for

the selection in the next

_generation.

Example

values of

Arc

for some rG and rp are shown in table II where we assume that

h

2

= 0.5 and the

_selection

rates are 0.1 in males and 0.5 in females.

From table

I,

it is found that

k

= -0.7337

approximately

in this case. Table II

contains

_{only the}

combinations of rp and rG that

satisfy

the condition that

A

i

> 0 and

_A

₂

> 0 where

_.!1

and

_a

_l

are defined in

(9).

This table shows that

always

Arc

< 0 as stated above and that the

_change

in rG is

generally large

in

spite

of

the moderate selection rates and

_only

one

_generation

of selection. Table III shows

the

difference,

denoted

by

Or!;,

between the initial value

of r

G

and its

_equilibrium

value attained after

_repeated

selections for _many

_generations

on the same condition

as in table II.

Although

the value of

Ar%

in table III is

approximate,

it is obvious

that

_OrC

has the same

_tendency

as

Ar

G

and its absolute value is

generally

_very

large.

From these

_facts,

we conclude that the

_genetic

correlation could be

changed

easily

in undesirable direction

_by

index selection.

Here we comment

_briefly

on a case where 2 traits have

_{antagonistic weights.}

Let us

_{put :}

and use the same P and G as in

(7).

Then,

in the same _way as the above

_example,

the

_change

in the

_genetic

correlation

_by

the selection

_using

this

_antagonistic

_weights

This

_equation

can also be obtained from

₍₈₎

_by

_affixing

minus

_signs

to all of _rG, rp

and

Arc.

Using

A

2

> 0 in

_(9),

it can be shown that

_Arc

in

₍₁₀₎

is

_always

_positive.

Thus,

in this case with the

_antagonistic

_weights,

the

_completely

reverse results are

obtained in

_comparison

with the above case with

_equal

_weights.

_However,

in both

cases, the

genetic

correlation is

_changed

in undesirable direction. If one wants the numerical

_examples

of

_Ar

_G

and

_t1rê

_using

the

_{antagonistic weights, they}

can be

obtained from tables II and III

_reverting

all the

_signs

of the value of rG, rp,

Arc

and

_t1rê.

EXAMPLE 2 : SELECTION ON ONE OF TWO TRAITS

Next,

we consider 2 traits

_again,

but selection is based on

_only

one trait. The traits

are assumed to be standardized to have unit

_phenotypic

variances as in

_Example

1.

_Then,

P and G before selection and b are :

where

_hf

_and

_hi

_{are the} heritabilities of the 2 traits and rP and rG are as in

Example

1. From

_(6),

the

_genetic

variance-covariance matrix in the next

_generation

after selection is :

It is

_interesting

that r!!s does not

depend

on _{rp at} all. From

(12),

we obtain

The maximum

_ofr

_cs/

_r

_c

is attained when rG =:1:1

irrespective of k

and

_hi,

and

its maximum is +1. On the other

_hand,

its minimum is attained when rG =

_0,

then :

Therefore,

the

_possible

_range of

_r

_G

_sIr

_G

is :

This indicates that the

_sign

of a

_genetic

correlation is not

_changed

_by

this

type

of selection and the

_genetic

correlation after selection is inclined toward zero.

Example

values of rGs and

Ar

G

for some

_r

_C

_’s

are shown in table IV where we assume that

h

2

= 0.5 in both traits and that the selection _{rates are} 0.1 in males

and 0.5 in females as in table II. Table IV also contains

&eth;.rG

which is the difference

between the initial value of

_genetic

correlation and its

_equilibrium

value. Note that this table does not have the column of rP. This is because the results does not

depend

on _rp as described above. From table

_IV,

we find that _rGS is inclined toward zero from its

original

value rG as stated above and that

Ar

G

and

Ar* G

are

not so

_large

_generally

as

compared

with those in table II and 111.

Changes

in

_h2,

denoted

_by

_Ah’,

induced

_by

selection on the first trait can be

It is obvious that

_.6.h!

_<

_0,

so that

h2

_always

decreases

_irrespective

of the values of

genetic

parameters.

This fact holds

_good

not

_only

in this case, but also in every case

of index selection. It is

_only

a

_{special example}

of the

_general

fact that heritabilities

always

decrease

_by

index selection

_irrespective

of the values of

_genetic

_parameters

and index

_weights

as stated above in this note.

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

Pearson K

₍₁₉₃₁₎

Tables

_for

Statisticians and

_{Biometricians,}

Part II.

_Cambridge

University

Press,

Cambridge

Robertson A

₍₁₉₆₆₎

A mathematical model of the

_culling

process in

dairy

cattle.

Anim Prod

_8,

95-108

Tallis MG

₍₁₉₆₅₎

Plane truncation in normal

_populations.

J R ,Stat Soc Ser B

_27,

301-307

Tallis MG

₍₁₉₈₇₎

Ancestral covariance and the Bulmer effect. Theor

_Appl

Genet

73,

815-820

Tallis

_{MG, Leppard}

P

₍₁₉₈₈₎

The

_joint

effects of selection and assortative

_mating