Restricted multistage selection indices

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Restricted multistage selection indices

C Xie, S Xu

To cite this version:

Restricted

_multistage

selection indices

C Xie,

S Xu

Department of Botany

and Plant

_{Sciences, University of California,}

Riverside,

CA

_92521-0124,

USA

(Received

12 November

_{1996; accepted}

3

_April

₁₉₉₇₎

Summary -

Multistage

selection index is an

_important

extension to

_single

_stage selec-tion index for

_{genetic improvement}

of

multiple

traits.

_Recently,

a

multistage

selection

procedure,

referred to as selection index

_updating,

has been

_developed.

This

_procedure

combines several desired _aspects of

_{independent culling}

and selection index. In this

_study,

we

_directly

extend selection index

_updating

to cases of restricted

_multistage

selection in-dices

_{by imposing}

restrictions for

_solving

index coefficients. The

_resulting

indices restrict

genetic changes

to zero or to some

_proportion

in the chosen characters or linear functions of characters. As

such,

it makes

_multistage

selection indices more flexible. The

_possibility

of

_imposing

restrictions on different _stages is also discussed. A numerical

_example

is

_given

to illustrate the calculation of restricted

_multistage

selection indices.

independent culling

/

multistage selection

_/

restricted selection index

_/

sequential

selection

Résumé - Index contraints dans la sélection à étapes. L’index de sélection à

_plusieurs

étapes

est une extension

_importante

de l’index de sélection à une seule

étape. Récemment,

Xu et Muir ont

_développé

une

_procédure

de sélection à

_{plusieurs étapes appelée}

index de sélection avec mise à

_jour.

Cette

_procédure

combine

plusieurs

aspects intéressants de la sélection sur index et de la sélection à nivéaux

_{indépendants.}

Dans cette

étude,

nous

étendons directement la mise à

_jour

des index de sélection au cas d’index de sélection à

plusieurs étapes

en imposant des _{restrictions pour} trouver les

_coefficients

des index. Les index

correspondants contraignent

les

_{progrès génétiques}

à être nuls ou à être dans des

rapports donnés pour des caractères ou des combinaisons de caractères. De la sorte, les index de sélection

_multiétapes

sont

plus souples.

La

_{possibilité d’imposer}

des restrictions à

_{différentes étapes}

est aussi discutée. Un

exemple numérique

est donné pour illustrer le

calcul des index de sélection

_multiétapes

contraints.

sélection à niveaux _{indépendants}

_/

sélection _multiétapes

_/

index de sélection

con-traint

_/

sélection séquentielle

INTRODUCTION

The

_genetic

merit of a

_plant

or an animal is often defined as a function of several

that maximizes the

_improvement

of an overall

_genetic

value for all traits.

_Later,

Kempthorne

and

_Nordskog

₍₁₉₅₉₎

introduced the idea of restricted selection

_index,

which holds certain characters constant while

_allowing

other characters to increase

freely.

Tallis

₍₁₉₆₂₎

and Harville

₍₁₉₇₅₎

extended this idea to the case where some traits can be

_changed

_by

amounts

_proportional

to a

_{predetermined}

selection

_goal.

Rao

₍₁₉₆₂₎

_presented

a method for

_computing

an index to

_improve

one trait while

_requiring

_changes

in other traits to be in

_specific

directions. James

₍₁₉₆₈₎

showed how restrictions could

_{simultaneously}

be

_imposed

on the

genetic

results

of selection and on the index

_weighting

factors. The

_theory

of restricted selection indexes was further extended

by

Niebel and Van Vleck

(1982).

They

introduced fixed and

_proportional

restrictions when more than one selection index is used in a

population.

A detailed review of restricted selection indices was

_given

_{by Brascamp}

( 1984) .

When traits have a

_{developmental}

_sequence in

_ontogeny

or there is a

_large

dif

ference in the cost of

_measuring

various

_traits,

_{independent culling}

or

_multistage

index selection for

_multiple

trait

_{genetic improvements}

is the most efficient proce-dure with

respect

to cost

_{saving owing}

to the

_possibility

of

_{early culling}

_(Young

and

Weiler, 1960;

Young,

1964;

Namkoong,

1970;

Cunningham,

1975; Ducrocq

and

Col-leau, 1989;

Norell et

_al,

_1991).

In most _cases, the extension of these

_developments

for

independent

culling

selection to the case of

selecting

combination of traits at

each

_stage

is

_{straightforward. Recently,}

Xu and Muir

_{(1991, 1992)}

developed

a

mul-tistage

selection

_procedure,

referred to as selection index

_updating,

that combines

several desired

_aspects

of

_{independent culling}

and selection index. Use of selection

index

_updating

allows breeders to determine the

_optimum

truncation

_points

for the maximization of

_genetic

_gain

in

_aggregate

_breeding

_value,

economic return in terms

of

_genetic

gain

per unit

cost,

or

_profit.

In all

_multistage

selection

_{procedures presented by}

those

_authors,

the selection

goal

was to

_improve

all characters without restrictions.

However,

a breeder _may be interested in

_increasing

the overall

_{genetic gain}

with restrictions that certain traits

remain constant or

_change

in a

_{predetermined}

amount. For

instance,

a

_poultry

breeder may feel that mean _egg size should be

_kept

at a constant intermediate level while

_using

a

_multistage

selection index to maximize overall economic value. The breeder may manage this based on

_{body weight}

_{and egg}

_weight

measured at

the first

_stage,

and

_production

at the second

_stage.

_Hence,

a

_two-stage

restricted

selection index _may arise in

practice.

The _purpose of the

present

note is to extend restricted

_single

_stage

selection indices to the case of restricted

_multistage

selection indices.

THEORY

In

_subsequent

derivations the

_assumptions

and notation will be similar to those of Xu and Muir

_(1992).

For

_convenience,

the relevant notation is summarized below:

y = an n x 1 vector of

_phenotypic

values of n traits with elements _Yi,

g = a k x 1 vector of

_genetic

values of k traits to be

_improved

with elements gi,

w = _a k x 1 vector of economic

_weights

with elements _wj,

H

_{= w!g,}

the

_aggregate

_breeding

value,

P =

_Var(y),

an n x n

_phenotypic

variance-covariance

_matrix,

G =

Cov(y,

g

T

),

_{an n x} k

_genetic

covariance

_matrix,

and

G

r

= an n x r matrix

_{corresponding}

to the r columns

_{(restrictions)}

of G.

Suppose

that n traits are to be selected in m

_stages

(m

<

_n).

The

_phenotypic

values of n

_traits,

_y, can be

partitioned

into m

_subvectors,

_{Yl, y2 ... Y.&dquo;,,.} For

example,

if there are four traits to be selected in three

stages,

yl and y2 are

measured in the first

_stage,

y3 is obtained in the second

_stage,

and _y4 is measured

in the third

_stage,

then y =

[

Yl

Y2 y3

y4!T ,

Yi =

[Y

l

y2)T !

Y2 =

[Y

l

y 2

y3)T !

and y3 =

!yl

y

2 Y3

y4!T

For

simplicity,

vector n =

{n

l

, n

2

, ... ,

I nn represents

the

number of traits measured _{up to}

_stage

_m, where ni denotes the number of traits

measured _{up to} the ith

_stage.

Let z =

[Z

l

2

Z

Z3!T

be a 3 x 1 vector of the

_updated

selection indices defined

_by

Xu and Muir

_(1992),

then the indices for the above

example

are

is a 4 x 3 matrix with the ith

column, b

i

,

representing

the

_weights

for the selection indices at the ith

_{stage. Thus,}

the selection index at the ith

stage

is

According

to _{Yl, y2, ... , ym,} P and G can be

_{correspondingly partitioned}

as:

- - -

-Let

_Q

be a submatrix of P and A be a submatrix of

G, ie,

_Qij

=

[P]!

and

A

i

=

[G]

i

for

i, j

=

1, 2, ... , m.

Note that

[G]

i

differs from

G

r

defined

before. Selection response based on an index that maximizes the economic values of selected individuals is

_proportional

to _pZH

_(Kempthorne

and

_Nordskog,

_1959),

the correlation between index

_(Z

_i

₎

values and

_genetic

merits

_(H

=

w

T

g),

that is

is a minimum

subject

to the restrictions

where _gris a r x 1 vector of

_genetic

values for r restricted

_traits,

k

r

is a r x 1 vector

of desired

_values,

_Z

₍

_i

_-

_l

_{) =}

_,...,

_i

_[Z

_Z!i-1!!

is a vector of selection

_indices,

and

_{B!i_1!}

is a submatrix of B that contains index coefficients _{up to}

_stage

i -1. To insure there exists a

_solution,

it has to be that r < ni,

ie,

the number of restrictions at a

_given

stage

should be less than the number of characters at the same

_stage.

The first restriction

_[6]

constrains some characters to be

_changed

in

_{predetermined}

amounts

(Tallis, 1962).

The second restriction

_!7!,

_requiring

that indices at different

_stages

are

_independent,

insures the existence of an exact solution for truncation

_points

without

_resorting

to

_multiple

_integration

_(Xu

and

_{Muir, 1991,}

_1992).

After

_{introducing Lagrange}

_multipliers

of A and T

(A

is an r vector and T is an

m - 1

vector),

the

_{optimum b}

_i

is found

_by

_minimizing

the

_{following quantity}

with

respect

to

bi:

Vector differentiation

_gives

where

_R

_i

₍

_i-1

₎

=

B!i-1)(a(2-1)i!

From constraints

_[6]

and

_!7!,

we obtain

where

_R(

₂

_{_}

₁

₎

_i

=

Qi(i-1)B(i-1)

=

RZ!i-i)!

After

_{solving equations}

_{[lla, b]}

for A and

T and

_substituting

them into

_[10],

we have

where I is an

identity

matrix with the same dimension as

Q

ii

,

and

This is similar to the restricted

_single

stage

selection index of

_Kempthorne

and

_Nordskog

_(1959).

When r =

0, ie, G

r

=

0,

equation

[14]

gives

the

ordinary

unrestricted

_multistage

selection index

_{presented by}

Xu and Muir

₍₁₉₉₂₎

Let

Oz

i

be the coefficient of standardized selection

intensity

for the ith selection

index and Oz =

[Az

i

...

Az

i

...

Oz.&dquo;,!,

then the _vector of

_{genetic gains,}

OG,

is

predicted by

where

_OG

_i

_,

the ith element of

_AG,

is the

_{genetic gain}

of the ith trait. The total

genetic change

in

_aggregate

_breeding

value is

where w are the economic values of the

_traits,

and the

_A

_i

is defined as before.

Note that

_equation

_[17]

_provides

the linear

_relationship

between OH and

_Az,

but does not indicate how to select the

optimal

set of Az so that OH is maximized. Let ui be the truncation

point

corresponding

to selection

_{intensity Oz}

i

and

q

2 =

1-!(ui)

be the

proportion

selected for the ith

index,

where 4) is the cumulative

distribution function of the standard normal distribution. The

interrelationship

among

Az

i

,

ui

and q

i

is

Substituting

equation

[18]

into

_[17]

and

_using

Newton’s

_method,

the

_optimum

AH can be found

_{by maximizing}

AH with

respect

to u =

[

Ul

_{u2 ...}

u

mV

under

the constraint

where _p is a

_{predetermined}

total

_proportion

selected

_(Xu

and

_{Muir, 1992;}

Xu et

_al,

1995).

NUMERICAL EXAMPLE

Data from a classical

_example

of index selection

_given

_by

Hazel

₍₁₉₄₃₎

and

Cunningham

(1975)

are used to illustrate the

_computation

of restricted

_multistage

selection indices. The information used in a swine selection scheme

_comprises

the

five traits:

_pig’s

own market

_weight

_),

_l

_(y

_pig’s

own market score

_(y

₂

_),

_productivity

of dam

_(y

₃

_),

_average market

_weight

of

_pig

and littermates

_(y

₄

_),

and average market

are w =

!1!3,1, 2!.

The estimated

phenotypic

and

_genotypic

variance-covariance

matrices for the five traits are:

The overall

_proportion

selected is _{set to p} =

6%,

having

_a standardized selection

differential,

Az = 1.98538.

A

_three-stage

selection is

_performed

with _yl and _y2 selected at the first

stage,

y3 at the second

_stage,

and _y4 and y5 at the third

stage.

This selection

procedure

is

simply

denoted as n =

{2, 3, 5}.

We _{now assume} that the selection

goal

is to

_keep

_y2 constant or

_change

at a desired

_level,

while

_allowing

_yl and y3 to increase without

constraints. The unrestricted

_three-stage

selection has index coefficients that are determined

_by

_equation

_!15!,

_ie,

A multidimensional Newton’s iterative

_equation

_system

is used to search for the

optimal

truncation

_points

_(u

_i

_{s) (Xu}

and

Muir,

1992).

The sets of truncation

_points

(u

i

),

standardized selection differentials

_(Az

_i

_),

and the

_proportions

selected in each

stage

(q

i

)

are:

The

_gains

for three traits

_predicted

from this index are AG =

[18.98569

0.63881

0.33699]

Assume now that we want to maximize the response from selection with the

restriction of no

_change

(ky

2

=

0)

in attribute

y 2

, where the

subscript

y2 is used to

denote the

_{corresponding}

values for the second trait

₍

_Y2

₎

and the same thereafter. For this case,

_Gy

₂

=

[13.5093

2.2391 0.0000 8.1056

1.3435]!.

That

_is,

Gy

2

consists of the second column of G. The restricted index coefficients

_computed

from

_equation

_[14]

are:

Using

the same sets of uis,

s

Az

i

and qis as those obtained from the unrestricted selection

_indices,

we obtain

_{genetic gains,}

AG =

[15.5069

0.0000

0.3247]

T

,

and

_aggregate

economic

_value,

OH = 5.81835.

Therefore,

selection on restricted

multistage

indices should result in zero

_gain

in y2 as

expected.

As a further

check,

it can be verified that

Cov(gy

2

,

Z

i

)

=

Gy b

2

=

_0,

when

ky

2

= 0.

As a final

example,

we demonstrate an index that maximizes economic

_gain

while

allowing

y2 to reach a

_{predetermined}

value

_(Tallis,

_1962,

1968).

Let

C

OV

(gy

2

,

_Z

_i

_{) =}

- 0.2. Then the restricted selection

_procedure

has index

_{coefficients,}

A _summary of

_{genetic gains}

and

_aggregate

_breeding

values from

_single-

or

multi-stage

selection with or without restrictions is shown in table I. The

genetic gain

from

_one-stage

selection is used as a standard for

_comparing

the two- or

_three-stage

selection

_procedure.

The unrestricted

_two-stage

selection

_presented

here is the same as

Cunningham’s

selection

procedure

4

_except

that the actual truncation

_point

in the first

_stage

selection is

_{predetermined}

in

_{Cunningham’s}

_procedure

_4(u

=

1.2338)

with a

_proportion

selected on

_h

of

38%

and is

_optimized

in selection index

_updating

(u

=

1.5091)

with _a

_proportion

selected _on

_Z

_l

of

6.56%.

This leads _to the

aggregate

breeding

value of 8.0773 in this note and of 7.8675

presented by

Cunningham

(1975).

For a

_given

_stage,

the index is

_computed

from

_Z

_i

=

bi

Y

i’

As selection

_stages

advance,

more and more information is

_incorporated

into the index. In

_practice,

it is _necessary that restrictions on some characters should be

_incorporated

into the index at a later

_stage

when the trait measurements become available. This

problem

the index is derived with

_Cov(Z

_i

_,

_Z!)

= 0

for i !4 j.

For

example,

suppose we desired no

_genetic

_changes

for _y2 and 3y in a

_three-stage

selection with n =

{2, 4, 5}.

_Thus,

the constraint on _y2 would

_begin

at the first

_stage

and would

_begin

on _{y3 at} the

second

_stage.

Under such a selection

_procedure,

the restricted selection indices

_give

genetic gains

of AG =

!15.28095

_0.0000

O.OOOO!T

_{whereas the unrestricted selection}

indices

_produce

_{genetic gains}

of AG =

[18.90569

0.65909

0.41079]

T

.

From this

it can be seen that selection on restricted

_multistage

indices where restrictions are

incorporated

into index at different

_stages

results in zero

_gains

as

_expected.

DISCUSSION

There are a number of

_practical

situations to which the

_techniques

of the

_previous

sections could be

_applied.

For

_example,

in the selection of

_dual-purpose

bulls for

use in artificial

_{insemination,}

the bulls are first selected

_using

information from

their

parents;

they

are

_subsequently

screened for their own

_performance,

and

_finally

selected on the basis of a _{progeny test} of their

_daughters.

In that _case, selection can

be

_regarded

as a stochastic _process in continuous time. For

_practical

and economic reasons the selection is

_performed

in several

stages

in such situations

_(Cunningham,

1975;

Niebel and Van

_{Vleck, 1982;}

Norell et

_{al, 1991;}

Xu et

_al,

_1995).

When selection uses all information collected from a number of traits

_performed

in several

_stages,

the selection

_{objective usually}

remains constant. That

_is,

we are interested in the

_quantity

associated with

_primary

_product.

In some

_situations,

certain

_growth

_patterns

_may be more

_economical,

or otherwise more

_desirable,

than others. For

_instance,

in the

_example

_given

_by

Tallis

₍₁₉₆₈₎

one desirable

characteristic in fat lamb

_production

is

rapid

and

_early

increase in

_{body weight.}

On the other

_hand,

if the

_body

_weight

increases too

_much,

some

_production

and

_marketing

_problems

_may arise.

_Hence,

selection is a _process to use available

information to

_push

breeds towards the

_optimum

_growth

_pattern

_(Tallis,

_1968;

Harville,

1975;

Niebel and Van

_Vleck,

_1982).

The

_technique

of restricted selection index is

_justified

as a means to achieve a

_prespecified

selection

_goal

_(Lin,

_1989).

The

_{procedure given}

here either restricts

_genetic

_changes

to zero or to

propor-tional

_changes

in chosen characters or linear functions of characters. In this

_note,

the restrictions are achieved

_{by calculating}

_b

_i

under certain constraints and are unrelated to economic

_weighting

factors. The restrictions also

_might

be achieved

_by

modifying

economic

_weighting

factors

_(James,

_{1968; Lin,}

_1989).

The effect of restricted

_multistage

selection indices is similar to restricted

sin-gle

stage

selection indices. For

_instance,

the restriction is

_obviously

achieved at

the _expense of total

aggregate

genetic gain

as

_compared

with unrestricted indices

(table I).

Selection on restricted

_multistage

indices

_produces

an even lower

_gain

than that on restricted

_single

_stage

index. This is similar to cases of unrestricted selection where the

_multistage

selection index would result in less

_{genetic gain}

than the

_single

_stage

selection index

_owing

to

_{early culling}

_(Xu

and

_Muir,

_1991).

However,

reduction in

_{genetic gains,}

as one referee

_pointed

_out,

is not

_necessarily

coupled

with restricted index selection. Because economic

_weights

in the numerical

knowledge

of economic values _{is vague,}

_especially

over the

_long

term

_(Pesek

and

Baker, 1969;

Niebel and Van

Vleck, 1982;

Itoh and

Yamada, 1988a,

b).

Under

these

_{circumstances,}

_{it may} be more efficient to

_give

relative values or to

_assign

predetermined

amounts of

_{genetic gains}

to the traits

_(Tallis,

_1962;

Itoh and

_Yamada,

1988a,

b).

The coefficients for the restricted index

_presented

in this note are found so that the correlation between

_Z

_i

and H at each

_stage

is maximum with constraints

Cov(Zi, Z!)

= 0

for i =1=

_j.

An index constructed in that _manner will have

ex-plicitly

determined solutions for the truncation

_{points. However, genetic gain}

will be somewhat less than that obtained without constraints

_(Xu

and

_Muir,

_1992).

But Xu and Muir

₍₁₉₉₁₎

also showed that under certain conditions the

_efficiency

of restricted

_multistage

index selection _may

_greatly

exceed that of conventional

multistage

selection because the latter dose not

_incorporate

information from

pre-vious

_stages

of selection into the current

_stage.

_Owing

to

_{computational}

_advantages

over best linear unbiased

_prediction

_(BLUP)

under selection

_models,

this

_procedure

can be used to

_{effectively design performance}

_testing

and selection programs that

will

optimize profit

in commercial

_breeding

_(Xu

et

_al,

_1995).

For situations with

phenotypic

variances and

_covariances,

_{heritabilities,}

economic

_values,

and costs of

measurement

_known,

the breeder can

_investigate

_many

_options

relative to choices of traits and

_stages

with

respect

to

_maximizing

either total economic value of

_genetic

change

or

_profit.

ACKNOWLEDGMENTS

The authors would like to thank the referees for

helpful

comments in an

_early

draft. This research was

supported by

the National Research Initiative

Competitive

Grants

Program/USDA

95-37205-2313 to SX.

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