Optimum truncation points for independent culling level selection on a multivariate normal distribution, with an application to dairy cattle selection

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Optimum truncation points for independent culling level

selection on a multivariate normal distribution, with an

application to dairy cattle selection

V. Ducrocq, J.J. Colleau

To cite this version:

Original

article

Optimum

truncation

_points

for

_{independent culling}

level

selection

on a

multivariate normal

distribution,

with

an

_application

to

_dairy

cattle selection

V.

_Ducrocq

J.J. Colleau

Institut National de la Recherche

_Agronomique,

Station de

_Génétique

Quantitative et

_Appliquée,

Centre de Recherches de

_{Jouy-en-Josas,}

78350

_{Jouy-en-Josas,}

France

(received

1 March 1988,

accepted

19

_{September 1988)}

Summary — Independent culling

level selection is often

_practiced

in

_breeding

_programs because extreme animals for some

particular

traits are

_{rejected by}

breeders or because records on which

genetic

evaluation is based are collected

sequentially. Optimizing

these selection

procedures

for a

given

overall

_{breeding objective}

is

_equivalent

to

_finding

the combination of truncation thresholds or

culling

levels which maximizes the

_expected

value of the overall

genetic

value for selected animals. A

_{general Newton-type algorithm}

has been derived to

_perform

this maximization for _{any number of}

normally

distributed traits and when the overall

_probability

of

being

selected is fixed.

Using

a

power-ful method for the

computation

of multivariate normal

_{probability integrals,}

it has been

possible

to undertake the numerical calculation of the

optimal

truncation

points

when _up to 6 correlated traits or

stages of selection are considered

simultaneously.

The extension of this

algorithm

to the more

com-plex

situation of

_maximizing

annual

_genetic

_response

_subject

to nonlinear constraints is demonstra-ted

_using

a

_dairy

cattle model

involving

milk

_production

and a

_secondary

trait such as type. Conside-ration is

_given

to three of the four

_pathways

of selection: dams of bulls; sires of bulls; and sires of

cows.

Independent culling

level selection -

_dairy

cattle -

multistage

selection -

_{genetic galn}

-multivariate normal distrlbution

Résumé — Seuils de troncature

_optimaux

lors d’une sélection à niveaux

Indépendants

sur une distribution multlnormale, avec une

application

à la sélection chez les bovins laitiers. Une sélection à niveaux

_{indépendants}

est souvent

_pratiquée

dans les programmes

génétiques,

parce que les animaux extrèmes pour certains caractères sont

_rejetés,

ou _{parce que les données}

qui

servent à l’évaluation

_génétique

des animaux sont recueillies

_{séquentiellement. L’optimisation,}

pour un

_objectif

_donné, de ces

règles

de sélection

équivaut

à la recherche des seuils de troncature

qui

maximisent

_{l’espérance}

de

_{I objectif}

de sélection _pour les animaux retenus. Un

_algorithme

géné-ral de _type Newton est établi pour effectuer cette _{maximisation pour} un nombre

quelconque

de caractères distribués selon une loi mulünormale et

_lorsque

la

_probabilité

finale d’être retenu est bxée. A

_partir

d’une méthode

puissante

de calcul

d’intégrales

de lois multinormales, il a été

_possible

d’entreprendre numériquement

le calcul des seuils de troncature

_quand

_jusqu’à

6 caractères ou

étapes

de sélection corrélés sont considérés simultanément. L’extension de cet

_algorithme

à des situations

plus complexes,

comme la maximisation du

_{progrès génétique}

annuel sous

_plusieurs

contraintes non linéaires, est illustrée à travers le calcul de

_{règles optimales}

de sélection des mères à taureau, pères à taureau et

pères

de _{service pour} la

_production

laitière et pour un caractère secondaire tel _que le

_pointage

laitier dans un schéma de sélection

_typique

des bovins laitiers. sélection à niveaux

_{Indépendants -}

bovins laltlers - sélectlon

_par

_{étapes - progrès}

Introduction

It is often

_possible

to describe selection

_{objectives through}

_{linear combinations -}

aggre-gate genotype -

of

_breeding

values on several

_(m)

traits. Then the

_optimal

selection

pro-cedure consists of

_using

a selection index which combines observed values for several

(n)

sources of information

(Hazel

and

Lush,

1942).

However,

this

approach

is

_occasionally

not used for two main reasons:

a)

Practitioners sometimes

emphasize

the need to cull extreme animals

_(deviants

for

some

_traits)

because

_they

are deemed undesirable.

Then,

the selection

_objective

is

impli-citly recognized

as

_being

nonlinear. The cost of such a

_practice

with

_respect

to a strict

application

of the

_optimal

_{linear index may} not be

_justifiable

when this

_nonlinearity

is

unimportant

or

_{questionable.}

b)

The records

required

to

_compute

the selection index are not

_always

available

simul-taneously

and/or their cost does not

_justify

their collection for all the candidates for selec-tion. Therefore selection schemes involve different

_stages

which

_correspond

to

trunca-tions on the

_joint

distribution of all

_possible

records.

Within these

constraints,

it is

_{potentially interesting}

to evaluate and

_improve

the

effi-ciency

of

_practical

selection

_{procedures by determining}

_optimal

conditions of

_application

of

independent

culling

level selection. As far as we

_know,

the

published

works on this

topic

have been

_greatly

limited

_by

the number of variables considered.

_Generally

speaking,

studies on the

_algebraic

derivation of

_optimum

truncation thresholds with

cor-responding

numerical

_computation

have dealt with no more than 2 variables

_(Namkoong,

1970;

Evans, 1980;

Cotterill and James, 1981; Smith and Quaas,

1982). Recently,

Ducrocq

and Colleau

₍₁₉₈₆₎

treated

_examples

with 3 variables to illustrate

_potential

uses

of a numerical method to

_compute

multivariate normal

probabilities.

In

_{practice, though,}

the number of traits or selection

stages involved may be

significantly

larger.

Moreover,

the

_{algorithms previously}

considered have been

_specifically

_developed

for a

_given

num-ber n of variables and their extension to any n is not obvious

_though

algebraic

conditions which must be verified at the

_optimum

have been

_{reported (Jain}

and

Amble, 1962;

Smith and

Quaas, 1982).

Indeed,

as a

_consequence

of the limitation of the number of variables

considered,

the

optimization problem

has been restricted to rather

_simple

_types

of

_objective

_functions,

which may not

_adequately

summarize the overall

_efficiency

of selection schemes. In

par-ticular,

authors have not considered functions such as the annual

_genetic

_gain

of the

selection

_{objective - computed using}

Rendel and Robertson’s

₍₁₉₅₀₎

formula - in

rela-tively complex

situations

(e.g. involving

the 4

_paths

of transmission of

genetic progress).

This paper

presents

basic

_yet

_general,

i.e. for any n

_algorithms

which can be used for the

_computation

of

_optimal

truncation

_points

for a broad class of

_objective

functions with

one or more constraints.

_Theory

is

_developed

and

_applications

are

_presented

for the

general

multivariate

problem

considered

by

Smith and Quaas

_(1982).

A

_practical

example

of

_application

in the

_dairy

cattle context is described.

_{Corresponding}

numerical

Solution of Smith and Quaas’

_problem

in the

_general

case

Statement of the

_problem

Let u, xl, ....x,, be n+1 random variables with

joint

multivariate normal distribution:

u is the

_breeding

_objective

_{and the xis} are the observed variables.

The

_problem

is to _{find cl, Cn&dquo;&dquo;C2}so as to maximize

_Ep

_(u)

_subject

to:

where P is

_given

and

_represents

the overall fraction of candidates selected.

Notations

Let

_!&dquo;

_{(x; R}

_n

₎

be the standard multivariate normal

_density

of dimension n with variance covariance matrix

_R

_n

_.

Let

We need the

_following

recursive definitions for distributions conditioned on

_q::;;n-1

Solution

Using

the

_general

result of Jain and Amble

(1962)

we have:

Since

_Q(c

₁

_{,... c}

_n

₎

is to be

_equal

to a constant

P,

the maximization of

_Ep

_(u)

is tanta-mount to the maximization of

N(c

l

,..., c

n

).

The constraint

Q(c,,... c

n

)

= P is

incorporated

using

the method of

_{Lagrange multipliers (Bass,}

1961, p. 928;

Smith and

Quaas,

1982):

the

_optimal

truncation

_points

are those for which the

_partial

derivatives of the function

f

_(w)

=

N(c

1’

&dquo; e

n

)

+

_{X(Q(c,... c}

_n

_{) - P)}

with

_respect

to w’=

₍

_C1’

_&dquo;

_’’

_cn,

_X)

are 0. Â. is called a

Lagrange

multiplier.

The

_resulting

_system

of nonlinear

_equations

in w’

_{= (}

_C1’

... cn.

X)

is solved

iteratively

using

the multidimensional Newton’s method

_(Dennis

and

_Schnabel,

_1983).

Denote as

wct> the

_approximate

solution at iteration t

_(w(o)

is a

_{given starting value).}

A better estimate w(t+1)is

_computed

from:

The final solution wIt) = w* is obtained when

is

_sufficiently

small,

where

I I

h I I denotes any

norm.

As

_long

as the

_starting

value w(o) is not too far from w*

_(generally,

w(o) = 0 seems to be

a robust initial

value),

convergence is very fast

(quadratic

convergence: Dennis and

Schnabel,

1983).

c* is a local maximum for E

_{(u), provided}

’

is

positive

definite,

but

_nothing

_guarantees

that w* is a

_global

maximum for f

_(w).

Now,

note that:

where:

Another method exists for the derivation of these

_expressions,

first

_reasoning

on

deri-vatives of

_integrals

and then

_using

Jain and Amble’s

₍₁₉₆₂₎

formula on conditional

considera-tions on several u

_{(general problem}

with several constraints on several

_{objectives) (see}

Appendix).

Expressions

(3)

to

₍₁₁₎

include all the elements

_required

for the

_computation

of the

vector

_8f(w)/Bw

and the matrix

₍₈

₂

_f(w)/Bw

_Sw

_*

₎

in

_(1).

In

_particular,

It can be observed that the

equations (6f(w)/6c

i

)

= 0 in

(12)

for 1 !i!n _are linear in X.

The size of the

_system

of

_equations

to be solved can be

easily

reduced

by absorption

of the

_Lagrange

_multiplier.

For

_example,

we have:

is equivalent to:

Derivatives with

respect

to _{the cis} of the

_equations

in

₍₁₇₎

are

_required

for the

applica-tion of Newton’s method as in

_(2).

_They

are

_readily

derived

using (6)

to

(8).

Numerical

_applications

Studies on

_independent

_culling

level selection have been

mainly

limited to 2-trait selec-tion

_probably

because

_general

_{and efficient programs} to

compute

the multivariate normal

probability integrals

in Jain and Amble’s formula were not available for dimensions > 2.

and Soms

_{(1976) proposed}

a

_general

method characterized

_{by good precision}

when

cor-relation coefficients and truncation

_points

are not

too

extreme. For more details on this

method,

its

precision

and

_computation

times,

see

_Ducrocq

and Colleau

_(1986).

Dutt’s .

technique

is well suited for numerical

_applications

of the

_{optimization algorithm}

_presented

in _{this paper when up} to 6 selection

_stages

or traits are considered.

In this

_particular

case,

expressions (7)

to

₍₁₁₎

are

_simpler,

since

_Q

_jj

= 1 and

Q

ijk

= 0.

Algebraic

and numerical results are

_equivalent

to those

_given

_by

Smith and Quaas

(1982).

In

_(9),

_Q;ik

= 1. The

algorithm

described here leads to the same results as _those

pre-sented in

Ducrocq

and Colleau

(1986).

3) . n = 4 to 6.

Consider for

_example,

n traits with

_r;

_i

=

(-1

(j/20i)

for

1 <_i<_j<_n

and with economic

weight

mi= 1+i/20 1!i!n.

Table

_{I presents}

the truncation

_points

_qs on these traits which maximize

Ep (u

I c1,... c&dquo;)

when the overall selected fraction is P =

0.25, 0.025,

0.001. At iteration 0,

the cis were taken

_equal

to 0. The

_stopping

criterion for the Newton’s iteration was:

where

_E

_i

was the i th left hand side of

_system

(17).

Convergence

was fast and

_depended

on how far the initial value of the truncation

points

was from the solution.

_{Note, however,}

that in the

_{examples presented}

in Table I,

correlations between variables are not _very

_high

Q

_{r;! !}

1 S 0.3)

and the

_weights

of the

diffe-rent traits are of the same

_magnitude.

When this is not the case, the

optimal

selection

procedure

may involve no selection at all on one or several of these traits. The same

observation

_applies

to small overall selection

_{intensity (Young,}

1961;

Namkoong,

1970;

Smith and

Quaas, 1982; Tibau I Font and

Ollivier, 1984;

Ducrocq

and

Colleau,

1986).

In

limiting

cases

_(with

_{very low} or _very

_high

selection

_intensity

on one or several traits or

when correlations are

_extreme),

it should be remembered that the

_precision

of Dutt’s

algorithm

for

_computation

of multivariate

_{probability integrals}

may be

unsatisfactory

(Ducrocq

and

Colleau,

1986).

Then alternative methods may have to be used

_(e.g.,

Rus-sell et

_al.,

_1985).

An

_application

in the

_dairy

cattle context

Assumptions

Dairy

cattle selection is

_{performed through}

a _{sequence of}

_stages

which characterizes the

transition from one

_generation

_(g)

to the next

_{(g+1 ).}

In the additive

_polygenic

situation which is assumed for most of the traits selected in domestic

animals,

it is

_possible

to describe these

_stages

_through

truncation selection

These first include selection criteria

_{corresponding}

to _{the transition between}

genera-tions g and

g+1

(reproductive stage),

followed in the course of time

_by

those criteria used

during generation

g+1,

before the next

_{reproductive cycle.}

Our

_approach

for the

optimiza-tion of these successive selecoptimiza-tion

_stages

relies on the

_assumption

of multivariate

norma-lity

for these 2 criteria when candidates for selection are born. Such an

_assumption

is

plausible

in the additive

_polygenic

context,

especially

when

_heritability

values are low

(Bulmer,

1980, p.

154;

see also Smith and

Hammond, 1987,

for a discussion on this

point).

A more strident

_assumption

is that the

_dispersion

_parameters

of the

_joint

multiva-riate distribution remain constant

_through

the different selection

_cycles.

Breeding objective

and selection

_stages

Assume that the selection

_objective

in a

_dairy

cattle breed is a linear combination of 2

traits: &dquo;milk

production&dquo;

and a

_secondary

trait such as

_{&dquo;type&dquo;}

_(both

of these _{traits may be} themselves linear combinations of more

_{specific characters).}

A

_possible

_{sequence of} selection

_stages

which

approximates

what is often done in

_practice

is the

_following

(Figure 1):

1)

Dams of bulls

(DB)

of

generation

g are selected based on their estimated

_breeding

values X1

(for milk)

and _X2

_{(for type),}

with

respective

thresholds C1 and c2on the standar-dized variables. These dams of bulls are mated to sires of bulls

(SB)

of

_generation

g.

2)

The sons of these cows are _{progeny tested. Sires of} cows

_(SC)

and sires of bulls of

milk)

and X4

(for type).

Truncation thresholds on these 2 variables are different for SC

and SB

_(c

₃

_,

_c4and _{c5 c6,}

_{respectively).}

Selection of DB can be modelled as if it were

_performed

at birth of the male calves. This is essential in order to be able to invoke the

_restoring

of multivariate

normality

at

each

_generation.

Let RP be the

_{registered (with}

known

_pedigree)

and recorded

popula-tion of cows _{and let y denote the}

_proportion

of these cows

which

can be

_potential

dams of bulls

(e.g.

!y= 0.53 if Al sons are selected from cows with at least 2 known

lactations).

If it is assumed

_(as

in

_Ducrocq,

₁₉₈₄₎

that an _average of _nd = 6

potential

dams must be selected in order to obtain one male calf

entering

progeny test, it can be considered that

DB selection is

_performed

_by

truncation on the estimated

_breeding

values x, and X2 of the dams of nb=

(y RP)/n

d

male calves.

The

_expression

of the annual

_genetic

_gain

_{given by}

Rendel and Robertson

₍₁₉₅₀₎

is:

Selection on the dam of cow

_path

is

_{ignored (I!}

=

0).

Constraints

Three constraints are added here:

1)

The fraction

_Ty

of the

_population

bred to _{young sires is considered} as constant,

since in

practice

this is

_usually

the

_limiting

factor for the extension of _progeny test. In this

example,

the number of recorded

_daughters

_{per young sire}

_n

_v

_y

is also assumed

constant:

then,

the number

_ny

_{of young sires}

_{progeny-tested}

_{each year is fixed,} as well as their

_{repeatablity.}

2)

The number of sires of cows selected each year is determined

_by

the number of cows

_{(= (T-RP)}

+

_(1-Ty)

_RP)

to be bred to proven sires in the whole

population (T)

and the total number of doses

produced by

a

_given

sire

_during

his lifetime

_(AI).

- - ,--,

3)

The number of sires of bulls retained each year is constrained to be

_equal

to nsB, the number below which

_problems

of

_inbreeding

and reduction of

_{genetic variability}

are

feared.

Numerical methods and results

When constraints

_{(22), (23)}

and

₍₂₄₎

are

satisfied,

equations (18)

to

₍₂₁₎

lead to the

follo-wing

result:

where

L,

the sum of the

_generation

intervals over the 4

_paths,

is a constant in our case.

The combination of truncation

points

c;,

i=1,...

6 which maximizes

(25)

with the constraints

_{(22), (23)}

and

₍₂₄₎

is obtained

_{by equating}

to 0 the derivatives of f

_(w)

with

respect

to w’ =

(c

l

,..c,,

X,

_/.1,

v)

where:

and

X, p,

v are

_Lagrange

_multipliers.

The first and second derivatives of f

(w)

are

_readily

obtained

_using

the

_general

formu-lae

_given

in the

preceding

sections. The 3

_Lagrange

_multipliers

are eliminated

_through

tri-vial

_absorption.

The nonlinear

_system

to solve then involves 6 unknowns: the 6

trunca-tion thresholds. Solutrunca-tions obtained

_using

Newton’s method are

_presented

in Table

III,

where

_parameters

take the values

_given

in Table II. The

_stopping

criterion for the

where

_E

_i

_*

was the ith left hand side of the absorbed

_system

of

_equations.

Convergence

was fast -

_always

less than 8 iterations - as

_long

as the

_starting

value

c 0) (of c =

{c

;

6

a

,

’

was not too far from the solution. In contrast with the numerical results

presented

in Table

I,

c !o! = 0 does not lead to _{convergence, because the values of} c in

the next iterates are found to be

_{extremely large}

or

low; i.e.,

totally

unrealistic and in

regions

where the

_precision

of Dutt’s method is not

_good.

This behavior

_is,

at least

indi-rectly,

a _{consequence of the very}

_large

selection

intensity

of proven sires. To avoid this

situation,

more realistic

_starting

values must be chosen. For

_example,

c (0) can be

com-puted

as the truncation

_points

for which the conditions

_{(22), (23)}

and

₍₂₄₎

are satisfied with a

_given

distribution of selection efforts between milk and

_{type, i.e.,}

c (0) is such that

Q

(c!O)

=

d,/a,

Q

(c,co!! c!<0»

=

d!!

Q

(c,tot,

c!<o>,

C3

(o))

=

d

2

la,

etc... for _{some a.} In most

cases

_presented

in Table

_III,

a = 0.75. For

strongly

unbalanced

weights

for the 2 traits

(5:1),

convergence is obtained

only

for even

_larger

values of a

_(e.g.,

a =

0.99).

It should be noted that selection on

_type

was considered

_posterior

to selection on milk

production.

This

_assumption

has been chosen because this

_corresponds

to what is done in

_practice

for dams of bulls in France:

only

the best cows for milk

_production

are

evalua-ted for

_type.

This does not influence the values of

_optimal

thresholds but

_only

their use

The results in Table III underline the

sensitivity

of this

type

of

_computation

to the

eco-nomic

_weights

_assigned

to the 2 traits and their

_genetic

correlation.

_According

to the rela-tive economic

_weight

for

_type,

the

_optimal

_{situation may}

_correspond

to

_virtually

no selec-tion on

_type

_{(weights 5:1)}

or to a

_culling

on

_type

evaluation of between a

_quarter

and a

half of the candidates

_{(weights 2:1)}

on each

_path.

When the

_genetic

correlation between milk and

_type

varies from

_slightly

_{positive (0.15)}

to

_{slighly negative (-0.15) (thus covering}

the _range of values most often indicated in the

_{literature) optimum}

selection intensities and overall

_{genetic gain}

for the

_objective

function are not

_dramatically

modified. But

_then,

the

_genetic

trend for

_type

varies from a _{very favorable increase}

_(for

a

_{positive correlation)}

even when no selection on

_type

is

performed,

to a not

_negligible

decline

_(for

a

_negative

correlation and

_weights

3:1 or

_{5:1 ).}

These features were also

_pointed

out

_{by Ducrocq}

(1984).

Conclusion

The

_{previous example clearly}

demonstates

that,

at least in certain

_{situations, it is}

pos-sible to

_{algebraically}

and

_{numerically develop}

_algorithms

to

_compute

_optimal

selection

intensities in

_{relatively complex multistage breeding}

schemes and with several nonlinear constraints.

They

are not limited

by

drastic conditions such as

_2-stage

_selection,

uncor-related traits and/or very

simple optimization

criteria.

Indeed,

more

_complex

situations than that

_presented

in the

_example

can be

_envisaged.

For

_example,

the

_overlap

_of

gene-rations of bulls and cows can be accounted for in the determination of

_genetic

in Hill

_(1974),

Elsen and

_{Mocquot (1974, 1976)}

and

_Ducrocq

and Quaas

_{(1988): dividing}

each

_population

of candidates into

_homogeneous

cohorts of animals of same _age, sex

and

reproductive

role,

intracohort selection differentials can be

_computed

under the

assumption

that selection is

_performed

_{by retaining}

all the individuals whose estimated

genetic

values are above a

_unique

set of truncation thresholds for the n traits or

_stages

considered

_{(Ducrocq, 1984).}

_Then,

_computation

of elements in

₍₁₈₎

and their derivatives is tedious but

_perfectly

feasible.

However,

before

_{undertaking optimization}

studies of

complex

schemes,

it should be remembered that troublesome

_problems

of convergence may occur such as those found in the

_application

described. In the

_particular

case of the

dairy

cattle context, convergence

problems

were also found when we tried to relax the

assumption

of fixed number of young sires

sampled

(ny)

or the number of

_daughters

per young sire

_(n!),

i.e.,

their

_{repeatability}

*.

_Obviously,

further research is needed in a

wide-ning

field.

Finally,

the determination of an

_optimal

selection

_policy,

which is

_{&dquo;optimal&dquo; only}

from a

purely genetic standpoint,

should be followed

_by

a

_{sensitivity analysis.}

_{It appears} that the

annual

_{genetic gain AG,}

varies

_{only slightly}

over a _{wide range of} values for some

para-meters. Situations close to the

_optimum

_{may be} of

_greater

interest because

_they

are

simple,

more

_practical

or less

_expensive

to

implement.

The

_knowledge

of the true

opti-mum is nevertheless

_required

to assess this

_{&dquo;suboptimality&dquo;.}

References

Baas J.

_{(1961 )}

Cours de

_{Mafnomatfque,}

Masson, Paris, vol. II

Bulmer M.G.

₍₁₉₈₀₎

The Mathematical

Theory

of Quantitative Genefics. Oxford

University

Press,

Oxford

Cotterill P.P. & James J.W.

(1981) Optimizing

two-stage

independent culling

selection in tree and animal

breeding.

Theor.

_Appl.

Genet. 59, 67-72

Dennis J.E. & Schnabel R.B.

₍₁₉₈₃₎

Numerical Methods for Unconstrained

_Optimization

and

Nonli-near

_Equations.

Prentice-Hall,

Englewood

Cliffs

Ducrocq,

V.

_{(1984) Cons6quences}

sur le

progr6s g!n8tique

laftler d’une sélection sur des

carac-t6res secondaires chez les bovins. Genet Sel. Evol. _{16, 467-490}

Ducrocq

V. & Colleau J.J.

₍₁₉₈₆₎

Interest in

_{quantitative genetics}

of Dutt’s and Deak’s methods for numerical

_computation

of muftivariate normal

_{probability integrals.}

Genet Sel. Evol.18, 447-474

Ducrocq

V. & Quaas R.L.

₍₁₉₈₈₎

Prediction of

_genetic

_response to truncation selection across

gene-rations. J.

_Dairy

Sa. 71, 2543-2553

Dutt J.E.

₍₁₉₇₃₎

A

_{representation}

of muttivariate

_{probability integrals by integral}

transforms. Biome-trika 60, 637-645

Dutt J.E.

₍₁₉₇₅₎

On

_computing

the

_{probability integral}

of a

_general

multivariate t. Biometrika 62,

201-205

Dutt J.E. & Soms A.P.

(1976)

An

_{integral representation technique}

for

calculating general

mulfivarl-ate

_{probabilities}

with an

_application

to multivariate

_x

₂

_.

Commun. Stat 5, 377-388

Elsen J.M. &

_Mocquot

J.C.

(1974)

Recherches pour une rationalisation

technique

des sch6mas de sélection des bovins et ovins. Bull. Tech.

_Dept.

Genet. Mim., INRA 17 7

Elsen J.M. &

_Mocquot

J.C.

_{(1976) Optimisation}

du renouvellement des femelles dans les troupeaux

laitiers soumis au croisement terminal. Ann. Genet. Sel. Anim. _{8, 343-356}

Evans R.

₍₁₉₈₀₎

Constrained selection

_{by independent culling}

levels. Mim. Prod. 30, 79-83 *

In the latter case, derivatives of the objective function with _respect to _ndly are extremely complex to derive

Hazel L.N. & Lush J.L.

₍₁₉₄₂₎

The

efficiency

of three methods of selection. J. Hered. 33, 393-399

Hill W.G.

₍₁₉₇₄₎

Prediction and evaluation of response to selection with

overlapping generations.

Anim. Prod. 13, 37-50

Jain J.P. & Amble V.N.

_{(1962) Improvement through}

selection at successive stages. J. Indian Soc.

Agric.

Stat. 14, 88-109

Namkoong

G.

_{(1970) Optimum}

allocation of selection

intensity

in two _stages of truncation selection.

Biometrics 26, 108-115 s

Rendel J.M. & Robertson A.

₍₁₉₅₀₎

Estimation of

_{genetic gain}

in milk

_{yield by}

selection in a close herd of

_dairy

cattle. J. Genet. 50, 1-8

Russell N.S., Farrier D.R. & Howell J. (1985) Evaluation of mulfinormal

_{probabilities using}

Fourier series

_{expansions. AppL}

Stadst. 34, 49-53

Smith S.P. & Quaas R.L.

(1982) Optimal

truncation

points

for

independent culling

level selection

involving

two traits. Biometrics 38, 975-980

Smith S.P. & Hammond K.

₍₁₉₈₇₎

Assortative

_mating

and artificial selection: a second

_appraisal.

Genet Sel. Evol. 19, 181-186

Tibau I Font J. & Ollivier L.

(1984)

La selection en _{station chez le porc. Bull.}

Techn. Dept.

Genet.

Anim., INRA, 37 !

Young

S.S.Y

(1961 )

A further examination of the relative _efficiency of three methods of selection for

genetic gain

under less restricted conditions Genet. Res. _{2, 106-121}

Appendix

Another method for

calculating

the derivatives of N. Besides the notations defined in the text, let us define:

R

2

,ij

= _square of the

multiple

correlation coefficient between u and the

_predictors

_{xi, xj.}

By

first

_{differentiating}

N and then

using

Jain and Amble’s formula on the

_resulting

conditional

dis-tributions, it can be shown that:

This

approach

was shown to be

strictly equivalent

both

theoretically

and

_{computationally}

to that