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Increasing long-term response to selection
Nr Wray, Me Goddard
To cite this version:
Original
article
Increasing long-term
response
to
selection
NR Wray
ME
Goddard
Livestock
Improvement Unit,
Victorian Instituteof
AnimalScience,
475,
MicklehamRoad, Attwood,
Victoria3049,
Australia(Received
8September
1993;accepted
25May
1994)
Summary - Selection on estimated
breeding
value(EBV)
alone maximises response toselection observed in the next
generation,
butrepeated
use of this selection criterion does notnecessarily
result in a maximum response over alonger
time horizon. Selection decisions made in the currentgeneration
have at least 2 consequences.Firstly, they
influence the immediategenetic
response to selectionand, secondly, they
influence theinbreeding
of the next andsubsequent generations.
Accumulation ofinbreeding
has anegative impact
on futuregenetic
responsethrough
reduction in future genetic variance and anegative
impact on futureperformance
ifinbreeding depression
affects the selected trait.Optimum
selection decisionsdepend
on the time horizon of interest. If this isknown,
then a
breeding objective
can be defined. A selection criterion isproposed
in which thepositive
contributions of a selected group of parents to immediate genetic response(determined
by
their averageEBV)
is balancedagainst
theirnegative
contribution tofuture
genetic
response(determined
by
their contribution toinbreeding).
The valueassigned
to the contribution toinbreeding
is derived from thebreeding objective.
Selection of related individuals will be restricted if the detrimental value associated withinbreeding
is
high;
restrictions on the selection ofsibs, however,
is flexible fromfamily
tofamily
depending
on theirgenetic
merit. A selectionalgorithm
isproposed
which uses the selection criterion to select sires on 3 selectionstrategies,
to select oni)
a fixed number ofsires;
ii)
a variable number of sires each allocated anequal
number ofmatings;
oriii)
a variable number of sires allocated anoptimal proportion
ofmatings. Using
stochasticsimulation,
these selectionstrategies
for sires arecompared
with selection on EBV alone. Whencompared
at the time horizonspecified by
the selectiongoal,
theproposed
selection criterion is successful inensuring
ahigher
response to selection at a lower levelof inbreeding
despite
the selection of fewer sires. The selection strategyiii)
exploits
random year-to-yearvariations in the
availability
of individuals for selection and is successful inmaximising
*
Correspondence
and reprints:c/o
PMVisscher,
Roslin Institute,Roslin, Edinburgh
EH25
9PS, Scotland,
UK.t Present address: Animal Genetics and
Breeding
Unit,University
of NewEngland,
response to the selection
goal.
The derivation of the valueassigned
toinbreeding
isnot exact and cannot guarantee that the overall maximum response is found.
However,
simulation results suggest that the response is robust to the detrimental value
assigned
toinbreeding.
artificial selection
/
selection response/
inbreeding/
BLUP/
computer simulation Résumé - Accroissement de la réponse à la sélection dans le long terme. La sélection sur la valeurgénétique
estimée(VGE)
considérée seule maximise laréponse
à la sélection observée dans lagénération
qui suit, mais l’utilisationrépétée
de ce critère de sélection ne garantit pas nécessairement laréponse
maximaLe sur unelongue période.
Les décisions de sélection prises àchaque génération
ont au moins 2conséquences.
Ellesinfluencent
d’abord laréponse génétique
immédiate à lasélection,
et ensuite elles déterminent le niveau deconsanguinité
dans lagénération
suivante et lesgénérations
ultérieures. L’accumulation de laconsanguinité
a uneffet négatif
sur laréponse future
en réduisant la variancegénétique
et uneffet négatif
sur laperformance future
si le caratère sélectionné subit unedépression
deconsanguinité.
Les décisions de sélectionoptimales dépendent
de laperspective considérée. Si celle-ci est
déterminée,
alors unobjectif
de sélection peut êtredéfini.
On propose ici un critère de sélection danslequel la
contribution positive d’un groupe de parents sélectionnés à laréponse génétique
immédiate(déterminée
par leurVGE
moyenne)
est contrebalancée par leur contributionnégative
à laréponse génétique
future
(déterminée
par leur contribution à laconsanguinité).
La valeur de la contribution à laconsanguinité
est dérivée del’objectif
de sélection. La sélection d’individusapparentés
entre eux sera soumise à restriction sil’effet
nuisible de laconsanguinité
estfort ;
les restrictions à la sélection de germains peuventcependant
varier d’unefamille
à une autreen
fonction
de leur valeurgénétique.
Unalgorithme
de sélection est proposé pour établirle critère de sélection des
pères
enfonction
destratégies :
sélectionneri)
un nombrefixe
depères, ii) un
nombre variable depères
à chacundesquels
on attribue un nombreégal d’accouplements,
ouiii)
un nombre variable depères
entrelesquels
onaffecte
lesaccouplements
d’une manièreoptimale.
À
l’aide de simulationsstochastiques,
cesstratégies
de sélectionpaternelle
sontcomparées
à la sélection sur VGE seule.Quand
on compareles résultats au terme de la
période spécifiée
dansl’objectif
desélection,
le critère de sélectionproposé
réussit à assurer uneréponse
à la sélectionaugmentée
et un niveau deconsanguinité
diminué endépit
d’un nombreplus faible
depères
sélectionnés. Lastratégie
de sélectioniii)
exploite
lesfluctuations
aléatoires des nombres depères disponibles
d’une année à l’autre et maximise laréponse
pourl’objectif
de sélection. le calcul de la valeur attribuée à laconsanguinité
n’est pas exacte et ne peut pas garantir que laréponse globale
maximale est obtenue.
Cependant,
les résultats de simulationsuggèrent
que laréponse
prédite
est robuste vis-à-uis deseffets
nuisibles attribués à laconsanguinité.
sélection artificielle
/
réponse à la sélection/
consanguinité/
BLUP/
simulation surordinateur
INTRODUCTION
When
breeding
programmes areconsidered,
it iscommonly
assumed that newparents are selected on the criterion of
highest
estimatedbreeding
values alone. This criterion results in maximum response to asingle
generation
ofselection,
but
repeated
use of this criterion does notnecessarily
result in maximumgenetic
response over a
longer
time horizon. Selection decisions made in the currentto
selection,
theimpact
of which is seenimmediately
in thegenetic
merit of theiroffspring
born in the nextgeneration.
Secondly, they
influence theinbreeding
ofthe next and
subsequent
generations.
Accumulation ofinbreeding
has anegative
impact
on futuregenetic
responsethrough
reduction in futuregenetic
variance anda
negative impact
on futureperformance
ifinbreeding depression
affects the selectedtrait.
Dempfle
(1975)
showed that selection limits achieved with mass selection couldbe
surpassed
by within-family
selectionparticularly
when selection intensities andheritability
werehigh; within-family
selection caused lower levels ofinbreeding
and hence ensured
higher
genetic
variance in thelong
term. Best linear unbiasedprediction
(BLUP)
is now thepreferred
method for calculation of estimatedbreeding
values(EBVs).
The EBVs of relatives arehighly
correlatedespecially
ifBLUP is
applied
under an animalmodel;
selection on BLUP EBVs alone can resultin
higher
rates ofinbreeding
than under massselection,
and hence availablegenetic
variance is more
quickly
reduced.Indeed,
in some circumstances it has been foundthat mass selection can result in
higher long-term
genetic gain
than selection onBLUP EBVs
(Quinton
etal, 1992;
Verrier etal,
1993),
and thepractice
of selectionon BLUP EBV alone has been
questioned.
Some authors haveinvestigated
theconsequences of
ignoring
records on some relatives( eg,
Brisbane andGibson, 1993,
scheme
SUBOPT)
but thisimplies
thatignorance
can sometimes bepreferred
toknowledge.
Others havesuggested
that anartificially
high
heritability
could beused in the BLUP
equations
(eg,
Toro andPerez-Enciso, 1990;
Grundy
andHill,
1993)
whichgives
moreweight
to individual rather than relativesrecords,
but this confuses the method ofprediction
ofbreeding
values with theselection
criterion. Theintuitively
attractive answer must be to combine the EBVs(calculated
in theoptimal
way)
into a selection criterion thattruly
reflects theunderlying
selectiongoal, thereby
increasing,
rather thandecreasing,
the amount of information includedto make selection decisions.
Several authors have
investigated
selection criteria thatattempt
to ensurehigher
genetic
response over alonger
time horizon. These includeimposing
restrictions onthe numbers of sibs selected from any
family
(eg,
Toro andPerez-Enciso, 1990;
Brisbane and
Gibson, 1993;
Grundy
andHill,
1993),
selection on a criterion whichalters the
emphasis
given
towithin-family
andfamily
information( eg,
Dempfle,
1975;
Toro andPerez-Enciso,
1990;
Verrier etal,
1993;
Villanueva etal,
1994),
selection of an increased number of
parents
butallocating
morematings
tohigher
ranked
parents
so that the overall selectionintensity
is the same as if a smaller number of sires had been selected(Toro
andNieto,
1984;
Toro etal, 1988,
Lindgren,
1991),
selection on a criterionEBV
i
-weight
X,
where X is the averagerelationship
of the individual with the other selected
parents
(Goddard
andSmith, 1990a;
Brisbane andGibson,
1993),
or linearprogramming
to determine the set ofmatings
out of all
possible
sets that maximises response to selection under agiven
restrictionfor
inbreeding
(Toro
andPerez-Enciso,
1990).
All of these alternatives have met withsome success in
gaining
higher
genetic
response at lower levels ofinbreeding
oversome time horizon. The methods all
aim,
in an indirect way, to maintaingenetic
variance and restrict
inbreeding,
but the actual criterionby
which this is achieved isperhaps arbitrary.
Noguidelines
have beenpresented
whichmight
ensure thatIn
general, investigation
ofbreeding
programmes assumes themating
of a fixed number of sires with a fixed number of damsgenerating
a fixed number ofoffspring
each
generation.
Theexpected
optimum proportion
ofparents
to select is a functionof the ratio of the time horizon of the
breeding
programme and the number of animals available for selection(Robertson,
1970;
Jodar andLopez-Fanjul,
1977).
Inpractice, however,
the number of females selected is constrainedby
the femalereproductive
rate andtesting
facilities foroffspring. By
contrast,
restrictions onthe number of sires are
likely
to be muchbroader,
ifthey
exist at all(particularly
when artificial insemination is
used).
Thegenetic
merit of individuals available forselection each
generation
ispartly random,
thereforeoptimum
selection decisions thatexploit
this randomness may result in different numbers of siresbeing
selectedat each
generation
and differential usage of the sires.In this paper, an
attempt
is made toprovide
a selection criterion which isexplicit
in itsgoal
ofmaximising
response to selection over aspecified
time horizon. As wellas
reducing
genetic variance,
inbreeding
may cause adepression
inperformance.
Selection
goals
are considered for which the aim is to maximisegenetic
response less the cost ofinbreeding
depression
over some time horizon. Selection rules arepresented
which aredyanmic
in theirattempt
toexploit
thegenetic
merit ofparents
which ariserandomly
(in
part)
eachgeneration.
METHODS
The aim is to find a selection criterion that
weights
selection response versusfuture
inbreeding
in alogical
way. The relevantweights
mustdepend
on abreeding
objective,
and therefore the definition of thebreeding objective
is ourstarting
point.
The derivation of the selection criterion is based on the maximisation of response
to the
breeding objective.
However,
since the selection criterion affectsinbreeding
and the level ofinbreeding
influences theoptimum
selectioncriterion,
it is notpossible
to find a selection criterion which is constant eachgeneration
and whichcan
guarantee
maximisation of thebreeding objective. Therefore,
the selection criterion is notexpected
to ensure maximisation of thebreeding objective,
butit is
expected
to result inhigher
response to thebreeding objective
than selectionon EBV alone.
Finally,
the selection criterion is used inconjunction
with differentselection
algorithms
which may allow different numbers ofparents
to be selectedor allocate different numbers of
matings
to eachparent
in order to maximise the effectiveness of the selection criterion. Forsimplicity,
we consideronly
selection onmales and the selection of females is assumed to be at random.
Breeding
objective
where
AG
j
is the increase ingenetic
merit of animals born ingeneration j,
F
j
is their averageinbreeding
coefficient and D is thedepression
inperformance
per unitof
inbreeding.
F
j
can beexpressed
aswhere OF is the rate of
inbreeding
pergeneration.
AG
j
can beapproximated by
where
AG
L
is theasymptotic
rate ofgain
pergeneration
expected
in an infinitepopulation
afteraccounting
for the effects of selection(the
’Bulmereffect’, Bulmer,
1971).
Thisapproximation
forAG
j
arisesby
assuming,
firstly,
thatAG
j
ispredicted
by
ir! _ 1QG,!-1
where i is the selectionintensity
eachgeneration,
and rj-and
U2
,
-1
are the accuracy ofselection
andgenetic
variance,
respectively,
pertaining
toanimals born in
generation j-1.
Secondly,
it is assumed thatQG,!_1 .a !c,L(1-F!-1)
and rj-1 a5r
L
(1 -
F!-1)1/2.
Thus,
it is assumed that rate ofgain
(and
itscomponents)
are reduced eachgeneration
by
the level ofinbreeding
achieved.Substituting
theexpressions
forAG
j
andF!
into(1!,
R
t
can be written as:Ignoring
terms ofhigher
order than linear in OFthen,
which is the same as the
expression
usedby
Goddard and Smith(1990b).
The linearapproximation
to OF should besatisfactory
if OF <1%
as it is in many livestockpopulations
(if
OF = 0.01 and t =30,
the first formula forR
t
is 26.OOG -0.26D,
while the formulausing
the linearapproximation
isR
t
:
25.7AG -0.30D).
Forsmall, intensely
selectedpopulations
that havehigher OF,
theapproximation
maybecome less
acceptable;
to check the effect ofthis,
the simulations to bereported
have OF as 1 -
3%.
Thebreeding objective
for eachgeneration
can be written aswhere
Equation
[2]
implies
that in each of the tgenerations
of selection there is apositive
contribution to the
breeding
objective
ofgenetic
response and there is a detrimentalcontribution to the
breeding objective
as a function of the rate ofinbreeding.
Selection criteriongeneration
of selection decisions iswhere sm
is
a vectorcontaining
theproportion
ofoffspring
born to each sire andb&dquo;,,
is the vector of estimatedbreeding
values(EBVs)
of sires deviated from theoverall mean of EBVs of all available sires and dams
prior
to their selection. s /and
b
are definedanalogously
for dams. The averagecoancestry amongst
theparents
weighted by
their contribution to the nextgeneration
represents
the effecton
inbreeding
inducedby
the selectiondecisions,
that iswhere
A
mm
,
A
m!
and
A
f
represent
the additivegenetic
relationship
matricesbetween
sires,
between sires anddams,
and between damsrespectively.
The rate ofinbreeding
is(w! - w! _ 1 ) / ( 1 - w! _ 1 ).
Assuming
thatwj -
1 is
small,
as it is in mostcommercial livestock
populations,
the rate ofinbreeding
isapproximated by
For
example,
when sires and dams are unrelated and arenon-inbred, A
mm
andAf
f areidentity
matrices,
A&dquo;,,
f
is null and if allN&dquo;,,
sires andN
f
dams are usedequally
(ie,
8m =1 N;!
and
s f =1N
f
where
1 is a vector ofones)
then w!_1 =0,
OF = Wj =
1/8N!I
+ 1/8NiI
(Wright,
1931).
Substituting
theexpressions
[4]
and[5]
intoAG
L
and AF of thebreeding objective
(equation (2!)
gives
the selection criterion(V).
The aim is to choose 8m and s so that the selection criterion is maximised.
However,
w!_1 is determinedby
selection decisions made lastgeneration,
whichis unaffected
by
8m and sbecause
they
specify
selection decisions made thisgeneration.
Therefore,
the selection criterion can besimplified
toIf our interest is restricted to decisions
regarding
male selection(ie
choosing
Smand
assuming
that females are selected atrandom,
so that all available femaleshave
equal probability
offeaturing
ins f),
thensjAffsf
is not affectedby
theselection decisions and can be
ignored.
s!A!s!
represents
the averagerelationship
between selected males and the
randomly
chosenfemales;
we assume that this isThe
approximations
invoked in the derivation ofequation
[7]
mean that it mustbe considered as a heuristic selection criterion whose usefulness will be tested
by
the simulation results.The aim of the selection criterion is to determine which sires to select
amongst
the males available for selection and whatproportion
ofmatings
should be allocatedto each. The
optimum
value of Sm can be foundby differentiating
V withrespect
to
s!
afterincluding
the restriction that themating proportions
must sum to1,
s!l
=1,
via aLaGrange multiplier,
A:Solving
for 8mgives
and since
s!
=1,
thenSelection
algorithm
The selection criterion V can be used to determine the
optimum
number of sires(n)
to select under theprevailing
circumstancesusing
thefollowing algorithm.
1. Rank sires on EBV and select the best n = 1.
2. For the
remaining
sires,
calculateYn+1!
for eachsire,
whichdepends
on thegroup of n sires
already
selectedplus
the individual sire to be considered.3. Rank the sires on their individual
Un+1!
values,
select the best sire if(V
[nH]
-V
[n]
)
> 0 thenrepeat
fromstep
2(n
= n +1),
otherwisestop
the search andselect
only
the first n sires nominated.This
algorithm
can be used to allow different sire selectionstrategies
eachusing
the selection criterion[7].
Strategy
1: Fixed number of sires(N&dquo;,,)
used each year, each allocated anequal
(as
far as
possible)
number ofmatings
sm of orderN
m
and 8m =N,
;
11;
repeat steps
2 and 3N&dquo;! -
1times,
always
selecting
the sire with thehighest
Yn+1!
value instep
3.Strategy
2: Selection of a variable(optimum)
number of sires eachgeneration,
each allocated anequal
number ofmatings
8m =n-
1
1.
Strategy
3: Selection on a variable(optimum)
number of sires with a variable number ofmatings
allowed/sire,
8m definedby
equation
!8!.
If the
algorithm
is used to select a variable number of sires eachgeneration
(strategies
2 or3),
the selection criterion balancessuperiority
ingenetic
merit withinbreeding
considerations. The aim of the selectionprocedure
is toexploit,
in ancurrent
generation.
Thisalgorithm
does not ensure that ’the’ best group of sires isselected.
However,
in simulations of smallpopulations
where it has beenpossible
to
subjectively
compare the group chosenby algorithm
versus ’the’ best group outof all
possible
combinations,
thealgorithm
hasperformed
well. Thealgorithm
maynot
perform
as well forlarger
populations,
but is is stilllikely
to be close to theoptimum.
To
gain
insight
into the selectioncriterion,
assume that sires are usedequally
(strategy 2).
Fromequation
(7!,
it can be shown that an n + lth sire is selected ifwhere
b
i
are the elements ofb&dquo;,,
and a2! are the elements ofA&dquo;,,.&dquo;,.
Presented in thisway, it is
apparent
that the contribution of the n + lth sire togenetic
merit of the selected group ofsires,
is balancedagainst
his contribution toinbreeding.
When thesires are
completely
non-inbred and are not related to eachother,
the contributionto
inbreeding
ofselecting n
sires is1/8
s
m
A&dquo;,,&dquo;,s&dquo;,
=1/8n
and an n + lth sire is selected ifAt the other
extreme,
if thepopulation
iscompletely
inbred(all
elementsof A
mm
are
2)
then the contribution toinbreeding
of selection n sires is1/8 s£Ammsm
=2,
and an n +1th sire is selected if
This is an artificial
example,
because when thepopulation
istotally
inbred,
there is noremaining genetic
variance and the EBVs of all the sires are the same.However,
the
implication
is that as thepopulation
becomes moreinbred,
the criterion for selection of sires becomes morestrict,
implying
a reduction in the number of siresselected.
However,
this is counteractedby
a reduction in the variance of EBVs sothat values on the left-hand
side
ofequation
[9]
also become smaller.Value of
Q
Implementation
of thealgorithm proposed
above for selection of siresdepends
onthe definition of
Q
which,
inturn,
isdependent
on the definition of thebreed-ing
objective.
A value forQ
can be foundby substituting
aprediction
forAG
L
Linto
equation
[3],
which in turndepends
onpredictions
ofi,
rL and aC,L’ Un-der the variable number of siresoptions,
theoptimum
number of sires(assuming
If selection is based on
phenotypes alone,
for a trait withheritability
h
2
andphenotypic
variance in the basepopulation
unity,
thenor2, G
o
= h2
and(Bulmer,
1980)
where k is the variance reduction factorappropriate
to the selectionintensity
(averaged
over the 2 sexes, for each sex k =i(i -
x),
xbeing
the standardnormal
deviate),
andr
=Œ&,L (Œ&,L
+ 1 -h
2
) -
I
.
Alternatively,
if selection is onBLUP EBVs then a lower bound to the accuracy of selection before
accounting
forthe Bulmer
effect
is:and
(Dekkers, 1992).
This lower bound to accuracy of selection for BLUP assumesthe
only
informationcontributing
to an individuals EBV is its own record and itsparental
EBV s. When an individual has many sibs withrecords,
the accuracymay be
considerably
underestimated. Indeed theOG
L
predicted
when selection ison BLUP EBVs
using
this lower bound accuracy may not besignificantly higher
thanAG
L
predicted
for mass selection.However,
theseequations
provide
asimple
deterministic
approximation
with which to attain aball-park
prediction.
The definition for
Q
canonly
beapproximate,
since theoptimum
value ofQ
is aniterative balance between selection response and
inbreeding,
particularly
when thenumber of sires is allowed to vary; the value of
Q
influences the selectiondecisions,
and the selection decisionschange
theoptimum
value ofQ.
Infact,
the valueassigned
toQ
(equation
!3!)
assumes that the selectiongoal
isalways
tgenerations
into the future. If the selection
goal
is cumulative net response togeneration
t withno interest in response in
subsequent
years, thenQ
inequation
[2]
should take onsubscript j representing
the selection criterion ingeneration j
(j = 0, t - 1)
withUnder this
definition,
the selection decisions made ingeneration
t - 1give
nodetrimental
weighting
to the effect of selection on futuregenetic
variance becauseunder the selection
goal
it is assumed that selectionstops
ingeneration
t. This definition isquite
unlikely
inpractice.
We would recommendQ
to be defined as inequation
[7]
where t takes on a medium time horizon value. SimulationsPopulations
are simulated with discretegenerations
in whichN
m
males are mated toN
f
females and each femalegives N
sex
offspring
of each sex.N
and
A!
arefixed each
generation.
In the basegeneration N
m
=N
f
,
but thereafterN&dquo;,,
may befixed or
variable, depending
on the sire selectionstrategy.
Thephenotype
(
Pj
)
ofindividual j
is simulated as pj = uis the
environmental
value of the individual. For a trait withphenotypic
variance ofunity
andheritability
ofh
2
,
an infinitesimal model ofgenetic
effects is assumed. Inthe base
population
Uj issampled
from a normal distributionN(0,
h
2
),
and in latergenerations
Uj issampled
from a normal distributionN(0.5(u
s
+u
d
),
0.5(1 -
f )h2),
where us and ud are the true
breeding
values of the sire and dam ofindividual j
andf
is their averageinbreeding
coefficient. Eachgeneration
ej issampled
from anormal distribution
N(0,1 -
h
2
).
Dams are selected at random.EBVs are calculated
by
true- orby pseudo-animal
model BLUP. In thetrue-BLUP,
theonly
fixed effect is the overall mean, basepopulation
variances are usedand all
relationships
between animals are included. In thepseudo-BLUP,
EBVs arecalculated
using
an index ofindividual,
full and half sib recordsplus
EBVs of thedam,
sire and mates of the sire(Wray
andHill,
1989)
:
The selection indexweights
change
eachgeneration
depending
on the availablegenetic
variance(o, 2,j),
which is calculated as(Wray
andThompson,
1990a),
whereF
j
andF
j
-
are the actual averageinbreeding
coefficients over all individuals born in
generations j
and j -
1 andr?
=0
,
2
,
I
jl
o, G,
j
2
where
a;,
j
is theexpected
variance of the index ingeneration j
(calculated
from the indexweights
andgenetic
variance); k
j
_
1
is half the variance reduction factorappropriate
to the number of males selected ingeneration j -
1(since
dams areselected at
random).
When the number of sires andmatings/sire
arevariable,
thevariance
reduction factor is based on an effective number of sires calculated asN
f/
m,
where m is the average number ofdams/sire,
m =sn snNf
1
,
wheres*
is
theinteger
vectorSn
N
f
of actual numbers ofmatings/sire.
Whenmatings/sire
are
variable,
all individuals have EBVs calculatedusing
the same index which assumes the same average number ofdams/sire,
m. The use ofpseudo-BLUP
isvery efficient on
computing
timecompared
withtrue-BLUP,
particularly
whenconsidering
schemes over manygenerations.
Simulations based on true-BLUP areused
only
as a check that thepseudo-BLUP
results in similar selection decisions.Selection continues for 30 discrete
generations
(20
fortrue-BLUP)
and results arethe average of 200 simulation
replicates. Response
to selection ingeneration
t,
R
t
,
is calculated as the mean over all individuals born ingeneration
t of pj -D f
j
,
wheref
j
is theinbreeding
coefficient for individualj;
when D =0, R
t
represents
theaverage
genetic
merit. Note that when D >0,
the recordsanalysed
in the BLUPare still the pj, thus we assume exact
prior
correction of records forinbreeding
depression.
Theunderlying
genetic
model couldrepresent
a trait controlledby
alarge
number of additive lociplus
a group of loci with rare deleteriousrecessives,
which make anegligible
contribution to the additive variance. Thisgenetic
model is one of several which could be chosen to simulateinbreeding
depression,
but this modelcorresponds
to the way in whichinbreeding
depression
is accounted for inthe
genetic
evaluation of livestockpopulations.
Summary
statistics are calculated within thesimulations,
these include:R
t
,
F
t
calculated as the mean of allf
j
,
rateof
inbreeding,
OF =(F
t
-
F
t
-¡)/(1 - F
),
t
i
-
averaged
from t = 2 calculated asE
Tij
[Nm(Nm -1)]
where
Tij
= 1 if the sires iand j
are sibs and 0 otherwise.Population
structure alternativesBasic:
N
=100,
A!
=4, h
2
=0.4,
D =0,
the selectiongoal
isR
30
(equation
!1!),
Q
is defined inequation
[3],
andAG
L
is calculatedassuming
mass selection.Alternative 1:
Q replaced by Q
*
,
whereQ
*
=cQ,
where c is a constant. Thisalternative allows
investigation
of therobustness
of theprediction
of the value ofQ
and values used in different simulations arec = 0.5, 0.8, 0.9, 1.0, 1.1, 1.2,
1.5.Alternative 2:
N
f
= 25.Alternative 3:
N
sex
= 2.Alternative 4:
h
2
= 0.1.Alternative 5: D
= 3.33, equivalent
to1%
inbreeding
depression/%
inbreeding
fora trait with coefficient of variance of
15%.
Alternative 6: D =
3.33,
selectiongoal
R
io
(equation
!1!).
For each
alternative,
simulations for each of the 3-sire selectionstrategies
arecompared
with selection on EBV alone. For selection on EBV and sire selectionstrategy
1,
fixed values ofN
m
used areN
m
=3, 6, 9, 12, 15, 18, 21,
24 for allalternatives
except
alternative 5 whereN
m
=6, 9, 12, 15, 18, 21,
24,
30.RESULTS
In table I response to the
breeding objective,
R
t
,
and level ofinbreeding
F
t
fort =
10,
20 arepresented
forpseudo-BLUP
and true-BLUP for the basicpopulation
structure
(h
2
= 0.4 or0.1)
withN&dquo;,,
= 9 when selection is on EBV orstrategy 1,
and
using
anexpected
number of sires of 9 for the calculation ofQ
for sire selectionstrategies
2 and 3. Agood
agreement
was found between results forpseudo-BLUP
andtrue-BLUP,
particularly
forstrategies EBV,
1 and 2. Instrategy 3,
where anaverage index is used for all
offspring
in thepseudo-BLUP
based on an averagenumber of full and half
sibs,
the response from thepseudo-BLUP
isslightly
lessthan is found with true-BLUP when
h
2
= 0.1. On the basis of theseresults, only
pseudo-BLUP
is used forinvestigations
of the full range of schemes.In
figure
1 response to selectionR
io
andR
30
areplotted
against
level ofinbreeding
F
lo
andF
30
respectively
for the basicscheme,
for sire selection onEBV,
and for
strategies
1-3,
where thebreeding objective
isR
30
.
Thisrepresentation
of the results demonstrates the success of different simulations inachieving high
R
t
but low
F
t
.
Thehighest
response atgeneration
10 is achieved with a low numberof sires selected on EBV alone.
However,
by
generation
30,
the time horizon of thebreeding
objective,
quite
a differentpicture
is seen: when the same number of siresare
selected, R
30
for sire selectionstrategy
1 isalways
greater
than for sire selectionon EBV alone and
F
30
isconcurrently
less. The maximumR
30
for selection on EBV alone is 10.75 ! 0.029 which occurs with 12 sires(amongst
the sire combinationsconsidered)
atF
30
of 0.491 ! 0.0025. Whilst the maximumR
30
for selection on V(equation
[7])
with a fixed number of sires(strategy 1)
ishigher
at 11.1O::f::0.030,
which assumes the selection of 9 sires each
generation.
Forstrategy 4,
the actual number of sires selected is 9.6 f 0.07yielding
anR
3o
of 11.18 ! 0.033 at anF
3o
of 0.403 t 0.0018. Forstrategy 3,
the actual number of sires selected .is 14.0 !0.09,
which when differential usage is accounted forcorresponds
to an effective number ofsires selected of 9.6f0.08
yielding
anR
30
of 11.39::1::0.032 at anF
30
of 0.402±0.0020.Selection of a variable number of sires
(strategies
2 and3)
results in ahigher
R
3o
value than selection of a fixed number of
sires,
although
theoptimum
occurs withapproximately
the same number of sires used on average and atapproximately
thesame level of
inbreeding.
Allocation of a variable number ofmatings/sire
results in ahigher
R
30
than allocation ofmatings
equally.
The standard errors ofR
30
arehigher
for
strategies
2 and 3 for selection onEBV,
despite
the lower level and standarderror of
inbreeding implying
the variable selectionstrategies
may be associated with morerisk,
if risk is measuredby
variance of response.A number of
assumptions
areemployed
to obtain the value ofQ
for sire selectionstrategies
1-3.Alternative
1investigates
theimportance
of accurateprediction
ofF
t
,
AF and theprobability
ofco-selecting
sibs alldecrease,
asexpected.
In thisexample,
aslightly higher
R
30
could be achievedby
using
Q
*
=0.9Q. However,
thestability
ofR
30
over the wide range ofQ
*
is more notable. Similar results were foundfor selection
strategies
2 and 3. In theseexamples, Q
has been calculatedusing
aAG
L
appropriate
to massselection,
Q
= 6.11. IfQ
were calculated asappropriate
to BLUP based on the lower bound accuracy then
Q
=6.42,
equivalent
to c =1.05,
and if an accuracy based on the actual number of full and half sibs is used thenQ =
6.89,
equivalent
to c = 1.13. From tableII,
it isapparent
that the selectionresults are robust to the method used to
predict AG
L
.
Similar results are foundfor the number of sires assumed to
predict Q
forstrategies
2 and3,
assuming
the selection of 3 or 15 sires instead of 9 sires isequivalent
to c = 1.16 or 0.92respectively.
TheQ
aredependent
on the timehorizon,
t,
eg, t = 20 isequivalent
to c = 0.65. Since the
optimum
c for thisexample
is 0.9despite
the fact that EBVs are BLUP and theprediction
ofOG
L
assumes massselection, implies
thatthe
underprediction
of rLQG,L is counterbalancedby
theoverprediction
ofi,
the selectionintensity,
which is calculated from normal distributiontheory
assuming
that the best sires on EBV are selected.
Despite
theproposal
in the Methods section that adecreasing Q
each
generation,
Q!,
would be moreappropriate
formaximising
a selection of a fixed future timehorizon,
the use ofQ!
for thisexample
resulted inslightly
lowerR
30
than fromusing
a fixedQ
(results
notpresented).
This reflectserrors in
prediction
Q
rather thancontradicting
theprinciple
that aQ
j
should be moreappropriate.
Qualitatively,
the results for alternativepopulation
structures 2-5 are similar tothose for the basic
scheme, generating
graphs
similar inshape
tofigure
1. Results are tabulated in table IIIusing
theN
m
(out
of the alternativesexamined)
whichgenerates
maximum response toR3!
for selectionstrategies
EBV and 1. The averageeffective number of sires used in
strategy
3 isapproximately
the same as the average actual number of sires used instrategy
2. In allalternatives,
there is atendency
for a2 and
3,
eg, for the basic schemestrategy 2,
the number of sires selected declinesfrom 10.4 to 9.3. The
optimum
number of sires when selection is on EBV isalways
higher
than forstrategies
1 and 2(and
effective number of sires ofstrategy
3),
therefore each individual has less half sibs available for selection.
Despite this,
theprobability
of coselection of sibs ishigher
forstrategy
EBV.Comparing
each alternative to the basic scheme for theoptimal
sire selectionstrategies
2 and3,
thefollowing
observations can be made: in alternative2,
N =
25,
response is less and
inbreeding
ishigher,
due to the smaller numberof both female and male parents.
Despite
the smalleroptimum
number ofsires,
the
mating
ratio(dams/sire)
is decreased. In alternative3, N9e!
=2,
the lowerinbreeding encouraged by
the smallerfamily
size is counterbalancedby
the increasein
inbreeding
causedby
the smalleroptimum
number of sires andencouraged by
the smallerQ
value. In alternative4, h
2
=0.1,
strategies
1-3 are allsuperior
tostrategy
EBV,
but there is little to choose between them. Forstrategy
EBV,
AF ishigher
with 15 sires than it is with 12 sires in the basic scheme.Strategies
2 and 3 choose a loweroptimum
number of sires than in the basic scheme and resultsin a
higher optimum
rate ofinbreeding.
In alternative5,
D =3.33,
the value ofQ
is increasedby
a factor of8.9,
whichdiscourages
the coselection of sibs andresults in
considerably
lower rates ofinbreeding.
In thisalternative, R
t
represents
genetic
merit ingeneration
t lessDF,
whereas in the other alternativesR
t
issimply
genetic
merit;
forstrategy 3,
the averagegenetic
merit is 11.25 ± 0.030compared
In
figure
3 response to selectionR
lo
andR
3o
areplotted
against
level ofinbreeding
F
lo
andF
30
respectively
for alternative6,
D =3.33,
breeding
objective
R
l
o,
for sire selectionstrategies
EBV and1-3,
which can becompared directly
with
figure
2,
where thebreeding objective
isR
3o
.
Asexpected,
for alternative6,
sire selection
strategies
rank 3 > 2 > 1 > EBV forR
lo
,
withoptimum
number of sires forstrategies EBV,
1-3being
12, 12,
8.3 t 0.07 and 12.2 t 0.10respectively.
However,
if the same selection criterion is continued untilgeneration
30,
then thesuperiority
ofstrategies
2 and 3 is lost.DISCUSSION
Selection on the criterion V
(equation
[7])
always
results in ahigher
responsecriterion
R
30
and lower level ofinbreeding
F
30
when thebreeding objective
isR
3o
,
than selection on EBV alone for the simulation
examples
considered. When thenumber of sires selected is
fixed,
achieving
the maximum response toR
30
depends
on thejudicious
choice of the number of sires. Thealgorithm
to select a variablethe same
F
30
as with thestrategy
using
theoptimum
fixed number of sires. Selectionon criterion V with
Q
as defined inequation
[3]
does notnecessarily
result inthe absolute maximum response to the
breeding objective.
This is because thederivation of
Q
contains severalapproximations:
i)
theequality
ofequations
[1]
and!2J;
ii)
prediction
of rL and UG,1&dquo; andiii)
calculation of selectionintensity
asif the sires have been selected on EBV alone. Of
these,
iii)
islikely
to be mostcritical,
but it is difficult to see how toimprove
on thisapproximation
as thereis a
dynamic
interaction eachgeneration
between the value ofQ
used and theselection decisions made
(in
whichgenetic
merit is balancedagainst
relatedness of the selectedgroup)
and hence the selectionintensity
achieved.Fortunately,
the simulation resultssuggest
that thealgorithm
forselecting
sires isfairly
robust tothe value of
Q
chosen,
and it appears that the methodproposed
here topredict
Q
results in response close to the maximum. In theprediction
ofQ,
it islikely
that the
underprediction
inapproximation
ii)
counterbalances to some extent theoverprediction implied
by approximation
iii).
When selection is on EBV alone the rate of
inbreeding
cannot beaccurately
predicted
fromsingle
generation probabilities
of coselection of sibs(or
equivalently
variance of
family
size) (Wray
etal,
1990)
which can beexplained
through
theconcept
ofpartial
inheritance of selectiveadvantage
(Wray
andThompson,
1990)
across
generations.
In the V selectioncriterion,
thetendency
for an ancestor ofhigh
genetic
merit to leave more descendants in eachgeneration
is limitedby
thecontinual reevaluation of the
relationship
information in eachgeneration’s
selection decisions.The
advantage
of the selection criterionproposed
here is that it isclearly defined,
with the
goal
ofmaintaining genetic
variance over along
time horizon. In othermethods,
maintenance ofgenetic
variance is theunderlying
goal
but it isindirectly
achieved
by
criteria for which theoptimum
values are not known. Forexample,
if the selection criterion includes a restriction on the number of individuals toselect from any
sibship
(eg,
Toro andPerez-Enciso, 1990;
Brisbane andGibson,
1993),
what should the restriction be? The criterionproposed
hereautomatically
places
restrictions on the number selected perfamily
ifinbreeding
isperceived
as a
problem,
but will be flexible in itsrestrictions,
placing
less restrictions on afamily
which ishighly
superior
ingenetic
merit.Alternatively,
if selection is on acriterion which alters the
emphasis placed
on individual andfamily
information,
EBVi
-1
/2EBV sire -1 l/2EBVdam
+ weightsireEBV sire +
weight
d
amEBVdam
(eg,
Toroand
Perez-Enciso, 1990;
Verrier etal, 1993;
Villanueva etal,
1994)
what valuesshould be attributed to
weight
s
;
re
andweight
dam
and should theweights
be constant overgenerations ?
The methodproposed
here could be viewed as a flexible version ofthis criterion with
weights
given
tofamily
informationdiffering
for each individual. Inaddition,
theweights
may differ overgenerations,
where one couldspeculate
that,
initially
it may be favourable to eliminate thegenetically
poorerfamilies,
whilst inlater
generations
within-family
selection from each of thegenetically
similar familiesmight
beoptimal.
Even when the selection criterionspecifies directly
a restriction onrate of
inbreeding
(eg,
Toro andPerez-Enciso,
1990),
what is theoptimal
restrictionto
place
on rate ofinbreeding?
Theoptimal
rate must bedependent
on the valueattributed to