Optimum family size for progeny testing in populations with different strains

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Optimum family size for progeny testing in populations

with different strains

Sr Miraei Ashtiani, Jw James

To cite this version:

Original

article

Optimum family

size for

_progeny

_testing

in

_populations

with

different strains

SR Miraei Ashtiani

JW James

University of New South _{Wales, Department of} Wool and Animal Science,

PO Box _1, Kensington NSW, 2033 Australia

(Received

25 _September 1992; accepted 11 March

₁₉₉₃₎

Summary - The _design of progeny tests to _identify the best 1 or 2 sires tested is considered

for _{populations consisting} of a _large number of _genetically different _strains, such as the

Australian Merino. A fixed number of studs enter sires in the test, for which a fixed number

of _{progeny in} total are recorded. Evaluation is by best linear unbiased prediction

(BLUP)

with strain effects _being taken as random. When there is little variation between strains the results are similar to the well-known results of _Robertson, but when between-strain variation is _high the _optimum number of sires to be tested is _higher and _family sizes are

smaller, because information on sires from the same strain _provides information on each

sire.

progeny testing / family size _/ variation between strains _/ Australian Merino _/ selection

Résumé - Taille _optimale de famille pour l’épreuve de descendance dans des

popu-lations composées de _lignées différentes. La _{planification} des _épreuves de descendance pour choisir le meilleur ou les 2 meilleurs _pères est étudiée dans le cadre de _populations

comprenant un _grand nombre de _{lignées génétiquement différentes,} comme c’est le cas _par

exemple pour le Mérinos australien. Un nombre donné de lignées soumettent des pères à

l’épreuve de descendance, avec un nombre total _fixé de descendants contrôlés. L’évaluation des pères se _{fait par} la meilleure prédiction linéaire sans biais

_{(BL UP)}

avec des effets lignée

considérés comme aléatoires. _Quand la variation entre _lignées est _faible, les résultats sont similaires à ceux bien connus de Robertson, mais _quand la variation entre _lignées est _forte, le nombre _optimal de _pères à soumettre à _l’épreuve de descendance est _augmenté et les tailles de _famille sont diminuées. La raison en _{est que l’information} sur l’ensemble des

pères d’une même souche _fournit une _information sur chacun des _pères de la souche.

épreuve de descendance _/ taille de famille _/ variation entre _{lignées /} mérinos australien _/ sélection

*

On leave from: _Department of Animal _{Science, College} of _{Agriculture, University} of

Tehran, Karaj, Iran_**

INTRODUCTION

Progeny

testing

of bulls has been _very

_extensively

used in

_dairy

cattle

_breeding

for

many years, and more

_recently

has been

_widely

used in beef cattle

_breeding.

In

contrast, it has been very little used in Australian Merino

sheep breeding, though

there has been some increase in its

_application

in the last few _years. This use has

largely

been in sire reference

_schemes,

_aspects of the

_design

of which were discussed

by

Miraei Ashtiani and James

_(1991).

In this work attention was concentrated on

the

_design

of _systems to minimise

_prediction

error variances of differences between estimated

_{breeding values,}

in a similar _{way to} that of

Foulley

et al

_(1983).

This

_is,

however,

not

_necessarily

the best criterion for

_design

of such schemes.

It was

_pointed

out

_by

Robertson

₍₁₉₅₇₎

in the

_dairy

cattle context

_that,

when the aim is to select a fixed number of sires and the total number of _progeny available

is also

_fixed,

there is an

_optimum

_family

size which will

_give

the

_greatest

_expected

response. This

optimum

is a

_compromise

between _{greater accuracy} of estimated

breeding

values and _greater selection

_intensity.

When there is

_prior

information on

breeding values,

the

_optimum

structure is

_altered,

as shown

_by

James

(1979).

It seems useful to

_adapt

Robertson’s

_approach

to the

_design

of Australian Merino

sire

_evaluation,

but one feature of this breed needs to be taken into account. Short

and Carter

₍₁₉₅₅₎

showed that the breed is divided into several strains which are

much more differentiated than in most livestock

_breeds,

in which strain formation has

_usually

been

_slight.

Mortimer and Atkins

₍₁₉₈₉₎

have

_recently

demonstrated that there are substantial

_genetic

differences between studs within a division of

the Merino breed such as the

_Peppin

strain. Thus in

_considering

an

optimal design

to

_identify

_(say)

the best 1 or 2 sires from those

_evaluated,

it is _necessary to take

account of both between strain and within strain _variation, where here we use strain to mean _any

_genetically

different group, so that different

_Peppin

studs are referred

to as strains.

In this _paper, a _progeny test at a

_single

location is

_assumed,

and rams from a

given

number of studs are to be evaluated with a view to

identifying

the best 1 or

2 of those tested. The total number of _progeny available is fixed. The

_problem

is to

determine how _many rams from each strain

_(stud)

should be tested in order that

the true

_breeding

values of the 1 or 2 with the best estimated

_breeding

values are as

_high

as

_possible.

It will be assumed that the studs involved in the _{program may}

be

_regarded

as a random

_sample

from a

_{large population}

of such

_studs,

and that

sires within strains can be taken as unrelated.

THEORY

In the progeny

testing

program there are s sires from each of b strains which are

mated to females from a common _source, and n progeny from each of the bs sires

are

_recorded,

so that the total _{progeny is T} = bsn. T and b _are

regarded

as

_fixed,

so that the

_problem

is to find

_optimum

values of s and n.

If

_Y

_ijk

is the record on the kth

offspring

of the

jth

sire of the ith

_strain,

our

where

B

i

is half the deviation of the mean of the ith strain from the

_population

mean

breeding

value

_(BV),

_S

_ij

is half the deviation of the BV of an individual sire from

its strain mean, and

_E

_ijk

is an individual _progeny deviation. We assume

_genetic

and environmental variances are the same for all _strains,

denoting

the

phenotypic

variance as

Vp

and the

heritability

as

h’.

Then

N

i

B

N(0, V

B

)

S2! ! N(O, V

S

) ;

i

E2!! ! N(0, V

E

)

where -

_{N(!,, V)}

means

normally

distributed with mean A and variance V. It is assumed twins are rare so that all

offspring

may be taken to be

half sibs. We have:

so that

f

_represents the ratio of the between-strains to within-strains

genetic

variance. We define the ratio k as:

The overall BV of a sire is

_2u2!

_{where uij} =

Bi-I-52!.

We want to consider BLUP estimates _{of uij.} These can be obtained in 2 _ways. In one

approach

_equation

[1]

is

used as the

_model,

and BLUP of

_B

_i

and

_S

_ij

are used to find:

This would

_give

a

_diagonal

variance-covariance matrix for the random variables

to be estimated

_by

_BLUP,

but the

_{prediction error variances}

_(PEV

_S

₎

would need

to be calculated for the

_i

_i

_ij

from those for

_i

_B

and

_S

_ij

_.

In the second

_approach

we

rewrite the model as:

with a _pattern of correlations _among the _u2!. We then have:

var(u

ij

)

=

V

B

₊

V

S

, cov

(u2!, u

ij

,)

=

V

B

and _cov

,

ij

(u

u

i

,

j

,)

= 0. We shall

adopt

the

second

_approach.

Writing

[2]

in matrix terms we have:

with

_{var (u)}

= G and

var(e)

=

IV

E

,

where G is block

diagonal,

consisting

of b

blocks,

each s

_by

_s, with

V

B

+

V

s

on the

diagonal

and VB elsewhere. All other

elements of G are zero. We let

_C

_s

be the s

by

s matrix and then G =

I

b

_®

C

s

where ® denotes the direct

_product.

Then the Henderson mixed model

_equations

are:

In these

_equations

X’X is a

scalar, bsn,

X’Z and Z’X are row and column

vectors of

_length

bs with each element

_equal

to n. Z’Z is a

diagonal

matrix with

where

IS

is an s

_by

s unit matrix and _Js an

s by

s matrix with all elements

_unity.

Therefore:

Therefore each block of Z’Z +

_V

_E

_G-’

has the form:

Inverting

the coefficient matrix in

_[4],

one finds that

_diagonal

terms for the bottom

right

corner are:

Then the PEV for _{any sire is}

_{given by:}

For sires from the same strain the

_prediction

error covariance

_(PEC)

is:

while for sires from different strains the PEC is:

Thus for a

_given

sire:

For two sires from the same strain:

while for 2 sires from different strains:

The correlation between true and estimated

_breeding

values which measures the

The

_response

to

_{selection, R,}

which is the criterion for the

optimum

number of

sires to be tested and thus the size of each sire

_family,

is obtained as:

Here 0&dquo; Ais the within-strain

genetic

standard

deviation,

and i is the standardised

selection differential

_{corresponding}

to the

_proportion

of sires selected.

We also need to consider the intra-class correlation of the estimated BVs. The

analysis

of variance involves

_{EEu !,}

_{., Eii,2. Is}

and

_{ii2 lbs.}

_Omitting

the factor

_Vs

we

can find:

zj

which

_gives

us:

The between strains _component is then

_{( f — k/!).}

Thus the intra-class correlation between strains is:

This intra-class correlation is

_important

because the standardised selection

differential may be

seriously

affected when t is

large, especially

if b is small.

Values of the standardised selection differential were

_approximated

as follows.

Burrows

₍₁₉₇₂₎

showed that the finite

_population

effect for

_selecting

S animals

from N can be

approximated

in the

following

way. Let P =

S/N

and let i* be

This assumes the animals are

_independent,

_ie,

all selection criteria are

uncorre-lated.

The case where the N animals are not

_independent,

but consist

of

g groups each

of size m so that _gm = N _was dealt with

by

Hill

₍₁₉₇₆₎

and

_Rawlings

_(1976).

As

well as

_giving

an exact _treatment, each _gave an

_{approximation.}

We

investigated

both

_{approximations}

and found

that,

although they

often _agree

_well,

on occasions

the Hill formula _{gave greater} selection differentials for

selecting

two sires than for

selecting

one. These were conditions outside the _range for which Hill

_suggested

his

approximation,

but as

_they

included conditions we wished to use we decided to use

the

_Rawlings

formula rather than that of Hill. If t*is the _{average correlation among} all

_pairs

of

_animals,

the

_Rawlings

formula is:

and on

_substituting

the value of t* we have:

To find the

_optimal

_structure, a

design

was evaluated

by calculating

RI

O

’A

_using

equation

[5]

for 1 and for 2 sires

_selected,

with i calculated

_using

_[6]

and

_[7].

The

design

giving

the

_highest

value was taken as

optimal.

Results of this

_approximate

calculation were checked

_using

a simulation _program with 2 000

_replications

for a

given

set of _parameters.

For each set of

_simulations,

the

_procedure

was as follows. For each _strain, a

random variable was

_sampled

from the

_appropriate

_population.

Then for each sire to be tested an

_appropriate

random variable was

_sampled

and another random

variable was

_sampled

to

_provide

the _progeny mean deviation. The BLUP

_procedure

was then used to _get EBVs for all _sires, and the known true

_breeding

values for the best 1 or 2 on EBV were used to calculate

_{genetic gains.}

This was

_replicated

2 000

times,

and the mean and SE of the

_{genetic gain}

were calculated. Standard errors were never more than 2.2% of the mean

_gain.

In the case of 1 strain or when the

_f

value is very

small,

the Robertson

(1957)

approach

would be a suitable solution. If _p is the _proportion

_selected,

x is the truncation

_point

of the standard normal distribution

_{corresponding}

to p, then the

optimal

structure should be

_given

_by:

RESULTS

Calculations were made with the total

_testing

facilities set at T = _300, 600 and

1000. The number of strains

_(b)

was taken as _3,

_5,

or

_{10, h}

2

was

taken as

_0.1,

0.3

or

_0.5,

and the

_f

ratio as

_0.5,

0.1 or 0.01. For all 81 combinations the values of s

which maximised the

_expected

_{genetic gain}

were located

by

a search _process and

The

maximum _responses for different combinations of the _parameters are illus-trated in tables

_I,

II and III. These were calculated

_using

the

Rawlings

approxima-tion.

In table I the value of

_f

has been taken as _0.1, so

V

s

= 10

V

B

.

When

the

_heritability

or the total number of _{progeny increases,} with other _parameters

the best 2 sires tested.

_Higher

h

2

and T result in a _greater number of sires

being

tested,

the _{greater response}

_coming

from

_higher

selection

_intensity

rather than

more accurate evaluation. The _accuracy of evaluation would be

_fairly

_low,

with 10 progeny per sire often

being

close to

_optimal

_{in many situations.} When two

sires rather than one are

_selected,

the number to be tested should be somewhat greater, but the ratio of _responses for the 2 selection

_regimes

is not _very sensitive

strain correlation in estimated

_breeding

values is x5 50% greater than the correlation

between true

_breeding

_values,

but is not

_{especially high.}

In table II it has been assumed that the variance between strains within the

population

is half the variance of BVs between sires within strains. In

_populations

like the Merino there is variation between strains for

_economically

important

traits

of this order of

_magnitude

_(Mortimer

and

Atkins,

1989).

If the results are

compared

with those in table I it is seen, as

_expected,

that the selection _response is _greater,

greater. The

_magnitude

of

_change

is

_dependent

on several

_variables,

but _responses are 20-25% greater, and the number of sires in the test is ! 50% greater. The differences between

_selecting

1 or 2 sires are _very similar to those when

_f

= 0.1

for selection _response and number of sires tested. The between strain correlations

in estimated

_breeding

value are > 3.5 times those in table

_I,

because of the

_large

change

in

_V

_B/

_(V

_B

_{+ Vs).}

Table III shows the results of an _{extreme case,} when

_genetic

variance is 100

times _greater within than between groups

( f

=

0.01).

Consequently,

the between strain correlations are _very small

and,

as

expected,

the results are less

_dependent

on the number of strains involved in the test and are almost uniform for each level

of

h

2

and T. The effects of

h

2

and

T on _responses are more or less the same as in

tables I and II. Another

_expectation

is that in this case the results should be

_fairly

close to those for an undivided

population

(Robertson, 1957).

We found reasonable

agreement between the 2 sets of results. Perfect _agreement could not be

_expected

because of differences in calculation

_procedures,

as well as the small differences in

underlying

assumptions.

The

_proportion

selected in all cases studied was never more than

11%,

so selection

differentials are

_reasonably

_high.

A

_comparison

of

_having

10 versus 3 strains in the test is shown in table

_IV,

where results are

_presented

as the

_change

in _response or number of tested sires as

a fraction of the value when b = 3. When

f

= 0.1 the _extra

response from

using

10

strains is !

_3%,

but when

_f

= 0.5 the extra

The

_{approximation}

results were checked

_by

simulation with 2 000

_replicates

per

parameter set, for a number of _parameter sets. In almost all cases the simulation

gave

significantly

greater responses than the

approximation.

Figures

1 and 2 show the

_{approximation}

as well as the simulation results

_plotted

_against

the number of

sires per strain. In

figure

1,

h

2

= 0.1 and in

figure

_{2, h}2= 0.3. In both

figures

T = 300 and different values of

f

and b _were used.

Although

for _a

_given

value of

b the simulation results for successive s values show

_{sampling fluctuations, they}

generally

show a

_fairly

flat _pattern near the

_optima.

The

_sampling

fluctuations

meant that determination of

_optimal

numbers of sires was rather

_questionable

from

the

_simulations,

and we

_preferred

to use

the approximation despite

its _apparent

bias in

_estimating

_response.

DISCUSSION

The results show the effect of

_genetic

variation between strains on the

_design

of

progeny

testing

programs. The calculations were based on the

_assumption

that

the number of strains taken from the

_population

is

_fixed,

and

_equal

numbers of candidates from each strain are tested. It is also assumed that the strains _may be

regarded

as a random

sample

of the

population

from which

they

come and that

sires are unrelated _apart from

_being

members of the same strain.

how many strains are available. This is

_currently

the situation in the Australian

Merino.

_However,

sometimes the number of strains included should also be

_regarded

as

variable,

and an

_optimum

for this variable would also be

_sought.

This would

significantly

increase the amount of

_computing

_required,

but would not involve _any

further theoretical

_{developments,}

and _presents an

_opportunity

for extension of these

analyses.

The _assumptions on

_{relationships}

_{among sires} are

_perhaps

_{oversimplified,}

but

are

_probably

not unreasonable for the

_early

_stages of such programs.

However,

once

a few

_highly

selected sires were

_widely

used the

_existing

differences among strains would be

reduced,

and some of the sires from different strains would be related.

Thus our conclusions would not be

_applicable

over a

long period.

We have assumed that the estimation of

_breeding

values is

_by

_BLUP,

with

random _group effects and known parameters. In some cases this led to

_{fairly high}

correlations between EBVs of sires from the same strain

_{(0.5 -}

_0.6).

If the _group

structure were to be

_ignored,

the estimates of

_breeding

value would be less accurate,

since

_{they ignore relationships}

between animals from the same strain. This

problem

does not arise in the same _way in most cattle

_populations

where distinct strains

have not

_formed,

and

_pedigrees

can be used to account for

_genetic

similarities between animals. If there are

only

a few

_genetic

_groups,

_they

_may be best treated as fixed

_effects,

but in the Australian Merino there are a

large

number of

studs,

and _{it appears} reasonable to assume their mean

_breeding

values can be

_regarded

as

random effects. The between-strain

_genetic

_parameters are,

however,

poorly

known

gain

to be achieved. The difficulties

_arising

from lack of

_knowledge

of between-strain

genetic

parameters have been discussed in another context

_by

Atkins

_(1991).

When the between-strain

_genetic

variation was

low,

our results turned out to be

rather similar to those of Robertson

_(1957).

The minor differences

_probably

arise

from our use of the finite

population

selection differential

adjusted

for correlation

between selection criteria on

_candidates,

and from our estimation of the overall mean

by

BLUP rather than

_assuming

it to be known.

When b = 1 and sires from

only

₁ strain _are

evaluated,

results _are the _{same as}

if there were no variation between strains

_{( f =}

_0).

In this case the

_optimum

_family

size is

_{approximated by}

n = 0.56

(K/h2)°!5,

where K is the number of

progeny tested _{per sire} selected

_{(Robertson, 1957).}

When there are more strains in the trial

and there is

_appreciable

variation between

_strains,

the total number of sires tested

increases with the number of

_strains,

and also with

_f.

The _{response to} selection also increases. This is attributable to the fact that with a _greater number of more

variable strains

_represented

there is more variation between tested _sires, so that not

so _{many progeny per} sire are

required

to differentiate between them. As a

result,

more can be tested to

allow

a _greater selection differential to be achieved.

For a

_given

set of

conditions, increasing

the total number of _progeny recorded

leads to an increase in both the

family

size and the number of sires tested.

The

_comparison

of the simulation results and the results of the

_{approximation}

appears to show a consistent downward bias in the estimates of _response

_given

by

the

_{approximation,}

_though

the location of the

_optima

_appears to be little affected.

Nevertheless it seems that better

approximations

for selection differentials than

those of Hill

₍₁₉₇₆₎

and

_Rawlings

₍₁₉₇₆₎

may be worth

searching

for. As is not uncommon, the curves of response

plotted

against

number of tested sires are rather

flat near the _optima, so that _precise location of the _optimum is not of

_practical

importance, since a value close to the

_optimum

will

_give

a _response

negligibly

less

than the best

_possible.

From a

_practical

_viewpoint,

the

_optimum

structures seem

_unlikely

to

_appeal

to

_breeders,

because the small

_family

sizes _{in many} cases would be considered too

small to

_give

reliable results. The small

_family

sizes are

possible

because of the use of

information on _progeny of other sires from the same strain.

However,

within-strain

comparisons

will not be _very accurate when

_f

is

_large.

Our results show that selection across strains can

_give

worthwhile

_improvements

over selection within _strains, as

_expected

on theoretical

_grounds.

Smith and

Banos

₍₁₉₉₁₎

have studied this

_question

under different conditions ana also found

worthwhile

_advantages

for selection across

populations

in some situations.

ACKNOWLEDGMENTS

REFERENCES

Atkins K

₍₁₉₉₂₎

Data

_processing

and

_linkages

within and between sire evaluation

schemes. In: Merino Sire Evaluation in Australia

_(Brien

_{FD, Kearins RD,}

Maxwell

WMC,

eds)

Wool Res Dev

_{Corp, Melbourne, Australia,}

27-33

Burrows PM

₍₁₉₇₂₎

_Expected

selection differentials for directional selection.

Bio-metrics

28,

1091-1100

Foulley JL,

Schaeffer

LR,

Song

H,

Wilton JW

₍₁₉₈₃₎

_Progeny

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