Power for mapping quantitative trait loci in crosses between outbred lines in pigs

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Power for mapping quantitative trait loci in crosses

between outbred lines in pigs

Luis Gomez-Raya, Erling Sehested

To cite this version:

Original

article

Power for

_{mapping quantitative}

trait

loci in

crosses

between

outbred lines in

_pigs

Luis

_{Gomez-Raya* Erling}

Sehested

Department

of Animal

_{Science, Agricultural University}

of

_Norway,

Box 5025, N-1432

_As,

_Norway

(Received

28

_{May 1998; accepted}

12

_May

₁₉₉₉₎

Abstract - Power for

_{mapping quantitative}

trait loci

_using

crosses between

segregat-ing populations

was studied in

_pigs.

Crossing

_generates

gametic disequilibrium

and

increases

_{heterozygosity.}

The condition for a

heterozygosity

_among Fl individuals to

be _greater than in either line at crossing is that allele

_frequency

should be lower than

1/2

in one line and

higher

than

_1/2

in the other line. Maximum

_expected

_power and

expected

risk were used to _compare hierarchical backcross

_(each

boar mated to several sows; contrast within

boars),

hierarchical intercross

(each

boar mated to several sows; contrast within both boars and

_sows)

and traditional intercross

_(each

boar mated to

one _{sow; contrast} within both boars and

sows).

The use of hierarchical

designs

(back-cross and

intercross)

increased _power for traits with low or intermediate heritabilities.

For small

_QTL

effects and low heritabilities the hierarchical backcross

design

gave the

highest expected

power but also the

highest

risk. There is not a

_{general design}

which

allocates resources in an

_optimum

fashion across situations

_{(heritabilities,}

_QTL

ef-fect,

heterozygosity).

A

compromise

between

_designs

with

_high

power and low risk

is

_suggested.

A hierarchical backcross

_design

of 400

_piglets

and

_{heterozygosity}

0.68

requires

between four

_(maximum

expected

power)

and

_eight

boars

_{(minimum risk)}

to

detect a

_QTL

of 0.5

_phenotypic

standard deviations. Selection of extreme individuals

in

_parental

lines increased power up to 21 %. Commercial crosses are

_proposed

as an

alternative to

_experiments

for

QTL

mapping. @

Inra/Elsevier,

Paris

gene mapping / quantitative trait locus

/

statistical power

/

hierarchical design /

pig

Résumé - Puissance de détection des loci à effets quantitatifs dans les croisements entre _lignées non consanguines chez le porc. On a étudié chez le porc la puissance

de détection des loci à effets

_quantitatifs

à partir de croisements entre

_populations

en

ségrégation.

Le niveau

_{d’hétérozygotie}

chez les individus FI est

supérieur

à celui de

*

Correspondence

and reprints

l’une ou l’autre des

_lignées

en croisement si les

fréquences alléliques

sont

_supérieures

à

_1/2

dans l’une des

lignées

et inférieures à

_1/2

dans l’autre

_lignée.

La

_puissance

maximale

_espérée

et le _risque

_espéré

ont été considérés _{pour comparer} les croisements

en retour

_{hiérarchiques}

_(chaque

verrat est

_accouplé

à

_plusieurs

truies : contrastes

intra-verrat),

l’intercroisement

_{hiérarchique}

_(chaque

verrat est

_accouplé

à une _{truie ;} contraste intra-verrat et

_{intra-truie).}

L’utilisation de ces deux _types de schémas

hiérarchiques

a

_augmenté

la _puissance de détection _pour les caractères à héritabilités

faibles ou intermédiaires. Pour les

CaTLs

à _petit effet et les héritabilités faibles, le schéma

_{hiérarchique}

avec croisement en retour a donné la _puissance

_espérée

maximale

et aussi le

risque

le

plus

élevé. Ce n’est pas le même schéma

expérimental qui

alloue

les ressources d’une

_{façon optimale quand}

on considère des situations différentes _pour

l’héritabilité, les effets de

_QTL

et le niveau

_{d’hétérozygotie.}

Un schéma

_{hiérarchique}

avec croisement en retour de 400

_porcelets

et une

_{hétérozygotie}

de _{0,68 requiert}

entre _quatre verrats

_(puissance

_espérée

_maximum)

et huit verrats

_{(risque minimal)}

pour détecter un

_QTL

avec un effet de _0,5

_{écart-type phénotypique.}

La sélection

d’individus extrêmes dans les

lignées parentales

a

_augmenté

la

_{puissance jusqu’à}

21 %. Les croisements entre

_lignées

commerciales sont

_proposés

comme alternatives

aux

expérimentations

de détection de

QTL. ©

Inra/Elsevier,

Paris

détection de _gènes / locus à effet quantitatif / puissance

statistique

/ schéma

hiérarchique / porc

1. INTRODUCTION

Crosses between inbred lines and

_analyses

within outbred lines have been

proposed

for

_quantitative

trait loci

_(QTL)

detection. In the first

_approach,

inbred lines are assumed to be

_fully

_homozygous

for alternative alleles at

both marker and

_{QTL. Consequently,}

F

l

individuals are

fully heterozygous

and

_gametic

_{disequilibrium}

is maximum.

_{High heterozygosity}

allows

_high

segregation and, therefore,

high

power. With maximum

gametic

disequilibrium,

a

specific

allele at the marker is associated with a

specific

allele at the

_QTL

in all families.

_{Consequently,}

increased _{power is} achieved because a lower number of contrasts is needed

using

across

family

analyses

based on a

larger

number of observations per contrast.

In the second

_{approach, analyses}

within outbred

_lines,

both marker and

_QTL

are

_segregating.

Power in crosses between inbred lines is much

higher

than in

analyses

within outbred lines. A combination of both

_approaches

has assumed that the lines are

_segregating

at the marker but fixed for alternative alleles at

the

_QTL

_{[1, 2!.}

Another

_alternative,

not

_explored

_yet,

is to consider

_parental

lines with different allele

_frequencies

at the

_QTL

because of different selection

history.

Conventional

_types

of crosses between inbred lines are intercross and back-cross. Power of an intercross versus a backcross

design

is increased because the number of

_segregating

rneioses in an intercross is twice as _many as in a back-cross.

_Experiments

conducted in

_pigs

have restrictions in

_family

structure to a

maximum of

_{approximately}

ten

_piglets

_per litter. A

small

number of full-sibs may result in low power

!9!.

A hierarchical structure, such as a few boars mated

to several sows

_each,

allows

_{larger subgroups}

of progeny

inheriting

alternative alleles from each

_boar,

which _{may increase} power.

It has been

_proposed

to use selective

_genotyping

to increase _power for

growing

progeny is less than the cost of

_genotyping.

Another use of

selection,

not considered

_yet,

would be to use extreme individuals for the

_quantitative

trait as

_parents

to

_produce

the

_F

_l

_.

This

_approach

increases the

_frequency

of

heterozygotes

in the

F

l

and, therefore,

statistical power.

The

_objectives

of this paper are:

1)

to

_investigate

the

_degree

of

disequilib-rium

_generated

when

_crossing

two

_{populations segregating}

at two

_loci;

₂₎

to

investigate

under which conditions the

_{heterozygosity}

_(and

_consequently

_power)

in the cross of two lines

_segregating

at a

_QTL

is

_higher

than within either

_line;

3)

to

_investigate

_power

_using

hierarchical

_designs;

₄₎

to

_{investigate optimum}

allocation of resources for a

_{given experiment}

_size;

and

5)

to

_investigate

the effect of selection of extreme individuals in the

_parental

lines on

_{heterozygosity}

and _power.

2. THEORY

2.1.

_{Underlying genetic}

model and

_assumptions

The _purpose of the

_{experimental crossing}

is the

_mapping

of

_{QTL by}

the use of

_genetic

markers such as microsatellites. It was assumed that recombination

events did not occur between marker and

_QTL

and that all

_offspring

were

informative for the marker. These

_simplifying

_assumptions

were made to reduce the number of

_parameters

to be shown in the results. The

_impact

of these

assumptions

in

_practical

scenarios is addressed in the discussion.

An

_underlying

mixed inheritance additive model was assumed with an

additive

biallelic locus

_segregating

at

_frequencies

of the favourable allele _pB and pc for lines B and

C, respectively.

The

QTL

was assumed to be in

Hardy-Weinberg equilibrium

and in

_gametic

_{phase equilibrium}

with the

_polygene

in each

_parental

line. The variance attributable to the

_QTL

in units of the residual

_phenotypic

variance in line i was:

_vb(i)

=

2p

i

(1 -

Pi)a2 /(u!

+ a

E

),

where _pi is the

_frequency

of the favourable allele in line

_i,

cr is the

_QTL

effect,

and

_{(a A 2}

+

0

,2 E

_is

the residual

_phenotypic

variance with

₀

_{,2 A}

and

_{OF2 E}

being

the additive variance attributable to the

_polygene

_component

and the environmental

_variance,

_{respectively.}

The

_heritability

_{comprising QTL}

and

polygene

variation in

_parental

line i was:

where

_{h! is}

the residual

_heritability

attributable to the

_polygene

_component,

and has value

_{a A 2 /(U}

₂

_A

+

_{0,2 E ). Parameters h! and}

_V

_{2 Q}

are

_usually

not known and

_only

estimates of

_h

₂

_(i)

are available for some traits. Different heritabilities

in the two

_{parental populations}

were not assumed because of the

_magnitude

of the

_sampling

variance of the estimates of

_heritability

and because

parental

populations

may have been raised under different environmental conditions. For

_simplicity,

_power

_computation

in this _paper was carried out for a

_given

QTL

effect and constant residual heritabilities among

F

l

individuals.

There-fore,

results are valid for a

_variety

of situations with

_respect

to the

_parental

with a different

_heritability

_(comprising

_QTL

and

_polygene).

The

_highest

pos-sible

_heritability

_(QTL

and

_polygene)

in the

_{parental populations}

_(denoted

as

h

2

(max))

leading

to a

_{given heterozygosity}

at the

_QTL

_among

_F

_l

individuals was

_computed

as an indication of the values of

_heritability

in

_practical

situa-tions. The

_impact

of this

assumption

on _{power is} addressed in the Discussion. Three levels of

_{heterozygosity}

in the

_F

_l

were considered:

_1,

0.68 and 0.32. A

heterozygosity

of 1 occurs when the

QTL

is fixed for alternative alleles. A

het-erozygosity

of 0.68 may occur for a _range of situations in

_{parental populations}

(e.g.

PB =

0.8;

pc = 0.2 and

p

B =

0.68;

pc =

0).

A

heterozygosity

of 0.32

represents

the situation where the two lines

_segregate

at the same

_frequency

(e.g.

pB =

_0.8;

pc =

0.8)

or when one allele is absent from one

_population

and

_segregating

at a

_frequency

lower than 0.5 in the other

_population

_(e.g.

p

B =

0.32;

p c

=

0.0).

The

advantages

of

crossing experiments

aimed at

_QTL

mapping

are:

1)

_{linkage disequilibrium}

between

_QTL

and marker alleles in the

F

l

;

and

₂₎

increased

_{heterozygosity among F}

_l

individuals.

2.2. Gametic

disequilibrium

in the cross of two outbred lines If the two

_populations

are fixed for alternative alleles at both marker and

QTL,

then

_{gametic disequilibrium}

is maximum in the

F

l

.

Therefore,

power is

increased with

_respect

to

_{within-family analysis.}

If the two

_{parental populations}

are

_segregating

at the same

_frequency,

then the

_resulting

_F

_l

_population

is in

linkage

equilibrium

and no benefits are

expected

from

_crossing.

If the _genes

are

_segregating

at different

_frequencies

in the

_{parental populations}

then some

degree

of

_{disequilibrium}

will be

_generated.

For a more

_general

_description

of this

problem, gametic disequilibrium

will be

_computed

at two

_loci,

M and

_N,

that can be either two markers or one marker and one

_QTL.

Gametic

_{disequilibrium}

between markers can be estimated if estimates of allele

_frequencies

in the

parental

lines at

_crossing

are available. Let

_f

_6M

and

_fb,!

be vectors of allele

frequencies

at locus M

_(with

k

_alleles)

and at locus N

_(with

_{g alleles)}

in

parental population

B.

_Similarly,

_f

_eM

and

_f!,V

are the

corresponding

vectors

of allele

_frequencies

at loci M and N in

_parental

line C.

_Assuming

that the two

loci are in

_{linkage equilibrium}

within the line then the matrix with

_gametic

frequencies

at each

_pair

of alleles for line B is:

The matrix of

_{gametic frequencies}

in line C is

_F

_eMjV

=

f

’M

f!!,.

The matrix

the cross between lines B and C is

_{given by: F}

beM

N

=

1/2(F

bMN

+

_F

_EMN

)-The allele

frequencies

for loci M and N in the

_F

_l

cross are

fbcm !

F

bcMN

1

1 and

_f6!N

= 1’

F

beMN

,

respectively.

In these

equations

1

_represents

the unit

vector. The

_{gametic disequilibrium}

matrix is then obtained

_{simply by}

_D

_MN

=

FB,MN - (fb,M

f6!NO

Elements of matrix

D

MN

are the

disequilibrium

values for all

_possible

combinations of alleles at loci M and N. Define the

_{disequilibrium}

parameter

between two loci M and

_{N, S2}

_MN

_,

as the sum of absolute values of all elements of matrix

_D

_MN

_{computed by 52,!,Ir,}

= 1’

abs(D

MN

)1,

where ’abs’

denotes the absolute value at each element of the matrix between brackets.

Therefore, S2

MN

measures the

general degree

of association between alleles at

loci M and N. The value of

52,,,1,!

ranges between 0 and 1. For

example, Q

MN

is 1 when lines at

_crossing

are fixed for alternative alleles and

_{disequilibrium}

is maximum and

_Q

_MN

is 0 when the lines at

_crossing

are

_segregating

at the same

frequency

for each allele and

_{disequilibrium}

is null.

2.3.

_{Heterozygosity}

in the cross of two outbred lines

Consider two outbred lines B and C

_segregating

at a biallelic

QTL

with alleles

_Q

and _q.

_Assuming

random

_mating,

the

_frequency

of

_heterozygous

individuals at the

_QTL

_among the

_F

_l

individuals is:

It is shown in

_Appendix

1 that the necessary condition for increased

heterozygosity

with

_respect

to either line at

_crossing

is that one line is

segregating

at a

_frequency

of the favourable allele

_greater

than

_1/2

and the other line at an allele

_frequency

lower than

1/2.

2.4.

_QTL

_mapping

_designs

for the cross between two outbred lines

Power

_computation

was carried out

_assuming

that allele

_frequencies

at

the marker were not _very different in the two

_{parental populations,}

and

consequently, gametic disequilibrium

was small and

_ignored.

_However,

the increased

_{heterozygosity}

in the

_F

₁

boars can be used to

_study

_segregation,

within

_families,

of a

_QTL

associated to a marker

_by

_{recording performance}

and inheritance of alternative marker alleles in the next

_generation

_(backcross

or intercross

_depending

on the

_mating

of

F

l

boars to one of the

parental

lines or

_F

_l

_sows,

respectively).

_Therefore,

the

_approach

that can be followed

is the same as in

QTL

_mapping

within outbred lines but with an increased

heterozygosity

among

F

l

individuals. Three alternative

_designs

are considered

in this section: hierarchical backcross

_design,

traditional intercross

design

and hierarchical intercross

_design.

Contrasts in hierarchical

_designs

can use

large

subgroups

of progeny of a mixture of half- and full-sibs within boar.

2.4.1. Hierarchical backcross

_design

In a hierarchical backcross

_design,

b boars from the

_F

_l

are mated to s sows

where y2!kc is the lth observation of

_phenotype

on a

piglet

with marker allele k inherited from boar i when mated to sow

_j,

_bo

_i

is the fixed effect of boar

_i,

_so2! is the fixed effect of

_{sow j}

_mated

to boar

_i,

_mijk is the fixed effect of marker allele k inherited from boar

_i,

_{and eijkl} is the residual random error. Model

(2)

is a three level hierarchical

_design

in which marker alleles are nested within

sows, which in turn are nested within boar. For

_simplicity,

sows and boars were assumed fixed. It is not

_strictly

correct because the model does not account for

relationships

between animals.

Power for model

₍₂₎

was

computed

following

the

x

2

_approach

of Gelder-mann

_[5]

and Weller et al.

_[14]

_comparing

the _square of the difference between the two _progeny

_{subgroups inheriting}

alternative alleles from their boar

_(SDP)

for each boar

_family

to the

_{expected squared}

difference under the null

hypothe-sis.

_Therefore,

the alternative

_hypothesis

is a

_QTL

linked to the marker and the null

_hypothesis

is the absence of a

QTL

linked to the marker. For a more de-tailed

_description

of the method and discussion of the

_assumptions,

see Weller

et al.

_!14!.

_Briefly,

the

_assumptions

are

_{complete linkage}

between a

_fully

infor-mative marker and a

_QTL,

and the

_analyses

are carried out within families. For

_simplicity,

it is assumed that there is no common environmental variance and litter size is fixed at 10.

The statistical tests to

_reject

or to

_accept

the null

_hypothesis

are based

on the distribution of the SDP. The summed SDP values divided

_by

the

squared

standard error of the contrast

_(SE

₂

₎

follow a central

_k

₂

distribution with b

_degrees

of freedom under the null

_hypothesis

for a

_{large sample}

size or when

_phenotypic

variance is known. For the alternative

_hypothesis

E(SDP/SE

2

)

N

x

2

(nc,

b),

where nc is the

non-centrality

_parameter

of a

non-central

_x2

distribution

with b

_degrees

of freedom. The value of the

non-centrality

parameter

is nc =

b he

BC

0!/SE!,

where 0 = _{a +}

6(l - 2 p

B

)

and 0 = c! +

_{6(1 - 2p}

_c

_),

for backcrosses with lines B and

_{C, respectively}

(Appendix

2),

a is half the difference between the two alternative

_homozygotes,

6 is the dominance

_deviation,

and

SE! =

_{4 [(1}

+

(1/4)shr - l/2h!)/(sp)]

[equation

(A3)

in

_Appendix

_3!.

Following

Weller et al.

_[14],

the _power to detect a

_{segregating QTL}

was

computed

as

_{1 - !}

= 1 !

p[x

2

(nc,

b)

_<

T],

where

₁

₃

_{is the}

probability

of

committing

a

_type

2 _error,

[p(x

2

(nc,

_b)

<

_T]

is the

_probability

_{of a x}

₂

value under the alternative

_hypothesis

_(non-central

_X

₂

_{with parameter}

nc and b

degrees

of

_freedom)

less than

_T,

with T

_being

the value of the central

_x

₂

_(b)

for a

_given

_significance

level of

_committing

a

_type

1 error.

2.4.2. Traditional intercross

_design

In the intercross

_design,

inbred lines B and C are crossed to

_produce

the

F

l

boars and sows that are intercrossed _among themselves. In the traditional intercross

_design

b boars are mated to one sow each. A linear model

_allowing

testing

for the

_marker-(aTL

effects is:

where Yijk is the hth observation on

_piglet

k with marker

_{genotype j}

_inherited

in

_family

_i,

_f

_z

is the fixed effect of

_family

_i,

_mij is the fixed effect of

and

_{heterozygous)}

and _eijk is the residual error. Model

(3)

would be the one to

use in the

_analysis

of real data.

However,

to take

_advantage

of the

_X

₂

_approach

of Geldermann

_[5]

and Weller et al.

_[14]

for _power

_calculation,

the

_following

linear model can be used:

where _!z!!l is the lth

_phenotypic

observation of

_piglet

inheriting

marker allele k from the sow and

_{allele j}

from the

boar,

mb

ij

is the fixed effect of marker allele

j

inherited from the

_boar,

m8ik is the fixed effect of marker allele k inherited

from the sow, and ei!xl is the residual error.

Model

₍₄₎

is

_{linearly equivalent}

to model

₍₃₎

under the

_assumption

of a

_fully

additive

underlying genetic

model and no sexual

_imprinting.

The distribution under the alternative

_hypothesis

of

E(SDP/SE

2

)

is

_{approximately}

_x

_z

_(nc,

_b)

and

X

2 (nc,

s)

for boars and _sows,

_{respectively.}

_Assuming

that the

_hypothesis

being

tested is the same either within boars or within sows, the distribution of the

sum of

’¿,(SDP /SE

2

)

for both boars and sows follows a

x

z

(2nc,

b+s)

under the alternative

_hypothesis.

The sum of variables

having

non-central

x

z

distributions

jointly independent

also follows a non-central

X2

distribution

with

degrees

of freedom

_equal

to the sum of the

_degrees

of freedom of the former non-central

X2

2

and

non-centrality

parameter

equal

to the sum of the

_{non-centrality}

_parameters

of the former variables

having

non-central

_x

_z

_.

The

_{non-centrality}

_parameter

of the

_z

_x

has a value 2nc =

(b

₊

s)

he

BC

,

(p /SE

2

where 0 = a +

_{6(l -}

PB -

p

C

),

and

SE!

=

_{4 !(1 -}

_(l/2)!)/p]

_[equation

_(A4)

in

_Appendix

_3].

_Similarly

to a hierarchical backcross

design,

power to detect a

_{segregating QTL}

was

computed

as 1 -

(3

= 1 -

p[x2(2nc,

2b)

_<

T],

where

/3

is the

probability

of

committing

a

_type

2 error,

!p(2(2nc,

2b)

<

_T]

is the

_probability

_ofax

₂

value under the alternative

_hypothesis

_(non-central

_X2

_{with parameter}

2nc and 2b

degrees

of

freedom)

less than

_T,

with T

_being

the value of the central

_X2

_(2b)

for a

_given

significance

level of

_committing

a

_type

1 error.

2.4.3. Hierarchical intercross

_design

In a hierarchical intercross

design,

b boars from the

F

l

are mated to s sows each from the

_F

_l

to

_produce

_p

_piglets

_per litter. Power calculation can be carried out

_using

the linear model:

where _Yijklm is the mth observation of

_phenotype

on a

_piglet

with marker allele k inherited from boar i mated to sow

_j,

bo

i

is the fixed effect of boar

i,

soij is the fixed effect of

sow j

mated to boar

i,

mb

ik

is the fixed effect

of marker allele k inherited from boar

_i,

_msijl is the fixed effect of marker allele I inherited from sow

_ij

and _eijklm is the residual error. Power calculation

using

model

₍₅₎

requires

different

_{non-centrality}

_parameters

within boars and

sows. The

_{non-centrality}

_parameter

in boars is: ncb =

b he

B

ce

2

/SE2,

where

0 = a + b( I - p

B

-

P

c )

(Appendix 2),

with

SE

2

=

_{4[(1 + (1/4)ph; -1/2h;)/(sp)]}

design.

The

_{non-centrality}

_parameter

in sows is nc,s =

bshe

BC

B

2

/SE

2

,

where

0 =

a+6(I-

PB

-p

c

)

(Appendix

2),

and

SE

2

= 4[(1 - ( 1 /2)h$) /p]

(Appendix 3).

Assuming

independence,

the distribution of the sum of

E(SDP/SE

2

)

for both boars and sows follows a

x

2

(nc

b

+nc.&dquo;

b(1+s))

under the alternative

hypothesis.

Under the null

_hypothesis,

the distribution of the sum of

E(SDP/SE

2

)

for both boars and sows follows a central

X

2 (b(

+

_s)).

Power to detect a

_segregating

QTL

was

computed

as

1 - (3

= 1 !

p!x2(ncb

_{+ , snc}

b(1

+

s))

_<

T],

where

/3

is

the

_probability

of

_committing

a

_type

2 error,

h

(nc

2

(p(x

+ nc,,

b(1

+

_s))

<

_T]

is the

_probability

_{of a x}

₂

value under the alternative

hypothesis

(non-central

x

2

with

_parameter

nci, + ncs and

b(1

+

_s)

degrees

of

_freedom)

less than

_T,

with T

_being

the value of the central

_x

₂

_(b(1

+

_s))

for a

_{given significance}

level of

committing

a

_type

1 error.

2.5.

_Design

of

_experiments

_using

crosses between outbred lines

Most

_experimental

costs are in

raising, genotyping

and

recording

of a

_given

number of

slaughter pigs

in the

_F

₂

or backcross.

Therefore,

the

_parameters

to be chosen

_by

the researcher are the number of

_boars,

the number of sows and the number of

_piglets

per sow. In

_practice,

it is convenient for the

handling

of the

_experiment

to fix the number of

_piglets

per sow. It is also economical since

the use of few

_piglets

_per sow would increase the cost in

raising

a

_large

number of sows. It will be assumed from now on that the number of

piglets

_per litter

in the

_experiment

is fixed at ten and the

_question

is how to make

_optimal

use of different numbers of boars and sows in the

F

l

for a

_{given experiment}

size

(total

number of

slaughter

pigs).

Parameters of interest are the

expected

_power

(EP)

and its standard

deviation

_(SD):

where

_pr(i)

is the

_probability

_according

to the binomial distribution of

having

i

_heterozygous

boars

_[assuming

a

frequency

of

heterozygous

boars

given by

equation

(1)],

and

_P

_i

is the _power with i boars

_{computed according}

to the

previous

sections.

_Expected

_{power is} utilized for each

_design

to account for random

_sampling

of the boars. SD can be used as a measurement of the variation in power.

Comparison

of alternative

_crossing

_designs

can be done

_by

comparing expected

power and its standard deviation at a fixed

_experiment

size. The

_approach

used in this paper for

optimum

allocation of resources is based on the

repeated computation

of

expected

power for all

possible

combinations of the number of boars and sows at a

_{given experiment}

size

_assuming

constant litter size of ten

_piglets.

The _power of the

_mating

structure

_having

the

_highest

expected

power will be called maximum

expected

power

(MEP).

The functions CINV and PROBCHI of SAS

_[12]

were used to

_compute

central and non-central

Another

parameter

of interest in

_{planning experiments}

is

_expected

risk

_(ER)

which we define as the

_probability

of

_having

_{power lower} than 0.5 due to

sampling

among

F

l

individuals in the

segregating

population:

2.6. Selection in

_parental

lines to increase _power for

_QTL

_mapping

Selection of

_{high ranking}

individuals in one

_parental

line and of low

_ranking

individuals in another

_parental

line can be used to increase statistical power for the

_experiment

when the lines are not fixed for alternative alleles. Assume a biallelic

_{QTL segregating}

in the two

_parental

lines. Consider a hierarchical

backcross

_design

in which

_high

and low

_ranking

individuals selected in the

parental

lines are

_randomly

mated to

_{produce F}

_l

_,

which are

_{again randomly}

mated to one of the unselected

_parental

lines. Two effects would occur in this scheme:

₁₎

an increase in the

_frequency

of

_heterozygous

individuals in the

F

r

which would increase the _power to detect

_QTL;

and

₂₎

an increase in

_genetic

variance of the backcross

offspring

due to

_{linkage disequilibrium}

_[3],

which would decrease power to detect

_QTL.

2.6.1.

_{Frequency of heterozygotes}

_among

_{offspring of high}

and low

ranking

parents

Computation

of the

frequency

of

_heterozygous

_F

_i

individuals was carried

out

_by

_using

_{integrals simultaneously}

of the three normal distributions

cor-responding

to the three

_genotypes

at the

_QTL.

The selected

_proportion

_(p,

₅

₎

for the

_high

line is related to the

_proportion

selected among individuals with

genotypes

QQ, Qq and qq

(p

QQ

,

poq and

pqq)

by

the

equation:

with

_f(xIG

_i

₎

_being

the normal

_density

_given

_genotype

_(Gi

=

QQ,

Qq

or

qq).

Values _{pQQ, PQq, and Pqq} can be found for a

_unique

truncation

_point,

t,

analytically.

In the

_examples

considered in this paper

Simpson’s

rule

_[13]

was used for

_increasing

values of the abscissa until a value of t was found

satisfying

the above

_equation

for a

_given

_ps.

Computation

of

expected

_genotype

The same

_procedure

was followed to obtain

expected

genotype

frequencies

among low

ranking

individuals in the low line

_(pQ

_Q

_,

_L

_,

_L’

_PQ

_q

_p

₉

₉

_,

_L

₎

_using

the

same

_proportion

selected,

_ps, as for the

high

line.

_Computing

the

expected

frequency

of

_heterozygous

_F

_l

_offspring

_resulting

from the cross of selected

parents

from each line was carried out with

_equation

₍₃₎

of

_Gomez-Raya

and Gibson

_[6].

It assumes random

_mating

and utilizes the

frequencies

_among

extreme individuals from each of the two

_parental

lines.

2.6.2. Increased variance _among the

_offspring

_{of F}

_l

individuals The effect of selection of

_high

and low

_ranking

individuals on

_genetic

variance and on the estimation of

heritability

is discussed

by

Bulmer

[4]

and

Gomez-Raya

et al.

_!7!.

_Briefly,

if selection of

_high

and low

_ranking

individuals is carried out

in the same

_population

then

_genetic

variance increases in the selected _group of

parents

_QAS

=

(1 + !s!)o’!;

where

QA

is the

_genetic

variance for the

polygenic

component

in the unselected

_{population, k,}

₅

=

[z 4>(z)]/ps,

z is the absolute

value of standard normal deviates at cutoff

_points,

_{4>( z)}

is the ordinate and r

is accuracy of evaluation. The above

_equation

can also be used when

_crossing

high

ranking

individuals from one

_parental

line with low

_ranking

individuals from the other

parental

line as

long

as the

parental populations

have the same

genetic

variance and the same

_{polygenic heritability}

and allowance is made for

the different selection intensities for each

_genotype

at the

_QTL

and for each line. For

_simplicity,

an

_{approximation}

to account for different selection _pressures

across

_genotypes

was made

_by

_weighting

the values of z and

4

>(z)

_according

to

the

_proportion

selected from each

_genotype

and line in the

_computation

of

k

8

by

where

_zi&dquo;

and

_4>(z;&dquo;)

are the absolute values of standard normal deviate and ordinate at truncation

_point

for

_genotype

i and line m

(i

had values

_1,

2 and 3 for

_genotypes

_{QQ, Qq}

and _qq,

_{respectively).}

An exact

_computation

would be feasible

_{by using}

all

_possible

_groups

_resulting

in the

F

l

offspring

and

_{by including}

the variation of the means

_{corresponding}

to

the different groups. This variation is

small,

particularly

for

_QTL

of small

effect,

and the

_approach

used accounts for the increased _{variance among}

_offspring

from selected

parents.

Following

Bulmer

_(3!,

the

_genetic

variance in the

_F

_l

_generation

is

_{0’ A}

_(Fl)

₂

(1

+

(1/2)ksr2)u!.

The

_{disequilibrium}

is halved in the next

_generation

since

parents

in the

_F

_l

are chosen

_randomly.

The

_genotypic

_{variance among} backcross

offspring

from

_randomly

_mating

F

l

individuals to one of the

parental

lines

(unselected)

is:

The

_heritability

_among backcross

_offspring

is also increased

by

selection of

high

Power calculation was carried out as described in the

corresponding

section but

_using

the increased

_frequency

of

_{heterozygotes}

and

_heritability

as

computed

above. The estimates of the _gene effect are biased if selection of

_parents

is used. Correction for selection bias in the estimates of the _gene effect can be achieved after

_accounting

for the increased variance among backcross

_{offspring by}

where a* is the estimate of the gene effect from the

experiment

using selection,

and a is the estimate of the gene effect after correction for selection bias. Both

&* and a are in

_phenotypic

standard deviations units.

3. RESULTS

Table I shows the

_gametic

_{disequilibrium}

parameter

for alternative number of alleles

_segregating

in the

parental populations.

The

_{disequilibrium}

is

_high

when alleles at each of the two loci are fixed in one

_population

but

_they

are

rare in the other

population.

If the alleles at the two loci are

_segregating

at a similar

_frequency

in the two

_populations

then

_{disequilibrium}

is small. In this

situation,

the

_analysis

could be

_performed

within families

allowing

for different alleles at the marker to be associated with the same allele at the

QTL

in different families.

Maximum

_expected

_power and its standard deviation for three different

crossing designs

in a

_variety

of situations

(QTL

effect,

residual

heritability,

experiment

size)

for

_{heterozygosity}

_1,

0.68 and

0.32,

are

_given

in tables

_II,

111 and

_IV,

_{respectively.}

The

_{heritability including QTL}

and

_polygenic

variation

(h

2

(max))

of the

parental population having

the

_{highest possible}

value is

given

in these tables. It can be observed that residual

heritability

has a small effect on _power for the _range of heritabilities in the

parental populations.

Therefore,

power

figures

can be considered as a

_good

_{approximation}

when

accurate estimates of

_heritability

in the

_{parental population}

are not available. On the other

hand,

a reduction in the

frequency

of

heterozygotes

_among

F

r

individuals diminished the power for any situation considered. For

example,

for residual

_heritability

0.2 and

_QTL

effect 0.5 in an

_experiment

with 200

piglets

in a hierarchical backcross

design,

the maximum

_expected

_power with a

_significance

level of 0.05 fell from 0.85 to 0.31 for

heterozygosity

1 and

0.32,

respectively.

Power

_using

hierarchical

designs

(backcross

and

_intercross)

increases in all cases with the

exception

of traits with

high heritability

when

compared

to traditional intercross

_designs

_(tables

_{77-7 V).}

Maximum

_expected

power

using

hierarchical backcross

_designs

is

_larger

than

_using

hierarchical intercross

_designs

when the

_QTL

effect is small. The

_opposite

occurs for

large

QTL

effects.

The use of a

larger

experiment

size increases power in all cases. For

_example,

the _{average maximum} power across

_QTL

size and residual

heritability

(table

II)

intercross

_designs,

_{average maximum power} is

_0.58,

0.71 and 0.81 for

_experiment

sizes of

_200,

400 and

_800,

_{respectively.}

Variation _{in power} due to the

_sampling

in the

_F

_l

should also be consid-ered. Hierarchical backcross

_designs

show a much

_larger

variation than either

intercross

_designs

_(tables

III and

_II!.

An

_example

of how power, standard devi-ation of _power and

_expected

risk

_change

for alternative

_mating

_designs

is

_given

in table V. In this

_example,

_{heterozygosity}

is

_0.68,

residual

_heritability

of the trait is 0.2 and

QTL

effect is 0.5

_phenotypic

standard deviations with

corresponding

standard deviation of _power and

_expected

risk are 0.27 and 10

%,

respectively.

A

_compromise

between

_expected

_power and risk can be taken. For

example,

the use of five boars has a low

_expected

risk of 4

%

with an

_expected

power of 0.74. It is more obvious with a hierarchical intercross

_design

where

expected

power

using

between one and ten boars _{is very} similar

_(0.81

to

_0.83).

However,

the

_expected

risk is

_only

low when

_using

four or more boars. Hierarchical and traditional intercross

_designs

_gave identical results at

her-itability

0.5

_{(tables 11-I!.}

This is because traditional intercross is an extreme

case of hierarchical intercross

(one

sow _per

boar)

which is the best allocation

of resources at

_high

heritabilities.

Table VI shows the number of

_{boars, expected}

_power and risk for

_designs

having

maximum

_expected

_power and minimum risk in hierarchical intercross and backcross

_designs.

The residual heritabilities are

_0.10,

0.20 and 0.50 and

the

_QTL

_{size ranges} from 0.3 to 0.6

_phenotypic

standard deviations in an

experiment

of 400

piglets

with

F

z

heterozygosity

of 0.68. It can be observed that the numbers of boars for maximum

_expected

power and minimum risk are

Heterozygosity

and power

using

phenotypic

selection of extreme individuals

in a hierarchical backcross

_design

with 400

_piglets,

residual

_heritability

0.20 and

_varying

_percentage

selected are

depicted

in

figures

1 and

_2,

respectively.

Parental lines were

_segregating

at

_frequencies

of 0.8

_(selected

to

_increase)

and 0.2

_(selected

to

_decrease).

A

_percentage

selected of 100 is

_given

for

_comparison

with the unselected case.

_Percentage

selected of 0.5

%

was the lowest

attempted.

Heterozygosity

increases with

_increasing

selection _{pressure in} each line and with the size of the

QTL. Following

the same

_pattern,

_power increases with

decreasing

percentage

selected

_being

up to 21

%

more

(figure 2).

The increase

is small if the _{power in} the unselected situation is close to 0 or 1.

4. DISCUSSION

Most

_previous

research in

_QTL

_mapping

has assumed that lines at

_crossing

are fixed for alternative alleles at the

_QTL

_{[1, 2, 8].}

The

_availability

of

divergent

Domesticated breeds of

_pigs

have been

_undergoing

artificial selection to increase

growth

rate and litter

_size,

for

_example.

_Only

for research _purposes, selection criterion has been low

_growth

rate or reduced litter size in

experimental

populations. Therefore,

it seems more reasonable to assume that the lines at

crossing

differ in allele

_frequency

at the

_QTL

rather than that

_they

are fixed

for alternative alleles.

The benefits of

_crossing

lines are to increase

_gametic

_{disequilibrium}

between marker and

_QTL

alleles and to increase

_{heterozygosity}

among

F

l

individuals. Gametic

_{disequilibrium generated}

at

_crossing

is

_high

when the allele

_frequency

at the two loci

_(either

two markers or one marker and one

QTL)

is very different

in the two

_parental

lines. Gametic

_{disequilibrium}

between two linked markers could be estimated in chromosomal

_{fragments by}

the use of the

disequilibrium

parameter

proposed

in this paper. It would

require

that markers are not

_tightly

linked so

parental

lines could be assumed to be in

_linkage

_equilibrium.

If the

out within families as considered in this _paper with

_only

a small loss _{in power.} If the

_gametic

_{disequilibrium}

is

_high

_(e.g.

S2 >

_0.8)

then a maximum likelihood

approach

could be

_developed

_allowing

the same marker allele to be associated with the same

_QTL

allele

_only

in some families.

_{Consequently,}

_power

_figures

as

_given

in this

study

represent a lower bound of the achievable power in those

cases. More work is needed to assess the

_gain

_{in power in} situations where

disequilibrium

parameter

has intermediate values.

Power for

_QTL

detection in

_{experiments involving}

crosses between two

outbred lines is

_higher

than

_using

within-line

_experiments

when the

_frequency

of the favourable allele is

_higher

than

_1/2

in one

_parental

line and lower than

_1/2

in another

_parental

line. The increased _power can be attributed to the

_higher

frequency

of

_heterozygous

_F

_l

individuals than in either

_parental

breed. The

larger

the difference in allele

_frequency

between the two

_{parental populations,}

the

_larger

is the _{increase in power.}

It was assumed in the

_computation

of _power that residual

_heritability

was

constant for

_{varying QTL}

effects and for a

given heterozygosity

in the

F

l

.

This

_represents

a

_variety

of situations in which

_{parental populations}

can be

segregating

for a

_QTL

at a different

_frequency

and

_having

a different

heri-tability.

The

_approach

taken in this paper was to

_compute

also the

_heritability

(comprising

QTL

and

_polygene

_variation)

of the

_parental

line with the

_highest

possible

value for a

_{given heterozygosity}

in the

F

l

.

At low residual

heritabil-ity,

the contribution of

_{large QTL}

to the

_heritability

is

_high.

For

_example,

for residual

_heritability

0.1 and

_QTL

effect of 0.6

_phenotypic

standard

_deviations,

the

_heritability

is 0.22 when the allele

_frequency

in the

_{parental population}

is 0.68

_{(table 111).}

In

_spite

of the

_large

contribution of the

_QTL

variation to the

heritability,

changes

in power for different

_heritability

values are small.

Discrep-ancies in power

comparing

residual

heritability

0.1 and 0.2 for the same

QTL

size range between 0.00 and 0.05

_{(table 111).}

_Therefore,

_power

_figures,

as

_given

in this _paper, are well

_approximated

when

_precise

estimates of heritabilities from

_parental

population

are not available. The

_{approximation}

is

_particularly

good

when the

_QTL

has a small effect.

On the other

hand,

it was assumed when

_computing

_power that

recombina-tion between

_QTL

and marker did not occur. In most

_instances,

recombination would occur. If the recombination fraction between the marker and the

QTL

is c then the number of

_offspring

should be increased in

_proportion

to

_{1/( 1- 2c?}

to

obtain the same _power as with

_{complete linkage}

!14!.

Power could be increased

by using

interval

mapping

instead of the use of a

_single

marker

_!10!.

Most of the relevant information in interval

_mapping

comes from the non-recombinant

individuals. The number of

_{offspring required}

to obtain _power as

_given

in this paper should be increased in a

_proportion

1/(1-c’),

where c’ is the

recombina-tion fraction between the two markers. That

_is,

around 25

%

more

_offspring

are needed to achieve _power as

_given

in this _paper for interval

_mapping

between

two markers with a recombination fraction of 0.20. A

larger

_experiment

size will also be needed to achieve the same _power

_figures

as

_given

in this _paper

when the marker is not

_fully

informative. The use of interval

_mapping

can

mit-igate

the lack of informative

_offspring

by

utilizing

information

corresponding

to

_nearby

informative

_flanking

markers.

The models used in this paper

ignore

the

possibility

of a common full-sib

pigs.

Power for

_mapping

_{QTL affecting}

those traits would be reduced

_!9!.

The

approach

used in this

_study

to

_compute

_power could be modified to account

for this

problem

by

incorporating

a term

_{corresponding}

to the common

family

component

in the standard error of the contrasts.

It has also been assumed that the heritabilities in

_{parental populations}

as well as within the

offspring

of

F

l

remain

unchanged.

The

frequency

of

heterozygotes

at other loci

_affecting

the trait could increase

_genetic

variance among

F

l

individuals

and, therefore, heritability.

One

generation

of random

mating

is

_enough

to restore

_{Hardy-Weinberg equilibrium}

as would occur after

intercrossing

F

l

individuals.

_However,

_changes

in the

_genetic

variance among

offspring resulting

from the hierarchical backcross

_design

_may occur. If there are other

QTL affecting

the trait

_showing

dominance or

_epistasis,

then

changes

in the

_phenotypic

variance and

_heritability

could also occur.

Therefore, departures

from the _power as

_computed

in this _paper would be

_expected

for traits that show heterosis such as

reproductive

traits.

For a

_given

_power, a backcross

_requires

about twice as _{many progeny} as an intercross

design

because the number of

_segregating

meioses is doubled

in the intercross

_[10].

_{Experimental design}

for

_{mapping QTL}

in

_pigs

has the limitation that the number of

_piglets

per litter has a maximum around ten.

This limits the size of the

_subgroups

of _progeny where the contrasts are carried

out. Hierarchical backcross and intercross

_designs

can,

however,

be used in

_pigs

and allow contrasts within boars with

_{larger subgroups}

of _progeny. Power of intercross versus backcross

depends

on both

_heritability

and

QTL

effect when

using

hierarchical

_designs.

At

_{high heritability levels,}

the hierarchical intercross

design

is more

powerful

than the hierarchical backcross

design.

The

opposite

occurs at low

_heritability

values.

An

_interesting

result was that _power for small

_QTL

effects was

_higher

in

a hierarchical backcross than in a hierarchical intercross. The

only

difference between the two

_designs

is that the hierarchical backcross

_ignores

meioses from sows. In a hierarchical

_intercross,

contrasts between alternative alleles inherited from sows are small with a

_large

variation

_given

the small

QTL

effect and the small _{progeny group}

_{(ten piglets).}

_However,

the total number of

_degrees

of freedom _may be

_high

because of the

_large

number of sows. On the

_contrary,

in a hierarchical backcross each of the few boars has

_{large subgroups}

of _progeny which means that contrasts have low variation and

_hypothesis

_testing

is carried

out with a low number of

degrees

of freedom. As a consequence of the

above,

power may be reduced in the intercross.

There is not a

_general

_{experimental design}

which can allocate resources in an

_optimum

_way if the

_experiment

size is fixed. Power in

_experiments

with

a low number of boars is

_higher

for small size of

_QTL

and

_high

_heritability

(table V7).

The

opposite

occurs for

QTL

with

large

size and low

heritability.

Risk measured as the

_probability

of _power lower than 0.5 due to

_sampling

of

F,

boars is

_high

with a low number of boars. If the

_frequency

of

_heterozygous

F

l

individuals is low then the risk increases and the _power decreases

_(results

not

_shown).

A

_compromise

between

_expected

_power and risk should be made. A

_possibility

for

_experiments

_searching

for

_QTL

_affecting

traits with known

heritability

is first to decide the

_potential

_QTL

size and

_F

_l

_{heterozygosity}

detectable for a

_given

_experiment

size

(tables 77-7 V).

Second is to choose a