Theoretical aspects of applying sib--pair linkage tests to livestock species

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Theoretical aspects of applying sib–pair linkage tests to

livestock species

Ku Götz, L Ollivier

To cite this version:

Original

article

Theoretical

_aspects

of

_applying

_{sib—pair}

linkage

tests to

livestock

_species

KU

Götz

L

Ollivier

1

Institut

_für Tierxucht und _{Haustiergenetik} der Universität _Göttingen, Albrecht-Thaer _Weg 1, D-3l00 Göttingen, Germany;

2

Institut

National de la Recherche _Agronomique, Station de _C9ngtique

Quantitative et _Appliquée, 78352 Jouy-en-Josas Cedex, Fmnce

(Received

26 March 1991; accepted 12 December

₁₉₉₁₎

Summary - The Haseman-Elston

_(HE)

_{sib-pair linkage} test in its _original form is

computationally simple but suffers from low power. With the advent of highly polymorphic markers, the exclusive use of _fully informative _matings

_(ie

_matings where the number of

genes identical by descent for any sib pair can be inferred without

error)

for the HE test

becomes feasible. This article examines the influence of _{highly polymorphic} marker systems

(5 alleles),

large

family sizes

_{(6 full-sibs)}

and hierarchical _breeding structures

_(mating

ratio of

₂₅₎

on the _power of the HE test _by means of simulation studies. Simulations are performed under the _assumption that the costs of marker _genotyping are a _limiting factor for _{marker-QTL linkage} studies. _{Consequently,} the total number of individuals

_(parents

and

_offspring)

_typed is fixed at 5 000 in each of the situations _compared. The results show

that the power of the HE test is _considerably increased when both _{highly polymorphic} markers and _large full-sib families are available. For example, for a locus _explaining 8%

of the _phenotypic variance the power of the test increases from 14 to 74% if the locus has

5 alleles instead of 2 and _sibship size is 6 instead of 2. Hierarchical _breeding structures

tend to further increase the _power of the _test, for the _{example given} from 74 to 79%. linkage / marker gene / quantitative trait locus / Haseman-Elston test _{/ power}

Résumé - Aspects théoriques de _{l’application} du test de liaison _génétique par les

cou-ples de _germains aux _espèces animales _domestiques. Dans sa _{forme originelle,} le test de liaison _génétique de Haseman-Elston

_(HE),

basé sur les _couples de _germains, est _simple

à _calculer, mais statistiquement peu puissant. Avec des marqueurs hautement

polymor-phes, l’utilisation excdusive _{d’accouplements} totalement _informatifs

_(ie

des accouplements permettant d’établir avec certitude le nombre de _{gènes d’origine identique pour} n’importe quel couple de

_germains)

peut être _envisagée. Cet article examine, à l’aide de simulations, l’effet d’un _{système génétique} hautement

_{polymorphe (5}

allèles également

fréquents),

d’une

grande taille de _fratrie

_{(6 germains)}

et d’une structure _{d’élevage polygynique}

₍₂₅

_femelles accouplées à _chaque

_mâle)

sur la _puissance du test HE. Les simulations sont _faites en

entre _{gènes marqueurs et} locus de caractères _{quantitatifs.} En _{conséquence,} le nombre

to-tal d’individus _typés

_(parents

et

_descendants)

est _fixé à 5 000 dans chacune des situations

comparées. Les résultats montrent que la _puissance du test HE est considérablement accrue quand on _dispose à la _fois de marqueurs hautement polymorphes et de

_grandes

_fratries.

Ainsi, pour un locus expliquant 8% de la variance phénotypique du caractère, la _puissance du test au seuil de 5% est de 0,7/ au lieu de 0, 14 quand on _passe de 2 à 5 allèles au locus

marqueur et de 2 à 6 frères par fro,trie. Les structures d’élevage-polygyniques tendent à

accroître encore la puissance du test, qui, dans l’exemple ci-dessus, passe de _0,74 à _0,79 avec une structure de !5 _fratries issues du même _{père, par} rapport à des _couples de _parents

indépendants.

liaison _{génétique / gène} marqueur / locus quantitatif / test de Haseman-Elston _/ puissance

INTRODUCTION

The

_{sib-pair linkage}

method of Haseman and Elston

₍₁₉₇₂₎

is a tool for the

detection of

_linkage

between markers and

_quantitative

trait loci

_((aTLs).

The

_major

advantage

of the Haseman-Elston method is its

_{computational}

ease,

allowing

fast

screening

of a

_large

number of marker loci and traits.

_Further,

the method is robust

for a

large

_variety

of continuous distributions of the

_quantitative

trait

(Blackwelder

and

_Elston,

_1982).

_However,

in its

_original

form the method suffers from low power

(Robertson, 1973),

except when the effect of the

_QTL

is _very

_high

and

_linkage

between marker and

_QTL

is

_tight

_(Blackwelder

and

Elston,

1982).

In

_typical

animal

_breeding

_situations,

the

_{possibilities}

for the estimation of effects are often more

advantageous

than in human

populations. Usually

one

has

_{larger families, complete pedigree}

information and markers are available for

parents and

_offspring.

The advent of new,

highly polymorphic

markers such as

minisatellites

_(Jeffreys

et

_al,

₁₉₈₅₎

and microsatellites

_(Weber

and

_May,

_1988;

Soller

and

_Beckmann,

₁₉₉₀₎

increases the

_probability

of informative

_matings

and because

of that also the

_probability

for the detection of

given

linkage relationships.

The

_objective

of this paper is to examine the power of the Haseman-Elston test

in animal

_breeding

situations under the

_assumption

of the

_availability

of

_highly

polymorphic

marker systems.

THEORY .

The Haseman Elston test

The

_linkage

test of Haseman and Elston

₍₁₉₇₂₎

is based on the idea that the

_greater

the number of alleles a

pair

of full-sibs shares identical

by

descent

(ibd)

at a marker

locus which is linked to a

QTL,

the smaller the difference in the values of the

quantitative

trait which is affected

_by

the

_QTL.

A

_{generalized description}

of the method is

_given

_by

Elston

_(1990).

The number of _genes ibd at the trait locus for 2

and sibs’ _genotypes at the marker locus. The

_proportion

of genes ibd at the marker locus for sib

_pair

_j(-

₇

_r

_j

_,,)

is estimated from the

_parents’

and

_{offspring’s}

_genotypes

and the

_regression

of the

_squared

difference of the sibs’

_phenotypic

values on _!r!,n

is calculated. If

_linkage

between the marker and a

_QTL

_exists,

this

_regression

is

expected

to be

_negative.

Haseman and Elston

₍₁₉₇₂₎

assume random

_mating

with _respect to the marker

locus, linkage equilibrium

and no effect of the marker on the trait locus. The

phenotypic

value of the ith sib of the

_jth

sib

_pair

is assumed to be of the form:

where p

is the overall _{mean, 9ij} is the

_genotypic

value at the trait locus and _eij

is the residual effect

_including

_genetic

effects due to all loci other than the linked

QTL.

It is assumed that _ej =

el! - e2i is a random variable with zero mean and

variance

_Q

_e.

In a random

_mating

_population

with a trait locus

showing

a

_given

additive

genetic

variance

_{(a a 2)}

and no

dominance,

the

_expectation

of the

squared

difference

of the 2 sibs’

_phenotypic

values

_(Y

_j

₌

_{!xl! -}

_x2!!2)

_given

the

_proportion

_{of genes}

ibd at the trait locus

_(!r!t)

is shown

_by

Elston

₍₁₉₉₀₎

to be:

In

_practice

_7rjt is not known but has to be estimated

_by

the number of genes ibd at the marker locus

₍

₇

_r

_jm

_).

Elston

₍₁₉₉₀₎

shows the

_expectation

_{of l}

_j

to be:

if there are

_only

2 alleles at the trait

_locus,

where 0 is the recombination

_frequency

between the marker locus and the

QTL.

This

_expectation

can be written in the

form:

...&dquo;....1... - 1.1

Blackwelder

_(1977;

cited in Blackwelder and

_Elston,

₁₉₈₂₎

has shown that

despite

the non-normal distribution of

_Y,

the distribution of the estimated

_regression

coefficient

_(,3)

is

_{asymptotically}

normal so that it is

possible

to use standard normal

theory

to test the

_hypothesis

_Ho:

_Q

= 0

against

the one-sided alternative Hl:

{

3

< 0. Amos et al

₍₁₉₈₉₎

showed that the estimator of

_Q

is unbiased even if there is dominance at the trait

_{locus, provided}

that information on the marker genotype of the _parents is available.

Effect of marker

_polymorphism

Elston

₍₁₉₉₀₎

_gives

formulae for the estimation _{of !r!&dquo;i} from the

_parents’

and sibs’ _genotypes at the marker locus. Since this

_study

deals

_mainly

with

_highly

is available. In animal

_populations

there are

_{usually large}

full-sib families which makes it worthwhile to _type the parents first

and,

according

to these

_{results, only}

the

_offspring

of

_fully

informative

_matings.

A

_fully

informative

_mating

in this sense is a

_mating

where both _parents are

heterozygous

and have at least 3 different alleles at the codominant marker locus which

_corresponds

to

mating

types VI and VII in table II of. Haseman and Elston

(1972).

Thus,

the number of _genes ibd for a sib

_pair

can be inferred without error.

The

_frequency

of these

_matings

in a

_given

_{population depends}

on the number of

alleles at the marker locus and their

frequencies.

In the

_general

case of n alleles and

unequal

gene

frequencies

(p

i

),

the

expected

proportion

of

_fully

informative

_matings

under random

_mating

_(PFIM)

can be written as:

Taking

into account the number of animals

usually

tested in

_mapping

experi-ments, it is

_unlikely

for a _system to be declared as

_{highly polymorphic}

if one or two

of the alleles show extreme

_frequencies.

_Therefore,

the case of

_equal

allelic

frequen-cies

_(p

=

1/n)

is considered here and the

_proportion

of informative

_matings

_is then

given

by:

Table I

_gives

the

_expected

PFIMs for various values of n. This

_proportion

Effect of

_breeding

structure

Most animal

_populations

have a _very different structure from the human

popula-tions for whom this test was

_{originally designed.}

In

_general

the

breeding

structure

for

_poultry,

_pigs

and fish is favorable for effective

testing

because

_large

full-sib fam-ilies are available.

_Sheep

and goat

populations

have an intermediate value for the

application

of the test whereas cattle

_populations

show a _very unfavorable structure.

Blackwelder and Elston

₍₁₉₈₂₎

show

_that,

under the null

_hypothesis

of no

_linkage,

the

_{s(s &mdash;}

_1)/2

sib

pairs

from a

_sibship

of size s can be treated as

_independent

without

_affecting

the _type I error rate, so that

_treating

sib

_pairs

as

independent

should

_provide

a test with

_good

_power, and a correct nominal value. In addition to increased power, the

study

of

large

full-sib families

requires typing

of fewer parents

resulting

in a reduction of overall costs.

_Here,

_comparisons

will be made for a

given

overall cost of

_{typing, assuming}

that there is no limitation of the number of

individuals measured for the trait of interest. In that _case, if a

_proportion

PFIM can be selected _among the families

_available,

the number of

_sibships

( f )

of size s

which can be measured

_given

a total number N of

typed

animals is:

Table II

_gives

the numbers of _parents and

_offspring

for the 3 variants which

will be considered in the simulations. For the first variant

_{( &dquo;standard&dquo; ),}

for which

all _types of

_matings

are

used,

1250 sib

_pairs

from 1 250 families are

generated,

giving

a total of 5 000

_typed

animals. From these 5 000 animals 2 500 have to be measured for the

_quantitative

trait. The second variant has a PFIM of 0.576

(see

table I for n =

5).

From 1587

_{typed couples f}

= 914 show a

_fully

informative

mating

type. These 914

_couples

have a total of 1 828 progeny,

resulting

in a total

of 5 002

₍₂

*

₍₁₅₈₇

+

₉₁₄₎₎

_typed

animals. In the case of families with 6

sibs,

3 168

offspring

from

_f

= 528 families can be

_measured,

an increase of 75% in the number

of

_offspring

as

_compared

to the second variant.

Up

to now, the families were assumed to be unrelated.

However,

in animal

breeding populations

one male is

usually

mated to several females. This

_gives

rise

the variable

₅

_J

is derived from the difference of the

_phenotypic

values of 2

_full-sibs,

it is clear that there are no covariances between the Y!s of half-sib families that

have one _parent in common.

Thus,

the _power of the

Haseman-Elston

test is not

directly

influenced

_by

the

_genetic

structure of animal

_populations.

However,

if the cost of marker

_genotyping

is a

_limiting

factor and the families to

be

_genotyped

are selected in a

_2-stage

_procedure,

the number of measured

_offspring

given

a fixed number of assays can be

_considerably

increased. We consider

_again

the case of a

_population

from which

_fully

informative

_matings

are selected for

genotyping

of the

_offspring

with a fixed overall number of

typed

animals

(N).

In a

first _step

_only

sires are

typed

and the

_{heterozygotes}

are selected for

_genotyping

of

their mates. In a second _step,

heterozygous

dams with at least one allele different

from their mate’s are selected and have their

_offspring

_genotyped.

Then,

the final

number of families

_{( f )}

measured for

_quantitative

traits

_depends

on:

_sibship

size

_{(s) ;}

mating

ratio

_(r

= number of dams

per

sire);

selection rate of sires

_(m

_l

_{) ;}

selection

rate of dams

_given

_heterozygous

sires

₍

_M2

_).

The number of families is then:

With n

_equally

_frequent

alleles

_[6]

can be written as PFIM = _MlM2 with

m

l =

(n &mdash; 1)/n

and _m2 =

((n - 1)/nl -

2/n

2

.

Note that

[8]

does not reduce to

[7]

when r = _1, because of the

_2-step

selection

implied

in

[8].

Table III

_gives

_an

example

of the values of

_f

for different

_mating

ratios

_(r).

For low values of r, the

number of measured families increases

_rapidly

with

_{increasing mating}

ratio. The

largest

effect of this _strategy can be observed for the case of a

_polymorphic

marker

in 2 sib-families.

_Beyond

a ratio of 10 the value of

f

_converges

rapidly

towards the

limit for r

equal

to

_infinity,

which is

_appropriate

if

_only

one male is used.

Simulation studies

Simulations were

_performed

to examine the

_impact

of 4 factors on the _power of

the Haseman-Elston test:

_i)

_fully

informative

_matings;

_ii)

_family

size of 6

_full-sibs;

structure. Other factors varied were:

₁₎

variance due to the linked

QTL

(relative

to

phenotypic

variance)

and

₂₎

recombination

_frequency

between the marker and the

QT

L

.

METHODS

Data were simulated

_according

to the

_following

model in which _eij of model

₍₁₎

is

developed

in 5 terms:

where

x!! =

phenotypic

value of the ith sib in the

jth sibship

p = overall _mean

9ij = effect of the

QTL-genotype

of sib i

bvs! =

polygenic breeding

value of the sire

bvd!

=

polygenic breeding

value of the dam

o

i

j

= effect of Mendelian

sampling

on

_polygenic

value

ce

j = effect of _common environment for the

jth sibship

e2! = random _error

Table IV

_gives

the _range of variation for the different _parameters. Each simulation

was

replicated

500 times. The sizes of the examined

sibships

were 2 and

_6,

respectively.

For the

_{larger sibship}

size the test was based on all

_possible

sib

_pairs

within the

_sibship,

as

_{proposed by}

Blackwelder and Elston

_(1982),

_resulting

in 15

comparisons

per

sibship.

The gene

frequencies

at the trait locus were p = q = 0.5

and additive gene action was assumed. At the marker locus the _gene

_frequencies

were 0.5 for the the standard method

_(assuming

2

_alleles)

and 0.2

_{(corresponding}

to 5

_alleles)

for the case of

fully

informative

_matings.

In the

polyallelic

case

only

the

_offspring

of

_fully

informative

_matings

as defined above were considered as

_being

typed

and thus included in the

_analyses.

All simulations were calculated for a total

To check the

_significance

level,

simulations were

_performed

under the null

hypothesis

(no

effect of

_QTL

or recombination

_frequency

=

0.5)

for all of the

variants

_presented

here. The

_{empirical significance}

level was determined as the

percentage of

_replications

_that,

under the null

_hypothesis,

exceeded the critical value

of a 1-sided t-test with _type I error of 5%. In none of the cases was this _percentage

significantly

different from

_5%,

in a test based on the binomial distribution. The

SAS univariate

_procedure

_{(SAS, 1989)}

indicated no

significant

departure

of the

regression

coefficients from

_normality.

RESULTS

In this section the

_impact

_{of fully informative}

_{matings arising}

from

_highly

polymor-phic

markers and of

_sibships

of size 6 on the _power of the Haseman-Elston test is

examined. Columns 3 and 4 of table V show the effect of

_using

_only

_fully

informative

matings

on the _power of the Haseman-Elston test. The column for the standard

version of the test shows the poor power of this method for the

QTL

effects

con-sidered here. When the

_QTL

contributes 16% to the

_phenotypic

_variance,

which is

equivalent

to 1.1

_phenotypic

standard deviations between the 2

_homozygotes,

the

power is

only

33%. The use of

_fully

informative

matings

hardly

increases _power,

except for

_{higher QTL}

effects.

However,

even for the

_largest

QTL

effect the power

of the test is below 50%.

Table

V. Effect of standard vs _fully informative _matings and effect of _family size on the

power

(in

%)

of the Haseman-Elston test, assuming a constant number of _typed animals

(N

= _{5 000,}

h

2

= _0.25, a = _{0.05, 500}

_{replications).}

For lower

_QTL

effects the _power shows more or less erratic fluctuations with

between 15 and 20 _percentage

_points

when the recombination rate increases from 0 to 0.1.

The use

of larger families

leads to a

_major

increase in the _power of the

Haseman-Elston test.

_QTL

effects of 8% can be detected with more than 50% power if the

recombination rate is 0 or 0.05

(column

_5,

table

V).

The effect of a

_{within-family}

environmental _component on the power of the test

is

_given

in table VI. This component reduces the

_{within-family}

variance and thus increases the power of the test. The average increase is 59% of power for the 2-sib

and 40% for the 6-sib families. In the latter _situation, a

QTL

effect of 4% can be

detected with

_nearly

50% power if a common environmental _component is _present

and there is no recombination between the marker and the

QTL.

In the simulations of hierarchical

_breeding

structures the numbers of families for

r = 25 _were used. The results _are

given

in table VII. It _can be _seen that in this

situation the test based on 2-sib families is still not

_competitive.

In the case of 6-sib

families one should be able to detect a

QTL

effect of 8% with _power between 48

and 79%. For smaller

_QTL

effects power is not sufficient unless there is additional

&dquo;support&dquo;

from common environmental effects.

Soller and Genizi

₍₁₉₇₈₎

_using

the method of

_Jayakar

₍₁₉₇₀₎

_presented

calcu-lations for half- and full-sib

_designs.

The method of Soller and Genizi

₍₁₉₇₈₎

for

a

_QTL

_contributing

4%

of the

_phenotypic

variance of the

_population

has been

compared

to our simulation

_results,

as summarized in table VIII. The base of the

comparisons

is an

_equal

number of

_preselected

_matings

for both tests

_(fully

infor-mative for

_{Haseman-Elston,}

intercross for Soller and

_Genizi).

The test of Soller and Genizi

₍₁₉₇₈₎

_always

has less _power than the Haseman-Elston test for the two

DISCUSSION

The _present

_study

confirms the

_findings

of Robertson

₍₁₉₇₃₎

that the Haseman-Elston

_{sib-pair linkage}

method in its

_original

form has _very low _power. This is

especially

true if the variance

_{explained by}

the

_QTL

is small as

compared

to the

should aim at a reduction of the residual variance. On the other

hand, systematic

environmental and sex effects that affect the difference between full-sibs would

have to be eliminated.

_This,

as well as the increase in _power due to common

environmental

_effects,

leads to the recommendation that full-sib families should be reared

_together,

as

long

as no

_competition

effects occur.

Preselection of

_fully

informative

_matings

leads to an increase of the _power of

the _test,

_especially

for

_higher

_QTL

effects.

_Furthermore,

the power estimates found

by

simulation agree well with the values calculated

according

to the

asymptotic

formulae

_given

_by

Blackwelder and Elston

₍₁₉₈₂₎

and Amos and Elston

_(1989).

Thus,

for a

_{given mating}

structure and marker

_{polymorphism,}

the necessary size of

experiments

for the detection of

_{marker-QTL linkage}

can be calculated in advance

for a

_given

QTL

effect. Power could also be increased

by

selective

genotyping.

Instead of

_selecting

informative

_matings

one would rather select

sibships

with at

least one extreme individual for the trait

_investigated

_(see

the 2-sib case

_recently

analysed by Carey

and Williamson

_(1991)).

Blackwelder and Elston

₍₁₉₈₂₎

concluded that a

_significance

level of 5% as chosen

in this

_study,

would not be _strong evidence for

_linkage

in view of the low

_prior

odds

in favor of _{2 genes}

_being

on the same chromosome.

However,

since this method is a

preliminary

instrument for the

_scanning

of

_segregating

populations,

a

_relatively

high

type I error could be

_accepted

in order to reduce the risk of

rejecting

possible

QTLs.

A crucial

_question

is whether power is influenced

by

using

non-independent

comparisons

in

_large

families. Results of

_Hodge

₍₁₉₈₄₎

suggest that the information

contained in a

_sibship

of size s

_approaches

2s &mdash; 3 for s > 4.

_However,

these

_findings

could not be confirmed in simulation studies of Blackwelder and Elston

₍₁₉₈₂₎

and Amos et al

_(1989),

who treated

_sibships

of sizes 3 and 5 as in the _present

study.

Their results indicate that neither _type I error nor _power are affected

by

treating

all the

_comparisons

in a

sibship

as

independent.

The same was true for the _present

study.

Furthermore, family

sizes would not be

_equal

in field studies. This is advanta-geous with respect to power since the

_expected

number of

_comparisons

per

sibship

increases with the variance of

_sibship

size. A random variation of

_family

size

_(s)

in

_pig

_populations

may be

approximated by

a Poisson distribution with

parameter

8 =

E(s)

=

V(s).

Thus,

_as

E[s(s - 1)/2] = [

-1)

8

8

(

₊

V(s)!/2

= !/2,

the _average

number of

_comparisons

per

family

is

always

greater than with constant

_family

size

and twice as much for s = 2. For

pig populations,

_{an average}

family

size of 6 is a

rather

_{pessimistic assumption;}

in

_practice

it would rather be between 8 and 9 and

one such

_family

would on _average be

equivalent

to 32-40 families of 2 sibs.

The extension to hierarchical

_breeding

structures increases the _power if the families to be

_typed

are selected in a

_{2-stage procedure. Relationships}

between

families do not induce covariances between

_{the 5}

_J

and therefore need not be

considered.

In a

typical

_pig

breeding

situation the method can be used to detect

_QTLs

of

effects between 4 and 8% of the

_phenotypic

variance with about 50% power for a

total of 5 000 assays. This should include all

_economically

_interesting

_QTLs

because

in the

_supposed

situation

_{(linkage equilibrium)}

the

_phases

must be determined

determination of

linkage

phases

causes additional _cost,

_only

_(aTLs

with

large

effects are of interest.

Another

_question

is whether the

_preselection

of

_fully

informative

matings

may

give

rise to false

_linkage.

Preselection is made

_only

with

_regard

to the marker

genotype.

Therefore,

false

_linkage

should not occur as

_long

as there is

linkage

equilibrium

between the marker and the

_QTL.

However,

this cannot be proven

until the

_QTL

_genotype can be determined

directly.

As

suggested by

an _anonymous

referee,

inferences

_suggesting

evidence in favor of

_linkage

in the case of

linkage

disequilibrium

should be correct.

_Furthermore,

_falsely

_positive

evidence should not

occur for unlinked loci which are in

_{linkage disequilibrium}

when data for the parents

are available.

However,

this has never been studied for

sib-pair

methods. The main reason for

_{linkage disequilibrium}

in animal

populations

is

hybridization

between

subpopulations

that have been

_kept

_separate for several

_generations

_(Lande

and

Thompson,

1990).

If this should be the case in the

_{population considered,}

one

would rather use the standard

methodology

of

multiple

regression

which

exploits

the

_existing

_{linkage disequilibria.}

A

_comparison

of the _present results with those from other workers is difficult

since

_only

few

_{investigations}

deal with the

_problem

of

_{detecting linkage}

within

segregating

populations.

The

_comparison

with the method of Soller and Genizi

(1978)

showed that the _power of this method is inferior to the Haseman-Elston

test. The reason is that the Soller and Genizi method favours half-sib families in

the order of 1 500 animals and is thus better suited to

_dairy

cattle

_populations.

Weller et al

₍₁₉₉₀₎

introduced the

_{granddaughter}

_design

which leads to a

considerable increase of _{power for} a

_given

number of _assays

compared

with the

Soller and Genizi

₍₁₉₇₈₎

_design.

It also

_depends

on

_{highly polymorphic}

markers

and its range of

_application

is limited to

_dairy

cattle

_populations

because this method

_requires

_large

numbers of sons per sire and

granddaughters

per son to be effective.

Beckmann and Soller

₍₁₉₈₈₎

considered crosses between

segregating

populations.

Their method is based on

F

2

crosses and thus

_requires

additional

testing

of

F

i

individuals,

with at least 2

_generations

and a

_special

experimental design.

It also

implies

2

_{parental populations}

close to fixation

_(p

>

_0.8)

for alternative

_QTL-alleles

and a difference between the 2

homozygotes

of at least 0.4 standard deviations. The authors _suggest that this is more

likely

for traits like disease resistance than for the

traditional

_performance

traits.

CONCLUSIONS

With the advent of multiallelic

_markers,

the Haseman-Elston

_sib-pair

_linkage

method becomes more

_powerful

for the detection of

linkage

between markers and

QTLs.

However,

the marker should at least have 5 or more alleles at intermediate

frequencies

in order to reduce the number of parent animals to be

_typed.

The

method allows the detection of

_linkage

within _any

_{segregating population.}

It can serve to indicate whether more

sophisticated

methods,

that estimate recombination

frequency

and allele effects but

_require

_special

_mating

_plans,

are

_appropriate.

The

The power of the method increases

considerably

with

increasing

family

size and

with hierarchical

_breeding

structures.

_Thus,

it is

_especially

useful for

_species

with

high reproductive

rates such as

_pigs

and

poultry.

Amos and Elston

(1989)

extended

the _{test to any type} of non-inbred relative

_{pair. Thus,}

it could also be

_applied

to

dairy

cattle where

_large

half-sib families are available.

_However,

on _average twice as

many half-sib

pairs

are needed as

compared

to full-sibs and at _present there exists no

_possibility

of

_combining

different _types of relatives in one

analysis.

ACKNOWLEDGMENTS

The leave of KU G6tz at the Station de _{G6n6tique Quantitative} et Appliqu6e at INRA, Jouy-en-Josas, was _{supported by} a _grant from the Deutsche _{Forschungsgemeinschaft,} which is _{gratefully acknowledged.} The authors wish to thank two

_anonymous

referees for valuable comments on the _manuscript.

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