Use of sib-pair linkage methods for the estimation of the genetic variance at a quantitative trait locus

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Use of sib-pair linkage methods for the estimation of the

genetic variance at a quantitative trait locus

H Hamann, Ku Götz

To cite this version:

Original

article

Use of

_{sib-pair linkage}

methods

for

the

estimation of

the

_genetic

variance

at

a

quantitative

trait

locus

H

Hamann, KU

Götz

Bayerische

Landesanstalt

f3r Tierzv,cht, Prof

Diirrwaechter-Platz 1,

85586

Grub, Germany

(Received

26

_{August 1994; accepted}

19 November

₁₉₉₄₎

Summary - Until

_recently,

the

_{sib-pair linkage}

method of Haseman and Elston could

only

be used for the detection of

_linkage

between a

_quantitative

trait locus

_(QTL)

and a marker locus. It was not

_possible

to estimate the amount of

_genetic

variance contributed

by

the

_QTL

or its recombination fraction with the marker locus. With the advent of dense marker maps for

nearly

every domestic

species,

every

QTL

should be located between 2

flanking

markers. In this

_situation,

the Haseman-Elston test can be modified to estimate

the variance of a

_{putative QTL}

as well as its recombination fractions with the 2

_flanking

markers. In the present paper, we derive 2 different estimation methods for the

_QTL

variance based on the

squared performance

of full sibs: in one

only

the

QTL

variance

is

_estimated,

while in the other both the

QTL

variance and the recombination fractions are estimated. The method that estimates

_only

the

QTL

variance turns out to be more

powerful

than the other. With respect to the estimation of

_QTL

variance both methods

give

results close to the true values.

_However,

the estimation of recombination fractions resulted in an overall underestimation of the true _parameters.

sib-pair linkage

/

quantitative trait locus

_/

genetic marker

_/

genetic variance

_/

recombination fraction

*

Correspondence

and reprints

Résumé - _Emploi des méthodes d’évaluation des liaisons génétiques par les couples

de _germains pour estimer la variance génétique à un locus de caractère quantitatif.

Jusqu’à

une

_{période récente,}

le test de liaison

génétique

de Haseman et

_Elston,

basé sur les

couples

de

_germains,

ne

_pouvait

être utilisé que pour la mise en évidence de liaisons entre un locus à

effet quantitatif

(QTL)

et un locus _marqueur. Il n’était _pas

possible

d’estimer la part de la variance

_génétique

totale liée au

QTL,

ni le tau! de recombinaison avec le locus _marqueur. Suite au

_{développement}

de cartes denses dans la

plupart

des

espèces

d’élevage domestiques, chaque QTL

est

_susceptible

d’être localisé entre 2 locus marqueurs

à

_{la fois}

la variance du

_QTL

et les taux de recombinaison avec chacun des locus _marqueurs

flanquants.

Dans le

_{présent article,}

2 méthodes d’estimation de la variance du

_QTL

basées sur les

_{différences quadratiques}

des

_performances

des

_germains

sont

_{développées :}

l’une n’estime que la variance du

QTL,

en revanche l’autre estime la variance du

QTL

et les 2 taux de recombinaison. Une étude de simulation permettant

d’apprécier

la puissance

et la

_qualité

des 2 méthodes d’estimation est

_présentée.

La méthode permettant d’estimer la variance du

_{QTL uniquement apparaît plus puissante}

que la seconde.

Chaque

méthode

donne des résultats assez

_proches

des vraies valeurs en ce

_qui

concerne la variance du

QTL.

Les taux de recombinaison sont en revanche

globalement

sous-estimés.

liaison génétique par couples de germains

/

locus de caractère quantitatif

/

marqueur

génétique

/

variance génétique

/

taux de recombinaison

INTRODUCTION

Haseman and Elston

₍₁₉₇₂₎

_developed

the idea of

_{detecting linkage}

between a

genetic

marker and a

quantitative

trait locus

(QTL)

by examining

the

squared

difference of the

performance

of full-sibs. Other studies

_(Blackwelder

and

Elston,

1982;

Amos and

_Elston,

₁₉₈₉₎

showed that the method is robust

_against

a

_variety

of distributions of the trait examined and that it can also make use of multivariate

data

_(Amos

et

_al,

_1989).

G6tz and Ollivier

₍₁₉₉₂₎

found that in animal

_populations,

especially

in

_pigs,

the power of the method is at least

comparable

to that of methods based on the

analysis

of variance.

_However,

the Haseman-Elston method in its

original

form could

_only

detect

_linkage

between a marker and a

_QTL,

but could not

estimate whether this was due to a

QTL

with

_large

effect at a

large distance,

or to a

_QTL

with small effect that is

_closely

linked to the marker.

Studies for the establishment of a

_{complete linkage}

_map of the

_pig

_genome are

under way

(Anderson

et

al, 1993;

Rohrer et

al,

1994).

This will lead to the situation

that in the near future _every

_QTL

of economic

_importance

will be in the

_vicinity

of

2

_flanking

markers. This article will show how

_{sib-pair linkage}

tests can be

_applied

for the estimation of the variance caused

_by

a

QTL

located between 2

flanking

markers. A simulation

_study

will be

_presented

to examine the power and

properties

of the method.

MATERIALS AND METHODS

Theory

Haseman and Elston’s test

₍₁₉₇₂₎

is based on the idea that the difference in the

performance

of full-sibs becomes smaller if the sibs share a

_larger

_proportion

of alleles identical

_by

descent

_(ibd)

at a

_QTL

with

_large

effect. Elston

(1990)

_gives

a

_{general description}

of the method that will

_{only briefly}

be outlined here for the

simplified

case of a

_QTL

with no dominance. The basic variable of the

Given the

proportion

of genes ibd at the

_QTL

_(!r!t),

Elston

₍₁₉₉₀₎

shows that the

_expectation

of

_Y

_j

is:

where

_a’

_is

the additive

_genetic

variance due to the

_QTL

and

_ae

_the

variance of

the difference of all other

genetic

environmental

components.

Since the

proportion

of genes ibd at the

_QTL

cannot be

_observed,

the

_proportion

of genes ibd at the linked marker locus

₍

₇

_r

_j

_m)

must be used to estimate _7rjt. The

_expectation

of

_Y

_j

given

!r!!

is:

where B is the recombination

_frequency

between

_QTL

and marker locus. This is a

general

linear

_{regression equation}

and can be written as:

The

_expectation

of the

_regression

coefficient is:

where b is an estimator of

!3.

This

_expectation

is zero if either

Q9

_is

zero or 0 is

equal

to 0.5. Blackwelder and Elston

₍₁₉₈₂₎

showed that the distribution of the estimated

_regression

coefficients is

_{asymptotically}

normal.

_Thus,

a

_simple

one-sided t-test can be

_applied

to test whether the

_regression

coefficient is

_{significantly}

negative. However,

it can also be seen from the

_expectation

of b that a

_{significantly}

negative

estimate can result from a

_{large 0 together}

with a

_{large QTL}

effect or from a small

_QTL

effect and

_{tight linkage.}

To estimate 0 and

_Q

_q,

we _suppose that there are 2 markers

flanking

the

QTL.

This

assumption

seems valid in the case where a

_complete

marker map

exists. The number of

parameters

to be estimated increases to 3: 2 recombination

frequencies,

which will be

_designated

0

1

and

₀

₂

_,

and the

_QTL

variance

_Q

_q.

The total recombination

_frequency

between the 2 markers

₍₀

_t

₎

can be

supposed

to be

known from a

_{mapping experiment}

or can be estimated

_directly

from the data.

Method I. Estimation

_using

2

_separate

tests of

_linkage

Two different

_approaches

can be taken to estimate

_{a§ in}

the case of 2 markers. The

first

approach

arises in a situation where

_separate

test

_linkage

for 2 markers lead to

significant

results. If the marker loci are known to be

_linked,

all 3

_parameters

can

be estimated

_using

the

_expectations

of the 2

_regression

coefficients

_(b

₁

and

_b

₂

_).

As

no interference between

₀

₁

and

₈

₂

_.

Since

B

2

can be inferred from

0

1

and

0

t

via

_equation

_[5],

solutions for the 3

unknowns can be found.

_However,

because the _{range of}

_possible

values for the 2

regression

coefficients is

_{theoretically}

between

_plus

and minus

_infinity,

there is not

always

a solution in the _range of real numbers.

Method II. Estimation

_using

the combined information of 2 markers The second

_approach

starts out from the fact that from

_equation

_[2]

the estimator

of the

_regression

coefficient divided

_{by —2}

is

_already

a biased estimator of

o- q 2.

In the

_single

marker _case, the bias increases

_rapidly

even for small recombination

frequencies,

rendering

the estimator

_practically

useless.

_If,

_however,

the data is restricted to

_sib-pairs

with the same

_proportion

of _genes ibd at both marker-loci

(

7

r

jml

=

7

r

j

&dquo;

2

)

1

then in the

_majority

of cases the

_proportion

of _genes ibd at the

_QTL

(

7

r

jt

)

is

_equal

to the

_proportion

_{of genes ibd} at the 2 marker loci. This will not occur in 2 rare situations:

_i)

in case of double recombination and the 2 recombinations

take

_place

on either side of the

QTL;

and

ii)

if 2

_separate

recombination events

in 2 sibs take

_place

on different sides of the

_{QTL. Consequently,}

the

_proportion

of

alleles ibd at both marker loci is a reliable estimator of !r!t. The

_price

to be

_paid

for this is that the

_proportion

of usable

_sib-pairs

is reduced

_by

a factor that can be

expressed

as:

where Tf

t

=

(1 -

₂₀

_t

₊

20

¡

).

In the _case of _a 20 cM marker

map and informative

matings,

55%

of the

_sib-pairs

would be

_selected,

and if the markers were at distances

of 4 cM that fraction would increase to

86%.

The

_expectation

_{of F}

_§

_given

a certain

_proportion

_(x)

of alleles ibd at the marker loci is:

which can

_again

be written as a linear function of _7rjml:v

where

1Jt

1

=

(1

-

201

₊

20i)

and

_!2

=

(1

-

202

₊

2

B

2)

.

Again,

_a9

=

-b

o/

2

is a biased estimator of the

_QTL

variance. Whether the bias is

_acceptable

or

_not,

it

_depends

on the size of the

biasing

factor

in the range of realistic values for

O

t

.

Figure

1 shows the value of k as a function of

₀

₁

for 4 different values of

O

t

.

The maximum bias

_always

occurs if

0

1

=

.

2

0

The maximum is _not

equal

_to

O

t/

2

because with no

_{interference,}

the 2 recombination rates do not act

_additively.

For

large

values of

_Bt,

k can take values down to

_0.93,

while for smaller values the bias

is

_negligible.

_Figure

2 shows that _range of

_possible

values and the

_expectation

of k

depending

on

_O

_t

_.

It can be seen that the

expectation

of k results in a bias of less

than

5%

over the whole _range considered.

The

_expectation

of k for a

given 0

t

can be

easily

calculated. Since k is

always

between 0 and

_1,

a second estimator for the

_QTL

variance can be derived

_{by dividing}

where

_E(k)

is

_{given by:}

Simulation

A simulation

_study

was conducted in order to examine the _power of the 2 methods and the

_goodness

of estimation. Data were simulated

_according

to the

_following

model:

’

where:

r

z

j

=

phenotypic

value of animal i in

family j

!’

f

1, = overall mean

q2! = effect of the

QTL

_genotype

of animal i

6f!

= sire’s contribution to

polygenic breeding

value

(without

QTL

genotype)

bv

dj

=

dam’s contribution to

_{polygenic breeding}

value

_(without

_QTL

_genotype)

!2!

= Mendelian

sampling

effect

ce

j = effect of common litter environment

eZ! = residual error

For the constant

_parameters

in the

_simulation,

the

_following

values were used:

(including

the

_QTL

_effect)

and common environmental variance was 0.2. The

population

structure simulated was that of a

typical pig-breeding

situation with

25

_sires,

10 dams _per sire and 8 _{progeny per} sire-dam

_{pair. Thus,}

a total of 2 000 progeny were simulated in each

_replication.

For a discussion of the effects of the

mating

structure and common

_environment,

see G6tz and Ollivier

(1992).

Gbtz and Ollivier

₍₁₉₉₂₎

found that the use of

fully

informative

matings

can

increase the power of the Haseman-Elston test for a

_given

number of

_genotypings.

Consequently, only

these

_matings

were used in the calculations. This has no consequence for the

validity

of the

results,

but it should be borne in mind that

the number of

_genotyped

individuals in

_practice

would be

_{slightly higher}

than

2 _{275. Within any}

_family,

all

_possible

differences between full-sibs were used for the

calculation of

_Y!s

as

_proposed

_by

Blackwelder and Elston

(1982).

This resulted in

28

_comparisons

_per

_family

and 7 000

comparisons

per round of simulation.

Variable

_parameters

in the simulation were:

(i)

the distance between the 2 markers

₍₀

_t

₎

(ii)

the

_position

of the

_QTL

between the 2 markers as

_expressed

_by

₀

₁

and

₀

₂

(iii)

the size of the

_QTL

effect

The distance between the 2 markers was varied

approximately

between 0.04 and

0.154, assuming

no interference. The combinations of

₀

₁

and

0

2

that were simulated are

_given

in table I.

Two codominant alleles with

_{equal frequencies}

were assumed at the

_QTL.

For

both marker loci 10 alleles with

_equal

_frequencies

were simulated and for the

_QTL

effect

_genetic

variances of

_40,

80 and 120 were assumed. This resulted in a total of

30 different

_variants,

each of them

_being

simulated with 1 000

_{replications.}

Analysis

of simulation results

The power of the methods was defined as the

_percentage

of

_replications

where

the null

_hypothesis

was

_rejected

at the

5%

level. For Method II this

_approach

is

unambiguous

while this is not the case for Method I. For the first method there

are 2 null

_hypotheses

of which 1 or both can be

_rejected

at a = 0.05. Since

nominal

_type

I errors for a

global

error of

5%

can

only

be determined

by

simulation

under the null

_hypothesis.

_However,

these

_type

I errors still

_depend

on the 2

recombination

_frequencies

so that the true state of nature must be known for an exact determination.

Therefore,

it was decided that a

_replication

was

significant

for

Method I if both null

_hypotheses

were

_rejected

at a

5%

level. For the

_{interpretation}

of the results it should be borne in mind that Method I has a

_{slight disadvantage.}

For the estimation with Method II

_{only sib-pairs}

with the same

_proportion

of

alleles ibd at both marker loci were selected from the same data that were used

for the estimation with Method I. As was

explained

previously,

this results in a

reduction of the number of effective

_sib-pairs

of between

15%

_(for

_B

_l/

_B

_Z

=

0.02/0.02)

and

42%

_{(for 0}

_1/

₀

₂

=

0.02/0.14).

To assess the

_goodness

of the

_estimation,

all

_replications

of a certain variant

(significant

and

_{non-significant)}

must be

_averaged.

As can be seen from

_equations

[3]

and

_[4],

the first method

_requires

the

_square-root

of the ratio of

_b

_i

and

_b

₂

for

the estimation of

_{a q 2.}

In

_{practical applications}

this is not

_likely

to cause

problems,

since

_significant

regression

coefficients

_always

have a

_{negative sign.}

In a simulation

with a low value for the

QTL effect, however,

this causes

_problems

because the

estimated

_regression

coefficients are

_normally

distributed and a certain fraction can

be

expected

with

positive

values. For these

replicates

a value for

_Q9

_cannot

be

estimated. Because

_regression

coefficients at

_positive

values are all

non-significant,

the

_missing

_QTL

variances cause an overestimation of this

_parameter.

RESULTS

Power of the 2 methods

Table II shows the power of the 2 methods of estimation for all simulated variants.

For a

_QTL

effect of

_40,

the _power is low for both methods and all variants.

_However,

it can be observed that Method II has

higher

power in all variants and that the

decrease _{in power} with

_{increasing 0}

_t

is less for the second method. For a

_QTL

effect

of

80,

the

superiority

of the second method is evident. The

superiority

is more

pronounced

if the values of

0

1

and

0

2

are

_unequal,

which is caused

_by

an

_increasing

proportion

of

_replicates

where

_only

one of the 2 tests in Method I

_gives

a

significant

result. If the

_QTL

effect is

_120,

both methods have

_high

power with differences

occurring

only

if the 2 recombination rates were of _very different size.

Estimation of

B

l

,

0

2

and

_a9

_using

Method I

The _average estimated values for

_Q

_q

_are

_given

in table III for the 2 methods. For low values of

_Q

_q

_an

overestimation _occurs, which is caused

by

the fact that a certain

number of

_replications

could not be calculated for reasons mentioned above. This

could be as much as

24%

of the

_replicates.

With

or2

_equal

to 80 and

_120,

the

percentage

of

_replicates

without result decreased to 10 and

3%,

respectively.

In accordance with these

_numbers,

the overestimation is less with

_increasing

_QTL

variance and

_decreasing

recombination fractions within

_QTL

variance. For a

QTL

fractions. In accordance with the fact that most of the

_dropouts

occur if the 2

recombination fractions are of _very different

_sizes,

the worst estimates are achieved

if the

_QTL

is located close to 1 of the 2

_flanking

markers.

The estimated values for the recombination fractions

_equally

suffer from the

problem

of

_replications

without solution. In contrast to the estimation of

_QTL

variance,

this leads to an underestimation of recombination fractions for small values

ofa q. 2

Table IV shows that for a

_QTL

variance of 40 the estimators are

heavily

biased

downwards. This

_improves

with

_increasing

values for

_{a q 2.}

A remarkable decrease of the standard deviation of the estimates can also be observed.

However,

none of

the estimates are _very

precise,

mainly

due to the low

expected

numbers of double

recombinants within the 2 000 progeny.

Estimation of

_a

_using

Method II

The estimates for

_a9

_using

the second method are also

presented

in table III. From

theory,

Method II is

_expected

to underestimate the true value of the

_parameter.

This

_expectation

is confirmed

_by

the results with a

single

exception.

Especially

for

_QTL

variance of 40 the estimates are

_clearly

_superior

to those of Method 7. For

larger

values

_{of B}

_t

the underestimation

_gets

_larger

but

stays

within the _range that can be

explained by

the

decreasing

value of k.

The results for the second estimator

_(i!2*)

are also

_given

in table III as Method IIc.

On _average, this

_manipulation

reduces the underestimation of

_{aq from}

about

3%

DISCUSSION

The

_present

_study

has shown that with 2

_flanking

markers for a

_given

_QTL,

the

principle

of

_{sib-pair linkage}

methods can be

_applied

to obtain estimates of the

_QTL

variance and to locate the

_QTL

in the interval. The simulation

_study

made use

of the results of G6tz and Ollivier

_(1992),

who showed that in animal

_breeding

the

_preselection

of

_fully

informative

_matings

is an

_appropriate

_{way to}

_improve

the power of

QTL-detection

for a

_given

number of

_genotypings.

If _many markers are to be

_examined,

the

_parents

will

_only

be informative for a fraction of the markers.

Since the

_major

costs in the

_given

_design

arise from the

_typing

_{of progeny,} these

should

_only

be

_typed

for the markers where their

_parents

are informative. With

Method II it should also be

_possible

to use

_multiple

markers as

proposed by

Haley

et al

_(1994).

However,

without modification this

_only

seems feasible if the

_linkage

phases

in the

_parents

are

known,

recombinant sibs are excluded from

the

analysis

and the markers are not so far

_apart

that double recombinants become

important.

The results on power for the detection of a

_given

QTL

_suggest

that a

_QTL

contributing

8%

of the

_phenotypic

variance can be detected with a

power

between 55 and

75%

for the

_{given design.}

In

_comparison

with the results of G6tz and Ollivier

(1992)

it must be taken into account that in the

_present

_study

the

_QTL

effect was

included in the total

_genetic

variance. In a

comparable

situation the power of both

methods

_presented

here is less than in G6tz and Ollivier

_(1992).

The reasons are

that the

_type

I error for Method I is not

_comparable

to the

5%

level in the

_previous

study

and that Method II uses fewer

sib-pairs.

The power of

Method

II is

superior

to that of Method I in all of the simulated variants. The reason is

_evident,

since Method I does not use the

prior

information that the 2 marker loci are linked with a

known recombination fraction for the test of

_linkage.

This information is

_only

used

Future research should be directed towards a _way of

_{incorporating}

the information

of

_linkage

between the markers in the detection of

_linkage

with a

_QTL.

One _way to do this has

_recently

been

presented

by

Fulker and Cardon

_(1994).

They

used the information of 2 markers to estimate 7rt and

regressed

g

on this

estimated value. Since the estimation

_only

works if

₀

₁

and

₀

₂

are

known,

they

use an

approach

similar to interval

_mapping

_(Lander

and

_Botstein,

₁₉₈₉₎

to

_plot

the

t-statistics

_against

the

_putative

_QTL

_position.

The authors also encountered

problems

in

_trying

to determine the correct t-value for a certain

_type

I error

_rate,

even for

the

_single

interval

case, which

they

solved

_by

simulation

under the null

_hypothesis.

However,

in every realistic scenario this

problem

will occur since

_usually

_many

markers will be examined at a time. In this

_situation,

none of the test statistics has

a

_simple

distribution

_and

one would

_always

have _{to escape to} simulation studies in

order to examine the distribution of the test statistic.

A

_comparison

of the results of Fulker and Cardon

₍₁₉₉₄₎

at

h

2

= 0.125 with our

results

_{(Method II)}

for a

_QTL

variance of 120 shows little difference. This indicates that sibs with different

_percentages

of alleles ibd at the 2 marker loci contribute

little

or

_nothing

to the estimation of 7rt.

In the estimation of the

_{QTL variance,}

Method I is characterized

_by

the overestimation of

_a

_§

if

the true value is small. In

_practice,

this is not

_likely

to

be a

_problem,

since

significant

_regression

coefficients are

_always

_negative,

but it

makes it difficult to prove the unbiasedness of the estimator.

_However,

for a

_QTL

variance of 120 no overestimations occurred and underestimations were in all cases

less than

3%.

Method II is a

_priori

a biased estimator. The

results

show that the bias is small if the

_QTL

is located close to one of the markers and that the estimation of

or2can

be

_improved

_{by dividing}

the initial estimator

_by

_E(k).

The maximum bias

can also be

_quantified

if the recombination fraction between the 2 markers is known.

However,

since Method I

_gives

similar estimates and information about the location of the

_QTL,

one could use Method II to detect simultaneous

_linkage

of 2 markers with a

_QTL

and then use Method I for the estimation.

_{Unfortunately,}

Fulker and

Cardon

₍₁₉₉₄₎

_give

little information about the

_quality

of their estimator of the

QTL

variance. The

_only

result

_{they give}

indicates that their

_algorithm

leads to an

overestimation of

_a

_q

₂

The estimation of recombination

_frequencies

between the

_QTL

and the markers leads to

_{unsatisfactory}

results. The

_majority

of recombination fractions were

underestimated,

although

for

_{higher QTL}

effects the estimators came close to the

true values. Non-estimable

_{replicates certainly}

influenced these results as well. The same observation can be made from the results of

_{Fulker and}

Cardon

(1994)

which show

_relatively

flat curves in the

_vicinity

of the true

QTL

location and a

_tendency

to

_place

the

_QTL

in the middle of the interval. In

_comparison,

our Method I tends to locate the

_QTL

closer to the marker with the smaller recombination fraction.

Knott and

_Haley

₍₁₉₉₂₎

examined the

_application

of maximum likelihood

_(ML)

in

_outbreeding

_populations

with a full-sib structure. The

_advantage

of ML is the fact that it is

_possible

to estimate the _gene effects at the

QTL

as well as the _gene

frequencies.

However,

the

computational

effort is much

_higher

for ML than for

the

_approach

in the

_present

_paper. The authors conclude that for the treatment

of realistic

population

structures and the inclusion of fixed

_effects,

numerical

Haley

and Knott

₍₁₉₉₂₎

_presented

a method for

the

_mapping

of

(aTLs by

regression

in crosses between inbred lines. This

_design

is

not

tractable with our

methods because of the

complete

linkage

disequilibrium

in an

_F

₂

derived from

inbred lines.

_However,

those authors found that there ’seemed little

_advantage

to

be

_gained

from resort to maximum likelihood methods for the

_analysis

of these

types

of data’. The method is similar to that of Fulker and Cardon

₍₁₉₉₄₎

since both methods are based on the idea of interval

_mapping

_(Lander

and

_Botstein,

1989).

CONCLUSIONS

The extension of Haseman and Elston’s

₍₁₉₇₂₎

method

_{of sib-pair}

_linkage

_presented

here allows for the estimation of

_QTL

variance and recombination fractions if a

relatively

dense and informative marker map is available. Since the method uses

only intra-family

comparisons

it does not need to take fixed effect into account so

long

as

_they

affect all sibs in a

_family

in the same _way.

However,

there are some limitations of the method that shall be mentioned here. The first is that the results

_presented

_{rely heavily}

on the

availability

of

highly

polymorphic

markers. If one or both of the markers is not very

polymorphic,

the

number of

parents

to be

_typed

increases

_{dramatically.}

For a discussion of this

topic

see Gbtz and Ollivier

(1992).

The second limitation is the

_dependency

of

sib-pair linkage

tests on the

_magnitude

of the residual variance. For traits with low

heritability

the _power of the method is

_low,

while

_high

_heritability

and common

environmental effects are favourable. In

_addition,

_large

_family

sizes increase the

power of the method

(G6tz

and

Ollivier,

1992).

In

_{outbreeding populations}

the detection of

_segregating

_(aTLs

is

_generally

more

difficult than in crosses between inbred lines

(Knott

and

_Haley,

1992).

_However,

the

detected

_(aTLs

are known to be

_segregating

while with

_(aTLs

detected in

_crossing

experiments

a

large

fraction of favourable

QTLs

will

already

be fixed in the

_superior

line. It is doubtful whether

_preselection

of

_fully

informative

_matings

will still work if _many markers are to be

_examined,

since

_parents

will be informative

_only

for a

fraction of markers.

_{Nevertheless,}

it would be

_possible

to

type

the progeny

only

for

those markers where their

parents

are informative.

ACKNOWLEDGMENTS

This work was

_{supported by}

a _grant of the ’Deutsche

_{Forschungsgemeinschaft’}

which is

gratefully acknowledged.

REFERENCES

Amos

_CI,

Elston RC

₍₁₉₈₉₎

Robust methods for the detection of

_genetic

_linkage

for

quantitative

data from

_pedigrees.

Genet

_Epidem

6, 349-360

Amos

_CI,

Elston

_RC,

Wilson

_{AF, Bailey-Wilson}

JE

₍₁₉₈₉₎

A more

_powerful

robust

sib-pair

test of

linkage

for

_quantitative

traits. Genet

_Epidemiol

_6, 435-449

Blackwelder WC,

Elston RC

₍₁₉₈₂₎

Power and robustness of

_sib-pair

_linkage

tests and extension to

_{larger sibships.}

Commun Statist-Theor Methods

_11,

449-484

Elston RC

₍₁₉₉₀₎

A

_{general linkage}

method for detection of

_major

_{genes. In:} Advances

in Statistical Methods

_Applied

to Livestock Production

_(K

Hammond,

D

_Gianola,

_eds),

Springer,

Berlin, Germany

Fulker

_DW,

Cardon LR

₍₁₉₉₄₎

A

_{sib-pair approach}

to interval

_mapping

of

_quantitative

trait loci. Am J Hum Genet

_54,

1092-1103

G6tz

KU,

Ollivier L

₍₁₉₉₂₎

Theoretical _aspects of

_{applying sib-pair linkage}

tests to

livestock

_species.

Genet Sel

Evol 24,

29-42

Haley CS,

Knott SA

₍₁₉₉₂₎

A

_{simple regression}

method for

_{mapping quantitative}

trait

loci in line crosses

_using

_flanking

markers.

_{Heredity 69,}

315-324

Haley CS,

Knott

_SA,

Elsen JM

₍₁₉₉₄₎

Multi-marker

approaches

to

_{mapping quantitative}

trait loci in livestock. Proc 45th

_{EAAP-Meeting, Edinburgh, UK,}

5-8

_September

1994 Haseman

_JK,

Elston RC

₍₁₉₇₂₎

The

_{investigation}

of

_linkage

between a

_quantitative

trait

and a marker locus. Behav Genet 1, 3-19

Knott

_{SA, Haley}

CS

₍₁₉₉₂₎

Maximum likelihood

_mapping

of

_quantitative

trait loci

using

full-sib families. Genetics

_132,

1211-1222

Lander

_ES,

Botstein D

₍₁₉₈₉₎

_Mapping

Mendelian factors

_{underlying quantitative}

traits

using

RFLP

_linkage

_maps. Genetics

_121,

185-199

Rohrer

_GA,

Alexander

_LJ,

Keele

_JW,

Smith

_TP,

Beattie CW

₍₁₉₉₄₎

A microsatellite