Mapping linked quantitative trait loci via residual maximum likelihood

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Mapping linked quantitative trait loci via residual

maximum likelihood

Fe Grignola, Q Zhang, I Hoeschele

To cite this version:

Original

article

Mapping

linked

_quantitative

trait loci via

residual

maximum

likelihood

FE

_{Grignola Q Zhang}

I

Hoeschele

Department of Dairy Science, Virginia Polytechnic

Institute and State

_University,

Blacksburg,

VA

_2l!061-OS15,

USA

(Received

2

_{September 1996; accepted}

19

_August

₁₉₉₇₎

Summary - A residual maximum likelihood method is

_presented

for estimation of the

positions

and variance contributions of two linked

_QTLs.

The method also

provides

tests for zero versus one

QTL

linked to a _group of markers and for one versus two

_(aTLs

linked. A

_{deterministic,}

derivative-free

algorithm

is

_employed.

The variance-covariance matrix of the allelic effects at each

_QTL

and its inverse is

_computed

conditional on

_incomplete

information from

_multiple

linked markers. Covariances between effects at different

(aTLs

and between

_CaTLs

and

_polygenic

effects are assumed to be zero. A simulation

_study

was

_performed

to

_investigate

_parameter estimation and likelihood ratio tests. The

_design

was a

_{granddaughter design}

with 2 000 sons, 20 sires of sons and nine ancestors of sires. Data were simulated under a normal-effects and a biallelic model for variation at each

QTL. Genotypes

at five or nine

_{equally spaced}

markers were

_generated

for all sons and

their ancestors. Two linked

(aTLs

accounted

_jointly

for 50 or 25% of the additive

_genetic

variance,

and distance between

QTLs

varied from 20 to 40 cM. Power of

detecting

a

second

QTL

exceeded 0.5 all the time for the 50%

QTLs

and when the distance was

(at

least 30 cM for the 25%

QTLs.

An intersection-union test is

_preferred

over a likelihood ratio _test, which was found to be rather conservative. Parameters were estimated

quite

accurately

except for a

_slight

overestimation of the distance between two close

QTLs.

quantitative trait loci

_/

_{multipoint mapping}

_/

residual maximum likelihood

_/

outcross

population

Résumé - Détection de _gènes liés à effets quantitatifs

(QTL)

grâce au maximum

de vraisemblance résiduelle. On

_présente

une méthode de maximum de vraisemblance résiduelle _{pour estimer} les

positions

et les contributions à la variabilité

_génétique

de deux

QTLs

liés. La méthode

_{fournit également}

des tests de l’existence d’un seul

_QTL

lié à un _groupe de _marqueurs

(par

_rapport à

zéro)

ou de deux

_QTLs

(par

_rapport à un

seul).

Un

_algorithme

déterministe sans calcul de dérivées est utilisé. La matrice de variance-covariance des

_{effets alléliques}

à

_{chaque QTL}

et son inverse est calculée conditionnellement

à

l’information incomplète

sur les _marqueurs

_multiples

liés. Les covariances entre les

effets

aux

_{différents QTLs}

et entre les

effets

aux

_QTLs

et les

effets polygéniques

sont

*

supposées

nulles. Une étude de simulation a été

effectuée

pour

analyser

les

paramètres

estimés et les tests de _rapports de vraisemblance. Le schéma

_{expérimental}

a été un schéma

«

_{petites-filles}

» avec 2 000

_fils,

20

_pères

des

_fils

et 9 ancêtres de ces

_pères.

Des données

ont été simulées avec un modèle de variation au

_QTL

de type Gaussien ou

_{biallélique.}

Les

_génotypes

pour cinq ou

_neuf

_marqueurs

_{également espacés}

ont été

_générés

_{pour tous} les

_,fils

et leurs anchêtres. Deux

_QTLs

liés

_expliquaient

conjointement 50 % ou 25 % de la variance

_génétique

additive et la distance entre les

QTLs

variait de 20 CM à 40 CM. La

puissance de détection d’un second

_QTL

a

_{dépassé 0,5,}

dans tous les cas _pour la situation 50

_%,

et

_quand

la distance entre

_QTLs

était

_supérieure

ou

égale

à 30 CM _pour la situation 25 %. Le test d’un

QTL

par rapport à deux

QTLs correspond

à la réunion de deux tests.

On l’a trouvé

_{plutôt conservatif.}

Les

paramètres

ont été estimés avec une

_{grande précision}

excepté

la distance entre deux

QTLs proches

qui a été

_légèrement

surestimée.

locus de caractère quantitatif

/

cartographie multipoint

/

maximum de vraisemblance résiduelle

_/

_population

_consanguine

INTRODUCTION

A

_variety

of methods for the statistical

_mapping

of

_quantitative

trait loci

_(QTL)

exist. While some methods

_{analyze squared phenotypic}

differences of relative

_pairs

(eg,

Haseman and

_{Elston, 1972;}

Gotz and

_Ollivier,

_1994),

most methods

_analyze

the individual

_phenotypes

of

_pedigree

members. Main methods

_applied

to livestock

populations

are maximum likelihood

_{(ML) (eg,}

_Weller,

_1986;

Lander and

_Botstein,

1989;

Knott and

_Haley,

_1992),

_{least-squares}

_(LS)

as an

_{approximation}

to ML

_(eg,

Weller et

_al,

_1990;

_Haley

et

al, 1994;

Zeng,

1994),

and a combination of ML

and LS referred to as

_composite

interval

_mapping

_(Zeng,

₁₉₉₄₎

or

_{multiple QTL}

mapping

(Jansen, 1993).

These methods were

developed mainly

for line

_crossing

and, hence,

cannot

_fully

account for the more

complex

data structures of outcross

populations,

such as data on several families with

_{relationships}

across

_families,

incomplete

marker

_information,

unknown number of

_QTL

alleles in the

_population

and

_varying

amounts of data on different

_(aTLs

or in different families.

Recently,

Thaller and Hoeschele

_{(1996a, b)}

and Uimari et al

₍₁₉₉₆₎

_implemented

a

_Bayesian

method for

QTL

_{mapping using}

single

markers or all markers on a

chromosome,

respectively,

via Markov chain Monte Carlo

_algorithms,

and

_applied

the

_analyses

to simulated

_{granddaughter designs}

identical to those in the

_present

study.

Hoeschele et al

₍₁₉₉₇₎

showed that the

_Bayesian

_analysis

can accommodate

either a biallelic or a normal-effects

_QTL

model. While the

_{Bayesian analysis}

was

able to account for

_{pedigree relationships}

both at the

_QTL

and for the

_polygenic

component,

and _gave

_good

_{parameter estimates,}

it was _very

_demanding

in terms

of

_{computing time,}

in

_particular

when

_fitting

two

_(aTLs

_(Uimari

and

_Hoeschele,

1997).

Therefore,

Grignola

et al

_(1996a)

_developed

a residual maximum likelihood

method,

using

a

_{deterministic,}

derivative-free

_algorithm,

to _map a

_{single QTL.}

Hoeschele et al

₍₁₉₉₇₎

showed that this method can be considered as an

approxi-mation to the

_{Bayesian analysis fitting}

a normal-effects

_QTL

model. In the

normal-effects

QTL

model

postulated

by

the REML

analysis,

the vector of

QTL

allelic effects is random with a

_prior

normal distribution. The REML

_analysis

builds on

it to the estimation of

QTL, polygenic,

and residual variance

_components

and of

QTL location, using incomplete

information from

multiple

linked markers.

Xu and

_{Atchley (1995)}

_performed

interval

_{mapping using}

maximum likelihood based on a mixed model with random

QTL

effects,

but these authors fitted additive

genotypic

effects rather than allelic effects at the

_QTL,

with variance-covariance

matrix

_proportional

to a matrix of

_proportions

of alleles

identical-by-descent,

and assumed that this matrix was known. Their

analysis

was

_applied

to unrelated full-sib

_pairs.

In order to account for several

QTLs

on the same

_chromosome,

Xu and

Atchley

(1995)

used the idea behind

composite

interval

_mapping

and fitted variances

at the two markers

_flanking

the marker bracket for a

_QTL.

This

_{approach, however,}

is not

_appropriate

for

_{multi-generational}

_pedigrees,

as effects associated with marker

alleles erode across

_{generations owing}

to recombination. It is also

problematic

for

outbred

_populations,

where

incomplete

marker information causes the

_flanking

and

next-to-flanking

markers to differ _among families.

In this _paper, we extend the REML method of

_Grignola

et al

_(1996a)

to the

fitting

of

multiple

linked

_QTLs.

While the extension is

_general

for _any number of linked

_QTLs,

we

_apply

the method to simulated

_{granddaughter designs by}

fitting

either one or two

_QTLs.

METHODOLOGY Mixed linear model

The model is identical to that of

_Grignola

et al

_(1996a),

_except

that it includes effects at several

_(t)

QTLs,

and it can be written as:

where y is a vector of

_phenotypes,

X is a

_{design-covariate matrix, j3}

is a vector of fixed

effects,

Z is an incidence matrix

relating

records to

individuals,

u is a vector

of residual additive

_(polygenic)

_effects,

T is an incidence matrix

relating

individuals

to

_alleles,

vi is a vector of

QTL

allelic effects at

_QTL

_i,

e is a vector of

residuals,

A is the additive

_{genetic relationship matrix,}

_c

₇

_’

_is

the

_polygenic

variance,

_Q,

₀

_{,2 V(}

_i)

is the variance-covariance matrix of the allelic effects at

_QTL

i conditional on

marker

_information,

_Qv!2!

is the allelic variance at

_QTL

_{i (or}

half of the additive variance at

_QTL

_i),

R is a known

_diagonal

_matrix,

and

Q

e

is

residual variance. Each

matrix

_Q

_i

_depends

on one unknown

_parameter,

the _map

position

of

QTL

i (d

i

).

Parameters related to the marker _map

_(marker

_positions

and allele

_frequencies)

are assumed to be known. The model is

parameterized

in terms of the unknown

parameter’s heritability

(h

2

=

_{0&dquo;!/0&dquo;2),}

with

_{aj_}

_being

_phenotypic

and

_U2

additive

effects at

_QTL

i

_(v?

=

o,’ v(i) /’a 2;

i =

_{l, ... ,}

_t),

the residual variance

_{0,2, e}

and

_QTL

map locations

d

l

, ... ,

di, ... , dt.

A model

_equivalent

to the animal model in

_[1]

is

_(Grignola

et

_al,

_1996a):

where W has at most two non-zero elements

equal

to 0.5 in each row in columns

pertaining

to the known

_parents

of an

individual,

F

i

is a matrix with _up to four

non-zero elements per row

_pertaining

to the

_QTL

effects of an individual’s

parents

(Wang

et

_al,

_1995;

_Grignola

et

_al,

_{1996a), Ap}

and

_Qp(

_j

₎

are sub-matrices of A

and

_{Q, respectively, pertaining}

to all animals that are

_parents,

and m and ei

are Mendelian

_sampling

terms for

_polygenic

and

_QTL

_{effects, respectively,}

with covariance matrices as

_specified

in

_equation

_!2!.

While

_Var(m)

is

_diagonal,

_Var(e

_i

₎

can have some

off-diagonal

elements in inbred

_populations

_(Hoeschele,

_{1993; Wang}

et

al,

1995).

Note that models

_[1]

and

_[2]

are conditional on a set of

_QTL

_map

_positions

(and

on marker

_positions

which are assumed to be

_known).

_Dependent

on the

map

positions

are the matrices

Q

i

in model

_[1]

and the matrices

_F

_i

and

_Qp(

_j

₎

in

model

_!2!.

Note furthermore that models

_[1]

and

_[2]

assume zero covariances between effects

at different

_QTLs,

and between

_polygenic

and

_QTL

effects.

_However,

selection tends

to introduce

_negative

covariances between

_(aTLs

_(Bulmer,

_1985).

A reduced animal model

_(RAM)

can be obtained from model

_[2]

_{by combining}

m,

the e

i

(i

=

1, ... ,

t) and

e into the residual. Mixed model

_equations

_(MME)

can be formed

directly

for the

RAM,

or

_by

_setting

_up the MME for model

_[2]

and

absorbing

the

_equations

in m and the ei

(i

=

with the A matrices defined in

_equation

_!2!.

Matrix

_D,

which results from successive

absorption

of the Mendelian

_sampling

terms for the

_polygenic

_component

and the

QTLs,

can be shown to be

_{always diagonal}

and _very

_simple

to

_compute,

even when

several

_{(aTLs (t}

>

₂₎

are fitted. Let

_6v!i!!!

_represent

the Mendelian

sampling

term

pertaining

to v effect k

(k

=

1, 2)

of

individual j

at

QTL

i,

and

6,,(

j

)

the Mendelian

sampling

term for the

_polygenic

effect of

_{j. Then,}

the element of D

_pertaining

to

individual j

(d

jj

)

is

_computed

as follows:

where

r

jj

is the

_jth

_diagonal

element of

R-

1

.

REML

_analysis

The REML

_analysis

was

performed using

interval

mapping

and a derivative-free

algorithm

to maximize the likelihood for _any

_given

set of

QTL positions,

as

described

_{by Grignola}

et al

_(1996a)

for a

_single

_QTL

model. The

_log

residual

likelihood for the animal model was obtained

by adding

correction terms to the residual likelihood formed

_directly

from the RAM MME

_(Grignola

et

_al,

_1996a).

The RAM residual likelihood is:

where N is the number of

_phenotypic

_{observations,}

NF the number of estimable fixed effects

_(rank

of

_X),

NR

RA

,yI

the number of random

genetic

effects of the

parents

((1

+

_2t)

times the number of

_parents),

CRAM

is the coefficient matrix in

the left-hand-side of

_[3],

P =

V-’ -

1

,

1

X’V-

1

X)-

X(X’V-

1

V-

V =

Var(Y)/(7!,

and

_GRA,!,I

is a

block-diagonal

with blocks

Ap(7! and

Qp(i)(7!(i)

for i =

1, ... ,

t (see

The RAM residual likelihood is modified to obtain the residual likelihood for the animal model as follows

(Grignola

et

_{al 1996a):}

where A is the

_{block-diagonal}

with blocks

A

u

and

_!v(i)

_(i

=

_{1, ... ,}

t)

from

!2!,

_Czz

is the

_part

of the MME for model

_[2]

_pertaining

to m

_{and e}

_i

(i

= 1, ... ,

t

),

_and

NR

is total number of random

_genetic

effects

_!(1

!-

_2t)

times the number of

_animals]

in the animal model.

The

_analysis

is conducted in the form of interval

_mapping

as in

_Grignola

et al

(1996a),

except

that now a t-dimensional search on a

_grid

of combinations of

positions

of the

_{t CaTLs}

must be

_performed.

More

_precisely,

we

_{performed cyclic}

maximization

_by

_optimizing

the

_position

of the first

_QTL

while

_holding

the

_position

of the second

_QTL

constant and

_{subsequently fixing}

the

_position

of the first

QTL

while

_optimizing

the

_position

of the second

_QTL,

etc. A minimum distance

was allowed between the

_QTLs,

which was determined such that the

_(aTLs

were

always separated by

two markers. Whittaker et al

₍₁₉₉₆₎

showed that for

_regression

analysis

and F2 or backcross

_designs,

the two locations and effects of two

_(aTLs

in

_adjacent

marker intervals are not

_jointly

estimable. With other methods and

designs,

locations and variances of two

_(aTLs

in

_adjacent

intervals should be either

not estimable or

_poorly

estimated. At each combination of

_d

₁

and

d

2

values,

the residual likelihood is maximized with

_respect

to the

_parameters

_h

_z

_,

_v2

_(i

=

_1,...,

t)

and

_er!.

Matrices

_Qp(

_j

_),

_F

_i

and

_Ov!2!

were calculated for each

_QTL

as described in

Grignola

et al

_(1996a).

Hypothesis testing

The _presence of at least one

QTL

on the chromosome

_harboring

the marker

_linkage

group can be tested

_by

_maximizing

the likelihood under the

_one-QTL

model and under a

_polygenic

model with no

QTL

fitted

(Grignola

et

_al,

_1996a).

The distribution of the likelihood ratio statistic for these two models can be obtained

via simulation or data

_permutation

(Churchill

and

_Doerge,

_1994;

_Grignola

et

_al,

1996a, b;

Uimari et

_al,

_1996).

_Here,

we consider

_testing

the

_one-QTL

model

_against

the

_two-QTL

model. This test is

_{performed by comparing}

the maximized residual likelihood under the

_two-QTL

model with

_(i)

the maximized residual likelihood under the

_{one-QTL model,}

_(ii)

the residual likelihood maximized under the

one-QTL

model with

_{QTL position}

fixed at the REML estimate of

_d

₁

obtained under the

_two-QTL

_model,

and

_(iii)

the residual likelihood maximized under the

_one-QTL

model with

_QTL

_position

fixed at the REML estimate of

_d

₂

obtained under the

two-QTL

model. The distribution of these likelihood ratio statistics is not

_known,

and

_obtaining

it via data

_permutation

would be difficult

_{computationally,}

as _many

permutations

would need to be

_analyzed,

and as the two-dimensional search took

1-2 h of run-time for the

_design

described below. The likelihood ratios

_{corresponding}

to

_{(i) (LR}

_d

_),

_(ii)

_(LR

_dl

_),

and

_(iii)

_(LR

_d2

₎

should have an

_asymptotic

_chi-square

distribution within 1 and 3

_degrees

of freedom. When

_using

_LR

_dl

and

_LR

_dz

_,

both ratios have to be

_significant

in order to

_reject

the null

_hypothesis

of one

_QTL.

This

for the first likelihood ratio the

_hypotheses

are:

_H

_o

_:

Œ!(l) -I-

0 and

Œ!(2)

= 0 _versus

H

1

:

_{Œ!(l) -I-}

0 and

_{Œ!(2) -I-}

0,

and for the second likelihood ratio the

hypotheses

are:

Ho:

_{Œ!(1) =}

0 and

_{Œ!(2) -I-}

0 versus

_H

₁

_:

_{Œ!(1) -I-}

0 and

_{Œ!(2) -I-}

0. The intersection-union test constructed in this way can be

_{quite conservative,}

as its size _may be

much less than its

_specified

value ce. For

_{genome-wide testing,}

the

_significance

level

should also be

_adjusted

for the number of

_independent

tests

_performed

_(the

number of chromosomes

_analyzed

times the number of

_independent

_traits).

SIMULATION

Design

The

_design

simulated was a

_{granddaughter}

_design

(GDD)

as in the

_{single QTL}

study

of

_Grignola

et al

_(1996b),

where marker

genotypes

are available on sons and

phenotypes

on

daughters

of the sons. The structure resembled the real GDD of the US

_public

_gene

_{mapping project}

for

_dairy

cattle based on the

_dairy

bull DNA

repository

(Da

et

_al,

_1994).

The simulated GDD consisted of 2 000 sons, 20

sires,

and nine ancestors of the sires

_{(fig 1)..}

The

_phenotype

simulated was

_daughter

_yield

deviation

(DYD)

of sons

(Van-Raden and

_Wiggans,

_1991).

DYD is an _average of the

_phenotypes

of the

_daughters

adjusted

for

_systematic

environmental effects and

_genetic

values of the

_daughters’

dams. For details about the

_analysis

of

_DYDs,

see

_Grignola

et al

_(1996b).

Marker and

_QTL

_genotypes

were simulated

according

to

_{Hardy-Weinberg}

fre-quencies

and the _map

_positions

of all loci. All loci were in the same

linkage

_group.

Each marker locus had five alleles at

_{equal frequencies.}

Several

_designs

were

con-sidered which differed in the map

positions

of the two

QTLs,

in the number of

markers,

and in the

_proportion

of the additive

_genetic

variance

_explained

_by

the

two

_QTLs.

These

_designs

are defined in table II. Also simulated was a

_{single QTL}

at 45 cM to test the

_{two-QTL analysis}

with data

_generated

under the

_{single QTL}

model.

Polygenic

and

QTL

effects were simulated

according

to the

_pedigree

in

_figure

1.

Data were

_{analyzed by}

_using

the

_pedigree

information on the sires. Note that

in the

_simulation,

no

_linkage

_{disequilibrium}

(across

families)

was

_{generated, ie,}

polygenic

effects were zero.

_Therefore,

an additional

_design

was simulated where

linkage disequilibrium

was

_generated

_by

_simulating

DYDs also for

_{sires, creating}

a

_larger

number of sires and

_culling

those sires with DYD lower than the 90th

percentile

of the DYD distribution.

_{QTL positions}

for this

_design

were 30 cM

(interval 2)

and 70 cM

_{(interval 3)}

with five

_markers,

and the

QTL

model was

the normal-effects model

_{(see below).}

Estimates of the simulated correlations

_(SE

in

_{parentheses),}

across 30

replicates,

were -0.20

(0.05),

-0.33

_(0.04),

and -0.32

(0.04),

between

_pairs

of v effects at

_QTL

1 and

_{QTL 2,}

between

_pairs

of v effects

at

_QTL

1 and

_polygenic

_effects,

and between

_pairs

of v effects at

_QTL

2 and

polygenic effects,

respectively.

The effects of one or several

_generations

_{of phenotypic}

truncation selection on additive

_genetic

variance in a finite locus model has been

QTL

models

Two different

QTL

models were used to simulate data. Under both

models,

phenotypes

were simulated as

where n

j

was the number of

daughters

of son

_j,

_gi!k was the sum of the v effects

in

_daughter

k of

_{son j at}

_QTL

_i,

_uj was a

_normally

distributed

_{polygenic effect,}

e

j was a

normally

distributed

residual, polygenic

variance

(

0

&dquo;)

was

equal

to the difference between additive

_genetic

variance

_{(afl )}

and the variance

_explained

_by

the

QTLs,

and

_{afl was}

environmental variance. Number of

_daughters

_per son was set to

50,

corresponding

to a

_reliability

_(Van

Raden and

Wiggans,

1991)

near 0.8. Narrow

sense

heritability

of individual

_phenotypes

was

h

2

=

_0.3,

and

_phenotypic

SD was

Q P = 100.

Note that the

_QTL

contribution to the DYDs of sons was

_{generated by sampling}

individual

QTL

allelic effects of

_daughters

under each of the two

_genetic

models described below. This

_sampling

of

_QTL

effects ensures that DYD of a

_heterozygous

son, or of a son with substantial difference in the additive effects of the alleles at

a

QTL,

has

larger

variance _among

daughters

due to the

_QTL

than a

_homozygous

son or a son with similar

_QTL

allelic effects.

Two different models were used to describe variation at the

_QTL,

which are

identical to two of the models considered

_{by Grignola}

et al

_(1996b).

Normal-effects model

For each individual with both or one

parent(s)

_unknown,

both or one

effect(s)

at

QTL

k(k

=

1, 2)

were drawn from

_N(O,

_{a v 2(k)).}

For the

pedigree

in

_figure

_1,

there

were 32 base

alleles,

and each

_QTL

was treated as a locus with 32 distinct alleles

in

_passing

on alleles to descendants. The

_parameter

_a

_v

_2(k)

was set to

_{0.125or or}

0.625(J&dquo;!,

ie,

QTL

k accounted for

25%

₍₂

_V2

=

0.25)

or

12.5%

(2v!

=

0.125)

of the

total

additive

genetic variance,

respectively.

k k

Consequently,

the two

_(aTLs

accounted

_jointly

for between 25 and

50%

of the additive

_genetic

variance.

Biallelic model

Each

_QTL

was biallelic with allele

_frequency

_pi =

p 2 =

P= 0.5. The variance _at

QTL

k was

where for p = 0.5 and

2v!

= 0.25 _or

2v!

=

_0.125,

half the

homozygote

difference _at

RESULTS

The

_designs

studied are described in table II and differ in the

QTL positions,

in the number of

_markers,

and in the

_proportion

of the additive

_genetic

variance

explained jointly by

two linked

_QTLs.

Overall,

the

QTL

parameters

were estimated

quite accurately

as in the

single-CaTL

_analysis

of

Grignola

et al

_(1996b),

_except

that there was a

_tendency

to overestimate the distance between the

CaTLs

with

decreasing

true distance.

Parameter estimates for all

_designs

in table II and for the normal-effects

_QTL

model used in the data simulations are

_presented

in table III. There

appeared

to

be a

_{slight tendency}

to overestimate the

_QTL

variance contributions

_(v

₂

_),

_but,

in most cases not

_{significantly.}

The

_QTL

map

positions

and the distance between

the

QTLs

were estimated

accurately

when the true _map distance between the

(aTLs

was 30 or 40 cM. When the true _map distance was

_only

20

_cM,

there was a

tendency

to overestimate the

_QTL

distance. This overestimation was

_{significantly}

more

_pronounced

when the number of markers was reduced from nine

(every

10

_cM,

designs IIIA,

B)

to five

_(every

20

_{cM, designs IVA,}

_B).

To

_investigate

whether the overestimation of the

QTL

distance was related to the search

_{strategy requiring}

a minimum distance between the

QTLs

such that these were

always

separated

by

two markers

_(with

the

_exception

of

_{designs IVA, B),}

the minimum distance

was reduced to 10 and 2 cM.

_However,

_parameter

estimates and likelihood ratios remained

_unchanged.

When the

_(aTLs

accounted

_jointly

for

_only

25%

of the additive

_genetic

variance

as

_compared

to

_50%,

there was little

_change

in the

_precision

of the estimates of the

QTL

variance contributions. Standard errors of the

_{QTL positions}

were

_higher,

and

overestimation of the distance between

_{(aTLs only}

20 cM

_apart

was

_slightly

more

pronounced.

Parameter estimates for

_designs

simulated under the biallelic

_QTL

model are

shown in table V.

Except

for the

_{QTL model,}

these

designs

are identical to

designs IA,

B and

IIIA,

B in table II. Parameters were estimated with an _accuracy

not

_noticeably

lower than for the normal-effects

_{QTL model,}

an observation in

agreement

with the

_single-(aTL

_study

of

_Grignola

et al

_(1996b).

When

_analyzing

the

_designs

in table II with the

_single-(aTL

_model,

the most

likely QTL

position

(d

in tables III and

_V)

was

always

somewhere in between the

the estimated

_QTL

_position

was _very near the mean of the true

_positions.

This result was

_expected,

as both

_(aTLs

had

_equal

variance contributions and on _average

equally

informative

_flanking

markers.

Likelihood ratio statistics for all

_designs

in table II and for the normal-effects

QTL

model are

_presented

in table IV. The _average values of the likelihood ratio statistics for

_testing

between the

_single-

and

_two-QTL

models declined as

_expected

with

_decreasing

distance between the two

_QTLs

_(designs

_IIA,

B versus

_{designs IIIA,}

B),

with

_decreasing

number of markers

_(designs

IA,

B versus

IIA,

B,

and

designs

IIIA,

B versus

_{designs IVA,}

B),

and with

_{decreasing joint}

variance contribution of the

_QTLs

_(A

versus

_B).

The _average value of the likelihood ratio statistic

_LR

_d

was

always considerably

lower than those of

LR

dl

and

LR

d2

.

The _power

_figures

in table IV were calculated

_assuming

that

_LR

_dl

_,

_LR

_d2

and

LR

d

follow either a

_chi-square

distribution with 1 df or 3 df and

_using

an cx value of

_0.05/29

= 0.0017. To allow for

any

interpretation

of these power

figures,

we

estimated the

_type-I

error

_{by simulating}

data with a

_single

_QTL

_explaining

either

25,

12.5 or

6.25%

of the additive

_genetic

variance. For the

_type-I

error

_estimation,

a was set to

_0.05,

and the number of

replicates

was 200. Estimates of

type-I

errors are in table

_VI,

for the two tests

_(LR

_d

and

_LR

_d

_,

and

_LR

_d2

₎

and for thresholds from

chi-square

distributions with

1,

2 and 3 df.

Type-I

errors tended to increase

_slightly

with size of the

_QTL

_variance,

and were

consistently

lower for the test

_using

_LR

_d

_.

Based on these

results,

the

_{empirical type-I}

error was close to the

_{pre-specified}

value of 0.05 for the

LR

dl

and

LR

d2

tests when

_using

the

_{xi-threshold,}

while it was

With this

_background,

the _power of

_rejecting

the

_single-(aTL

model based on

requiring

both

LR

dl

and

_LR

_d2

to exceed the

_significance

threshold was as

expected

always

higher

than or

_equal

to the power of the

LR

d

statistic. For the test based on

LR

dI

and

_LR

_d2

_,

_power declined as

_expected

with

_decreasing

distance between

_QTLs

and with

_decreasing

true

QTL

variance contribution. For the

_{joint QTL}

variance contribution of

50%

and the test based on

_LR

_d

_,

and

_LR

_d2

_,

_power was

_equal

to or

higher

than 0.5

_always

for the

_xi

threshold

and

always

except

for

_design

IVA for the

_X2

_threshold.

For the

_joint

_QTL

variance contribution of

25%,

a _power of at

least 0.5 was achieved

_only

for the 30 cM distance between

_QTLs

and the

X2

and

X3

thresholds.

This

_finding

must be

interpreted by keeping

in mind the choice of

a = 0.0017 and the fact that the

_test,

_as contructed

_here,

is rather conservative.

For the data simulated with a

_{single QTL explaining}

either 25 or

12.5%

of the

additive

_{genetic variance,}

the map

positions

estimated under the

two-QTL

model

were near the true

_position

and a

_ghost

_position

to either side of the true

_position

in most

_replicates.

This behavior of the REML

analysis

seems to

_support

the use

of

_LR

_dl

and

_LR

_d2

instead of

_LR

_d

_.

Likelihood ratio statistics for the biallelic

QTL

model and some of the

_designs

in

table II are

_presented

in table V.

_Overall,

likelihood ratios and _power

_figures

were

similar to those for the normal-effects

_QTL

_model,

with somewhat lower power for

design

IA but

_{slightly higher}

_power for other

_designs.

These differences are most

likely

due to the limited number of

_replications

_(30).

The

_cyclic

maximization

_strategy

for the

two-QTL

model took about 20 min

of serial wall-clock time on a

_21-processor

IBM SP2

_system

for a chromosome of

80 cM

_{length, compared}

with 1.5 h for a two-dimensional search. Run-time for the

single-QTL

analysis

was at most 8 min.

For the

_designs

in table

_II,

the two

_QTLs

had

_equal

variance contributions.

and 25 and 45 cM

_(nine

_{markers), respectively,}

were simulated

_using

the

normal-effect

_QTL

_model,

with

_QTL

1

_explaining

25%

and

_QTL

2

12.5%

of the additive

genetic

variance. The average estimates

(with

SE in

parentheses)

of

QTL

position

from the

_single

_QTL

_analysis

were 0.396 M

(0.023)

and 0.298 M

(0.010)

for the

30 and 70 cM and 25 and 45 cM

_{designs, respectively, being}

closer to the first locus with the

_larger

variance contribution. Estimated

QTL positions

from the

two-QTL

analysis

were 0.285 M

_(0.008)

and 0.720 M

_(0.010)

for the 30 and 70 cM

_design,

and 0.077

_(0.012)

for the second

_{design, respectively.}

For the 30 and 70 cM

_design,

power was 0.47

(0.40)

for the

Xi

2 (X’)

threshold and the test based on

LR

dl

and

LR

d2

.

For the 25 and 45 cM

_design,

_power was

_only

0.33

_(0.10)

for the same tests.

When

_linkage

disequilibrium

was

generated

by phenotypic

truncation selection

of sires for the

_design

with

_QTL

_positions

of 30 and 70 cM and

_{joint QTL genetic}

variance contribution of

_50%,

_QTL

_parameters

and their estimates were

_clearly

affected.

Heritability

was estimated low

(0.213 ! 0.020),

and the

vfl (k

=

1, 2)

_were

estimated

_{high (0.184 ±0.014,}

_{0.211 ±0.014),}

but

_{QTL positions}

were estimated

accurately

(0.292 t

0.010,

0.716 ±

_0.007).

Power

_appeared

to be somewhat reduced

compared

to the same

_design

without

_selection,

and was estimated at 0.86 and 0.73 for the

_X

_i

_{1 and X thresholds,}

_{respectively.}

Reduction _{in power} was

_probably

due to

the

_high

estimate of error variance

(1737.9 t 55.3).

CONCLUSIONS

The REML

_{analysis of Grignola}

et al

_{(1996a, b),}

based on a mixed linear model with

random and

_normally

distributed

_QTL

allelic effects and conditional on

_incomplete

information from

_multiple

linked

markers,

has been extended here to fit

_multiple

linked

QTLs.

This extension is necessary to eliminate biases in the estimates of the

_QTL

_parameters

_position

and

_variance,

which occur when

_fitting

a

_single

_QTL

and other linked

QTLs

are

_present.

For the

_present

study,

the

analysis

had been

implemented

for two

_QTLs

on the same chromosome

_using

a two-dimensional

search. When

_fitting

more than two linked

_QTLs

or additional unlinked

_QTLs,

a more efficient search

_strategy

may be

required,

or alternative

algorithms

(eg,

Meyer

and

_Smith,

_1996).

In the

_meantime,

a

_{cyclic optimization approach}

was

implemented,

with one

QTL position

held constant while

_optimizing

the

other,

and vice versa, which reduced the number of likelihood evaluations and hence CPU time

_considerably

relative to a two-dimensional search. As likelihood maximizations

at different

_position

combinations are

_independent

of each

_other,

use of

_multiple

processors, if

available,

would reduce run-times

substantially.

For the

_one-QTL

model,

relationships

between the REML

_analysis,

the

equiv-alent method of Xu and

_{Atchley (1995),}

the method of Schork

_(1993),

and the

Bayesian analysis

of Uimari et al

₍₁₉₉₆₎

were discussed in

Grignola

et al

_(1996a).

The method of Xu and

_Atchley

₍₁₉₉₅₎

_fitting

variances associated with the

next-to-flanking

markers to account for additional linked

_QTLs

would not have worked well for the

_designs

studied here. A first reason is the inclusion of ancestors of the

sires in the

_analysis,

as their method fits random effects associated with the marker

alleles in founders which erode across

_generations

due to recombination. Another

reason is the small number of families

differing

in the

flanking

and

next-to-flanking

markers,

resulting

in too little information for estimation of variances associated with the

_{next-to-flanking}

markers in the method of Xu and

_Atchley

_(1995).

For similar _reasons,

_composite

interval

_mapping

_{(Zeng, 1994)}

and

_{multiple QTL}

map-ping

(Jansen, 1993),

which are based on the inclusion of markers as

cofactors,

are

not suitable for the

_analysis

of

_{multi-generational pedigrees.}

REML

_analysis

under the

_two-QTL

model

yielded

fairly

accurate

_parameter

estimates.

_Map

distance between the

QTLs

was

_{overestimated,}

with

_decreasing

study,

the REML

_analysis

was robust to the number of alleles at the

_QTLs,

as there was

_relatively

little difference in

_parameter

estimates and likelihood ratio statistics

between

_{designs generated}

with the normal-effects and biallelic

_QTL

_{models, given}

the number of

_{replicates performed.}

Previous

_{linkage analyses}

_(eg,

Knott and

_Haley,

1992;

Haley

et

_al,

₁₉₉₄₎

lead to the conclusion that a minimum distance of 20 cM

was

_required

between linked

_QTLs

for their

_separate

detection. This result was

confirmed in the

present

study. However,

discrimination among different numbers

of

_QTL

_(eg,

one versus

_two)

_requires

additional

research,

including

an

_{investigation}

of alternative

_approaches

such as an

_adaptation

of

_composite

interval

_mapping

to

pedigree analysis.

Gains in power from

_fitting

additional unlinked

QTL

and selection of

_QTL

to be included in the

_analysis

are related

_{topics warranting}

further research.

If there is

_{linkage disequilibirum}

due to

_{selection, QTL positions}

will still be estimated

_accurately,

while variance estimates and power may be affected.

Accounting

for

_{disequilibrium}

in the

_analysis

should be

_{investigated.}

ACKNOWLEDGMENTS

This research was

supported by

Award No 92-01732 of the National Research Initiative

Competitive

Grants

Program

of the US

Department

of

Agriculture, by

the Holstein Association

_USA,

and

_by

ABS Global Inc.

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