Mapping quantitative trait loci in outcross populations via residual maximum likelihood. II. A simulation study

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Mapping quantitative trait loci in outcross populations

via residual maximum likelihood. II. A simulation study

Fe Grignola, I Hoeschele, Q Zhang, G Thaller

To cite this version:

Original

article

Mapping quantitative

trait loci

in

outcross

_populations

via residual

maximum likelihood.

II.

A

simulation

_study

FE

_Grignola

I

Hoeschele

_{Q Zhang}

G Thaller

2

Department

of Dairy Science, Virginia Polytechnic

Institute and State

_University,

Blacksburg,

VA

_2l!061-0315,

_USA;

2

Lehrst!ihl

_{fuer Tierzucht,}

Technical

_University

_of

Munich at

_{Weihenstephan,}

85350

_{Munich, Germany}

(Received

18 March

_{1996; accepted}

20

September

1996)

Summary - Position and variance contribution of a

_{single QTL together}

with additive

polygenic

and residual variance components were estimated

_using

a residual maximum

likelihood method and a derivative-free

_algorithm.

The variance-covariance matrix of

QTL

effects and its inverse were

computed

conditional on

incomplete

information from

multiple-linked

markers. Simulation was

employed

to

_investigate

the accuracy of

param-eter estimates and likelihood ratio tests. The

_design

was a

granddaughter design

with 2 000 sons, 20 sires of sons and 9 ancestors of sires.

Designs

with 1 000 and 600 sons were

also

_{investigated.}

Data were simulated under three different

_genetic

models for the

QTL,

a

biallelic

model,

a multiallelic model with ten alleles at

_{equal frequencies,}

and a model with

normally

and

_{independently}

distributed

_QTL

allelic effects for base individuals. The trait

analyzed

was

daughter yield

deviation or the

daughter

_average

adjusted

for environmental

effects and merits of mates of the sons.

_Genotypes

for five markers situated on the same

chromosome were

_generated

for all sons and their ancestors. Data were

_analyzed

with and without

_{relationships}

among sires. Parameters were estimated with

_good

_accuracy under

all three simulation models. The REML method was

_fairly

robust to the number of alleles

at the

_QTL

for the

_designs

studied.

quantitative trait loci

_/

residual maximum likelihood

_/

_mapping

_/

simulation

_/

granddaughter design

Résumé - La _cartographie de locus de caractère

_quantitatif

dans des _populations en

ségrégation à l’aide du maximum de vraisemblance résiduelle. II.

Étude

de simulation.

La position et la contribution à la variance d’un locus de caractère

_quantitatif

_(QTL)

ont

été estimées avec les variances

_{polygéniques}

additives et résiduelles par une méthode de maximum de vraisemblance résiduelle et un

_algorithme

sans dérivation. La matrice de

*

variance-covariance des

_effets

de

_QTL

et son inverse ont été calculées conditionnellement à

_{l’information incomplète}

relative à des ensembles de _marqueurs liés. Une simulation a été réalisée _pour déterminer la

_précision

des

_paramètres

estimés et les tests de rapport de vraisemblance. Le

plan d’expérience

était un schéma

petite-fille,

avec 2 000

fils,

20

pères

et 9 ancêtres de

pères.

Des schémas avec 1 000 et 600

_fils

ont

également

été étudiés. Les données ont été simulées sous trois modèles

génétiques différents

_pour le

QTL,

un

modèle

biallélique,

un modèle à 10 allèles

_{d’égale fréquence,}

et un modèle avec des

_effets

alléliques

distribués normalement et d’une manière

indépendante

chez les individus de base. Le caractère

_analysé

était la

_production

des

_{petites-filles}

en écart à la _moyenne, ou

leur moyenne

ajustée

pour les

effets

de milieu et les valeurs

_génétiques

des conjointes des

fils.

Les

génotypes

pour

cinq

marqueurs situés sur un même chromosome ont été

_générés

pour tous les

fils

et leurs

_ancêtres,

et les données ont été

_analysées

avec ou sans relation

de

_parenté

entre les

_pères.

Les

_paramètres

étaient estimés avec une bonne

_précision

dans les trois modèles simulés. La méthode était d’une robustesse

_{satisfaisante}

relativement au

nombre d’allèles dans les schémas étudiés.

locus de caractère

_quantitatif

_/

maximum de vraisemblance résiduelle

_/

cartogra-phie

/

simulation

_/

schéma petite-fille

INTRODUCTION

In a

_companion

_paper

_(Grignola

et

_al,

_1996),

a residual maximum likelihood

(REML)

method was derived for

_{estimating position}

and variance contribution of a

_single

_QTL

_together

with additive

_polygenic

and residual variance

_components.

The REML

_analysis

was

implemented

with a derivative-free

_algorithm.

The method

overcomes the

_shortcomings

of the traditional methods of linear

_regression

(eg,

Haley et al,

1994;

Zeng,

1994)

and maximum likelihood

_(ML)

interval

_mapping

_(eg,

Weller, 1986;

Lander and

Botstein, 1989;

Knott and

_Haley,

_1992).

ML and

_regression

methods cannot

_fully

account for the more

_complex

data structures of outcross

populations,

eg, data on several families with

_{relationships}

across

_families,

unknown

linkage

phases

in

_parents,

unknown number of

_QTL

alleles in the

_population,

and

varying

amounts of data information on different

_QTLs

or in different families. The REML method is based on a mixed linear model

including

random

polygenic

effects and random

_QTL

effects.

_Polygenic

effects

_represent

the sum of the additive

effects at all loci not linked to the markers. The

_QTL

allelic effects are assumed

to have a

_prior

normal distribution with variance-covariance matrix conditional on

information from

_multiple

linked markers. Because the true nature of the

_QTLs

is

unknown,

ie,

the number of alleles at a

QTL

in the

population

studied is

unknown,

the robustness of the REML

_analysis

to the number of alleles at the

_QTL

must be

evaluated.

One of the main

_{experimental designs}

for

_QTL

_mapping

in livestock is the half-sib

_design,

used in cattle in the form of

_daughter

or

_{granddaughter designs}

(Weller, 1990).

In this _paper, we evaluate the _accuracy of the REML

analysis

in

QTL mapping using

granddaughter designs.

The simulated

_designs

resemble actual

designs

for the US Holstein

_population.

Data are simulated under several

_genetic

models

_differing

in the number of

_QTL

alleles. The

_analysis

is carried out with and

without consideration of

_{relationships}

among sires.

Here,

we

_only

_present

simulation

results. The

analysis

is described in detail in the

_companion

_paper

_(Grignola

et

_al,

SIMULATION

Design

The most

_frequently

used

_design

for

_mapping

_QTL

in

_dairy

cattle is the

granddaugh-ter

_design

_(GDD),

where marker

_genotypes

are collected on sons and

_phenotypes

on

_daughters

of the sons. A GDD was simulated with a

_pedigree

structure

resem-bling

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 9 ancestors of the sires

_{(fig 1),}

and is identical to the

design

used in the

_Bayesian

_{linkage analyses}

of Thaller and Hoeschele

_(1996a,b)

and Uimari et al

_(1996a).

While this

_design

had 100 sons per

sire,

designs

with

_only

50 or 30 sons _per sire were also simulated.

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

_daughter’s

dams. Total variance of DYD

equals

Var(DYD)

=

0.25

0

&dquo;;/ R

and can be factored

into

where R is

_reliability

or

squared

_accuracy of the son’s estimated additive

_genetic

effect or

_breeding

_value,

and

0

&dquo;; is

the additive

_genetic

variance. In this

_{factorization,}

the first

component

is the variance among the sons’

transmitting

abilities or half of their additive

_{genetic values,}

and the second term is ’residual variance’ or the

variance _{of the average dam} and Mendelian

_genetic

effect of the

_daughters

and the average environmental effect. Variance of DYD can be rewritten as

where w =

(1 - R)/R

and

0

&dquo;;

=

a.

0.25

Q

Therefore,

when

analyzing

DYD with the

weight

w, DYD has an

expected heritability

of

_0.5,

because the

expected

value of

the estimate of

₀

_{&dquo;; is 0.25a2 a.}

_Hence,

there is the

_option

of

_{treating heritability}

as

known in the REML

_analysis

when

_phenotypic

data are DYDs.

Marker and

_QTL

_genotypes

were simulated

_according

to

_{Hardy-Weinberg}

fre-quencies

and the map

positions

of all loci. One

linkage

group was considered which

equal frequencies,

with the

_exception

of one

_design

where each marker had

_only

three alleles at

_{equal frequencies.}

The markers were

_spaced

20 cM

apart

and,

for

the results

_{presented here,}

the

_QTL

was located in interval 3 at 5 cM from the left

marker

_(other

_QTL

_positions

were simulated to

_verify

that the

_analysis

was

_working

properly).

Polygenic

and

_QTL

effects were simulated

_according

to the

_pedigree

in

_figure

1. Data were

_analyzed

(i)

_using

full

_pedigree

information and

(ii)

_assuming

that the

20 sires in

_figure

1 were

_unrelated,

as is common

_practice.

The

_QTL

contribution

to the DYDs of sons was

_{generated by sampling}

individual

_QTL

allelic effects of

daughters

under each of the

_genetic

models as described below. This

_sampling

of

QTL

effects assures that DYD of a

heterozygous

son or of a son with substantial

difference in the additive effect of its two

_QTL

alleles has

_larger

variance among

daughters

due to the

_QTL

than a

_homozygous

son or a son with similar

_QTL

allelic

effects.

Genetic

models

Three different

_genetic

models were used to simulate data. Common to all models

were the

parameters

narrow sense

_heritability

of individual

phenotypes

h

2

=

0.3,

phenotypic

SDa

p

=

_100,

and the order of and recombination _rates

among all loci. Under all three

_{models, phenotypes}

were simulated as

where ni was the number of

_daughters

of son

_{i, g}

was the sum of the v effects

in

_{daughter j}

of son

_i,

u was a

_normally

distributed

_polygenic

effect,

e was a

normally

distributed

_{residual, polygenic}

variance

_(or2)

was

_equal

to the difference between additive

_genetic

variance

_(oa

₂

₎

and the variance

_explained

_by

the

_QTL

(2

0

,2) ,

and

_or,2,

was environmental variance. Number of

_daughters

_per son was set to

_50,

_{corresponding}

to a

_reliability

(VanRaden

and

_Wiggans,

1991)

of near 0.8.

The ratio of the

_QTL

allelic variance

_(ov2)

to the additive

_genetic

variance

₍

₀

_,2)

is denoted

_{by v}

₂

below.

Model 1. Normal-effects model

For each individual with one or both

_parents

unknown,

one or both

QTL

effects,

respectively,

were drawn from

_N(O,

₂

_{) .}

_v

_a

For the

_pedigree

in

_figure

_1,

there were 32 distinct base

_alleles,

and the

_QTL

was treated as a locus with 32 distinct alleles in

passing

on alleles to descendants. The

_parameter

_U

_V2

was set to

_{0.250&dquo;! or 0.06250&dquo;!,}

ie,

the simulated

QTL

accounted for

50%

_(2v

₂

=

_0.5)

or

12.5%

(2v

2

=

0.125)

of

the total additive

_{genetic variance,}

_{respectively.}

In an additional

_{simulation, v}

2

was

set to zero to obtain the

_empirical

distribution of the test statistic under the null

hypothesis.

Model 2. Multiallelic model

The

_QTL

had ten alleles with

_equal

_frequencies.

For the biallelic

_QTL

with

2v

2

= 0.5

QTL,

means of the ten

_homozygous

_genotypes

_ranged

from -ga to Qa at

_equal

intervals. Means of

_{heterozygotes}

were calculated

_assuming

additive _gene action.

Given these means

(!,),

additive effects of alleles were determined

as where _pi = p

2 = ... = Pio = p = 0.1. The variance at the

QTL

was

which

_yielded

a value of

2v

2

=

0.204. Model 3. Biallelic model

The

_QTL

was biallelic with allele

_frequency

of 0.5. The variance at the

_QTL

was

where for _p = 0.5 and

2v

2

= 0.5 or

2

V

2

=

0.125,

QTL

substitution effect a was

determined and used to

_compute

the additive effects of the two

_QTL

alleles as _-pa

and

_{(1 - p)a.}

RESULTS

The REML

_analysis

for

_{single QTL}

_{mapping using}

all markers on a chromosome is decribed in detail in the

_companion

paper

(Grignola

et

_al,

_1996).

_Analyses

were

performed

with and without

_considering

_{relationships}

_among sires and with the marker

_{linkage phases}

of sires and ancestors known or unknown. For the

_designs

considered

_here,

the most

_probable

_(more

than

_90%)

_{linkage phase}

of the sires

was

_always

the true

_phase,

but

phases probabilities

calculated for the ancestors

indicated more

_uncertainty

about their true

_phases.

When

_phases

were treated as

unknown,

the

_{analysis employed}

_equation

_(11!

of

_Grignola

et al

₍₁₉₉₆₎

to calculate

the inverse of the variance-covariance matrix of the

QTL

effects of ancestors and sires. Contributions from sons were calculated

_assuming

that the most

_likely

sires

phases equalled

the true

_phases.

Analysis

of a

_single

data set took around 8 mins of

_computing

time on an

IBM SP2

_system

with RS6000 390 and 590 nodes. In a

_preliminary

_{investigation,}

the REML

_analysis,

_using

a derivative-free

_{Simplex algorithm}

was started from

very different initial values for the

parameters

to

_verify

convergence to the same

estimates. As an

_example,

a

_particular

data set simulated under the biallelic

_QTL

model

_(2v

₂

=

_0.5)

was

analyzed

_using

the two

_starting

value sets

_[0.9,

_{0.45, 0.6,}

200]

and

_[0.1,

_{0.05, 0.4,}

_1500]

for

_,

₂

_[h

_V

₂

_,

_{dQ, 0&dquo;;]}

with the

_parameters

defined in

table I. Parameter estimates and likelihood for the first

_starting

value set were

0.6380, 0.2790,

0.4434,

542.6,

and - 4 331.3.

_{Corresponding}

_figures

for the second

Parameter estimates and likelihood ratio statistics are shown in tables II-VI.

All results are based on 50

replicates.

In table

_II,

results for the normal-effects

QTL

model with

2v

2

= 0.5 are

_presented.

Several different

_analyses,

described in

table

_II,

were conducted.

_First,

sires were simulated as unrelated and

_analyzed

without

_{relationships.}

_Then,

_{relationships}

_among sires were simulated

_according

to

figure

1. In

_analyses

of these

_data,

sires were treated as

_unrelated,

treated as related

with known marker

_{linkage phases}

_(ie,

the true

_phases

were used for all sires and

ancestors),

treated as related with the most

_{likely linkage}

_phases

used in

_place

of the true

_phases,

or treated as related with

_{linkage phases}

considered as unknown

(by

using

equation

[11]

in

_Grignola

et

_al,

_1996).

Accuracy

of

parameter

estimates was

_{slightly higher}

when sires were simulated

and

_analyzed

as

_being

unrelated

_{(analysis I),}

_compared

to the case where sires were

simulated and

_analyzed

as related

(analyses

III-V).

When sires were simulated

related and

_{analyzed unrelated,}

the estimate of

_heritability

was

_slightly

lower and

the likelihood ratio statistic was lower than the

_{corresponding}

values obtained with sires treated as related

(analyses

II and

IV).

When sires were

_{analyzed using}

relationships,

treating

the most

_likely

marker

_linkage

_phases

of sires and ancestors

ratio statistics as

_{considering linkage phases}

as unknown

_{(analyses III-V).}

_Only

small differences between

_analyses

with

known,

most

_likely,

or unknown

_linkage

phases

are to be

_expected

for a

_{granddaughter design}

with marker information on

100 sons _{per sire.}

A further

_analysis

_(analysis

VI in table

_II)

was conducted on the same GDD

except

that markers had three alleles at

_{equal frequencies}

rather than the five alleles simulated for all other

_design

variations. Parameter estimates were not

_noticeably

affected

_by

the decline in marker

_{polymorphism,}

but the _average likelihood ratio statistic was reduced.

In the last

_analysis

of table II

_(VII),

_heritability

was fixed at 0.5.

_Accuracy

of estimates of the

QTL

parameters

and of residual variance was

improved

but the

value of the likelihood ratio statistic was almost

_unchanged.

Table III contains results for data sets

_generated

with

2v

2

= 0.125 and with

relationships

among

sires,

and

_analyzed

first

_by

_{ignoring relationships}

_among sires and

_{secondly by}

accounting

for

_{relationships}

with

_{linkage phases}

treated as

unknown

_(equation

_!11!

in

_Grignola

et

_al,

_1996).

_Analyses

I and II in table III were

conducted for the GDD with 100 sons _per sire. The estimate of

_heritability

and the likelihood ratio statistic were

_again

lower when

relationships

_among sires were

the likelihood ratios were

considerably

lower than those in table II. In the

analyses

of the

_{granddaughter}

designs

with

only

50

_(analyses

III and

_IV)

or 30

(analyses

between

_analyses

with and without

_{relationships}

in the likelihood ratio statistics

were rather small for the

_designs

with 30 and 50 _sons, where the likelihood ratio statistics were near the threshold values of 5.99

(0.05

_type-I

error

level)

and 9.21

(0.01

type-I

error

level)

when

_assuming

a

_chi-square

distribution with two

_degrees

of freedom.

Figures

3 and 4

_depict

residual likelihood

_profiles

for

_{single replicates generated}

with

2v

2

=

0.5 and

2

V

2

=

_0.125,

respectively,

and obtained from

_analyses

with

relationships

among sires and with

linkage phases

of sires and ancestors treated as

unknown. Both

_{figures display}

the marker

positions,

the most

_{likely QTL location,}

and a confidence interval

(CI)

for the

QTL

_position

calculated

_by

the LOD

drop-off method of Lander and Botstein

_(1989).

The limits of this CI were found

_by

determining

the map

position

at either side of the most

_likely

_position,

where the LOD score had fallen

_by

one unit

(this

calculation

_required

_converting

natural

logarithms

to base 10

_logarithms).

As

_pointed

out

_earlier,

information from all

markers was

utilized, leading

to smoother

_profiles

_(Knott

and

_Haley,

_1992;

_Georges

et

_al,

₁₉₉₅₎

than the

_original

interval

_mapping

method of Lander and Botstein

(1989).

Results for the multiallelic model

_(2v

₂

=

_0.204)

are

_presented

in table IV.

_Again,

data sets were

analyzed by

first

_treating

sires as unrelated and then

by utilizing

relationships

among sires with

linkage

phases

of sires and ancestors treated as

unknown. Parameters were estimated

quite

accurately,

and likelihood ratios were

significant

and similar to those for the normal-effects

_QTL

model with

2v!

= _0.125.

Results for the biallelic models with half of the

_homozygote

difference

_equal

to one additive

_genetic

SD

(2v

2

=

0.5)

or to one half of it

(2v

2

=

0.125)

are

in the

_simulation,

and data sets were

_{analyzed by}

_{ignoring relationships}

and

_by

accounting

for

_{relationships}

with

_{linkage phases}

unknown

_(equation

_!11!

in

_Grignola

et

al,

1996).

Ignoring

relationships

among sires

again

decreased the estimate of

heritability

but increased the estimate of

v

2

.

The least accurate estimate of

_QTL

position

was obtained under the biallelic model with

2v

2

= _0.125.

_Overall,

and at

least when

_{relationships}

among sires were considered in the

analyses,

parameter

estimates were not

_noticeably

inferior to those obtained under the

_correponding

normal-effects

_QTL

models

_(table

_II),

and likelihood ratios for the biallelic and normal-effects models were similar.

CONCLUSIONS

REML

_analysis

based on a mixed linear model with random

_QTL

allelic

_effects,

having

a

_prior

normal

_{distribution, provided}

_quite

accurate estimates of

_QTL

location and of the variance

_components,

in

_particular

of the

_QTL

variance contribution. Data were

_generated

under three different

genetic

models for the

QTL,

the

_biallelic,

the multiallelic

_{(ten alleles)}

and the normal-effects

_(two

effects per founder drawn from

independent

and identical normal

distributions)

model.

While the normal-effects _{model is very similar} to the

_{analysis model,}

the biallelic model with _gene

_frequency

_p and substitution effect a deviates the most from the

homozygous

for a _gene

_frequency

of _p =

0.5,

and variance among

daughters

of _a

homozygous

son is

_equal

to

while variance among

daughters of heterozygous

sons is

_higher

_{by 0.25a}

2

.

Therefore,

Thaller and Hoeschele

_(1996a,b)

and Uimari et al

_(1996a)

fitted two different

residual variances of DYD when

_performing

_{Bayesian analysis}

of

_linkage

of a biallelic

QT

L

.

Despite

the

discrepancies

between the biallelic model and the model of

_analysis,

the REML

_analysis

was

_quite

robust to the number of alleles at the

_QTL,

a result

which is in

_agreement

with

_findings

of Xu and

_Atchley

_(1995);

_ie,

_polymorphism

at the

QTL

did not

_strongly

affect

_parameter

estimates or

_hypothesis

tests. This

finding

confirms the usefulness of the REML

_analysis

as an alternative method of

analysis

which,

although

not

_{nonparametric, requires}

fewer

_{parametric assumptions}

(number

of

_QTL

alleles and their

_frequencies)

than maximum likelihood and

Bayesian analyses

based on biallelic

QTL

models.

_Furthermore,

REML is in

_general

known to be

_quite

robust to deviations from

_normality.

The REML

_analysis

can be considered as an

_{approximation}

to the

_Bayesian

analysis

of Hoeschele et al

₍₁₉₉₆₎

_fitting

a normal-effects

QTL

model. The

_Bayesian

analysis

has the

_advantage

of also

_being

able to fit a biallelic

_model,

but for the

the REML

_analysis

and

_requires

several hours of

_computing

time.

_Although

the REML

_{analysis fitting}

a

_{single QTL requires only}

around 8 mi of CPU

time,

this

requirement

still

_prohibits

the calculation of

_{genome-wide significance}

thresholds

for this method

_using

data

_permutation

_(Churchill

and

_Doerge,

_1994),

unless a

number of less

_stringent

_significance

levels are chosen to obtain threshold values and these are used to

_extrapolate

to the desired

_significance

threshold

_(Uimari

et

_al,

_1996b).

_Only

the

_{least-squares}

method allows direct

_computation

_of genome-wide

_significance

thresholds from a

large

number of

_permutations

(eg,

10 000 to

100000).

The REML

_analysis

has been

_implemented

in the Fortran _program

_SMREML,

which is available from the authors upon

request.

The _{program is}

_{currently being}

extended to fit two linked

_QTLs

_per

_chromosome,

rather than

_using

the

_approach

of Xu and

_Atchley

₍₁₉₉₅₎

_fitting

the variances associated with the

_{next-to-flanking}

markers to account for

_additional,

linked

_QTLs.

Their

_approach

is

_only

_approximate

as effects associated with marker alleles identified within founders erode over

generations.

Furthermore it

_requires

many additional

parameters

when the marker

polymorphism

is

_{limited, causing}

the

_flanking

and

_{next-to-flanking}

markers to differ among families. In the near

_future,

_{the program will be extended} to other

_designs

(eg,

full-sibships

within

_{half sibships),}

where the current

_computation

of the inverse of the variance-covariance matrix becomes

_approximate

due to uncertain

_linkage

phases

in

_parents

of final

_offspring,

and other ways of

computing

this inverse

_exactly

(eg,

Van Arendonk et

_al,

₁₉₉₄₎

will be

_implemented.

ACKNOWLEDGMENTS

This research was

_{supported by}

Award No 92-01732 of the National Research Initiative

Competitive

Grants

_Program

of the US

_Department

of

_Agriculture

and

_by

the Holstein

Association USA. I Hoeschele

_acknowledges

financial support from the

_European

Capital

and

Mobility

fund while on research leave at

_{Wageningen University,}

the Netherlands.

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