The power of two experimental designs for detecting linkage between a marker locus and a locus affecting a quantitative character in a segregating population

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The power of two experimental designs for detecting

linkage between a marker locus and a locus affecting a

quantitative character in a segregating population

Zw Luo

To cite this version:

Original

article

The

_power

of

two

_{experimental designs}

for

_detecting

_linkage

between

a

marker

locus and

a

locus

affecting

a

quantitative

character

in

a

segregating population

ZW Luo

University of Edinbu!gh,

Institute

_of

Cell Animal and

_Population

Biology, King’s Buildings, Edinburgh

EH 9

.!JT,

UK

(Received

14 November

1991; accepted

9

_February

₁₉₉₃₎

Summary - The statistical power of 2

_{experimental designs}

_{(backcrossing}

and

intercross-ing)

for

_{detecting linkage}

between a marker _gene and a _quantitative trait locus

(QTL)

in families derived from a

_{segregating population}

is

_{investigated.}

Formulae which relate

power to the recombination

frequency

(r)

between the genes, the

genetical properties

of

the quantitative trait controlled

_by

the

_QTL

and the

_design

_parameters are

_developed.

The

_reliability

of some

simplifying assumptions

was confirmed

_by

_computer simulations.

Application

of these formulae has shown that the power of the 2

designs

with

population

size of 1 000 was < 20% when r was 0.3 for all heritabilities of

single

gene

considered,

few

_large

families are better than many small

families,

and

backcrossing

is

generally

more

efficient than

_{intercrossing.}

The allele

_frequencies

and dominance _properties of the

_QTLs

have

_important

interactions in their effects on _power.

statistical power

/

marker - QTL linkage

/

backcross

_/

intercross

Résumé - Puissance de 2 _{plans d’expérience} pour détecter une liaison _génétique entre

un locus marqueur et un locus _influençant un caractère _quantitatif dans une

popu-lation en _{ségrégation.} Cet article étudie la

_puissance

_statistique de 2

_{plans d’expérience}

(rétrocroisement

et intercroisement de

_F

_I

₎

_pour détecter une liaison

_génétique

entre un

gène

marqueur et un locus de caractère

_quantitatif

_(QTL)

dans des

_familles

dérivées d’une

population

en

ségrégation.

Des

formules

sont établies pour

exprimer la puissance

en

fonc-tion du taux de recombinaison

_(r)

entre les

_gènes,

des

_{propriétés génétiques}

du caractère

quantitatif

contrôlé par le

QTL

et des

_paramètres

du

_{plan d’expérience.}

La

_fiabilité

de

’

quelques hypothèses simplificatrices

a été

_confirmée

par des simulations sur ordinateur.

L’application

de ces

_formules

_{montre que} la

_puissance

des

_{2 plans,}

pour une taille de

population

de

_{1 000,}

est

_inférieure

à 20%

_quand

r est

_supérieur

à

0,3

pour toutes les

héritabilités du

_{gène considéré, qu’un}

nombre limité de

_familles

de

_grande

taille vaut mieux

qu’un grand

nombre de petites

familles,

et que le rétrocroisement est

_{généralement plus}

ef-ficace

que l’intercroisement. Les

fréquences alléliques

et la dominance au locus du caractère

quantitatif interagissent fortement

dans leurs

_effets

sur la

puissance.

puissance statistique

/

liaison _marqueur-QTL

_/

rétrocroisement

_/

intercroisement

INTRODUCTION

With the

_{rapid development}

of molecular

_techniques

in the last

_decade,

their

application

to the

_{investigation}

of the

_genetical

basis of

_quantitative

characters has become a

_subject

of considerable

_activity

_(Botstein

et

_{al, 1980;}

Beckmann

and

_Soller,

_1986;

Lander and

_Botstein,

_1989).

The central idea of these new

investigations

was to use the

newly-discovered

molecular markers

(for

_example,

RFLPs)

at defined _map

_positions

for

_tracing

linked

_quantitative

trait loci

_((aTLs).

Methodologically,

this can be

accomplished by detecting linkage

between a

_genetic

marker(s)

and a

QTL(s)

_through

various

_appropriate

_{experimental designs}

_(Breese

and

_{Mather, 1957, 1960;}

_Thoday,

_1961;

_Jayakar,

_{1970; Hill, 1975; Weller, 1986;}

Luo, 1989;

Luo and

_Kearsey,

_1989;

Lander and

_Botstein,

_1989).

Hill

₍₁₉₇₅₎

demonstrated the use of

_analysis

of variance for

_{detecting linkage}

between a marker _gene and a

_QTL

_by

means of a nested

_backcrossing

or

intercross-ing experiment

and

_attempted

to work out the _power of these

_designs.

_However,

because of the

_varying

sizes of each of the nested groups, the numerator of the final

test statistic used in the

_analysis

of variance to detect the

_{marker-QTL linkage}

cannot be

_expressed

as a constant times a

random x2

variable. Therefore,

she was

unable to work out

_analytical

_expression

for the power of the

experimental designs.

Soller et al

_{(1976, 1978)}

_{suggested excluding}

the

_offspring

with

_heterozygous

marker

genotypes

in the _power

_analyses

of the intercross

_design

in order to increase the

power of the

designs.

This has also avoided the

complexity

caused

by

the

unequal

sample

sizes among the different marker

_genotypes

and allowed use of the normal

procedure

of hierarchical

_analysis

of variance so as to set _up an F-distributed test statistic.

_Obviously,

this results in the loss of useful information and

_artificially

inflates the

_expected

variance between

_offspring

marker classes.

The

_present

_paper will focus on

_exploring

a statistical

_approach

to work out the

experimental

power of the

designs suggested

by

Jayakar

(1970)

and Hill

(1975)

and

relate the power

directly

to

_genetic

_parameters

_{of the marker gene} and the

_QTL

and the relevant

_design

_parameters.

This will allow factors

_affecting

the _power to

THEORY

Basic

_assumptions

and

_{experimental design}

The method involves

_analysing

progeny from natural or controlled

_matings

in a

population.

Consider 2 autosomal

_loci,

one affects a

_quantitative

character

(QTL)

while the other is a codominant marker. The 2 loci are linked with a recombination

fraction of r

(r’

= 1 -

r).

Let the

_frequency

of allele

_Q

_I

at the

_QTL

be denoted

p

(p

= 1 -

q)

and the

phenotypic

distributions of the 3

_genotypes

at the

QTL,

ie

Q

1

Q

1

, Q

1

Q

2

and

Q

2Q2

are assumed to be

_N(

_IL

_+a,

_),

₂

_’

_J

₍

_N(

_M+

_d,

_(J’

₂

₎

and

_N(p-a,

_(J’

₂

₎

respectively,

where a and d

_represent

the additive and dominant effect at the

QTL

(Falconer,

1989).

With

_just

one

QTL,

0

2

will

be the environmental variance

alone,

but with other unlinked

_QTLs,

it will also include

_genetic

variance at these

loci. The

_phenotypes

of the 3 marker

genotypes,

viz

₂

_l

_{, M, M}

_{M, M}

and

_M

₂

_M

₂

are

distinguishable,

ie the marker locus is codominant and we assume that the

QTL

and the marker _gene are in

_linkage

_equilibrium

in the

_population.

One can score the

progeny of these families where

parents

are

M

1

M

1

x

₂

_M

₁

_M

or

M

I

M

2

x

_M

_l

_M

₂

_(ie

backcrossing

or

intercrossing)

and record the

_quantitative

_phenotype

and marker

genotype.

If,

for

_example,

we consider an

_experiment

_consisting

of s

_sibships,

within each of which there are m marker classes

(m

= 2 and 3 for

backcrossing

and

_{intercrossing}

_designs,

_{respectively).}

_{Let nZ!}

_represent

the number of sibs

within the

_jth

marker class within the ith

_sibship,

then the variation for the

quantitative

trait can be

_partitioned

into that between and within

sibships,

while

that of within

_sibships

can be further

_partitioned

into variation within and between marker

_genotypes.

For such unbalanced

_2-way

nested classification

_data,

variance

components

have been worked out

_by

Searle

_(1971,

p

475-477).

If it is further

assumed that each

_sibship

has a constant size of n then the total

_experimental

size

is s x n and

_analysis

of variance for both

_backcrossing

and

intercrossing

designs

is

illustrated in table

I,

in which:

Statistical model

In the

_analysis

of variance described in table

I,

the linear model for

_phenotypic

record of the

_quantitative

trait measured on the kth sib

_(k

=

i, 2, ... ,

n2!)

with the

jth

marker

_genotype

_(j

=

1,2,..., m)

within the ith

sibship

(i

=

1,2,..., s)

_can

be written as:

where _{ii is} an overall

population

mean while

_Q

_{i, /3}

_ij

and ez!! are contributions from

the

_sibship,

from the marker

_genotype

within

_sibship

and residual error

respectively.

They

are assumed to be

_{independently}

and

_normally

distributed with zero means

and variances

_{o, 2,}

_o

_l2

_and

_o,2

_{respectively.}

The

_frequency

distribution of the

_QTL

genotypes,

the

_expected

means and variances of the

_progenies

within the ith marker

genotypes

and within all

_possible

_sibships

were obtained

_by

IIill

_(1975),

and these

were

_carefully

rederived

_by

Luo

_(1989).

It was found that the

expected

variance

between marker

genotypes

within

_sibships

_(a2)

is:

and the

_expected

variance within marker

_genotypes

within

_sibships

₍

₀

_&dquo;)

is:

for the intercross

_design;

while the

_{corresponding}

variances for the backcross

_design

are:

It is

_easily

seen from

_equations

[3.lt

and

_[4.1]

that the

_expected

variance between

marker

_genotypes

within

_sibship

_(u

_M(I)

or

o,2 m(

B)

)

for either the intercross or backcross

_design

will be

_{statistically}

zero if the marker gene is not linked with the

_QTL,

ie r = 0.5. The

expected

variance could also be _zero if _one of alleles _at the

_QTL

is

_fixed,

_{ie p} = 0 or

_1,

but these situations are trivial. As

_pointed

out

_by

Jayakar

(1970),

under the null

_{hypothesis Ho :}

r =

_0.5,

the

_following

ratio of mean

squares:

is distributed as a central F-variable with

_expected

value of 1.

_However,

the ratio

The denominator of the

_right

side of

_[5]

is distributed as

12

However,

when the cell sizes

_(n

_ij

₎

are not constant over the marker

_genotypes,

the numerator

of the

_F-ratio,

cannot be

_expressed

as a linear combination of

chi-square

variables.

Therefore it is difhcult to determine the _power of the test

_directly,

_contrary

to a

traditional F-test when the null

_hypothesis

is false.

However,

under the

_assumption

of constant size of

_sibships,

the

_following

approximation:

can be

_incorporated

into

_equation

_[1]

for the

_{intercrossing}

_design

and

_[1]

can thus

be rewritten as:

- 1 !,

Similarly,

the

_following

_{approximation}

holds between sizes of 2

_subsibships

for

the

_{backcrossing design:}

which

_directly

results in:

1

therefore,

the

_expectation

of

_MS,,,

in

_equation

_[5]

can be

approximated by

a

general

form:

where

_a

_M

and

_<7{

_y

are

_respectively

defined

_by

!3.1!

and

[3.2]

for

_{intercrossing}

_design

or

_by

_[4.1]

and

_[4.2]

for

_{backcrossing design.}

If the marker

_genotypes

_[,3

_ij]

in model

[2]

are considered to be fixed effects in

_analysis

of variance described in table

I,

then the statistic for

_testing

the _presence of

_linkage

between the marker _gene and

QTL

is:

where F is a noncentral F-variable with

degrees

of freedom described in table I and the

_{noncentrality}

_parameter:

whose definition is the same as that in Kendall et al

_(1983,

_p

₃₇₎

and in Johnson

and Kotz

_(1970,

p

191).

By

definition,

the power function of the 2

designs

for

detecting

the

linkage

can

where

_Fv,,v2;

_{6 represents}

a noncentral F-variable with

_degrees

of freedom _vi and _v2

and noncentral

_parameter

6 while

_Fa;Vl;V2

stands for the upper a

_point

of a central

F-variable with

_degrees

of freedom VI and v2.

Power calculation

So

far,

the power for

detecting

the

linkage by

use of these

designs

has been shown

to be a function of the recombination fraction

(r)

and the basic

genetic

_parameters

at the

_{QTL, mamely}

the allelic

_frequency

p

(q

=

1 -p),

the additive and dominant

effects at the

_QTL

_(a

and

_d),

the residual variance

_{(or 2)}

as well as the

_experimental

design

parameters

s

(ie

the number

_{of sibships)}

and n

(ie

the size of the

sibships).

For a

_given

broad

heritability

(h’)

_b and dominance ratio

(f

= !)

at the

_QTL,

the

a

genetic

variance associated with the

_QTL

in an

F

2

population

is:

For

convenience,

let the

_phenotypic

variance of the

_quantitative

trait in the

F

2

population

be

_100,

the additive and dominant effect

_(a

and

_d)

can be solved as:

and the additive and dominance effects at the

_QTL

are obtained from:

Once the

_design

_parameters

_(s

and

_n)

and the

_genetic

parameters at the

QLT

(p,

f

and

_h’)

are

_given

_together

with the recombination

_frequency

between the

marker and

_QTL

_(r),

the value of the noncentral F-variable can be calculated

_by

using

equation

(9!.

For a

_given

significance

level a of the test, the _power of

_detecting

the

_linkage

can thus be worked out

_through

_equation

_[11]

_{directly by}

_using

the relevant statistical tables such as that

_{by Tang}

₍₁₉₃₈₎

or Tiku

(1967).

Although

these tables are available to

_provide

the power of an F-test

they

are restricted to

a limited number of

degrees

of freedom and to a limited _range of values of the

noncentral

_parameter.

_However,

several

_procedures

are available to

_approximate

the power of the F-test

(Patnaik,

1949;

Laubscher, 1960; Tiku,

1965,

1967).

For its

higher

accuracy, Tiku’s 3-moment common

_{approximation by using Laguerre}

series

was

programmed

in Mathematica

(Wolfram, 1991)

to evaluate the

_experimental

power in the

present

paper.

Power evaluation from simulations

Since

_{approximations}

_[6.2]

and

_[7.2]

were made in

_deriving

the power

function,

the

reliability

of these

_{approximations}

was checked

_by

_comparing

the theoretical

A Fortran-77

computer

programme was

_designed

for:

i)

_simulating

the inheritance

of the

_{marker-QTL linkage}

in the 2 nested

_experiments

as described above for _any

combinations of

_{experimental design}

and

_genetic

_parameters

_(Luo,

_1989);

_ii)

com-puting

F-value from

_analysis

of variance

_using

the simulation data

_following

the

algorithm

described

_by

Searle

_(1971);

and

_iii)

_calculating

the

_frequency

of

signif-icant F-values in

_replicated

simulation trials as in Carbonell et

_al,

_(1992),

which

gives

the

_empirical

_power.

RESULTS

Although

the _power of the 2

_designs

can be

easily investigated

_{at any} combinations

of

_{experimental design}

and

_genetic

parameters,

a total

experimental

size of 1 000 was

_only

considered here. The _powers of the 2

_designs

were evaluated

by

both

theoretical

_prediction

and

_computer

simulation for all

_possible

combinations of

2

_design

structures

_{(10 (sibships)}

x 100

_(sibs)

and 20 x

50),

heritability

h

2

=

0.01,0.05

and

_0.10,

allelic

_frequency

p =

0.25,0.5

and

0.75,

dominance ratio

f

=

_0.0,0.5

and 1.0 _as well _as recombination

frequency

between the marker

gene

and

_QTL

r =

_0.0,0.1

and 0.3. The

powers were evaluated at a

_significant

level

(a)

equal

to 0.05. For

_{simplicity, only}

_part

of the results were listed in table II for

demonstrating

an

_agreement

between _powers evaluated from theoretical

prediction

and simulation based on 500

replicates

(in parentheses).

The _powers of the 2

_designs

were also

computed analytically

for the

experimental

size of 1 000 but

_{realistically}

smaller size of

_sibsips

and were tabulated in table III.

It could be

_interesting

to _compare the

_present

_power

_predictor

to that of Soller and Genizi

_(1978).

Table III in Soller and Genizi

₍₁₉₇₈₎

listed the number of

_sibships

and the total

_experimental

sizes

_required

for

_achieving

a _power of

90%

when the

allelic

_frequency

_(p),

dominance ratio

_{( f )}

and contrast at the

_QTL

were

_0.5,

0.0

and 0.01

_(equivalent

to

1%

heritability

in the

_present

_study)

_{respectively,}

and the recombination

_frequency

between the marker and

_QTL

was zero. The _powers with

these

_population

structures and the same

_genetic

_parameters

were evaluated

by

use

of the

_present

method. The difference of the evaluated _{powers to}

90%

has been summarised in table IV.

Effects of recombination

_frequency

between the marker and

_QTL

_(r),

allelic

frequency

(p)

and dominance ratio

_{( f )}

at the

_QTL

on the _power of both

_backcrossing

and

_{intercrossing}

_designs

have been illustrated in

_figure

1 for a

_given

_heritability

of 0.1.

DISCUSSION

Derivations in the

_present

_paper have shown that the _power of the 2 kinds of

_designs

for

_detecting

_linkage

between a marker _gene and a

_QTL

can be

expressed

as function of

_design

_parameters

and

parameters

describing

genetic properties

of the marker

and

_QTL.

The _powers from theoretical evaluation agree very well with those from

stochastic simulation under consideration of a wide _range of situations

_{(table II),}

suggesting reliability

of the theoretical

_analysis.

lost

70%

of their power with an increase of r from 0.1 to 0.3.

_Moreover,

the

linkage

would be

_unlikely

to be detected

_(power

<

_20%)

when the

_QTL

would

be linked to the marker with a recombination

frequency >

0.3 when

h

2

6

0.1. It

has been

_pointed

out

_by

Risch

₍₁₉₉₁₎

and Collins and Morton

₍₁₉₉₁₎

that _{power is}

Woolliams

₍₁₉₉₂₎

studied the effect of recombination

_frequency

between marker and

QTL

on _accuracy of estimation of

genetic

_parameters

of the

QTL

with

heritability

of 0.1 and found that maximum likelihood estimates of these

_parameters

is

_usually

The power of both

designs

increased with

increasing

dominance ratio at low

allelic

_frequency

_(p

=

0.25) (fig la),

but decreased with

increasing

dominance ratio

at

_high

allelic

_frequency

_(p

=

0.75)

(fig

lc).

However,

there was little effect of

dominance on the _power of

_backcrossing

at the allelic

_frequency

of 0.5. At the same

allelic

_frequency

the power of

intercrossing

still increased with

_increasing

dominance ratio

_{(fig 1b).}

There was no evidence of effect of allelic

frequency

on the _power of both

designs

when gene effect at the

_QTL

was

_purely

additive

_{( f}

=

0.0) (fig

1d).

However,

the _power decreased with

_increasing

allelic

_frequency

when the allele

_displayed

dominance

_(fig

le,

f).

Soller and Genizi

₍₁₉₇₈₎

_published

the first

_{comprehensive}

theoretical

_study

of the same

designs

as addressed in the

_present

_paper but

through

investigating

sig-nificance of contrast between means of marker

_genotypes

of interest in

quantitative

trait.

_They

concluded that the effects of gene

frequency

and dominance level would

be

_important

when the number of families was small. Because when the number of

families is

_small,

the

_probability

that the contrast in each of the families be zero

is so

_large

that the _power

_requirement

will not be met for _{any size} of

_{family. They}

suggested

that the

_probability

of zero contrast would be 0.90 for backcrosses and 0.94 for intercrosses when a = d and

_2pq

= 0.3.

Therefore,

_at least 22 and 34 fam-ilies for the 2

_{designs respectively}

must be

_sampled

in order that on _average non

zero contrasts can be

_expected

in 2 of these families.

_However,

if the power of these

designs

is calculated in the way

developed

in the

present

paper, loss in the power

due to

_probability

of

_sampling

families with zero contrast can be

avoided,

since it

is

_likely

that those families with zero contrast will nevertheless contribute to

sig-nificance of the variance between marker

_types

within families. In

fact,

in the case

of

h

2

=

_0.01,

_f

=

_{1.0, 2pq}

= 0.3 and

experimental

size of 5 000

(10

x

_500),

a _power

of 0.76

_(0.70)

for the

_{backcrossing design}

or 0.64

(0.62)

for the

intercrossing

design

was obtained from the theoretical

prediction

(simulation)

in the

present

study.

Comparison

in table IV was made between the

_present

_power

predictor

and that

in Soller and Genizi

_(1978).

It can be seen that the powers of the

backcrossing

designs

were

_{slightly higher}

in the

_present

paper than in Soller and Genizi

(1978).

While the

present

method

_{yielded higher}

power of the

intercrossing

designs

with

small number

_{of sibships}

and

large

sibship

size than in Soller and Genizi

_(1978),

in

which

_offspring

class with

_heterozygous

_genotype

at the marker locus was excluded.

However,

with

_large

number of

_sibships

and small

_sibship

_size,

the power of the

intercrossing

designs

was lower in the

present

paper than in Soller and Genizi

(1978).

Several researchers

_(Hill,

_1975;

Soller and

_Genizi,

₁₉₇₈₎

have found that for a

given

total

_experimental

_size,

_design

with fewer

_sibships

but

_{larger sibship}

size

was more

powerful

than that with more

_sibships

but smaller

sibship

size. This was

confirmed

_by

the

_present

_study.

_Moreover,

it was found that decline in power due to smaller

sibship

size was more severe for

_{intercrossing}

_design

than for

backcrossing

design

(tables

II,

III).

The effect of the

_population

structure on the _{power is}

parallel

to its effect on

_degrees

of freedom of the residual

expected

mean _square

(table I).

For most animal

_species,

realistic full

_sibship

_{size is very}

small,

eg 5 to

_20,

but

half-sibship

size

_might

be _very

_large.

Weller et al

₍₁₉₉₀₎

_{investigated daughter}

and

in

_dairy

cattle

_populations

in which one sire

might

have several hundred

daughters

or

_{granddaughters}

if

_breeding

was

_by

AI.

_By

_organising

_experiment

of such animal

species

into

_{half-sibship population}

structure,

one

might

_expect

more _power since

the residual

_expected

mean _square would have more

_degrees

of freedom.

Comparison

of power of the 2

designs

revealed that

backcrossing

was

_generally

more

_powerful

than

_{intercrossing.}

This _agrees with the conclusion of Soller and

Genizi

_(1978).

The

present

studies have not

_{directly provided}

the total

_experimental

size

required

for a

_given

_power under a

_particular

_genetic

and

_design

situation. The

theoretical calculation in the

_present

_{paper can,}

_however,

be

_easily

used in the

procedure suggested by

Fox

₍₁₉₅₆₎

so as to obtain the size of

_experiment

for a

given

power in a

specific

situation.

ACKNOWLEDGMENTS

I thank MJ

_Kearsey,

R

_Thompson

and C

_Haley

for their useful comments and

helpful

discussions in this _paper and 3 anonymous reviewers for their many

constructive

_suggestions

and criticism.

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