Genetic evaluation for a quantitative trait controlled by polygenes and a major locus with genotypes not or only partly known

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Genetic evaluation for a quantitative trait controlled by

polygenes and a major locus with genotypes not or only

partly known

A Hofer, Bw Kennedy

To cite this version:

Original

article

Genetic

evaluation

for

a

quantitative

trait controlled

by polygenes

and

a

major

locus

with

_genotypes

not

or

only partly

known

A Hofer

BW

_Kennedy

₂

1

Department

of

Animal

_Sciences,

Federal Institute

_{of Technology}

_(ETH),

CH-8092

_{Zvrich, Switzerland;}

2

Centre

for

Genetic

_{Improvment of Livestock, University of Guelph,}

Guelph, Ontario,

N1 G 2W1, Canada

(Received

4 March 1992;

accepted

5

_August

₁₉₉₃₎

Summary - For a _quantitative trait controlled

by polygenes

and a

_major

locus with 2

alleles,

equations for the maximum likelihood estimation of

major

locus genotype effects and

polygenic breeding values,

as well as _major allele

frequency

and _major locus _genotype

probabilities,

were derived. Because the

resulting

expressions are

computationally

un-tractable for

_{practical application, possible approximations}

were

_compared

with 2 other

procedures suggested

in the literature

_using

stochastic _computer simulation.

_Although

the

frequency

of the favourable allele was

seriously

underestimated when _major locus

geno-types were

_{entirely unknown,}

the

proposed

method compares

favourably

with the 2 other

procedures

under certain conditions. None of the

_{procedures compared}

can

_{satisfactorily}

separate

major genotypic

effects from

_polygenic

effects.

_However,

the

_proposed

method has some

potential

for

_improvement.

major locus

_/

_genetic evaluation / segregation analysis

Résumé - _{Évaluation génétique} pour un caractère quantitatif contrôlé par des

polygènes et un locus majeur à _génotypes inconnus ou seulement partiellement

connus. Pour un caractère contrôlé par des

polygènes

et un locus

_majeur

à 2

allèles,

les

équations

pour l’estimation du maximum de vraisemblance des

effects génotypiques

au locus

majeur

et des valeurs

génétiques polygéniques

ont été

dérivées,

permettant aussi d’estimer la

_fréquence

de l’allèle majeur et les

_{probabilités}

des

_génotypes

à ce locus. Les _expressions

obtenues étant incalculables en

_pratique,

des

_{approximations possibles}

ont été

_comparées

par simulation

stochastique

à 2 autres

_{procédures proposées}

dans la littérature. Bien _que la

_fréquence

de l’allèle

_favorable

soit sérieusement sous-estimée

_lorsque

les

_génotypes

au

locus

_majeur

sont entièrement

_inconnus,

la méthode

_proposée

a

_quelques

_avantages sur

satisfaisante

pour

séparer

l’efJet

des

génotypes majeurs

des

_{effets polygéniques. Cependant,}

la méthode

_proposée

est

_susceptible

d’être améliorée.

locus _majeur

_/

évaluation _génétique

_/

_analyse de _{ségrégation}

INTRODUCTION

Statistical methods based on the infinitesimal

_model,

the

_assumption

of _many

un-linked loci all with small effects

_controlling

_quantitative

traits,

have been

success-fully applied

in

animal

_breeding.

An

_increasing

number of

studies, however,

have

reported

single

loci

_{having large}

effects on

_quantitative

traits. Such loci are referred

to as

_major

loci.

_Examples

are the

_prolactin

_(Cowan

et

_al,

₁₉₉₀₎

and the weaver

loci

_(Hoeschele

and

_Meinert,

₁₉₉₀₎

in

_dairy

_cattle,

and the halothane

_sensitivity

locus

_(Eikelenboom

et

_al,

₁₉₈₀₎

and a locus

_acting

on

&dquo;Napole&dquo;

_yield

(Le

_Roy

et

_al,

1990),

a

pork quality

_trait,

in

_pigs.

Only

in the case of the halothane locus has the

responsible

gene been identified and

procedures

for its

_genotyping

become available

(l!TacLennan

and

Phillips,

1992).

There is no

_difficulty

with

_genetic

evaluation for traits controlled

_by

a

_major

locus and

_polygenes

when

_major

locus

_genotypes

are known. A fixed

_major

locus effect has to be added to the linear model and

_major

locus effects and

_polygenic

breeding

values can be estimated

_by

the usual mixed model

_equations

_(Kennedy

et

al,

1992).

When

genotypes

are

unknown, however,

_satisfactory

statistical methods

are still

lacking.

Selection decisions could

_possibly

be based on animal models that

include the

_major

locus effects in the

_polygenic

part of the model. In cases where

the

allele

has some

_positive

effect on 1 trait but

_negative

effects on

_others,

it would

be desirable to have

_separate

estimates of the

_major

locus and

_polygenic

effects available. The 2 estimates would then be combined

_according

to the

_breeding

objective.

Because

_genotyping

of all the animals of a

_population

is

_likely

to be too

_expensive

if at all

possible,

statistical methods are

_required

that estimate

major

locus

_genotype

effects as well as

_polygenic

effects and

_major

locus

_genotype

probabilities

for each candidate.

Such a method was first

_proposed

in human

_genetics

_by

Elston and Stewart

(1971).

The unknown parameters of the model are estimated

_by

_maximizing

the

likelihood of the data. For models with both

_major

locus and

_polygenic

effects exact calculations are _very

_expensive

and become unfeasible for

_pedigrees

with more than

! 15 individuals. Several studies

compared

the

power of different

approximations

of the likelihood function to detect a

_major

locus in half-sib

_family

structures in animal

_breeding

data

_(Le

_Roy

et

_{al, 1989;}

Elsen and Le

_Roy,

_1989;

Knott et

al,

1992a).

Hoeschele

₍₁₉₈₈₎

_developed

an iterative

procedure

to estimate

_major

locus

_genotype

_{probabilities}

and effects as well as

_{polygenic breeding}

values. The

equations produced

for the estimation of

_genotype

_{probabilities}

were derived for

simple population

structures and were based on an

_{approximation}

of the likelihood

function.

_Kinghorn

et al

₍₁₉₉₃₎

used the iterative

_algorithm

of van Arendonk

regression

on

_genotype

probabilities.

A method was

proposed

to correct for the bias inherent in such

_analyses.

The

_objectives

of this

_study

were:

i)

to derive exact maximum likelihood equa-tions to estimate

_major

locus

_genotype

_{probabilities}

and effects for a

_quantitative

trait with mixed

_major

locus and

_polygenic

inheritance without _any

restrictions

on

population

structure;

ii)

to examine

_possible

_{approximations;}

and

_iii)

to _compare these

_{approximations}

with the methods of Hoeschele

₍₁₉₈₈₎

and

_Kinghorn

et al

(1993)

by

stochastic

_computer

simulation.

METHODS

Model

Consider a

_quantitative

trait which is controlled

by

1 autosomal

_major

locus with

2

_alleles,

A and a, and many other unlinked loci with alleles of small effects. Mendelian

_segregation

is assumed for all alleles at all loci. The allele with the

major effect, A,

has a

frequency

of _{p in} the base

_population,

which is assumed to be

unselected,

not inbred and in

_{Hardy-Weinberg}

and

_{gametic equilibria.}

In the base

population

the 3

possible

genotypes

at the

_major

locus

_(AA,

Aa and

_aa),

which will be denoted as

_1,

2 and 3

throughout

this _paper, are therefore

expected

to occur in

_frequencies

of p

2

,

2p(1-p)

and

(1-p)

2

,

_{respectively.}

Because

_genotyping

of animals

_might

be

_impossible

or too

_expensive,

we assume for the moment that the

_genotypes

at the

_major

locus are not known. With 1 observation per animal the

_following

mixed linear model can be formulated:

where y = observation vector

b = vector of

_non-genetic

fixed effects

g = vector of fixed

_major

locus

_genotype

effects

_[g

₁

₉₂

_g3!!

a = vector of random

_polygenic

_breeding

values

e = _vector of random _errors

X,Z =

known incidence matrices

T = unknown incidence matrix

_indicating

true

_major

locus

_genotypes

of all the animals in the

population

The

expectation

and variance of the random variables are assumed to be:

action of the

_polygenes

is assumed but dominance is allowed for at the

_major

locus. In order to

_keep

the model

_simple,

it is further assumed that the variance

components

Q

a

and Q

e

are

known. This

_assumption

_implies

that the

_genetic

variance caused

_{by polygenes}

is known but not the

_genetic

variation caused

_by

the

_segregating

_{major allele,}

which is determined

_by

the

_major

_genotype

effects and

_frequencies.

This critical

_assumption

has to be

_kept

in mind when

_discussing

tlte simulation results.

Likelihood function

The likelihood for mixed model

_[1]

was first discussed

by

Elston and Stewart

(1971).

The likelihood can be written as:

is a normal

density

and

Pr(Tlp)

is the

probability

of T

given

the allele

frequency

p and the

pedigree

information. Because variance

components

are assumed to be

known,

cl =

(27r)&dquo;°’!&dquo; -

!V !

.ol e 21-1.1,

with _no as the number of

observations,

is a constant.

_Following

Elston and Stewart

_{(1971), Pr(Tlp)}

can be

_computed

as a

product

of

_{probabilities:}

,,

where N

is

the total number of animals in the

_population

and

_{Pr(! !s!d)}

is the

probability

of animal

_{i having}

_genotype

indicated

_by

_t

_i

_,

the ith row of

_{T, given}

the

_genotypes

of its

_parents

s and

d,

and is assumed to be known. Elston and Stewart

₍₁₉₇₁₎

_give

_Pr(ti!t9,td)

for autosomal and sex-linked loci. When the

parents

are unknown

Pr(tz!ts,td)

is

replaced

by

the

frequency

of the

genotype t

i

in the

base

population.

Known

_major

locus

_genotypes

can be accomodated

_{by setting}

Pr(! !,!)

to zero whenever ti conflicts with the known

_genotype

of animal i. With the base

_population

_(animals

with unknown

_parents)

in

_{Hardy-Weinberg}

equilibrium,

Pr(Tlp)

can be written as:

where nl, n2 and n3 are the number of base animals of

_genotype

AA,

Aa and aa,

respectively,

and nb = _{nl + n2 + n3} is the total number of base animals.

With 3

_possible

_genotypes

the sum in

_[2]

is over

3

N

elements. For 20 animals the sum is

_already

over 3.5 x

10

9

_possible

incidence matrices T. Whenever T conflicts with the

_pedigree

information

_Pr(Tlp)

is zero.

Therefore,

depending

on the

_pedigree

structure,

a

large

number of the elements to sum are _zero, but there remains a

As

_pointed

out

_by

Elston and Stewart

₍₁₉₇₁₎

the 3 likelihoods conditional on an

animal’s

_{genotype t}

_i

are

_proportional

to the

_{probabilities}

of animal

_{i having}

1 of the 3

_possible

genotypes.

The conditional likelihoods can be obtained

_{by skipping}

animal i in the summation over all

_possible

incidence matrices T. Maximum likelihood estimation

In order to maximize

_L(y),

we need the first derivatives with

_respect

to

_b,

_{g and p:}

The

_probability

of T

_given

the data and the

_parameters

of

the

model will be denoted _wT and can be

_computed

as

where c2 is the

product

of cl and a

_scaling

factor such that

_E

_WT

= 1. Note that

T

without

_scaling

this sum is

_equal

to the likelihood

_L(y).

After

_setting

to zero and

rearranging

we

_get

the 2

_following

_equations:

Solving

for

_p

in the last

_equation

leads to:

This

_equation

can be rewritten

_by

_replacing

_2n

₁

+ n2

by

v!.

T.

_{[2 1 0!’,}

with

_v’

a row vector of

_length

N with ones for base animals and zeros for the other animals.

Because _mT

_depends

on

_b,

_g and _p,

_equations

[3]

and

_[4]

have to be solved

true values and

_{Q’ =}

_L

_wTT.

Note that the ikth element of

_Q!

at convergence is

T

an estimate of the

_{probability that}

animal i is of

genotype

k

_given

the data and the estimates for the fixed

effects b,

the

_major

locus effects

_g

and the allele

_{frequency p.}

As mentioned

_above,

the same estimate can be obtained

_{by calculating}

likelihoods

conditional on an animal’s 3

_genotypes.

Using

these

definitions,

_equations

[3]

and

[4]

can be written as:

The solutions for

b

T

,

_i’

and

_p

_r

_converge to maximum likelihood

_(VIL)

estimates. Local maxima in

_L(y)

could _pose a

problem

and will be discussed later. Hoeschele

(1988)

estimated the allele

_frequency

from the

_genotype

_{probabilities}

of all animals with records whereas

_[6]

considers

_only

base

_animals,

which is in

_agreement

with Ott

_(1979).

Because

_genotype

_{probabilities}

of base animals take information from their descendants into account, all information on the allele

frequency

in the base

populations

is

_properly

used

_by

_!6J.

Animal breeders are not

_only

interested in

_{estimating major}

locus effects _g and allele

_frequency

p but also in

predicting polygenic breeding

values a. This is

_usually

done

_by

_regressing

_phenotypic

observations corrected for fixed effects:

where

_Q

is

_Q!

at _convergence.

_Using

V-

1

=

1

[ZAZ

>.-

,

₊

1]!!

= I -

ZMZ’,

where

M =

[Z’Z

₊

A-

I

>.]-

1

(Henderson, 1984),

a _can also be

computed

_as:

The same solutions for

_b,

g

and a are obtained

by

_iterating

on the

following

equations

together

with

_[6]

instead of

_using

_{(5!, [6]}

and

_!7!:

Note that

_{2.:: wTT’Z’ZT}

=

diag(v§ .

q[)

=

D

r

,

where

vb

is a row vector

T

difficulty

with this

_approach

is that it is not feasible to

_compute

_Q’

and

_!

_{tUy -}

*

T

T’Z’ZMZ’ZT

for

_{large populations.}

Approximations

Above

_Q

_r

was defined as:

There are 2

_problems

associated with the

computation

of

C!’’. Firstly,

the

summation is over all

_possible

incidence matrices T

_and,

_secondly,

a

quadratic

form

_involving

V-’

has to be

_computed

for each element in this sum. It can be

shown that the

_following

is an

equivalent

expression

not

involving

V-

1

:

where

_£11

=

MZ’(y -

Xbr -

)

r

ZTg

(Le

Roy

et

al,

1989).

Because

aT

depends

on

T,

we would have to

_compute

fill

for every

possible

T,

which is not feasible. In order to

_simplify

the

_{computations,}

we could

replace

*11

by

M which does not

depend

on T. Note that âr =

L wT’

âT.

This

_{approximation}

was also considered

T

by

Hoeschele

_(1988).

The

_approximated

Q!

is then:

Instead of

_using

a

_single

estimate of the

_polygenic

breeding

value for each animal

irrespective

of its

_genotype,

we could use 3 values for each animal

depending

on

its

_genotype

but

_independent

of the

genotypes

of all the other animals. A similar

approximation

was considered

_by

Elsen and Le

Roy

(1989)

and Knott et al

(1992a,

1992b)

for a sire model and was found to be

_superior

to

_[9].

We considered the

following approximation:

where xi and

t

ik

are the ith rows of X and

ZT,

a

?3

is the

_ijth

element of

A-

1

,

and _cii is the

_diagonal

element of the coefficient matrix in

_[8]

_pertaining

to the ith animal

_equation.

The summation over all

_possible

incidence matrices T in

_[9]

or

[10]

can be avoided

by

using

algorithms developed

to estimate

_genotype

_{probabilities.}

_Here,

the iterative

algorithm

of van Arendonk et al

₍₁₉₈₉₎

was

applied.

This

procedure

will be

briefly

described in the next section.

As with

_Q!

the

_difficulty

with

_expression

_{E w’ -}

T’Z’ZMZ’ZT

is

_two-fold;

the sum is over all

possible

T,

and the

_computation

of each element in that sum is

expensive.

Let _m2! be the

_ijth

element of

_Z’ZMZ’Z,

and

_t

_ik

_(t

_jl

₎

be the elements of T for animal

_i(j)

and

_genotype

_/c(l).

_Now,

the klth element of

_L

_{wTT’Z’ZMZ’ZT}

can be calculated as:

Note that at _convergence

W’ -

tik .

_{<_,;}

is an estimate of the

_probability

that

T

animal i is of

_genotype

k and

_{animal j}

_of

_genotype

_L,

_given

the data. For

_independent

animals this

_quantity

is

_equal

to

_q’

_ik

_qj’l

the

_product

of the

_{corresponding}

elements in

Q’’

and, therefore,

the contributions of

_{L wTT’Z’ZMZ’ZT}

and

Q&dquo; Z’ZMZ’ZQ’

T

to B’’ cancel out. For

_dependent

animals the contributions to the klth element of B’ are:

Now if we

_neglect

the

_dependencies

between animals for the

computation

of

L

w2..

tik .

_t

_jl

we

_get:

T

and

_[8]

becomes identical to the mixed model

_equations

_given

_by

Hoeschele

_(1988).

Another _way to

_approximate

B’’ is to assume that A = I. We then

_get:

Estimation of

_genotype

_{probabilities}

Van Arendonk et al

₍₁₉₈₉₎

_developed

an iterative

algorithm

to estimate

_genotype

probabilities

for discrete

_{phenotypes. Kinghorn}

et al

₍₁₉₉₃₎

_applied

this

_algorithm

to continuous traits. The

_comparison

of this

_algorithm

with non-iterative methods revealed some errors in the formulae

_given

in the

_original

_paper

_(LLG

Janss and JAM van

Arendonk, 1991;

C

Stricker,

_1992;

_personal

communications).

We

_applied

a corrected version of this

_algorithm.

For each

_animal,

_genotype

_{probabilities}

from 3 different sources of information are

_{computed using approximation}

_[9]

or

[10].

One round of iteration involves 3

steps.

First

_genotype

_{probabilities}

are

_computed

_using

information from

_parents

and

collateral relatives

_proceeding

from the oldest to the

_youngest

animal. In the second

step, genotype

probabilities

are calculated

_using

information from the _progeny

proceeding

from the

_youngest

to the oldest animal.

_Finally,

_genotype

_{probabilities}

using

information from each individual

_performance

are calculated and the 3 sources

of information combined. The _{iteration process is}

_stopped

when the solutions for

genotype

probabilities

reach a

_given

_{convergence criterion.}

The

_algorithm

works for

_{simpler pedigree}

structures as simulated in this

_study

but does not allow for

loops

in the

_pedigree,

also known as

_cycles

(Lange

and

Elston,

1975).

Loops

in a

pedigree

occur

_through

_{genetic paths}

_{(inbreeding loops),}

_mating

paths,

or a combination of the 2

_{(marriage loops),}

eg, a sire mated to 2

_genetically

related dams. Both

_inbreeding

and

_marriage

_loops

are common in animal

breeding

data. A non-iterative

_algorithm

for

_pedigrees

without

_loops

was

recently proposed,

which should be more efficient than the one used in this

_study

_(Fernando

et

al,

1993).

Method of Hoeschele

₍₁₉₈₈₎

Hoeschele

₍₁₉₈₈₎

used a

_{Bayesian approach}

to derive an iterative

_procedure

to estimate

_genotype

_{probabilities Q,}

allele

_frequency

_p and

_major

locus effects g for

simple

pedigree

structures. The

_genotype

_{probabilities}

were estimated

_by

formulae that were

_developed

for the

_{specific pedigree}

structures considered

_using

approximation

[9].

In contrast to

_[6],

Hoeschele

₍₁₉₈₈₎

estimated p from the

genotype

probabilities

of all animals with records:

where no is the number of animals with records and

vo

is a row vector with ones for

animals with records and zeros otherwise. The

_equations

that estimate the effects

of model

_[1]

are the same as

[8]

_approximated

with

[11].

We

_applied

this method

in the simulation

_study

_using

the iterative

_algorithm

described above but with

approximation

[9]

to estimate

_genotype

_{probabilities}

instead of the formulae

_given

by

Hoeschele.

Method

_{of Kinghorn}

et al

₍₁₉₉₃₎

the

_{least-squares}

estimates are biased

_(see,

for

_example,

_{Johnston, 1984,}

_p

_428).

Kinghorn

et al

₍₁₉₉₃₎

treated the unknown incidence matrix T as the unknown

true

_independent

variable and the

_genotype

_{probabilities}

_Q

as an estimate for T associated with some errors.

_Using

_Q

instead of T in the model leads to biased estimates of

_g

_*

_.

_Kinghorn

et al

₍₁₉₉₃₎

derived a correction matrix

_W,

such that

g

=

W!!§* .

Given _certain

assumptions,

they

showed that W =

V!V(,

where

V

t

is a 3 x 3 covariance matrix of elements in the 3 columns of T and

_V

₉

is the

_{corresponding}

covariance matrix of elements in the 3 columns of

_Q.

Because

(co)variances

in

_V

_Q

are

generally

smaller than

_{(co)variances}

in

_V

_t

_,

_major

locus effects are overestimated in absolute terms when

_using

_Q

instead of T. The

(co)variances

in

_V

₉

were calculated from the actual solutions for estimates of

genotype

probabilities

of all animals with records. Covariances in

_V

_t

were

_computed

as:

where

_q

_.k

is the _average

_genotype

_probability

for

_genotype

k of all animals with records and can be

_regarded

as an estimate of the

_frequency

of that

_genotype

in the

_population.

_Genotype

_{probabilities}

were estimated with the

_algorithm

of

van Arendonk et al

_(1989).

This

_algorithm

_requires

the allele

_frequency

_p as an

input

parameter.

Kinghorn

et al

₍₁₉₉₃₎

_kept

the initial value for _p constant over all

iterations,

ie

_regarded

the initial _p as the true value. But if _p was

known,

Cov(t

k

,t¡)

could also be derived from the

_expected

_frequencies

of the 3

_genotypes.

In our

implementation

Cov(t!,tl)

was

_computed

with

[14]

and the allele

_frequency

_p was

estimated with

_(13!,

which is a natural deduction from

_!14!.

The linear model can be written in matrix notation as:

Kinghorn

et al

₍₁₉₉₃₎

assumed that

_Var(a

_*

₎

=

Var(a)

= A -

Q

a

and

Var(e

*

) =

Var(e)

= I -

Q

e.

The matrices

Q

and W are _not known and have

to

be estimated

from the data as described above.

Therefore,

the

_following

_system

of

_equations

has to be solved

_iteratively:

Estimates for g should be unbiased but estimates for b and a are still biased. We

attempted

to correct for

_{the bias}

in

b by

adding

(X’X)-

l

X’ZQ(W -

I)g’’

+1

,

the

expected

difference between

b

r+1

and

b

*r+1

under the

_assumptions

_E(T)

=

E(Q),

Simulation

The methods of Hoeschele

₍₁₉₈₈₎

and

_Kinghorn

et al

₍₁₉₉₃₎

were

compared

with

the method

_developed

in this

_{study applying}

approximations

[10]

and

_[12]

_using

stochastic

computer

simulation.

_Phenotypic

observations were

generated by

using

the

_following

mixed model:

where

_hys

_i

is the fixed effect of herd x _year x sex

_i,

_g! is the fixed effect of

major

locus

_genotype

_j,

_a2!! is the

_polygenic

breeding

value _{and e2!!} is the random residual effect. The effects in the model were

_sampled

as follows:

f hys

i

N(0,I

J

fI)

fa

ijk

} -

N(0,A

J§

)

and

_{{e2!! } N}

_N(0,I

_J§

_{) .}

_Major

locus

_genotypes

were simulated

with 2

_segregating

alleles.

Genotypes

of base animals were

generated

by sampling

2 alleles from a uniform distribution between 0.0 and 1.0 with threshold p, the

frequency

of allele A.

_Genotypes

of _progeny were determined

according

to mendelian

segregation.

The effect of

_genotype

3 was set to zero as there is a

dependency

between fixed herd x _year x sex and

_major

locus effects.

Three different sets of

_parameters

were used

(table I).

Only

additive effects of

the

_major

locus were

considered,

although

all of the methods

compared

allow for

dominance. In the first set of

_{parameters, 50%}

of the

_phenotypic

variance

_(variance

due to

_major

locus +

polygenic

variance + residual

_variance)

is due to

_genetic

effects,

75%

of the

_genetic

variance is due to the

_{major locus,}

and

25%

is due to the

_polygenes.

The

frequency

of allele A with

_major

effect is

25%

in the base

population,

which results in an allele substitution effect a of

_1.0,

ie

_genotype

effects of 2.0

_(AA),

1.0

_(Aa)

and 0

_(aa).

In

_parameter

set

_2,

the allele

frequency

p is

0.5,

but the

_genotype

effects and all the other

parameters

are the same as in set 1. Thus the variance due to the

_major

locus is increased from 0.375 to

_0.5,

and the

phenotypic

variance

_changes

from 1.0 to 1.125. In

_parameter

set

3,

the allele

_frequency

_{p is}

0.25 and

50%

of the

_phenotypic

variance is due to

_{genetic effects,}

as in

_parameter

set

_1,

but the

_proportion

of

genetic

variance due to the

polygenes

is increased from 25 to

40%,

which results in an allele substitution effect a of

0.8.

simple.

In each of 10

_herds,

20 base dams each had a record in year 1. A _group of 20 base sires each with their own record in a common herd x _year

_(eg

test

_station)

was mated with these base dams. Each sire was

_randomly

mated with 1 dam in each herd. Each

_mating

_produced

5 progeny in year 2. The sex of each _progeny was determined

_{by sampling}

from a uniform distribution between 0.0 and 1.0 with

threshold 0.5. The

_population

size was

1220,

made _up of 220 base animals and

1 000 progeny.

In each of the

alternatives,

the same _sequence of random numbers was used.

Therefore,

identical data sets were

_analysed

with each of the 3 methods considered. Each alternative was

replicated

25 times.

With each of the 3

_methods,

final solutions are obtained

_by

_repeatedly

_computing

genotype

probabilities

and

_solving

a

_system

of

_equations

to

_get

new solutions for

major

genotype

effects and

_{polygenic breeding}

values. A

_stopping

criterion of the form:

was used for

_major

_genotype

effects _{g and} the allele

frequency

_p.

RESULTS

When the

_genotypes

of all animals with records are

known,

the estimates for

_major

locus effects _g are identical for all 3 methods considered

(table II).

Estimates for the

allele

_frequency

p,

however,

differed

slightly.

Using

formula

[13]

(Hoeschele,

1988;

Kinghorn

et

at,

1993)

the standard deviations

_(SD)

of estimated p were

_larger

than estimates

_by

_[6].

The estimates for _g and p agree well with the true values. Estimates of _g across

_parameter

sets are

_{consistently slightly larger}

than the true

values,

which can be

explained by sampling

effects and the fact that for each of the

25

_replicates,

data for the 3

parameter

sets were

_generated

with the same set of random numbers. As

_expected

from the

_{heritabilities,}

the correlations between true and

_{predicted breeding}

values were the same for

_parameter

sets 1 and 2 and

_slightly

higher

for

_parameter

set 3. The correlations between

_{predicted breeding}

values and estimated

_major

locus effects were close to _zero,

_showing

that the 2 effects were

well

_separated

in all cases.

Table III shows the simulation results for the 3

_parameter

sets

_using

all 3

procedures

when

_major

locus

_genotypes

were unknown. For

_parameter

sets 1 and

2,

estimates of

_major

locus effects g were close to the true values or

slightly

underestimated with

_approximated

maximum likelihood

_(AML),

underestimated

by

about

20%

with the method of Hoeschele

₍₁₉₈₈₎

and overstimated

_by

25 to

30%

with the method of

_Kinghorn

et at

_(1993).

For

parameter

set

_3,

estimates of

_major

locus effects _g were zero for 2

_replicates

_using

AML and for 21

_replicates

using

the method of Hoeschele

_(1988).

Non-zero estimates

_{of g}

were biased

upwards

by

14%

with A1VIL and

_by

47%

with the method of

_Kinghorn

et at

_(1993).

Both ANIL and the method of Hoeschele

₍₁₉₈₈₎

showed a

_{large variability}

of the non-zero estimates of

_major

locus effects for

_parameter

set 3. When the true allele

AML,

but estimated

_quite

well with the other 2 methods. Correlations between true and

predicted

breeding

values were similar for AML and the method of Hoeschele

(1988),

but zero for the method

of Kinghorn

et al

(199_3).

For

_parameter

sets 1 and

2,

the correlations between true

_(Tg)

and estimated

(Qg)

_major

locus effects were

similar for all 3 methods. When

_major

locus effects were smaller

(parameter

set

₃₎

these correlations were

_largest

with the method

of Kinghorn

et al

(1993).

_Predicted

breeding

values were

positively

correlated to estimated

_major

locus effects

Qg

with AML and to a

larger

extent with the method of Hoeschele

_(1988).

Using

the method of

_Kinghorn

et al

₍₁₉₉₃₎

these correlations were

strongly

_negative.

Because _{poor estimation} of p also affects all the other

estimates,

additional simulations were done with the allele

_frequency

fixed at the true

_(expected)

value. Results are

_reported

in table IV for A1VIL and the method of Hoeschele

₍₁₉₈₈₎

for

parameter

sets 1 and 3. All other results were close to those of table III and are

therefore not shown.

_Major

locus effects g were underestimated less with AML and

the correlations were similar for both

methods.

For

_parameter

set

_3,

the number of

replicates

with estimates of zero for

_major

locus effects was

_again

much

_larger

with

the method of Hoeschele

_(1988).

Table V _compares the 3 methods for the case where all sires and

50%

of the

dams are

gendtyped

at the

_major

locus. There was still a

_tendency

for AML to underestimate the allele

frequency

p when the true

_frequency

was 0.25. The method of Hoeschele

₍₁₉₈₈₎

underestimated

_major

locus effects

considerably

more than

AML

₍₉

to

31%

ver.sus 1 to

_11%),

whereas these effects were overestimated

by

22 to

43%

with the method of

_Kinghorn

et al

_(1993).

The accuracies of

_predicted

breeding

values were

_again

similar for AML and the method of Hoeschele

(1988)

but

much lower for the method of

_Kinghorn

et al

_(1993).

The accuracies of estimated

genetic

values at the

_major

locus were similar for all 3 methods with a

tendency

of

lower accuracies for the method of

_Kinghorn

et al

_(1993).

When all the sires but

intermediate between the 2 cases of no animals and all sires

_plus

50%

of the dams

genotyped.

So

_far,

final solutions have been

_reported

for iterations where

_starting

values

were

equal

to true

_(expected)

values. Table VI shows the number of

replicates

that

converged

to the same solutions

_using

different

_starting

values. Low

_starting

values

were half the true values and

_high

_starting

values were 1.5 times the true values of

_major

locus effects _g and allele

_frequency

p. When

major

locus

genotypes

were

not

_known,

none to a few

_replicates

_converged

to a

_single

set of solutions with all 3 different

_starting

values. For the method of Hoeschele

₍₁₉₈₈₎

with

parameter

set

_3,

most of the

_replicates

that

_converged

to the same solutions

_converged

to an estimate of zero for

_major

locus effects _{g. For AML} and the method of Hoeschele

(1988),

all

the sires

_(but

none of the

dams)

were known. The

_largest

number of

_replicates

with all 3 solutions different was found with the method

of Kinghorn

et al

_(1993).

DISCUSSION

The method

_proposed

here

_(AML)

_generally

_slightly

underestimates

_major

locus effects _g and

_seriously

underestimates allele

_frequency

p when the true

_frequency

is 0.25. The underestimation

_{of p}

leads to increased estimates

_{of g, although}

not to the extent that the variance

_{explained by}

the

_major

locus

_stays

constant

_(tables

III and

_IV).

This variance is

_higher

when the allele

_frequency

is fixed at the true value. The allele

_frequency

was still

_considerably

underestimated for

_parameter

set 1 when the

_pppulation

size was 10 times

larger

than considered here

(results

not

shown).

The allele

_frequency

was estimated

_by

(6!,

which was derived

by

_maximizing

the likelihood of the

data,

whereas the other 2 methods used

_[13].

Additional simulation runs with

_parameter

sets 1 and 3 and

_{approximations}

_[9]

and

_[11]

together

with

_[6]

showed

_considerably

lower estimates of p and

higher

estimates of _g than results for the same 2

_{approximations}

_{applied together}

with

_[13],

the method of Hoeschele

_{(1988) (results}

not

_shown).

There seems to be a

problem

in

applying

[6]

together

with

_{approximations [10]}

and

_[12]

or, to a lesser

_extent,

with

[9]

and

_(11!.

Nevertheless

_[6]

is the correct

_equation

for the estimation of the allele

The method of Hoeschele

₍₁₉₈₈₎

_consistently

underestimated

_major

locus effects g which is in

_agreement

with the simulation results of the same author. For smaller

allele effects

_(parameter

set

_3),

_although

still

_quite

_large,

most of the estimates of _g were zero,

_indicating

that the

genotype

effects have to be

_large

in order to be

_recognized.

The same is true for

_A1VIL,

but to a lesser extent. There was a

tendency

for the

_accuracies

of

_{predicted polygenic breeding}

values

_(a)

and estimated

major

locus effects

_(6g)

to be

_slightly

_higher

with AML than with the method of Hoeschele

_(1988).

In an unselected

population

as simulated here the

expected

correlation between true

_polygenic

and

_major

locus effects is zero. The correlations

(gametic disequilibrium)

which will make

_separation

of the 2 effects more difficult.

For AML and the method of Hoeschele

_(1988),

the mean correlations _ra,a were

lower and

r- o-

were

_higher

when the allele

_frequency

was 0.5

_(parameter

set

₂₎

than when the same allele had a

frequency

of 0.25

(parameter

set

_{1) (tables}

III and

_V).

_Although

the

_proportion

of variance

_{explained by}

the

_major

locus is

_higher

with

_parameter

set 2 it seems to be more difficult to _separate

_polygenic

and

_major

locus effects with intermediate allele

_frequencies.

This was also found

by

Knott et

al

_(1992a)

for similar

_{approximations.}

For

_parameter

sets 1 and

2,

both methods showed a

large

reduction of 35 to

40%

_{for ra,a} and 25 to

32%

for

7

-

T

g,

Q

g

when

genotypes

were unknown rather than known

_(tables

II and

_III).

With the method

_{of Kinghorn}

et al

_(1993),

estimates of the allele

_frequency

_{p were}

generally

closer to the true values than with the other 2

_{procedures. However, major}

locus effects were overestimated and the correlations between true and

_predicted

breeding

values were close to zero which is in

_agreement

with their simulation results. The method

_attempts

to correct for the bias inherent in

_major

locus estimates

_by

_regression

on the

independent

variable

ZQ!,

an estimate from the

data,

which is associated with some error. The term

_ZQ!

is

_{postmultiplied}

_by

the correction matrix W!.

_ZQ’’W

_r

is then used the same _way as a usual incidence

matrix in the mixed model

_equations.

_{Multiplication by}

wr increases the variance of the

_independent

variable to the variance

_expected

for the unknown term ZT. Because wr is calculated over all animals with

records,

the new variance is correct

only

on the _average. For an animal with known

_genotype,

the elements in

_Q!

are

closer to an

identity

matrix in

_comparison

to dams. In

_{addition, breeding}

values

estimated

_by

_[15]

are still biased. These 2

_problems

are

probably responsible

for

the overestimation of g and very poor

prediction

of

_{polygenic breeding}

values. The

_performance

of the method was,

however,

less affected

by

smaller allele effects

(parameter

set

₃₎

than the other 2

_procedures.

For all 3

_procedures

there was a

problem

of different solutions with different

starting

values when

_genotypes

were unknown. For AML and the method of

Hoeschele

₍₁₉₈₈₎

the cause could be the

multimodality

of the likelihood function.

It seems to be necessary to

_compute

_approximated

likelihoods which then can be

used to select the solutions with the

_highest

likelihood. This could of course also

be done with the method of

_Kinghorn

et al

₍₁₉₉₃₎

but this method has no direct

relationship

with maximum likelihood.

In this

_study

variance

_components

were assumed to be known but in

_practice

have to be estimated.

_Using

incorrect values could lead to biased estimates of

_major

genotype

effects and

_frequencies.

For

_{example, using}

an underestimated

_genetic

variance

_might

result in an overestimation of the

_major

_genotype

effects. If a

_major

allele is known to be

_segregating

variance _{components free of}

_major

_genotype

effects would have to be estimated with model

_!1!.

This could be _very difficult because even

when the true variance

_components

were

used,

all 3 methods

performed poorly

when

no animals were

_genotyped.

Clearly,

none of the methods is

satisfactory

for a

_separate

_genetic

evaluation for

the

_major

locus and the

_polygenes.

In this

_{study only}

_large

effects were considered.

AML

and,

especially,

the method of Hoeschele

₍₁₉₈₈₎

were unable to detect smaller effects than used with

_parameter

set 3. For

_example,

the effects estimated for the

prolactin

locus in a Holstein sire

_family

_(Cowan

et

_al,

₁₉₉₀₎

were much smaller

than considered here. The method

proposed

has some

potential

for

_improvement.

Future research should focus on the

_development

of

_algorithms

to estimate

_genotype

probabilities

without _{any restriction} on

_pedigree

structures. The estimation of

joint

genotype

probabilities

for _{any 2}

_pairs

of animals

_together

with sparse matrix

techniques

to

_compute

the elements of M could avoid the need for some of the

approximations

made in this

_study.

ACKNOWLEDGMENT

This research was conducted while AH was a

_visiting

scientist at the

_University

of

Guelph.

Financial _support from the Schweizerischer

_{Nationalfonds,}

_Switzerland,

is

_gratefully

acknowledged.

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