Marker assisted estimation of breeding values when marker information is missing on many animals

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Marker assisted estimation of breeding values when

marker information is missing on many animals

Theo H.E. Meuwissen, Mike E. Goddard

To cite this version:

Original

article

Marker assisted estimation of

_breeding

values when marker

information

is

_missing

on

_many

animals

Theo

H.E. Meuwissen

Mike E.

Goddard

Research Institute of Animal Science and

Health,

Box 65,

8200 AB

_Lelystad,

the Netherlands

b

Institute of Land and Food _{Resources, University} of Melbourne,

Parkville,

Victoria _3052, Australia

(Received

15 December

_1998;

_accepted

4 June

₁₉₉₉₎

Abstract - Two methods are

_presented

that use information from a

_{large population}

of commercial

animals,

which have not been

genotyped

for

genetic

markers, to

calculate marker assisted estimates of

_breeding

value

_(MA-EBV)

for nucleus animals,

where the commercial animals are descendants of the marker

_genotyped

nucleus

animals. The first method reduced the number of mixed model equations per

commercial animal to

_{one, instead}

of one

_plus

twice the number of marked

_quantitative

trait loci in conventional MA-EBV equations. Without this reduction, the time taken

to solve the mixed model _equations

_including

markers could be very

large especially

if the number of commercial animals and the number of markers is

_large.

The solutions of the reduced set of _equations were exact and did not

_require

more iterations than the conventional set of

equations.

A second method was

_developed

for the situation where the records of the commercial animals were not

_directly

available to the

nucleus

_breeding

_programme but conventional non-MA-EBVs and their accuracies

were available for nucleus animals from a

_large

scale

(e.g. national)

breeding

value

evaluation,

which uses nucleus and commercial information.

_Using

these

non-MA-EBV,

the MA-EBV of the nucleus animals were

approximated.

In an

_example,

the

approximated

MA-EBV were _very close to the exact

_{MA-EBV. ©}

_{Inra/Elsevier,}

Paris

marker assisted selection

_/

breeding value estimation

_/

quantitative trait loci /

DNA markers

Résumé - Évaluation _{génétique assistée par marqueurs quand} l’information sur

les marqueurs est rare. On

présente

deux méthodes d’utilisation de l’information

provenant d’une

grande population

d’animaux commerciaux, non

_typés

_pour des

marqueurs, en vue de l’évaluation

génétique

d’animaux

_typés

dans les _noyaux de

*

sélection qui sont à

_l’origine

des _populations commerciales. La

_première

méthode limite à une seule

équation

du modèle mixte pour

chaque

animal commercial au lieu de une

_plus

deux

fois,

le nombre de loci

marqués, quand

on utilise les

équations classiques

du BLUP assisté _{par marqueurs.} Ceci permet de réduire substantiellement le temps

de calcul

_quand

le nombre d’animaux commerciaux et le nombre de marqueurs sont

grands.

Les solutions de ce

_système

réduit sont exactes et ne demandent _pas

_plus

d’itérations que le

système classique d’équations.

La seconde méthode est

_proposée

quand

les données des animaux commerciaux ne _{sont pas} directement accessibles aux sélectionneurs du _noyau de sélection alors _que leurs évaluations

_classiques

(non

assistées par

marqueurs)

le sont. Ces évaluations tiennent alors _compte des données des animaux du noyau et hors noyau. Dans ce _cas, la méthode est

_approchée.

Sur un

exemple,

cette

_{approximation}

a été trouvée très

_proche

de l’évaluation exacte assistée

par marqueurs.

©

Inra/Elsevier,

Paris

sélection _{assistée par}

_marqueurs

_{/ évaluation génétique}

_{/ loci}

à caractères

quantitatifs

/ marqueur

à ADN

1. INTRODUCTION

Fernando and Grossman

_[3]

_presented

a method to calculate the best linear unbiased

_{predicted-estimates}

of

_breeding

values

_(BLUP-EBV)

_using

the information that DNA markers are linked to a

_quantitative

trait locus

_((aTL).

Goddard

_[4]

extended the method to the use of

_flanking

marker information.

Although,

these methods are

relatively

_{easy to use,} the number of

_equations

rapidly

becomes

_large

when there are _many animals. Even with

_only

one marked

QTL,

there are three

_equations

_per animal: two

_estimating

both

_gametic

effects at the

_QTL

and one for the

_polygenic

effect

(the

_joint

effect of the

background

genes).

Every

extra marked

_QTL

increases the number of

_equations

per animal

by

two.

_Moreover,

when the

_flanking

markers are close to the

QTL,

the

_{probabilities}

of double cross-overs become small and the

_equations

close to

_singular,

and thus difficult to solve

_[13].

Meuwissen and Goddard

[8]

avoided these

_{singularity problems by assuming}

a

negligible probability

of

double recombinations within the

_flanking

markers.

As

_genetic

markers become more

_frequently

used in comnrercial

_breeding

programmes, the situation will

commonly

arise where

_only

a small fraction of

the animals have been

_genotyped.

The

_phenotypes

of

_{non-genotyped}

animals

may,

however,

be vital to the calculation of the effects of marked

QTL

as,

for

_instance,

in a

granddaughter design

where

only

bulls are

genotyped

but

only

cows are

_phenotyped.

Calculation of two

_QTL

effects for each marker for

many

non-genotyped

animals is wasteful and may inhibit the

implementation

of marker assisted selection. Hoeschele

_[7]

_greatly

reduced the number of equa-tions _{in very}

_general

_population

_structures,

but this method is

_complicated

and therefore difficult to

_apply

in

_practice,

_mainly

because it eliminates as

many

equations

as

_possible.

A more

_{simple breeding}

structure such as a

geno-typed

nucleus and

_{non-genotyped}

commercial

_population

structure can

greatly

simplify

the elimination of

_equations.

In some situations the

_organisation

con-trolling

the nucleus

_breeding

_{programme may} not have access to the records

The aim of this _{paper is} to

_present

a method that reduces the number of

marker assisted

_breeding

value estimation

_equations

in a

_population

where the

nucleus animals are marker

_genotyped

and the commercial animals are not

genotyped.

The reduction

_mainly

eliminates the

_equations

of

_{non-genotyped}

animals.

_Furthermore,

an

_approximate

method of

_calculating

MA-EBVs on

nucleus animals is

_presented,

which uses

only

the conventional non-MA-EBVs

of nucleus animals from a national

_genetic

evaluation to

_represent

the data from commercial animals.

2. METHODS

2.1.

_Reducing

the number of

_equations

The

_population

was

split

into nucleus and commercial animals.

Here,

the

definition of a commercial animal is: an animal that is not marker

_genotyped

and has no descendants that are

genotyped.

The nucleus animals are all

marker

_genotyped

animals

_plus

their ancestors. The method will still work if a commercial animal is

_erroneously

considered as a nucleus

_animal,

_although

the number of

_equations

will not be reduced for such an animal. The method

will

_{fail, however,}

if a nucleus animal is

_erroneously

considered as a commercial

animal. For

_simplicity

we

ignored

fixed effect

_equations,

but

including

them is

straightforward. Similarly,

we assumed here

_only

one marked

_QTL,

since the

inclusion of more marked

_QTL

is

_{straightforward.}

_Partitioning

the

_population

into nucleus and commercial

animals,

the model can be written as:

where yi

(y

2

)

is the vector of

_phenotypic

records of nucleus

_(commercial)

animals;

al

(a

2

)

is the vector of

_polygenic

effects of nucleus

_(commercial)

animals;

ql is the vector of marked

_QTL

effects of the nucleus

animals;

q2

(q

3

)

is the vector of

_paternally

_(maternally)

derived

_QTL

effects of the commercial

animals;

el

(e

2

)

is the vector of environmental effects of nucleus

_(commercial)

animals;

Z

l

is the incidence matrix of

polygenic

effects of nucleus

animals;

Z

2

is the incidence matrix of

_QTL

effects of the nucleus

_animals;

and

_Z

₃

is the incidence matrix of

_polygenic

effects of the commercial animals. Note that

_Z

₃

is also used as the incidence matrix of the

paternally

and of the

maternally

derived

_QTL

effects of the commercial

_animals,

because these effects have the

same incidence matrix as the

_polygenic

effects of the commercial animals. The

Z

2

matrix can differ

substantially

from

_Z

_l

when the inheritance of

QTL

effects

is traced from

_parent

to

_{offspring by}

the markers

_[8].

In order to solve the BLUP

_equations,

we need the inverses of the

(co)variance

matrix of

[a’ a’]

and of

_[q’

_q’

_q’],

which are obtained

_using

the methods of

_Quaas

[10,

11!

and

Fernando and Grossman

_!3!,

_{respectively.}

In order to reduce the number of

_equations

of the commercial

_animals,

the ’reduced animal model’

_approach

of

_Quaas

and Pollak

_[12]

was

_adopted.

This

the

_equations

of

non-parents

in a model with

QTL

and

_polygenic

effects. We

re-write

_equation

₍₁₎

as:

where U2 ! az + qz + q3. For the mixed model

equations

that follow from

equations

(2),

we need the inverse of the

_(co)variance

matrix of

_{[a’ q!}

_{1 U/ 2}

_1.

Following Quaas

[10, 11!,

we will assume that the animals within the nucleus

and within the commercial are sorted from old to _young.

_Next,

we _{write every}

element of

_[at

_{1 qf}

_{1 uf 2 in}

terms of its

_{’parental’}

elements

_plus

an

_independent

deviation from the

’parental’ elements,

where

’parental’

elements denote the

ai, ql or U2 elements of the

parents

of the current animal:

where P is an indicator matrix of the

_parents

of _ai, such that

_P

_ij

= 0.5 if

animal j

is a

_parent

of animal

_i,

and otherwise

_P

_ij

=

0;

Q2! = B2!

if

QTL,

is with

probability

O

ij

a direct _copy of

QTL,,

where

_QTL

_J

was one of the two

_{’parental’}

QTL

alleles of

_QTL,,

with

_{’parental’ denoting}

that

_QTL

_j

was involved in the

Mendelian

_sampling

process that resulted in

QTL

I

,

and for all other i and

j:

Qij

=

_0;

Rij

= 0.5 if nucleus

_{animal j is}

_{a parent} of commercial animal

_i,

and

otherwise

_Ri!

=

0;

Si!

= 0.5 if one of the two

_QTL

of commercial animal i is a

direct copy of the nucleus

gamete j

with a

probability

of 0.5

(the

probability

is

always

0.5 because commercial animals are not marker

_genotyped),

otherwise

S

j

=

0;

T

ij

= 0.5 when commercial

animal j

is a

_parent

of

_i,

otherwise

_T

_ij

= 0.

The elements

_{of E}

_l

_,

E2 and E3 are all

_independent,

unless the markers are not

completely

informative,

i.e. it is not

_{always possible}

to trace which marker is

inherited from the sire and which from the dam. In the latter case, the elements

of E2 may be correlated and the method of

Wang

et al.

_[14]

can be used to set up

(the

inverse

_of)

the

_(co)variance

matrix of the

_QTL

effects of the nucleus animals. The calculation of the

_(co)variance

matrix of the

QTL

effects of the nucleus animals becomes even more

complex

when ancestors of nucleus animals have

_missing

marker

_genotypes;

_however,

for this

_situation,

_Wang

et al.

_provide

an

_approximate

method _{to set up} the

_(co)variance

matrix of

_QTL

effects. We

will

_ignore

these

_{complications}

of

_obtaining

the inverse of the

_(co)variance

matrix of the

_QTL

effects of the nucleus animals

_here,

because the method that is used to obtain the inverse of this

_(co)variance

matrix does not affect the

_setting

_up of the inverse of the

_(co)variance

matrix of the u2

equations.

This is because the situation of uninformative marker information and

_ungenotyped

ancestors of

_genotyped

animals did not occur within the _group of commercial

animals,

since none of the commercial animals were

_genotyped.

where

D

l

is a

_diagonal

matrix with

_D

_iii

_equal

to

_Q

_{a, 0.75}

_Q

_a

_or

_0.5

_Q

_a

_when

no, one or both

_parents

are known of nucleus animal

_i,

respectively; D

2

is a

diagonal

with

_D

_2ii

_equal

to

_{a) or}

_{2Bi!(1 -}

_{0g )a) when}

_gamete

i is a founder

gamete

or is derived from

_{gamete j}

_with

_probability

_Bi!

_!3!,

_{respectively;}

and

D

3

is a

_diagonal

with

_D

_3ii

_equal

to

_Q

_{u, 0.75}

_Q

_u

or

0.5u!,

when no, one or both

parents

of commercial animal i are

_{known, respectively,}

where

₀

_’

=

_a2

₊

₂

_Q

_q.

Next we solve

_equation

(3)

for v’ =

[a’ q’ u’]

to obtain:

Taking

variances on both sides

_yields,

Finally

the inverse of

_Var(v)

is

G-

1

which is obtained as:

Similar to

_Quaas

_{(10, 11!,}

the

following

rules can be found to set _up

G-

1

.

1)

For the

_polygenic

effects of the nucleus animals

_part

of

G-

1

:

follow

_Quaas’

rules

_(multiply

_by

_I/or2

_to

account for the different variances in different

parts

of

G-1).

2)

For the

_QTL

effects of the nucleus animals

part

of

G-

1

:

follow the rules of Fernando and Grossman

_[3]

_(multiply

_by

_1/

_Q9

_).

3)

For the

_{genetic effects,}

u2, of commercial animal i:

- if both

parents

are unknown: add

1/

Q

u

to

_position

(i, i);

-

if one

_parent

s is known with

QTL

alleles al and a2 add to the indicated

If there are no

_equations

for the

QTL

alleles al and a2, i.e. s was a commercial

animal,

the additions to their

_positions

are

cancelled,

and the additions

simplify

to the

_original

rule of

_Quaas

_{[10, 11!;}

- if both

parent

s and d of animal i are known with

_QTL

alleles al and a2

of s and alleles a3 and a4 of

d,

add to the indicated

_positions:

If there are no

_equations

for the

_QTL

alleles _{ai, , a2 a3}

and/or

a4 the

additions to their

_positions

are cancelled. When all alleles _{ai, a2, a3}

and/or

a

4 have no

_equations,

the additions

simplify

to the

original

rule of

Quaas

[10, 11].

As can be seen from the above

additions,

the commercial animals add the

same values as in

_Quaas’

rules to the elements of their

_parents,

but if the

parents

are nucleus animals these values are added to their

_polygenic

and

_QTL

effects.

After

_setting

up the

G-

1

matrix,

we can _{set up} and solve Henderson’s

[6]

mixed model

_equations:

and

_Q

_e

is the environmental variance.

These

_equations

will

_yield

exact solutions of the estimates of

_polygenic

_(a

_l

₎

and

_QTL

effects

_(q

_l

₎

of the nucleus

_animals,

and of the sum of the

_polygenic

and

QTL

effects of the commercial animals

_(u

₂

₎

_(unless

approximations

have

to be

_applied

for

_setting

_up the

_(co)variance

matrix of the

_QTL

effects of the nucleus animals

_owing

to

_missing

marker

_genotypes

of ancestors of nucleus

animals).

A small

_example

of the calculation of the

G-

1

matrix is

_given

in

Appendix

A.

2.2. The use of conventional EBV to

predict

MA-EBV

In the case of cattle

_breeding

schemes

_especially,

the commercial animals

may not be owned

_by

the

_breeding

_organisation

and this

_organisation

_may not

national

_breeding

value evaluation. We would like to use this information to

improve

the _accuracy of the marker assisted

_breeding

value estimates in the nucleus. This

_problem

is similar to that of

incorporating

AI sire evaluations into intraherd

_breeding

value

_{predictions by}

Henderson

_(5!,

and our

approach

will therefore also be similar to that of Henderson.

The first _step is to absorb the commercial animal

_equations

into the nucleus

equations,

which will reveal which information from the commercial animals is needed. The full mixed model

_equations

are

_[writing

out

_equations

₍₈₎

and

_(5)]

see

(8bis)

in the

_following

_page.

Absorption

of the commercial animal

equations

(u

z

)

yields equation

(9),

shown in the

_following

_page, where B =

D-’ - D3l(I -

T)(Z3Z

3

+

(I

-T

/

)D3

l

(I -

T)]-l(I - T/)D3l,

and b =

D3

l

(I -

T)(Z3Z

3

₊

(I -

T

/

)D3

l

(I-T)]!Zgy2.

Note that

_equation

₍₉₎

reduces to the MA-EBV

_equations

of the nucleus animals without

_accounting

for _any information of commercial

_animals,

if B and b are set to zero. The term R’BR leads to additions to the

_equations

of the nucleus

_parents

of the commercial animals.

Similarly,

S’BS leads to

additions to the

_equations

of the

_QTL

that are carried

_by

the nucleus

_parents

of the commercial animals.

Further,

R’BS leads to additions to the animal *

QTL

block of the

_equation

₍₉₎

of the nucleus

_parents

_(of

commercial

_animals)

and their

_QTL

effects. The terms R’b and S’b result in additions to the

_right

hand side of the

equations pertaining

to the

_parents

of nucleus animals and their

_{QTL effects, respectively.}

We will

_approximate

these terms

_{R’BR, S’BS,}

R’BS,

R’b and S’b

_using

the results from a conventional national evaluation

of

_breeding

values.

The solutions of EBV of nucleus animals of the conventional national evaluation should

equal

the solutions from the

_equations

of the nucleus animals after

_absorption

of the commercial animals. The conventional

_equations

for nucleus animals after

_absorption

of commercial animals are:

where EBV is a vector of conventional EBV of nucleus animals

_(known

from national

_evaluation),

M =

[Z’Z

l

₊

(I -

P)’D-’(1 -

P)!e u!/u!],

which

is the coefficient matrix of the conventional mixed model

_equations

when

only

information from nucleus animals is used

_(note

that

_{(I -}

_P)

_/

_{D1l (I}

-P)

l

a§

equals

the inverse of the

_relationship

matrix of the nucleus

_animals).

Note also that the additions R’BR and R’b are the same as those in the MA-EBV

_equation

_(9).

Hence,

if we obtain

_{approximations}

for R’BR and

R’b

in

_equation

₍₁₀₎

we can

_{approximate equation}

(9).

We know the EBV

and their

accuracies,

ri, which result from

equation

(10).

Let the matrix

C =

(M

₊

R’BR)-

1

,

then the

diagonal

elements of C are:

where

_only

the

_diagonal

elements of C are known. The

_diagonal

elements of

A,

!ii, yield

the effective number of records that should be added to a nucleus animal

_i,

such that the _accuracy of its EBV is

_equal

to the _accuracy when the commercial animals were included. A similar effective number of records was

derived

_by

Henderson

_!5!,

but in his situation the animals within the herd did

not contribute

_{significantly}

to the EBV of the sire.

_Here,

we used the

following

iteration scheme to

_disentangle

the information that came from the nucleus

animals,

which is

_{represented by}

the matrix

_M,

and the information that comes

from the commercial

_animals,

which is

_{represented by}

the matrix A.

Newton’s iteration

_algorithm

was used to calculate the

_diagonal

matrix

A such that

_diag((M

+

_0)-

₁

₎

=

diag(C),

where

diag(X)

denotes a vector

containing

the

_diagonal

elements of the matrix X. Let the vector 6 =

diag(A).

The iteration scheme estimates b

_by:

step

1: a first

_{approximation}

_A

_[O]

_or,

_{equivalently,}

₆

_[o]

is obtained from:

step

2:

_improve

6

_by

_{Newton-Raphson}

iteration:

where

_[p]

denotes the

_pth

_iteration;

and H is a matrix of derivatives of

_diag((M

+

_D)-’)

with

_respect

to

_b,

which can be shown to

_equal

- (M

+

_A)-’

*

(M

+

_A)-’,

where * denotes element

by

element

multiplica-tion.

Given the

_approximated

mixed model coefficient matrix of the nucleus animals after

_absorption

of the commercial

_animals,

M

_{+ A,}

an

_{approximation}

of the

right

hand side of

equation

(10),

is obtained from:

where ARHS is an

_{approximation}

of the term R’b in

_equation

_(10).

_Since,

EBV and

_Ziy

_l

are

_known,

ARHS can be calculated from the above

_equation.

Next we will calculate the absorbed coefficient matrix of the marker assisted mixed model

_equation

_(9),

and their

_right

hand side. From the

_previous

section

we concluded that we could

_approximate

_R’BR

_I

_{by D}

_ii

_,

where

_R

_i

is the ith column of R. The vector

_R

_i

indicates which commercial animal is an

offspring

of nucleus animal

_{i by containing}

a

1/2

if the commercial animal is an

_offspring

of i or a 0 otherwise. If _al is one of the

_QTL

alleles of nucleus animal

_i,

the

a

l

th

column of

S, S

al

,

contains a

1/2

if the commercial animal is an

offspring

of

animal i. If _every nucleus animal has two

_unique

_QTL

_alleles,

as in the model of Fernando and Grossman

_!3!,

it follows that

_R

_i

=

S

al

=

S

a2

,

with _al and _a2

denoting

the

QTL

alleles of animal i. Hence:

where ax denotes al or . 2a

Thus,

the addition

D.ti

to the

_diagonal

of the

polygenic

equation

of the nucleus animal i should also be added to the

off-diagonal

of the

_{polygenic equation}

i and

_QTL

allele

_equation

al and a2; to the

diagonal

of both

_{QTL equations}

_al and _a2; and to the

_off-diagonal

elements of

a

l and a2. And the term

ARHS

I

should be added to the

_right

hand side of the

_equation

of animal

_i,

and of the

_QTL

_equations

al and a2. In

conclusion,

to account for the information of commercial

_animals,

for _every nucleus animal

i we add to the coefficient matrix of the MA

_equations

of the nucleus animals that

_ignores

information of commercial animals:

where al and a2 denote the

equations

for the

_QTL

effects of animal

_i;

and we

add to the

_right

hand side of these nucleus

_equations

for _every nucleus animal i:

Thus,

the additions

₍₁₁₎

and

₍₁₂₎

result in an

_{approximation}

of the marker assisted nucleus

_equations

₍₉₎

_{using only}

the EBV and accuracies to account

for the information of commercial animals.

The

_equality

of

R

i

to

_S

_a£

_requires

that the

_QTL

allele _a!. is

_only

_present

in one animal i.

_However,

in the model of Meuwissen and Goddard

_!8!,

_QTL

alleles

_might

be traced from

parent

to

_offspring

with

_certainty,

because

_flanking

markers were used and double recombinations were

_ignored.

In this model different animals _{may carry} the same

_QTL

allele a!, and

S

ax

=

Ei,

7

Ax

Ri,

where the summation is over all animals i that _carry

_QTL

allele ax. This

complication

of

S

ax

being

the sum of several

_R

_i

terms does not affect the additions in

_equations

₍₁₁₎

and

₍₁₂₎

which are due to terms that are linear in

_S

_ax

_,

because the correct additions are still

_performed

as all the animals

contributing

to

_S

_ax

are evaluated.

_However,

the additions to the

_QTL

allele

*

_QTL

allele block of

_equation

_(11),

are due to second order terms of

_S

_ax

_,

which

_implies

that more off

_diagonal

terms of the

_absorption

matrix B have

to be added. We will

_ignore

these extra off

_diagonal

terms of

_B,

which are due

to the second order terms of

_Sa,!,

and

_perform

the additions as described in

equation

(12),

which adds another level of

_{approximation}

to this method.

In the

_above,

the fixed effect structure of the nucleus animal data was

ignored,

but can be accounted for

_{by absorbing}

the fixed effect

_equations

into the

_equations

of the nucleus

_animals,

i.e. the matrix M would be the conventional mixed model coefficient matrix after

_absorption

of fixed effects.

Alternatively,

if

_absorption

of fixed effects is

_{computationally}

too

_demanding,

the

_following

_steps

can be undertaken to account for fixed effects:

(M

+

_{![1’]) -1}

is

_{replaced by}

the animal * _{animal block of:}

where X is the

_design

matrix of the fixed effect structure of the nucleus

_data;

step

2: if the fixed effect solutions are not available from the national

_breeding

value

_evaluation,

solve for the fixed effect

_{solutions, /3, using:}

step

3: calculate ARHS

_using:

The above methods that account for fixed effects assume that different fixed

effects are estimated in the nucleus than in the commercial

animals,

which will be the case in most situations. A brief

_example

of the use of non-MA-EBV in

the estimation of MA-EBV of nucleus animals is

_given

in

_Appendix

A.

2.3. Simulation

A data set was simulated to test whether the reduced number of marker

as-sisted mixed model

_equations

indeed

_yielded

the same solutions as the

_original

full set of

_equations,

to _compare the number of iterations needed to solve the reduced set of

_equations

and the

_{original equations}

_(which

_might

be more

diag-onally

dominant),

and to test the

_{approximate absorption}

of all

_equations

for commercial animals. The data set resulted from five

_generations

of simulation of a nucleus and a female commercial

_population,

where the nucleus animals are

selected on conventional

BLUP-EBV,

and the unselected commercial females are mated to the selected sires of the nucleus. The

_parameters

of the simulated data set are

_presented

in table I. MA-EBVs were calculated in a manner similar

to that of Meuwissen and Goddard

_[8]

in which it is assumed that if markers

cannot trace the inheritance of

_QTL

alleles from

_parent

to

_offspring,

then the

QTL

allele inherited is treated as

equally likely

to be either of the two alleles in

the

_parent

_(the

_possibility

that a

_segregation

_analysis

of the marker data

_might

improve

this

_prediction,

was

ignored).

The

_probability

that the markers could

not trace the inheritance of the

_QTL

was assumed to be

_0.1,

which occurred in

158 instances in the nucleus. In these instances a new

_QTL

effect was

postu-lated and estimated.

_Including

the 400 founder

_QTL

effects

₍₌

2 * 200 founder nucleus

_animals),

the number of

_QTL

effects of nucleus animals was 558. The

commercial animals were not marker

_genotyped

and so no

QTL

effects could be

traced. If no

_equations

were

_eliminated,

this would result in 10 000

_{(= 2 *}

5 000

commercial

_animals)

commercial

_QTL

effects.

The

_equations

were solved

by

Gauss-Seidel iteration. The _convergence

where SS is the sum of _squares of the deviations of the left hand side from the

right

hand side of the

_equations;

SST is the sum of _squares of the solutions. The

number of iterations needed to reach this convergence criterion is a

(imperfect)

measure of how

easily

the

equations

could be solved. This measure is not

_perfect

because the solution vectors of both methods are not the same, and SS may be

3. RESULTS

Table II shows the results of the EBV calculations. Without any reduction in

the number of

equations,

the total number of

equations

is: 16 558

_{(6 000}

animal and 10 558

_QTL

_effects).

When the

QTL equations

of the commercial animals

were

eliminated,

the number of

equations

was reduced

_by

10 000. In

practice,

this

_figure

will often be much

_larger,

because the commercial

_population

will be much

_larger

than in the simulation.

_Furthermore,

the number of iterations that

was needed to reach the _convergence

_criterion,

was smaller with the reduced

set of

_equations.

This

_suggests

that the reduced set of

_equations

was not _any

harder to solve than the

_{original large}

set of

_equations.

The solutions to both

sets of

_equations

were

_virtually

identical

(result

not

_shown),

_except

that the reduced set did not

_yield

estimates of individual

_QTL

effects of commercial animals. If the estimates of the

_QTL

effects of the commercial animals were

required,

they

could be obtained

_by

back

_solving

_(see

Appendix

B).

Table III

shows the results of the

_approximate

method

_using

conventional EBV of nucleus animals to estimate the MA-EBV of the nucleus animals. When we want to

predict

the total

breeding

value of the

animals,

ui, the use of conventional EBV

of nucleus

_animals,

instead of the

_phenotypes

of commercial

_animals,

leads to

very accurate

_predictions

of MA-EBV in the simulated data set. The

_predictions

of the individual

QTL effects,

qi, were also

_accurate,

i.e. the correlation and

regression

is close to 1. In this

_approximate

method the number of

equations

was

only

1 558.

4. DISCUSSION

4.1. Reduction in the number of

equations

A

_simple

method was

_presented

that reduced the number of

equations

of non-marker

_genotyped

commercial animals from three to one, where the latter

_equation

estimates the total

_breeding

value

_(polygenic

_plus

_QTL)

of the commercial animals. The reduction in the number of

_equations

was

_large

when

the number of commercial animals was

_large,

and the reduced set of

_equations

An alternative to this method is the use of a reduced animal model

_{[1, 2!,}

which would eliminate

_equations

for commercial animals

_only

if

_they

are

not

_parents.

Thus the method

_presented

eliminates more

_equations

than the reduced animal model

_approach

but less than Hoeschele

_(7!.

The

_importance

of

_using

the data on commercial animals when

_calculating

MA-EBVs for the nucleus animals is illustrated

_by

the case of a

_{granddaughter design}

where all

phenotypic

data come from the commercial

_{granddaughters.}

The method

proposed

could be used in a national

_genetic

evaluation where the number of additional

_equations

would be

_proportional

to the size of the nucleus.

Extension of the method to more marked

_QTL

is

_{straightforward.}

Let the

parental QTL

alleles of the first marked

_QTL

be denoted

_by

ai, az

(a

3

,

a

4

),

and those of the second marked

_{QTL by b}

_l

_,

_b

₂

_(b

₃

_,

_b

₄

_),

where the elements between the brackets are needed when a second nucleus

_parent

is known. Now

extra rows and columns for

_b

_l

_,

_b

₂

_,

₃

_(b

_b

₄

₎

are

_augmented

to the additions in

equations

(6)

and

_(7),

where the values in the

_augmented

rows and columns are the same as those in the rows and columns of _{, ia a2}

(a

3

,

_a

₄

_).

_Also,

the off

_diagonal

elements between

_aj

and bh are the same as those between _al

and _a2 for

_{all j}

and k.

_Hence,

with n marked

_QTL,

the number of

_equations

of commercial animals reduces from 2n, + 1 to

_just

one. _{In many} marker assisted selection

_schemes,

there _may also be _many nucleus animals that are

not

_genotyped,

_namely

the old ancestors of the nucleus that were born before marker

_genotyping

started. Some

_{cryo-conserved}

semen of old sires _may still

be available for

_genotyping,

but _many old ancestors will remain non-marker

genotyped.

In

_situations,

where the old ancestors result in a

_{computationally}

unmanageable

number of

_equations,

the

_approach

of Hoeschele

_[7]

eliminates all

QTL

equations

of

_{non-genotyped}

ancestors that are not on a

_genetic

_pathway

between two marker

_genotyped

animals.

_Although

more difficult to

_apply,

this method can result in a substantial reduction in the number of

_QTL

_equations

of

non-genotyped

ancestors. In the

present

simulated data

_set,

all nucleus animals

were

_genotyped

and a further reduction in the number of

_equations

was not

obtained

_{by using}

the

_approach

of Hoeschele

_(7!.

The rules

_presented

for

_setting

_up the

G-

1

matrix,

did not account for the

inbreeding

of the animals. If

_inbreeding

is not

_{negligible, Quaas’}

_{[10, 11!}

rules

can be used to account for

_inbreeding

in the nucleus animal * _{nucleus animal}

part

of the

G-

1

matrix,

which results in

_reducing

the elements of the

_D

₁

matrix

by

a fraction

_equal

to the average

inbreeding

coefficient of the

_parents.

_Also,

Wang’s

[14]

method accounts for

_inbreeding,

where the

_inbreeding

coefficient

is calculated at the

_{QTL given}

the marker information. In the

_equations

of the commercial

animals,

the _average

_inbreeding

coefficients can be accounted for

_by

reducing

the

_o,2

_term

in additions

₍₆₎

and

₍₇₎

_by

a fraction

equal

to the _average

inbreeding

coefficients of the

_parents.

The latter will be

_slightly

biased because the _average

_inbreeding

coefficient of the

_parents

will be different at the

QTL.

This bias can be corrected

_{by using}

a

weighted

_average

_{inbreeding coefficient,}

where the conventional

_inbreeding

coefficient

_averaged

over the

_parents,

the

inbreeding

at the

QTL

of the

_sire,

and that at the

QTL

of the

dam,

are

_weighed

4.2. Use of conventional EBV to

_predict

MA-EBV

A method was

_developed

that uses the information of conventional EBV of

nucleus animals and their accuracies instead of the data on commercial animals

to increase the accuracy of MA-EBV of nucleus animals. In the simulated data

set,

the

_approximate

_MA-EBV,

which used conventional

_EBVs,

were very close to the

_original

MA-EBV based on the full set of

_equations

which used the

_phenotypic

records from the commercial animals. The method was

also tested in a

_{granddaughter design}

_!15!,

where a

_{genotyped grand}

sire has

genotyped

sons which have conventional EBVs based on

_daughter

records. In

this

_situation,

the

_prediction

of the

_QTL

effects of the sons and the

_grand

sires was identical to when the

_original

records of the

_daughters

had been used

_(result

not

_shown).

This method could be used

_by

a

_breeding

_organisation

controlling

the nucleus

_breeding

_programme without

_including

_any information

on commercial animals. The method can be

_compared

to the use of

_daughter

yield

deviations to represent data on the

(commercial)

_daughters

of a nucleus

bull.

_However,

this method uses all commercial descendents of a nuclear animal

(via

its conventional

_EBV)

and avoids double

_counting

of the information from descendents in the nucleus. The method could be extended for

QTL

detection studies based on REML estimation of the variance due to a

_QTL,

bracketed

by

the markers.

The

_approximate

method to

_incorporate

EBVs from the commercial animals

into the nucleus MA-EBV is similar to the use of

_foreign

EBV in the national evaluation of a

_country.

_Except

that the

_foreign

EBV calculation does

_(almost)

not use local

_information,

such that the

_foreign

EBV

_{yield independent}

extra

information.

_Hence,

the _accuracy of the

_foreign

EBV can be

_directly

converted into an effective number of records

_{(or daughters)}

that is added to the

_diagonals

of the coefficient

_matrix,

and into

_{deregressed proofs,}

i.e. extra records

_(or

daughters)

are invented that contain the information of the

_foreign

EBV.

Here,

the situation was more

_complicated

because the conventional EBV of the nucleus animal

_already

contained the information of the commercial

_animals,

which made it more difficult to determine the extra information.

The calculation of the MA-EBV

_using

conventional EBV of commercial animals relies

_strongly

on the _accuracy of the conventional EBV. In the

_present

simulation

_study,

the accuracies were calculated

_by

inversion of the conventional mixed model

_matrix,

i.e. exact accuracies were used. In the case of a national

evaluation of

_EBV,

the number of

_equations

is too

_large

for direct inversion and the accuracies have been

_{approximated.}

These

_{approximations}

are often

_good

(e.g. !9!).

However,

poor

approximations

of the accuracies of the conventional

EBV will

_probably

reduce the accuracies of the MA-EBV

_{substantially.}

In _any

case, the method

presented

here seems to make as much use as

_possible

of

the conventional EBV of national evaluations to

_improve

the accuracies of the

MA-EBV of the nucleus animals.

REFERENCES

[1]

Bink

_{M.C.A.M., Quaas R.L.,}

Van Arendonk

_{J.A.M., Bayesian}

estimation of

dispersion

parameters with a reduced animal model

_{including polygenic}

and

_QTL

[2]

Cantet

_R.J.C.,

Smith

_C.,

Reduced animal model for marker assisted selection

using best linear unbiased

prediction,

Genet. Sel. Evol. 23

₍₁₉₉₁₎

221-233

[3]

Fernando R.L., Grossman M., Marker-assisted selection _using best linear

un-biased

_prediction,

Genet. Sel. Evol. 21

₍₁₉₈₉₎

467-477.

[4]

Goddard

M.E.,

A mixed model for

_analyses

of data on

multiple genetic

markers, Theor.

_Appl.

Genet. 83

₍₁₉₉₂₎

878-886.

[5]

Henderson

C.R.,

Use of AI relatives in intraherd

prediction

of

breeding

values

and

_producing

abilities, J.

_Dairy

Sci. 58

₍₁₉₇₅₎

1910-1916.

[6]

Henderson

_{C.R., Applications}

of linear models in animal

breeding,

Can. Catal.

Publ. Data, Univ.

_{Guelph, Guelph,}

1984.

[7]

Hoeschele

_I.,

Elimination of _quantitative trait loci

_equations

in an animal

model

_{incorporating genetic}

marker data, J.

_Dairy

Sci. 76

₍₁₉₉₃₎

1693-1713.

[8]

Meuwissen _T.H.E., Goddard M.E., The use of marker

haplotypes

in animal

breeding

schemes, Genet. Sel. Evol. 28

₍₁₉₉₆₎

161-176.

[9]

Misztal L,

Wiggans G.R., Approximation

of

_prediction

error variances in

large-scale

animal

models,

J.

_Dairy

Sci. 71

_{(Suppl. 2) (1988)}

27-32.

[10]

Quaas

R.L.,

Computing

the

diagonal

elements of a

large

numerator

relation-ship

matrix, Biometrics 32

₍₁₉₇₆₎

949-953.

[11]

Quaas

R.L., Additive genetic model with _groups and

_{relationships,}

J.

_Dairy

Sci. 71

₍₁₉₈₈₎

1338-1345.

[12]

Quaas

R.L., Pollak E.J., Mixed model

methodology

for farm an ranch beef

cattle

_testing

programs, J. Anim. Sci. 51

(1980)

1277-1287.

[13]

Ruane J., Colleau J.J., Marker-assisted selection for a sex-limited character in a nucleus

breeding population,

J.

Dairy

Sci. 79

(1996)

1666-1678.

[14]

Wang T.,

Fernando

_R.L.,

Van der Beek

_S.,

Grossman M., Van Arendonk

J.A.M.,

Covariance between relatives for a marked quantitative trait locus, Genet.

Sel. Evol. 27

₍₁₉₉₅₎

251-274.

[15]

Weller

_J.L,

Kashi Y., Soller

M.,

Power of

daughter

and

granddaughter

de-signs

for

determining linkage

between marker loci and quantitative trait loci in

_dairy

cattle,

J.

Dairy

Sci. 73

₍₁₉₉₀₎

2525-2537.

APPENDIX A

An

_example

_illustrating

the calculation of the inverse of the

_genetic

covariance

_{matrix, G,}

for the reduced set of

_equations

The

_pedigree

and marker

_genotypes

of five animals are

_given

in table AI. The marker

_genotype

of animal 5 is

_unknown,

and this animal has no descendants

with known marker

_genotypes.

_{Hence, given}

the definition in the main

_text,

animal 5 is a commercial

_animal,

and its

_QTL

allele

_equations

will be absorbed.

Let the

_polygenic

and

QTL

allelic variance be

_equal

to _{one, i.e.}

_Q

_a

₌

_Q9

₌

_1,

and

_a

=

_o

_l2

₊

2a)

= 3. The first

_step

is to set

up the inverse of the

polygenic-nucleus animal

_part

of the

_genetic

covariance

_matrix,

_G-

₁

_,

where the nucleus animals are

_{1, 2,}

3 and 4. This

_part

of the

G-

1

matrix is obtained

by using

The second

_step

is to _{set up} the

_QTL

effects

_part

of the

G-

1

matrix. For

simplicity,

we will assume here that the recombination rate between the marker

and the

_QTL

is zero.

(This

situation is similar to the situation where

_flanking

markers are used to trace the

_{QTL alleles,}

and the double recombination rate

is assumed

_negligible.)

Because the recombination rate is zero, the inheritance

of the

_QTL

alleles is identical to that of the marker

alleles,

i.e. there are four

QTL

alleles ql, q2, q3and , q4and the presence of marker allele

A

x

also indicates

the presence of

QTL

allele q!, where x =

1, 2,

3 or 4. The

_QTL

allele

part

of the

G-

1

matrix is calculated

_using

the rules of Fernando and Grossman

_[3]

or

Wang

et al.

_!14!,

and

_yields

an

identity

matrix here because the

QTL

alleles

q l

, q2, q3 and q4 are all founder alleles. After this

_step,

the

polygenic

and

QTL

allele

_part

of the nucleus animals of the

G-

1

matrix is:

The third

step

is to add the

_equations

for the commercial animals to nucleus animal

_equations

_using

additions

₍₆₎

or

(7)

of the main text.

_Here,

_only

animal

5 is a commercial animal and it has both

_parents

known

(namely

animals 3 and

4)

such that addition

₍₇₎

_applies.

Note that

_1/

_Q

_u

_is

_1/3,

such that the

_complete

Use of conventional EBV to

_predict

MA-EBV

Let us consider

_again

the

example

of table

AI,

and make use of the

conven-tional non-MA-EBV that are calculated for all

animals,

but are

_only

available

on the nucleus animals 1-4

_together

with their accuracies. These EBVs and their accuracies are calculated

assuming

an error variance of

Q

e

₌ 3. The first

step

is to obtain a first

_{approximation}

of the extra information due to the commercial animals

by calculating

_b

_[ol

=

diag(C-

1

) -

diag(M),

where the ith

diagonal

element of

C-’

is

_{!/(1 -}

_r?),

with A =

U2/0

,2

=

_1;

and M is the

coefficient matrix of the conventional animal model

equations

for the nucleus animals:

Hence,

₆

_[o]

_{= diag(C-}

1

)-diag(M)

_

[-0.6326

-0.6326 -0.2222

_-0.2222]’.

In

_step

2 we first _{set up} the H - matrix of derivatives of

diag((M

+

!) -1 )

with

_respect

to 6:

where

_0!!!

= a

_diagonal

matrix with the elements of

_6!0!

on the

diagonals.

Next we calculate:

The

_update

of 6 is now obtained from:

After three more

_updates

of

_{6 by}

_equations

(A2)

and

(A3)

the values of 6

converged

to

_[0.03

0.03 0.36

_0.36!’,

which are

_equal

to the

_A,

in addition

_(11).

Next we set _up the marker assisted mixed model

equations

of the nucleus

of these

_equations

is obtained from

_(Al),

such that the coefficient matrix of these

_equations

is:

where

_Z

_l

is the

_design

matrix of

_{polygenic effects,}

i.e. a

(4 * 4)

_{identity matrix;}

Z

2

is the

_design

matrix of the

_{QTL effects,}

i.e.

Z

2

=

[1

1 0

0; 0 0 1 1; 1 0 1 0;

0101];

and the factor 3 is due to

_Q

_e

₌ 3.

_Using

the estimates of 6 or,

equivalently, A

ii

,

we next

_perform

the additions in

_equation

₍₁₁₎

to obtain the coefficient matrix of the mixed model

_equations

that does account for the information of commercial animal 5:

Next,

the

_polygenic

*

_polygenic

_part

_{of these}

_equations,

i.e. the

_{(1:4, 1:4)}

block,

should be

_{multiplied by}

the non-MAS-EBV

_(see

table

_AI)

to obtain the

new

_right

hand side. This new

_right

hand side deviates from the

_original

hand

side,

i.e.

_Z!y,

_by

ARHS =

[0.03 -4.03 0 4.36]’.

This ARHS is used _to

perform

additions

₍₁₂₎

to obtain the

_right

hand side of the MA-mixed model

_equations

of the nucleus animals:

The solutions from the coefficient matrix

_(A4)

and the

_right

hand side

_(A5)

are

_{[0.887 -}

0.887 0.177 0.823 - 0.759 1.20 - 0.20 -

_0.241]’,

which are the

APPENDIX B

Back-solving

for the

_QTL

effects of the commercial animals in the reduced set of

_equations

The reduced set

_of

_equations

_yields

estimates of v’ =

_[a!

_{ql u2! (see text).}

Given the estimate

V,

we can solve for the Mendelian

sampling

_components:

s =

Vv,

where V and E are defined in

_equation

_(4).

The

_equations

of the commercial animals

_yield

no information to

_separate

the Mendelian

sampling

components

of the _u2

_effects,

_E3, into the

_components

due to

_QTL

effects and due to

_polygenes.

_Hence,

the

_splitting

of these

_components

is

_proportional

to

their variance

_components,

i.e.

where

_e

₃₁

and

_e

₃₂

are the Mendelian

sampling

_components

of a2 and q2 effects

of the commercial

animals,

with _q2

_denoting

the

_QTL

effects sorted such that the

_paternal

_QTL

effect of an animal is

always

followed

by

its maternal

QTL

effect. Next the estimates of a2 and q2 are obtained

_{by solving:}

where

_S

_*

_(T

_*

₎

is matrix with element

_{(i, j)}

_equal

to

_half,

if nucleus

_(commercial)

QTL,

is a

’parental’ QTL

of commercial

QTLi,

and zero

otherwise; a

l

and

q

l

are known from the MA-EBV evaluation of the reduced set of

equations;

the

elements of

a

2

and

_q

₂

_,

which are needed in the

right

hand side of the above

equations,

have been calculated before

_they

are needed when the animals are