Genetic grouping for direct and maternal effects with differential assignment of groups

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Genetic grouping for direct and maternal effects with

differential assignment of groups

Rjc Cantet, Rl Fernando, D Gianola, I Misztal

To cite this version:

Original

article

Genetic

_grouping

for direct

and maternal

effects

with

differential

assignment

of

_groups

RJC Cantet

RL Fernando

2

D

Gianola

3

I Misztal

1

Facultad

de

_Agronomia,

Universidad de Buenos

Aires,

Departamento

de

_Zootecnia,

Av San Martin 4453

(1l!17!,

Buenos

_{Aires, Argentina}

2 De

P

artment

of

Animal

Sciences, University of

Illinois,

Urbana,

IL 61801 ;

3

University of Wisconsin, Department of Dairy Science, Madison,

WI

_53706-128/!,

USA

(Received

13 March

_1991;

_accepted

26 March

₁₉₉₂₎

Summary - Genetic

_grouping

in additive models with maternal effects is extended to cover

differential

_assignment

of groups for direct and for maternal effects. Differential

grouping

provides

a means for

_{including genetic}

_{groups in} animal evaluations

when,

for

example,

genetic

trends for additive direct and for maternal effects are different. The cxll’lIsion

is based on

_including

the same animals in both vectors of additive direct and additive maternal

_effects,

and on

_exploiting

the

_resulting

Kronecker structure so as to

_adapt

the rules of

_Quaas

when maternal effects are absent.

_Computations

can be

_performed

while

reading pedigree data,

and no matrix

_{manipulations}

are involved. An

_example

is

_presented

to illustrate the computations. Extension of the

_procedure

to accommodate

_multiple

traits is indicated.

genetic groups

/

direct and maternal effects

/

animal

model / best

linear unbiased prediction

(BLUP)

Résumé - La constitution de groupes génétiques avec affectation différentielle selon les effets directs et maternels. _{La constitution de groupes}

_génétiques

dans le cadre de modèles

_additifs

avec

_effet

maternel est

_{généralisée}

à une

_{affectation différentielle}

à des

groupes selon les

effets

directs et maternels. Ce regroupement

différentiel fournit

une

méthode pour inclure les groupes

génétiques

dans l’évaluation des animaux

quand,

par

exemple,

les

_{progrès génétiques}

_pour les

effets

directs et maternels sont

différents.

Cette

généralisation

est basée sur l’inclusion des mêmes animaux dans les 2 vecteurs

_d’effets

additifs

directs et maternels et sur

_{l’exploitation}

des structures de Kronecker

_{résultantes,}

en

_adaptant

les

règles

données par

Quaas

(1988)

quand

il

n’y

a _pas

_{d’efj‘ets}

maternels.

*

Les calculs _peuvent être réalisés au cours de la lecture du

_fichier

des

_pedigrees,

et aucune

manipulation

matricielle n’est néce.ssaire. Un

_exemple

est

_présenté

pour illustrer les calculs.

L’extension de la méthode au cas multivariable est

_évoquée.

groupe génétique

/

effet direct et

_{maternel / modèle}

animal

_/

BLUP

INTRODUCTION

Genetic

_grouping

is a means for

_dealing

with

_{incomplete pedigree}

information in

genetic

evaluation

_{(Quaas, 1988).}

The

_{theory developed}

for additive effects

_(Quaas,

1988;

Westel et

_al,

₁₉₈₈₎

was extended to accommodate maternal effects

_by

Van Vleck

_(1990),

but this author considered

only

the situation where the unknown

parents

are

assigned

to the same groups for direct and maternal effects. He also warned about

_{possible singularities}

introduced

_{by grouping}

when

_solving

the mixed model

_equations

_{(Henderson, 1984).}

Grouping

animals is often a

subjective

_process

(Quaas

and

Pollak, 1981;

Hender-son,

1984;

Quaas,

1988)

in which individuals are

_assigned

to different

_populations

(groups)

based on some

attribute

such as _year of birth. Genetic

grouping

is

some-what less

_arbitrary

in the sense that

_only

unknown

_parent

animals are

_assigned

to _groups

_{(Quaas, 1988).}

When there are maternal

effects,

every unknown

parent

must be

_assigned

to a group for direct effects and to a group for maternal

effects,

and there may be situations in which the criteria for

_constructing

_{groups for the} direct effects differ from those used for the maternal effects. An

_example

is when

genetic

trends for direct and for maternal effects are different. The

objective

of this

study

is to extend the

_theory

of

_{genetic grouping}

in models with maternal effects

so as to allow for differential criteria to be used when

_assigning

_groups for directs effects and _groups for maternal effects.

THEORY

Let _yj be a record made

_by

individual i with dam

_j.

After Willham

(1963)

and

Quaas

and Pollak

_(1980),

an additive model for the

_maternally

influenced record Yi

j is:

where

_x’

is a row of the incidence matrix

_relating

the record of individual i

to an unknown

_{vector p}

of fixed

_effects,

_aoi is the direct

_breeding

value

_(BV)

of i for direct

_effects,

_amj

_is

the BV of

_{dam j}

_for

maternal

effects,

emj is an

environmental contribution common to all _progeny raised

_{by j}

and eoi is an

environmental deviation

_peculiar

to the record made

_by

individual i. The model is such that aoi, amj, eoi are random variables with

_Var(a

_o

_{i) _}

_{o-2 A}

_o

_{, Var(a}

_mj

₎

!A!m cov(aoi, anx!)

=

rijUAoAm! Var(en,!) _

u2

,

m

and

_Va

_{r (}

_eoi

₎

_{_}

_o,20;rij

is the additive

_relationship

between i and

_j,

_aij

_{being equal}

to

_1/2

in this case. All random

variables are assumed to be

_{mutually independent,}

with the

_exception

of aoi and

a mj

The BV’s for direct and maternal effects of any individual can be described as

the average of the BV’s of its

_parents

_plus

an

_{independently}

distributed Mendelian

sampling residual 0

(Bulmer, 1985).

Letting

k be the sire

_{of i,}

the direct BV of i is:

In the same _way, the maternal BV of i is:

Following Quaas

(1988),

in the absence of

_inbreeding

_Var(<p

_o

_i)

=

1/2 cr!,

and

Var (§

mj

) =

1/2lT!m’

Also:

because k

_{and j}

are unrelated. From the

_preceding,

_{cov(<!ot)!mt) ==}

1/2(TAoAn7,-Let the

_{positive-definite}

matrix

_Go

be:

The animal model with groups and

relationships

(Robinson,

1986; Quaas, 1988;

Westell et al

_1988;

is based on

_arranging

BV’s of all animals into 2 different

vectors,

a and ab.

_Every

identified individual in the

_pedigree

has a direct BV

in the a x 1 vector _ao and a maternal BV in the a x 1 vector _am such that a’ =

_{[a!, a’}

₎

_.

Unknown animals

_(parents)

from which individuals in a are derived

have their BV’s

_represented

in the 2b x 1 vector

_ab

=

[a

0

,

ab&dquo;1!.

These are the

&dquo;base&dquo;

_{population animals,}

and

_they

are assumed each to have a

_single

_progeny

represented

in a. Let

P

6

(of

order a x

b)

and P

(of

order a x

a)

be matrices

relating

BV of progeny to BV of unknown and identified

_{individuals, respectively.}

If base animals were

_known,

a matrix version of

_[2]

and

_[3]

would be

_{given by}

a =

(1

2

₀

P)a

+ !, where.

is _{a vector} that results from

stacking

the Mendelian residuals for direct and maternal effects. As in

_Quaas

_(1988),

it will be assumed that Mendelian residuals have

_expectation

_E(!)

= 0

and,

because _no

inbreeding

is

_assumed,

_{Var (!)}

=

1/2

Go

₀

I

a

.

This variance-covariance matrix follows from

matrix with elements

_d

_ii

=

1/2 - (F

si

₊

)/4,

D:

F

where

Fs

i

and

F

Di

are the

inbreeding

coefficients of the sire and the dam of individual i.

The vector a is better

_{represented conceptually}

_{(Quaas, 1988)}

_by

the

_expression:

Rearranging:

and

_solving

for a:

The base animals are assumed to be drawn at random from the distribution

where

_Q

₆

relates base animals to the &dquo;base&dquo;

_population

_{means, g.}

_Hence,

base animals are unrelated but do not

_necessarily

have the same mean. More

_explicitly:

where go and gmare the &dquo;base&dquo; mean vectors for direct and maternal

_effects,

respectively,

and the matrices

_Q

_bo

and

Q

bm

relate base animals to their

_respective

population

means. In

_general,

_Q

_bo

and

_Qb!

_may be

_different,

even

_{though including}

the same animals in ao and in am forces abo and abm to

correspond

to the same

base animals.

To

_exemplify,

consider the

_following

_pedigree:

Capital

letters denote identified individuals and lower case letters the unknown

or

_{&dquo;phantom&dquo;}

_parents.

_Symbols

in

_parentheses

indicate group

(direct, maternal)

of the unknown

_parents.

There are 2 groups for direct effects

_(D

₁

and

_D

₂

₎

and 2 groups for maternal effects

(M

i

and

M

2

);

note that some unknown

_parents

_{(a, d)}

have been

_assigned

to different _groups for direct and maternal effects. The matrix

The matrix

_Q

₆

is:

This formulation allows the rules of

_Quaas

to be extended

₍₁₉₈₈₎

for

_writing

the mixed model

_equations

for an animal model with

_genetic

_{groups for direct} and maternal effects in a

_simple

_way.

Expectation

of a

Using

[5],

we have:

for

_Q

= 11

2

&reg;

_(I

_a

_-

_{P) !P;)]Qb.}

The

_rectangular

matrix

_Q

is made of 2 blocks:

(I

a

-

P

_P

_bQbo

and

_(I

_a

_-

_,,,.

_P)-’P

_bQb

The first block is the same as in

_Quaas

(1988)

for a model without maternal effects.

Variance of a

Since

_{Var (a)}

=

_Go

_&reg;

_A,

_it follows that:

(auaas

(1988) pointed

out that

_P

_b

_Pb

₌

_{Diag {0.25 md,}

for mi =

_0,1,2

2 = the number of base

_parents

of the ith individual. Hence:

where D =

Diag

{0.25

mi+

_0.5}.

Hence:

Mixed model

_equations

A matrix version of model

_[1]

is:

where _y,

_p,

_{ao, am,} em and eo are the vectors of

records,

and of unknown

_fixed,

direct and maternal

_BV’s,

maternal environmental and direct environmental

_effects,

respectively.

In the same _way, the incidence matrices

_{X, Z}

_o

_,

_Z

_m

and

_E

_m

relate records to fixed

_effects,

direct and maternal BV and maternal environmental effects. Correct

specification

of the coefficients for cr!o!m. and

0

,

2

Am

in the

variance-covariance matrix of y in

[11],

for any

pair

of individuals with

records, requires

the additive

_{relationships}

between each individual and the dam of the other and the additive

_relationship

between the dams

_{(William, 1963).}

If an animal with a record in y has an unidentified

_{(&dquo;phantom&dquo;)}

_dam,

_{mis-specification}

of

_{Var (y)}

results due to

_taking

those additive

_{relationships}

as if there were zero. One solution

is to include the BV for direct and maternal effects of the

_{&dquo;phantom&dquo;}

dam

of

the individual in ao and a!,

respectively.

Note that this has the effect of

_increasing

the size of the

system

of

_equations

_by

twice the number of unknown dams of individuals with records in y. Maternal environmental effects of

_{&dquo;phantom&dquo;}

dams may also be included in em to force

_{u5!__}

to be

_present

in the variance of animals with a record

in _y and with an unknown

_dam,

as discussed

_by

Henderson

_(1988).

This

procedure

also increases the number of

_equations

to be solved. A more efficient

_strategy

is to

Therefore,

E(y)

=

Xp +

ZQg

and Var

(y)

=

Z(Go

₀

A)Z’

+

E .. E’

m

oE

2

m

+

i,

,

o,2 Eo*

Using

the

_QP

&dquo;transformation&dquo;

_(Quaas

and

_Pollak,

_1981);

modined mixed model

equations

for

_[11]

are:

!, ,.¡

where ae

=

(J’1

o/

(J’1

m

’

On

defining

W =

[X:0:Z:E!],

6 =

[#’:§’ :£’:6£]

and:

the above

_equations

can be

expressed

as

[W’W

+

A

*

]9

=

W’y.

Rules for

_calculating

A*

For the method to be

_{computationally}

feasible A* must be calculated without

performing

matrix

_{multiplications.}

The last block in A* is

_diagonal

and meets this

requirement.

The central block is the

_{&dquo;genetic&dquo;}

_part

of A* and can be calculated

by

simple

rules which are an extension of the work of

Quaas

(1988).

A referee

pointed

out a

_simple

_way of

_deriving

these rules and his

proof

is

_adapted

here. Observe that the central block in A*

_(without

_{o, E 2 !)}

is:

and,

on

_{using Q}

=

(I

2

&reg;

P

b

)Q

b

1

-P)-

a

(I

and

G-

1

as in

_(10),

the above

_expression

is

_equal

to:

In absence of maternal

effects,

[13]

reduces to the

_expression

obtained

_{by Quaas}

(1988;

page

1343)

for direct effects

only.

Let

<!

be element

i, j

of Go

1

.

Then, using

[13]

on

(12!,

we have that the

_{&dquo;genetic&dquo;}

_part

of A* is

equal

to:

where

_d-

₁

_{is diagonal}

element k of matrix

D-

1

and

_h!:k

_{(see 14)}

is the kth row

of

H

i

.

Most elements in each of these rows are zeroes

except

for 2

_negative

halves

(corresponding

to a sire or a sire base _group and to a dam or a base dam

group)

and a one

_{(corresponding}

to the

_individual).

The first a rows

_correspond

to direct effects and the rest to maternal effects.

Expression

[14]

shows that the 3 non-zero elements in each row of H make each

known individual to &dquo;contribute&dquo; 36 times

₍₌

3

2

x

2 x

₂₎

to the

_{&dquo;genetic&dquo;}

_part

of A

*

. The contributions can be described

letting i, f, j, k,

I and m

_represent

the row or column or A* associated with:

i = direct effect of _an

_individual;

f

= maternal

_genetic

effect of the same

individual;

j

= direct effect of the sire of the individual if the sire is

_known,

_or

group for direct effects of the unknown

_sire;

k = direct effect of the dam of the individual if the dam is

known,

_or

group for direct effects of the unknown

dam;

I =

_genetic

maternal effect of the sire of the individual if the sire is

known,

_or group for maternal effects of the unknown

sire;

m =

genetic

maternal effect of the dam of the individual if the dam is

known,

or

Therefore,

the 36 contributions result from all

pairwise

combinations of the above

subscripts.

As in

_Quaas

_(1988),

_{let ,}

₌

_0,

1 or 2 be the number of unknown

parents

of i and x =

4/(!,

₊

2)

and

_put:

Then,

each known individual makes the

_following

contributions which are added to

the

_{&dquo;genetic&dquo;}

_part

of A&dquo;:

Using

these rules

_plus

_!15!,

the contributions of each animal to elements of the

&dquo;genetic&dquo;

part

of

A&dquo;,

for the

_example,

are

displayed

in table I.

The

_algorithm

can,,,be extended to

_{multiple.traits, <as pomted=}

_out

_’

_{-by -a}

referee,

as follows. Let:

i

i

=

_equation

number of individual i for the lth

_trait;

j

i

=

_equation

number of the sire of _{i or its} sire’s group

(if

base

_sire)

for trait

_{l ;}

k

i

=

_equation

number of the dam of i or its dam’s

group

(if

base

dam)

for trait I. Let s =

_{d, l, 2,}

be the number of base

_parents

of i. For each animal calculate

- ! =

41(s

+

2).

Finally, letting

t be the number of

traits,

for m = 1 to t and

n = _{1 to}

t,

add to A* the

_following

9 contributions:

where

_gmn

is element

_{(m, n)}

of the inverse of the t x t matrix of additive variances

and covariance among the t traits. Note that for t = 2 there are 4 _passes

_through

the

_loops

of m and _n,

_resulting

in 9 x 4 = 36

contributions,

as in the case of direct and maternal effects.

DISCUSSION

The

_procedure

_presented

here allows for different criteria to be used when

_assigning

genetic

groups for direct and for maternal effects. If groups for direct and maternal effects are

assigned using

the same

_criterion,

our formulation

_gives

the same results as those of Van Vleck

_(1990).

The method can be

implemented

by

a

simple

modification of

_{existing algorithms}

for direct effects

_only.

The modification

_requires

different

_addressing

for

_genetic

_groups. This can be

accomplished

by writing

extra

columns on a file

_{containing pedigree}

information

_indicating

the group

_assignment

for maternal effects of the

_{&dquo;phantom&dquo;}

_parents.

Assigning

different groups may be used to account for different

_genetic

trends on a

_maternally

influenced trait. For

_{example, Benyshek}

et al

₍₁₉₈₈₎

_{analyzed weaning}

weight

records of beef calves and found a

_positive

_genetic

trend for direct

_effects,

whereas the trend for maternal effects was

_practically

null. In this _case, unknown

animals _may be

_assigned

to

just

one _group

(or none)

for maternal effects while

being

assigned

to _{several groups} for direct effects. Differential

_{genetic grouping}

can also

be

_employed

when

_genetic

trends

_{display genetic}

_(piecewise)

_patterns

_throughout

the years. For other

situations, assigning

different groups to direct and maternal effects _{may not} be feasible.

Quaas

(1988)

warned about

_{using complex strategies}

to

_assign

groups to

_missing

individuals so that

_confounding

between

_genetic

groups and other effects in the model is avoided. If _groups for direct and maternal effects are to be fitted there

is_

also the

_possibility

of

_confounding

between

_genetic

_groups for both

types

of effects. Consider the matrix

_[Z

_o

_o

_o

₍

_Z

_m

_om]

that relates records to

_genetic

_groups.

Z

m

(which

relates records to maternal

_BV).

However,

if

_Q

_o

=

Q

m

,

ie the same

criterion is used

_{to assign genetic groups}

for direct and

maternal

effects,

the risk of

confounding

both

_types

of effects or with other

efFects

in the model is

higher

than

the case of

_Q

_o

different from

_Q

_m

_.

I

’!,

A referee

_pointed

out an

example

indicating

that

lack

of

estimatibility

may not

_always

be

_{produced by confounding}

but also due to lack of

_expression

of the maternal effects. The

_problem

arises when there are animals with records and unknown sires and males and females are

_{grouped separately.}

Whereas the direct

effect for the

_{&dquo;phantom&dquo;}

sire group would be estimable the maternal effect would

not,

because none of these sires has female descendents with recorded progeny. As direct effects are

expressed

long

before maternal

effects,

direct _group effects will

be estimable well in advance of maternal group

effects,

the referee indicated. He

goes further to

_suggest

_that,

in this case, one can resort to form _groups with both males and females or have the last maternal _group

correspond

to a much

longer

time

_period.

In the

_present

work

_breeding

values for direct and maternal effects of

_missing

or

_{&dquo;phantom&dquo;}

dams of individuals with records are

_suggested

to be included in the vector of solutions to

_{correctly specify}

the variance-covariance matrix of the

observations,

as in Van Vleck

(1990).

As a _consequence the number of

_equations

to

be solved increases.

_However,

the

_procedure

of differential

_grouping

is

_independent

of

_enlarging

the vector of

_breeding

values to include those of the

_{&dquo;phantom&dquo;}

dams. If other methods of

_specifying

the variance of the records are

_found,

the

_procedure

presented

here may still be

applicable.

ACKNOWLEDGMENTS

The referees

_provided

_{many comments} that

_{greatly improved}

the

_original

version of this _paper.

_Any

_remaining

mistake is the

_{responsibility}

of the first author. RJC Cantet wishes to thank the

University

of Illinois and Universidad de Buenos Aires for financial

support

throughout

his

_graduate

studies.

REFERENCES

Bulmer MG

₍₁₉₈₅₎

The Mathematical

_{Theory of Quantitative}

Genetics. Clarendon

Press,

Oxford

Benyshek

LL,

Johnson

MH,

Little

DE,

Bertrand

JK,

Kriese La

₍₁₉₈₈₎

_Applications

of an animal model in the United States beef cattle

_industry.

J

_Dairy

Sci 71

(suppl

2),

35-53

Henderson CR

₍₁₉₈₄₎

_Applications

_{of Linear}

Models in Animal

_Breeding.

University

of

_{Guelph, Guelph, Ont,}

Canada

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