Generalizing the use of the canonical transformation for the solution of multivariate mixed model equations

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Generalizing the use of the canonical transformation for

the solution of multivariate mixed model equations

V Ducrocq, H Chapuis

To cite this version:

Original

article

Generalizing

the

use

of the

canonical

transformation

for

the

solution of

multivariate mixed

model

_equations

V

_Ducrocq

H

_Chapuis

1

Station de

_génétique

_quantitative et

appliquée,

Institut national de la recherche

_agronomique,

78352

_{Jouy-en-Josas cedex;}

2

Betina

_S61ection,

Le Beau

Chene, Tr6dion,

56250

_Elven,

France

(Received

4 December

_{1996; accepted}

27 March

₁₉₉₇₎

Summary - The canonical transformation converts t correlated traits into t

phenotypi-cally

and

_{genetically independent}

traits. Its

application

to

_multiple

trait BLUP

_genetic

evaluations decreases

_{computing requirements,}

increases the convergence rate of iterative

solvers and

_{simplifies programming.}

This paper presents alternative ways to

retain,

at least

partly,

these desirable characteristics in situations where the canonical transformation is

theoretically impossible:

when some traits are

_missing

in some animals

_(including

when a reduced animal model is

used),

when more than one random effect is included in the

model and when different traits are described

by

different models.

genetic evaluation

_/

mixed

_{model / computing}

_algorithm

_/

_multiple trait

_/

animal model

Résumé - Généralisation de l’utilisation de la transformation canonique pour la résolution des équations du modèle mixte multicaractère. La

_{transformation}

canoni-que

remplace

t caractères corrélés par t caractères

génétiquement

et

phénotypiquement

indépendants.

Son

application

dans des évaluations

génétiques

de type BL UP multi-caractères diminue les besoins

_{informatiques,}

accroît la vitesse de convergence

d’algo-rithmes de résolution itérative et

_simplifie

la

_{programmation.}

Cet article

_{présente di}

_f

fé-rentes manières de conserver au moins

_{partiellement}

ces

_{caractéristiques favorables}

dans les situations où la

_{transformation}

canonique est

_{théoriqv,ement impossible,}

c’est-à-dire

quand

certains caractères sont manquants pour certains animaux

_(y

_compris

_lorsqu’un

modèle animal réduit est

_v.tilisé),

_{quand plus}

d’un

_effet

aléatoire est inclus dans le modèle

et

_{quand différents}

caractères sont décrits par

différents

modèles.

évaluation génétique

/

modèle mixte

_/

algorithme de calcul

_/

évaluation

INTRODUCTION

In routine

_{genetic evaluations,}

theoretical considerations

_suggest

that in situations where records can be described as linear functions of fixed and random

effects,

best linear unbiased

_prediction

_(BLUP)

of

genetic

effects based on a

_multiple

trait

animal model should be used

_(Henderson

and

_Quaas,

_1976;

_Foulley

et

_al,

_1982;

Quaas,

1984;

Schaeffer,

1984).

The inclusion of the known

_relationship

between traits in a

joint

analysis

of these traits increases the amount of information available

and as a

result, improves

the _accuracy of

prediction

and corrects

potential

biases

resulting

from selection. van der Werf et al

₍₁₉₉₂₎

and

Ducrocq

(1994a)

review

the benefits to be drawn from a

_multiple

trait BLUP

_genetic

evaluation. Flexible

general

purpose

packages,

eg, PEST

(Groeneveld

et

_{al, 1990;}

Groeneveld and

_Kovac,

1990)

exist and are

_successfully

used to solve

_{complex multiple}

trait evaluations.

However,

the

_simple

iterative

_{algorithms commonly implemented}

in such

packages

can be

_extremely

slow to _converge when traits are

missing

for some animals or

when several random effects or _{groups of unknown}

_parents

are defined in the model

(Groeneveld

and

Kovac, 1992; Ducrocq, 1994a,

b).

Although generally acceptable

for data files of moderate

_size,

slow convergence can become a

limiting

factor for routine national evaluations.

In the

_particular

case when the same model with

_only

one random

_(genetic)

effect

_applies

to all traits and no records are

missing,

a canonical transformation

of the t records of each animal into uncorrelated records

_replaces

the

_large

system

of

multiple

trait mixed model

equations

with a set of t

_simpler

univariate

_systems

(Foulley

et

_al,

_1982;

_Quaas,

_1984;

_Arnason,

_1986;

_Thompson

and

_Meyer,

_1986;

Jensen and

Mao,

1988;

Ducrocq

and

_Besbes,

_1993).

The

_resulting

reduction in

computing

costs is often drastic.

However,

the restrictions on the model and data

structure

_required

for the

_{implementation}

of the canonical transformation are

rarely

fulfilled in

_practice.

Other transformations have been

_proposed

when some

traits are

_missing

(Pollak

and

Quaas,

_1982;

Quaas,

1984)

but it was found that a

_strategy

where

missing

values are

_iteratively

_{replaced by}

their

_expectation

and

therefore

_retaining

the

_possibility

to

_implement

the canonical transformation is

clearly

superior

(Ducrocq

and

_{Besbes, 1993; Ducrocq, 1994a,}

_b).

The _purpose of this _{paper is} to demonstrate that the basic

_objective

of the canonical

transformation, ie,

the reduction of a

_large

linear

_system

of

_equations

into

sets of

smaller,

sparser

systems

can be achieved in even more

_general

_situations,

_eg,

with different models for each trait or with more than one random effect other than

the residual. For the sake of

completeness,

the

simple

canonical transformation is

briefly

described with and without

_missing

values on some traits. An extension of

the above-mentioned

_strategy

for the

_missing

values case to reduced animal models is also

_presented.

MULTIPLE TRAIT MIXED MODEL

_EQUATIONS

where yi is the vector of records for trait

_{i; b}

_i

and ai are vectors of fixed and random effects and

_X

_i

and

Z

i

are the

_{corresponding}

incidence matrices.

_Here,

the

only

assumption

is that no more than one random effect other than the residual

e

i is considered in the model. The variance-covariance structure for the random

effects is summarized as follows:

Concatenating

the random

_(genetic)

effects and the residuals for all traits into

vectors a and _e,

_{respectively,}

the

_G

_ij

and

_RZ!

blocks are

_grouped

into matrices

G =

Var(a)

and

R

=

Var(e).

The

_{(i, j)}

blocks of the inverse matrices

G-’

and

R-1

are denoted

G

ij

and

R

ij

,

respectively.

The submatrices

_G

_ij

and

_R2!

are functions of the

_pedigree

and data structures

and of

_Go

and

_R

_o

_,

the

_genetic

and residual variance-covariance matrices between

traits. The

_general

form of the mixed model

_equations

is:

The number of

_equations

and the memory

requirements

for such

_systems

increase with t and

_t

₂

_,

_{respectively.}

Iterative solvers can be used but

_they

are

_relatively

complex

to

_implement

in the

_general

case. More

_importantly,

convergence rate can

be

_extremely

slow

_(Arnason,

_1986;

Groeneveld and

_Kovac,

_1992;

Reents and

_Swalve,

CANONICAL TRANSFORMATION

In this

_section,

we consider the

particular

case where there are no

_{missing records,}

ie,

each one of the recorded animals has a record on each of the

_{t traits,}

and the

same model

applies

to all traits. Let _y =

(

Y

’

Y

’ ...

_y’)’

be the vector

_including

all records for all traits and b =

(b ...

bt)’

be the vector of fixed effects. Each

vector _{yi is} of size N

Define

_Q

to be a matrix such that

’

oQ

QG

=

D -

1

,

where D is a

_diagonal

_matrix,

and

_QRoQ’

=

_It.

Such _{a matrix}

always

exists. A _way to

_compute

_Q

can be

_found,

eg, in

Quaas

(1984)

or

_Ducrocq

and Besbes

(1993).

Quaas

(1984)

described different ways to

_simplify

the multivariate

_system

_[3]

transforming

its coefficient matrix into a

_{block-diagonal}

matrix. A first

_approach

consists in

_applying

a linear transformation:

to the data vector _y and to

_manipulate

the model of

_{analysis accordingly.}

This leads to the transformed model:

where

_b

_Q

=

(Q

_Q9

I

B

)b,

a

Q =

(Q

_Q9

I,

V

* )a

and

e

Q =

(Q

_Q9

I

N

)e.

B and N* _are the dimensions of

_b

_i

and _ai,

_{respectively.}

Then:

Since D and

_It

are

diagonal

t x

_{t matrices,}

the

_resulting

_system

of mixed model

equations

is

_{block-diagonal}

_and,

_therefore,

the solutions for the fixed and random effects for each transformed trait i can be obtained

_solving

the univariate

_system:

where d

i

is the ith

_diagonal

element of D. The solutions on the

_original

scale are

obtained

_by

_simple

back-transformation:

mixed model

_{equations corresponding}

to model

_!4!:

Rewrite

_system

_[12]

as

_C

_u

=

W’y

and define

S

=

(

Q

Q9 0

IB

Q

Q9

0

IN*

) and

S

* =

_Q

_Q9

IN.

Premultiply

both sides of the

_system

by

S-’

=

(S-’)’

and insert

I(

B+Ar*

)

t

=

S-

1

S

in the left-hand side and

I

Nt

=

S

*

-’S

*

in the

right-hand

side. This results in:

and is

_equal

to:

which

simplifies again

into univariate

_systems

_!14!.

Canonical

transformation

and reduced animal model

The canonical transformation

_applies

without modifications to multivariate reduced animal models

_(RAM;

Quaas

and

_Pollak,

_1981).

This will be illustrated here in order

to introduce notations for later use. Let the indices

(p)

and

(n)

refer to

_parent

and

non-parent

animals

_(N

_P

+

N

n

=

N

*

).

One can rewrite model

[1]

as:

where

_K!n!

is a matrix

relating

records of

_non-parent

animals to their

_parents.

A

typical

row of matrix

_K(!)

has two non-zero elements

equal

to 0.5 in the columns

corresponding

to

_parents.

For the

_{t traits,}

with records ordered within trait: The

_part

of e*

_{corresponding}

to

_non-parents

includes the residual effect

_e(

_n

₎

as well

as the mendelian

_sampling

contribution

_!!n!.

Let

)

*

Var(e

= R*. If _em

_represents

the t elements of the residual vector e* for a

particular

animal _m, we have:

diagonal

element

₆

_m

If

_App

is the

_relationship

matrix between

_parents,

we have:

Then,

after transformation of the data y !--> _yQ

(or

after matrix

manipulations

similar to

_!13!),

_system

_[21]

can be

_partitioned

into t univariate ’RAM’

_systems

to

solve. For the transformed trait i and

_defining

_SZ!!!

=

INn

₊

diD

n

:

MISSING VALUES

As

_{previously indicated,}

the transformation

_[6]

or the matrix

_manipulation

[13]

require

identical incidence matrices X and Z for each trait. Therefore

_they

cannot

be

_{directly implemented}

when some recorded animals have

_missing

values for some

traits. A

_simple

_strategy

to avoid this constraint has been

_{proposed by}

_Ducrocq

and Besbes

₍₁₉₉₃₎

and

_Ducrocq

_(1994a,

_b)

and is

_briefly

reviewed here. The

underlying

idea is to

_iteratively

_replace

the

_missing

values

_by

their

_expectation

given

our current

_knowledge

of all

_parameters

and to solve the

_resulting

_sytem

as if

they

were not

_{missing, ie,}

_applying

the canonical transformation. It can be

algebraically

shown that this

_technique

leads

to

the same solutions for fixed and random effects as the usual

general

approach.

A formal

justification

results from

the use of the

_{expectation-maximization}

(EM)

_algorithm

of

_Dempster

et al

_(1977).

Using

subscripts

a and

_{

₃

for observed and

_{missing observations,}

_{respectively,}

and

assuming

that,

given

a and

b,

the

complete

(= augmented)

data _{vector y} =

(y!,

y) ) ’

follows

a multivariate normal distribution with mean

(It (9 X)b+ (It 0 Z)a,

the estimation of b and the

_prediction

of a

_require

the

_knowledge

of the vector of

The vector _y,3

_being

_unknown,

we

replace

T

(y)

at iteration k

_by

its

_expectation

(E step):

where for observed

records: .

k

(k)

=

y

a and for animal m with

_missing

records:

In the above

_formula,

_R

_om

_,

_aa

and

_¡3a

_R

_,

_Om

are obtained from

Ro

by choosing

the rows and columns

corresponding

to

missing

and observed traits for animal m.

_X

_m

_¡3

(respectively, X

ma

)

are obtained from

(It

0

_X)

_{by choosing}

rows

_{corresponding}

to

missing

(respectively, observed)

traits for animal m.

_Similarly,

_am, and a&dquo;La are

the elements of am

corresponding

to

_missing

and observed traits.

The M

_step

consists in

_solving

the mixed model

_equations

in order to obtain

new values

b!!+1!

and

a!!+1!

for b and a. This is much

_simpler

to

_implement

than

in the

_general

case because now a canonical transformation is

possible:

from the

records

_actually

observed and the

_prediction

of the

_missing

ones at the current

EM

_iteration,

transformed records on the canonical scale can be

_computed.

After

solution of the mixed model

_equations

on the canonical scale and backsolution on the

original

scale,

new

_predictions

for the

_missing

values are made and this

iterative scheme is

_repeated

until convergence. In

practice,

it is not necessary to go

back to the

_original

scale as

_updating

can be done on the canonical scale. Consider

that all traits for animal m have been ordered such that observed traits

precede

missing

ones:

Ym¡3

C y&dquo;’’a J .

If this is not the case, re-order ym,

Q

and R. Partition

Q

=

((aa

Q

¡

3

)

and

Q-

1

= C Q

a

J .

Then,

the vector of observations for animal m

Q,3

on the transformed scale at iteration

_(k)

is:

but we have:

where

_b

_Q

_.&dquo;,,

_is,

on the transformed

scale,

the vector of fixed effects

_pertaining

to

where

_X

_m

_represents

the rows of

_It

0X

_pertaining

to animal m. The matrices

_JQ!

(of

size t x

_t

_a

₎

and

_J

_Q2

_(of

size t x

_t)

_depend

on the

_missing

_pattern

_only,

and

_they

are

computed

_only

once for use at each iteration.

Furthermore,

each EM

step

can

be interlaced with the iterative

_procedure

used to solve the mixed model

_equations.

This results in

_large

_savings

in

_computing

time.

Application

to reduced animal models

_(following

a

suggestion

from

R

_Thompson)

With the

_previous

_approach,

in RAM

_situations,

_{it is necessary} to

_predict

y;

;,b

for all

non-parent

animals. This

requires

in

_[25]

the

_knowledge

of

a!!>.

This is in contradiction to the

_original

_purpose of

_using

a reduced animal

_model,

which is to

solve a smaller

_system

of mixed model

_equations

with the additive

_genetic

values of

the

parent

animals

_only.

To avoid the

computation

of

non-parent

animals’

_genetic

values,

one can

_replace

the

_{E step:}

L 1. 1

1

Then,

for a

_non-parent

m with

_{missing records,}

and with

_parents

’sire’ and ’dam’:

Again,

Rp.&dquo;,,

aa

and

_7!.oTn,/3a;

defined in

_[17]

and

_[18]

are obtained from

7Zo&dquo;,

_by

choosing

the rows and

columns

_{corresponding}

to

_missing

and observed traits. These

_{predicted missing}

values influence the

_right-hand

side of the RAM

equa-tions,

which after canonical transformation is of the form:

After each solution of the reduced

_system

of

_equations,

or after each iteration

completed,

the

_missing

terms in

_YQi(P)

and

_YQi(n)

of

_[30]

are

_{computed again given}

the current values of

b

&dquo;

_Q

and

_a

_(’)

MORE THAN ONE RANDOM EFFECT

The second _necessary condition in order to

_apply

the

_regular

canonical transfor-mation is the existence of

_only

one random effect other than the residual. In this

section,

this

_requirement

will be relaxed. Consider for

_example

a model with a

di-rect additive

_genetic

effect and a maternal

_genetic

effect. Assume the same model

for all traits

_(no

_missing

_values):

where m is the vector of maternal

_effects,

M the

_{corresponding}

incidence matrix and:

The

_{corresponding}

mixed model

_equations

can be written:

Simultaneous

_{diagonalization}

A

_{straightforward}

extension of the canonical transformation was

_{suggested by}

Lin

and Smith

₍₁₉₉₀₎

in a

particular

situation: if

G!,&dquo;,,

=

G&dquo;,,

a

= 0 and

G

aa

,

Gm

m

and

_R

_o

are

_{proportional,}

then it is

_possible

to find a matrix

_Q

such

_that,

after a

The

resulting

system

of mixed model

_equations

is

_{block-diagonal}

and

_simplifies

to

t univariate

_systems:

Again,

this result can be obtained via a

_manipulation

of the

_system

of

_equations

as in

_[13].

The conditions

_required

to

_diagonalize

three

_{(or more)}

covariance matrices

are rather drastic. Misztal et al

₍₁₉₉₅₎

_clearly

showed that accurate results can still

be obtained when the true covariance matrices are

_replaced

with

_{simultaneously}

diagonisable

approximations

of these matrices.

However,

this

_approach

is not

applicable

when the random effects are correlated

(G

am

-

I-

0).

Block-iterative canonical

transformation

A more

general

_strategy

consists in

solving

[34]

block-iteratively

(Hackbusch, 1994).

The

_diagonal

blocks are chosen such that the canonical transformation can be

applied.

Let

_Q

and P be the transformation matrices such that:

Using

these

_matrices,

a

_manipulation

similar to

_[13]

can be

performed

to

sim-plify

[34].

Define:

Premultiplying

both sides

_by

_SQP

and

_inserting

the

_appropriate

_identity

matrices between the coefficient matrix and the vector of unknowns on the one hand and

Equation

[40]

simplifies

to:

At each

_iteration,

one can solve the univariate

_systems:

DIFFERENT MODELS FOR DIFFERENT TRAITS

A third

_requirement

for the

_{applicability}

of the canonical transformation is the

use of the same model for each trait.

_Again,

several

_strategies

exist to at least

partly

retain the benefits of the transformation for

_{computational simplicity.}

For

simplicity,

assume here that in model

[1],

the incidence matrices

Z

i

= Z are the same for all traits

_{i (no}

missing

values on recorded

animals)

but that the incidence

matrix

X

i

can _vary from one trait to another. Block-iterative

_approach

multiplication by

the current solutions of the

_{corresponding}

effects. For the fixed effects

part,

the

_system

remains multivariate:

But for the random effects

_block,

it is

_possible

to take

_advantage

of the fact that the incidence matrix is the same for all traits:

where

y¡

k+1J

=

yi -

Xb¡k+1J.

Applying

the

regular

canonical transformation

(as

in

_!6!)

to

_(47!,

the random effect block becomes a set of t univariate

_systems:

Gengler

and Misztal’s

_approach

Gengler

and Misztal

₍₁₉₉₆₎

_proposed

an

_approach

that makes use of the

algorithm

described for the

_missing

values case

(note:

this

_approach

was also found and used

by

Goddard

_(1995,

pers

comm)).

Their method is better illustrated

using

a two-trait

example:

Here,

the vector

_f

₁

_(with

associated incidence matrix

_F)

refers to fixed effects

influencing

trait 1

_only,

_h

₂

_{(matrix H)}

refers to those

influencing

trait 2

_only

and b

(matrix

X)

includes effects

_affecting

both. Consider for

_example

an animal m with

two recorded traits:

Gengler

and Misztal

₍₁₉₉₆₎

_duplicate

all records and define a

complete

model

For the animal m

_above,

this means the use of four

_records,

two of them

_being

considered

_missing:

Here,

dum refers to a

dummy

level defined for

_missing

records

only

and different

from _any level

_{corresponding}

to observed records. After this artificial

modification,

the same incidence matrices

_apply

to all traits

_(for

the first and second records for each trait in

_{[52], respectively)}

and a

_regular

canonical transformation can be

performed

using

the

_approach

described in the

_previous

section to include

_missing

records. No additional

_coding

is

_required

as

long

as the initial code

_accepts

multiple

records. All records are

_analyzed

with the _proper model.

An

_{approach using}

constraints on _unnecessary effects

We will

_slightly

generalize

Gengler

and Misztal’s

_example

described in

_[49).

Consider

two _groups of traits:

The idea here is to define the

_complete

model

_[51]

for all traits and to solve the

resulting

mixed model

_equations

under the constraints:

The full

_system

of mixed model

_equations

is:

This

system

has

_exactly

the same form as described in

[12]

and can be

trans-formed into a set of t smaller

_systems

_by

canonical transformation. The other two

blocks to solve are:

The constraints will be

_applied

on these smaller

_systems.

For

example,

for the f

block,

we want to solve:

Note that the set of constraints can be written as:

To

is a

_diagonal

t x t matrix with

_diagonal

element

_equal

to one for each effect to be

constrained to 0 and

_equal

to 0 otherwise.

_Solving

_[60]

is

_equivalent

to

_minimizing

the

expression:

under the constraint

_(To

0

_If)

f = 0. Constrained minimization

problems

can be

solved

_using

_{Lagrange multipliers.}

Let 71 be a vector

of Lagrange

multipliers.

Taking

the derivatives of U +

a’(T

0

_I

_f

_)f

with

respect

to f and

71,

we

_get

the

following

linear

_system:

Now,

premultiplying

both sides

_by:

the

_system

becomes:

System

[64]

is of the form

and has a solution

_equal

to:

Given the form of C and T and in

_particular

C-’T’

=

Q-

T

To

_Q9

(F’F)-’

=

T’C-’,

it follows

_{C-’T’(TC-’T’)-T}

=

T’(TT’)

-

T

and

[65]

leads to the

expression:

which can be written more

_simply

as:

where

_W

_o

_is

a t x t matrix

_computed

_only

once and

_f

_oQ

are the unconstrained

solutions. These unconstrained solutions are obtained

solving

t simple

univariate

systems.

If more than one effect is included in

f,

the inverse of a full rank submatrix

of F’F is used in

_[66]

or an iterative

approach

is used to obtain

_f¿!+1).

.

A numerical

example

of the

_algorithm

is

_given

in the

_Appendix.

DISCUSSION

The

_systematic

use of a canonical transformation to solve

_multiple

trait mixed

model

_equations

is desirable for several reasons, all

relying

on the

computational

advantages

it offers.

The

_resulting

_system

is much sparser. Increased

sparsity

has a beneficial effect on the _convergence rate of iterative methods. This has been

_repeatedly

found in

univariate evaluations

_(eg,

when an

equivalent

model is used to increase the

sparsity

of the coefficient matrix when groups of unknown

_parents

are defined

(Quaas,

1988))

as well as for multivariate

analyses

(Pollak

and

Quaas,

1982; Ducrocq

and

Besbes,

1993).

convergence in univariate

(eg,

with reduced animal

models)

as well as in multivariate

situations

_(Arnason,

_{1986; Ducrocq, 1994a,}

_b).

Programming

is

_easier,

since each

_system

is

_equivalent

to a

_single-trait

analysis.

Programming

can be made more efficient: iteration on data

(Schaeffer

and

Kennedy)

may be

replaced

by

(possibly partial)

storage

of the coefficient matrices in

core. The

_system

being

smaller and sparser, a direct solution _may become feasible in

some _cases,

possibly

on

_parts

of the

_system

only.

The search for

optimized

iterative

algorithms

(eg,

with

_optimum

relaxation factors

_(Misztal

and

_Gianola,

₁₉₈₇₎

may

become worthwhile.

The estimation of variance

components

is

_{greatly simplified}

and less

_computer

intensive

_(Meyer,

_1985;

Jensen and

_Mao,

_1988).

In this _paper, it was shown that there are _{ways to} retain at least

_partly

these

_{computational advantages}

when the three main

requirements

for the

_regular

canonical transformation to be feasible are not fulfilled.

When data are

_missing,

one can

_replace

the

_missing

records at each iteration

_by

their

_expected

values

_given

the current

parameters;

the extra

_coding

is

_minimal;

the

potential drawbacks,

such as the need to backsolve for each

non-parent

solution when a reduced animal model is used

(Ducrocq

and

Besbes,

1993),

can be avoided.

When more than one random effect other than the residual is included in the

model and a simultaneous

diagonalization

of all the covariance matrices between

traits is not

_possible,

a canonical transformation can still be

_applied

to the

_diagonal

block

_{corresponding}

to each random effect. Block iteration is a well-known extension

of iterative

techniques

(Hackbusch, 1994).

Although

the

_procedure

_proposed

here is

relatively

simple,

it is not

_guaranteed

that such a

_strategy

is

always

more efficient

than when other

partitions

of the coefficient matrix in

_[33]

are used.

Quaas

(pers

comm)

suggests

than in some _cases, it _may be better to sort the

equations

in

[33]

by

traits within animal and to treat

_together

all the

_{equations referring}

to the same

animal in a block-iterative

procedure.

Direct and maternal effects of each animal are then

_{computed jointly}

instead of

_separately

as in

[42]

and

(43!.

The behavior of

both

_strategies

needs to be

_compared

in real-life situations. This was not considered here.

When the models

describing

each trait

differ,

three alternatives are

_proposed.

The first one relies on the same idea

_developed

in the case with more than one

random effect: one can isolate in the mixed model

equations

a block on which

the canonical transformation can be

_applied.

Most

often,

this will

_represent

by

far

the

largest block, corresponding

to additive

_genetic

values.

_However,

this

_implies

a

specific coding

for the solution of the

_remaining

multivariate

part

[46].

The second

_approach

_(Gengler

and

Misztal,

1996)

has the

_advantage

of

_being

applicable

without additional

coding.

Its drawbacks are the

large

increase in the

size of the data file

_(the

initial size is

_{multiplied by}

as _{many times} as there are

distinct

_models)

and the slower convergence

resulting

from the

large

number of

missing

records whose values have to be

_computed

at each iteration. It is not known whether convergence is

always

guaranteed

in

practice.

The third alternative consists of

_creating

a

complete

model

including

all fixed

effects of interest and

constraining

to zero the solutions of the unwanted effects

for each trait.

_Equation

_[67]

illustrates the fact that this constraint is _easy to

the same rate as when the

_complete

model is

_applied

without constraints. It should be

_{noted, however,}

that this

_{approach imposes}

a block-iterative solution of the

fixed effects one block at a

_time,

which sometimes _may be much less efficient than a

simultaneous solution of all fixed

_effects,

for

_{example using}

direct methods.

_Again,

it was not our intention to

_actually

_compare the

_{practical performances}

of these

approaches

because each one _may be valuable in a

_particular

context.

CONCLUSION

Multiple

trait BLUP

_equations

have been known for a

_long

time

_(Henderson

and

_Quaas,

₁₉₇₆₎

but

_despite

the desirable theoretical

_properties

of the

_resulting

predictions

and the

_huge

progress in

computing

technology,

their

_large

scale

application

is far from

_systematic,

_mainly

because

_computing

time remains a

limiting

factor. Some of the

_{strategies presented}

here _may lack the

_flexibility

or

simplicity

required

for _easy

_{implementation}

in a

_general

_purpose

_package

for the solution of multivariate mixed model

_equations,

but

_they

_may lead to invaluable

savings

in

_computing

time

_(Ducrocq

and

_Besbes,

_1993;

_Ducrocq,

_1994a,

_b),

_eg, for routine national

_genetic

evaluations. These

_savings

can be further increased

_by

the

use of more efficient

_single-trait

solvers

_(Misztal

and

_Gianola,

_1987;

_Ducrocq,

_1992;

Carabano et

_al,

_1992).

Obviously,

these different _ways to

_generalize

the use of the canonical

transfor-mation can be extended to more

_general

situations with

_{simultaneously missing}

values,

different models and more than one random effect. The

_{strategies proposed}

are

_{complementary}

and can be combined in a

_{straightforward}

manner. One

excep-tion that was not dealt with here is the case when the number of random effects

considered may vary between

traits,

for

_example,

when

_only

some traits _may have

repeated

observations

_(implying

the

prediction

of a

_permanent

environment

_effect)

or _{may be under the} influence of maternal effects.

_However,

the basic idea

_developed

can be extended to such a case: in the initial

_system

of mixed model

_equations,

the blocks

_{corresponding}

to each

particular

random effect can be isolated and a

canon-ical transformation can be

_applied

to each of these

_blocks,

_through

the matrix

manipulation

described in

_!13!.

ACKNOWLEDGMENTS

The authors are

grateful

to R

_Thompson,

RL

_Quaas,

M Goddard and JJ Colleau for their useful comments and

_suggestions

on the

_topic.

REFERENCES

Arnason T

₍₁₉₈₆₎

Practical _aspects in the estimation of

breeding

values for multi-trait selection. World Rev Anim

_{Prod, 22,}

75-84

Carabafio

_{MJ, Najari S, Jurado,}

JJ

_{(1992) Solving}

_iteratively

the Mixed Model

_equations.

Genetic and numerical criteria. In:

_43rd

Annual

_{Meeting of}

the

_{EAAP, Madrid, Spain,}

14-17

_{September 1992,}

vol I

_(Abstr)

Ducrocq

V

₍₁₉₉₂₎

_Solving

animal model

_{equations through}

an

_{approximate incomplete}

Cholesky decomposition.

Gen Sel Evol

_24,

193-209

Ducrocq

V

_(1994a)

_{Approches algorithmiques}

du mod6le animal. In: Seminaire Modèle

animal, Department

de

_{g4!n!tique animale, Inra,}

La

_{Colle-sv,r-Loup, France,}

26-29

septembre,

(JL

Foulley,

M

_Mol6nat,

_eds),

_{Inra, France,}

13-24

Ducrocq

V

_(1994b)

_Multiple

trait

_{prediction: Principles}

and

_problems.

In: 5th World

_Congr

Genet

_Appl

Livest

_{Prod, Guelph,}

7-12

_August

1994, University

of

Guelph, Guelph,

vol

XVIII,

5WCGALP

Organizing Commitee,

455-462

Ducrocq V,

Besbes B

₍₁₉₉₃₎

Solution of

_multiple

trait animal models with

_missing

data on some traits. J Anim Breed Genet

_110,

81-92

Foulley JL,

Calomiti

_S,

Gianola D

₍₁₉₈₂₎

Ecriture des

equations

du

_blup

multicaract!res. Ann G6n6t Sel Anim 14, 309-326

Gengler

N,

Misztal I

₍₁₉₉₆₎

_{Approximation}

of

_reliability

for

_{multiple-trait}

animal models with

_missing

data

_by

canonical transformation. J

_Dairy

Sci

_79,

317-328

Groeneveld

_E,

Kovac M

₍₁₉₉₀₎

A

_{generalized computing procedure}

for

_setting

up and

solving

mixed linear models. J

Dairy

Sci

73,

513-531

Groeneveld

E,

Kovac M

₍₁₉₉₂₎

Performance characteristics of different

_{solving strategies}

in multivariate mixed models. Livest Prod Sci

30,

319-331

Groeneveld

_E,

Kovac

_{M, Wang}

T

₍₁₉₉₀₎

_PEST,

a

_general

_purpose BLUP

_package

for

multivariate

_prediction

and estimation. In:

_l!th

World

_Congr

Genet

_A

_PP

_I

Livest

_Prod,

Edinburgh, UK,

23-27

_July

1990

_(WG

Hill,

R

_Thompson,

JA

Woolliams,

eds),

vol

_XIII,

488-491

Hackbusch W

₍₁₉₉₄₎

Iterative Solution

_{of Large Sparse Systems of Equations.}

Springer-Verlag,

New-York

Henderson

_{CR, Quaas}

RL

₍₁₉₇₆₎

Multiple

trait evaluation

_using

relatives’ records. J Anim Sci 43, 1188-1197.

Jensen

_J,

Mao I

₍₁₉₈₈₎

Transformation

_algorithms

in

_analysis

of

_single

trait and of

multiple

trait models with

equal design

matrices and one random effect. J Anim Sci

_66,

2750-2761

Lin

_CY,

Smith SP

₍₁₉₉₀₎

Transformation of multitrait to unitrait mixed model

_analysis

of data with

multiple

random effects. J

_Dairy

Sci

_73,

2494-2502

Meyer

K

₍₁₉₈₅₎

Maximum likelihood estimation of variance components for a multivariate mixed model with

equal design

matrices. Biometrics 41, 153-165

Misztal

I,

Gianola D

₍₁₉₈₇₎

_{Extrapolation}

and _{convergence criteria} with Jacobi and Gauss-Seidel iteration in animal models. J

_Dairy

Sci

70,

2577-2584

Misztal

I, Weigel K,

Lawlor T

₍₁₉₉₅₎

_{Approximation}

of estimates of

_(co)variance

com-ponents with

_{multiple-trait}

restricted maximum likelihood

_{by multiple diagonalization}

for more than one random effect. J

_Dairy

Sci

_78,

1862-1872

Pollak

_{EJ, Quaas}

RL

₍₁₉₈₂₎

Alternative _strategy for

_{building multiple}

trait mixed model

equations.

J

_Dairy

Sci 65, 102

_{(suppl 1)}

Quaas

RL

₍₁₉₈₈₎

Additive

genetic

model with groups and

relationships.

J

Dairy

Sci

71,

1338-1345

Quaas

RL

₍₁₉₈₄₎

Linear

prediction

in BLUP school handbook. In: Use

_of

Mixed Models

for

Prediction and

_for

Estimation

_of

_(Co)variance

_Components,

5-7

_{February 1984,}

Animal Genetics and

_{Breeding Unit,}

Univ of New

England,

New South Wales

Quaas RL,

Pollak EJ

₍₁₉₈₁₎

Mixed model

_methodology

for farm and ranch beef cattle

testing

programs. J Anim Sci 51, 1277-1287

Reents

R,

Swalve H

₍₁₉₉₁₎

_Convergence

in multi-trait animal models. In:

_42nd

Annual

Meeting of

the

_{EAAP, Berlin,}

8-12

_February,

vol

I,

160

_(Abstr)

Schaeffer

_{LR, Kennedy}

BW

₍₁₉₈₆₎

_Computing

solutions to mixed model

_equations.

In:

3rd World

_Congr

Genet

_Appl

Livest

_{Prod, Lincoln, Nebraska, USA,}

16-22

_July

1986,

vol

_XII,

382-393

Thompson R, Meyer

K

₍₁₉₈₆₎

A review of theoretical _aspects in the estimation of

_breeding

values for multi-trait selection. Livest Prod Sci

_15,

299-313

van der Werf

J,

van Arendonk

J,

de Vries A

(1992)

_Improving

selection of

_{pigs using}

correlated characters. In:

43rd

Annual

_{Meeting of the EAAP, Madrid, 14-17 September}

1992,

vol

_I,

14

_(Abstr)

APPENDIX

Numerical

_example

for the

_algorithm

_using

constraints on _unecessary effects

Consider a

_joint

analysis

of

_growth

traits and leanness in

_{turkeys. Body weight}

is measured at 12

_(BW12)

and 16

_(BW16)

weeks of age. Leanness is assessed

through

the measure of backfat thickness

_using

an ultrasonic

_probe

(ultrasonic

backfat

_thickness,

_UBT).

The

_description

model used for

_{body weights}

includes a

contemporary

group

(hatch)

as

only

fixed

effect,

while the UBT measure is also

affected

by

the

operator

effect. Consider the

_following

records for four

_{turkeys A,}

B,

C and D:

The incidence matrix F for the

_operator

effect is:

The matrix

_To

is

1

Applying

a block-iterative

_strategy,

the solutions of the batch and animal effects are obtained

through

a

_regular

canonical

decomposition

and the solution of the

operator

effect

_f

_Q

on the transformed scale is

_{given by}

_[66]:

The solutions on the

original

scale are obtained

_by

_simple

back transformation.

In

_particular

f=

(

Q -

0

I

f

)

f

Q

: