A simple method of computing restricted best linear unbiased prediction of breeding values

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A simple method of computing restricted best linear

unbiased prediction of breeding values

Masahiro Satoh

To cite this version:

Original

article

A

_simple

method

of

_computing

restricted

best linear unbiased

_prediction

of

_breeding

values

Masahiro

Satoh

Department

of Animal

Breeding

and

Genetics,

National Institute of Animal

_Industry,

P.O. Box

_5,

Tsukuba

_{Norin-kenkyudanchi,}

Tsukuba-shi 305,

Japan

(Received

6

_{May 1997; accepted}

12

_February

₁₉₉₈₎

Abstract - Restricted best linear unbiased

_prediction

_{(restricted BLUP)}

is derived

_by

imposing

restrictions

_directly

within a

_multiple

trait mixed model. As a

_result,

the

restricted BLUP

_{procedure requires}

the solution of

_high

order simultaneous

_equations.

In the _present paper, a

_simple

method for

_computing

restricted BLUP of

_breeding

values is

_presented.

The

_technique

is

_{valuable, particularly}

when a

_large

number of restrictions are

_imposed

in a

_multiple

trait mixed model such as constraints of

_{achieving predetermined}

relative rates of

_genetic

improvement for all traits.

_©

_{Inra/Elsevier,}

Paris

mixed model method

_/

restricted BLUP

Résumé - Méthode _simple de calcul d’un BLUP restreint. Le BLUP restreint est calculé directement en _posant les contraintes en sus des

équations correspondantes

à un modèle

linéaire mixte multivariate. En

_{conséquence,}

la

procédure

du BLUP restreint demande la résolution d’un

_{plus grand}

nombre

d’équations

que dans le cas habituel. Dans cet

_article,

on

_présente

un

_{reparamétrage qui}

_permet d’aboutir à un

_{système plus simple}

et de taille réduite. Cette

_technique

est

_{particulièrement}

intéressante

_quand

un

_grand

nombre de

restrictions est

_imposé

dans un modèle mixte

_{multivariate,}

comme

_quand

on cherche à

obtenir des _rapport

_{prédéterminés}

de

_{progrès génétiques}

pour tous les caractères.

©

Inra/Elsevier,

Paris

modèle mixte multivariate

_/

BLUP restreint

1. INTRODUCTION

Kempthorne

and

_Nordskog

_[9]

_gave the basic derivation of restricted selection

indices. The model assumed was that the observation _{vector y,} is y = f _{+ u +}

e, u

and e are multivariates

normally

distributed with

E(u)

=

E(e)

=

0,

and f is fixed

and assumed known.

_Var(u)

_{= IQ}

_9Go

_,

_var(e)

=

o

,

9

R

IQ

and

cov(u, e’)

=

_0,

where

Go

is the

_genetic

variance-covariance

_matrix,

_R

_o

is the environmental variance-covariance

_matrix,

and Q9 is the direct

_product

operation.

They

were interested in

maximizing improvement

in

_m’u

_i

_,

but at the same time not

_altering

the

_expected

Cou

i

,

in the candidate for

_selection,

where m is a vector of relative

_weights,

ui is

the subvector of u

_pertaining

to the ith

_animal,

and

C’0

has r

_linearly

_independent

rows.

_{They proved}

that such a restricted selection index is

b’y

i

,

where _yi is the

subvector of _y

_pertaining

to the ith

_animal,

and b is the solution to

Index

_theory

was further extended to include various restrictions

_by

Mallard

_!11),

Harville

_{[3, 4),}

among others. In

practical

applications,

however,

large

data sets with unknown means and related animals render restricted selection index

_predictors

of

breeding

values

impossible

to

compute.

(auaas

and Henderson

_{[13, 14]}

extended the

BLUP

_procedure

of Henderson

_[5]

to allow estimation of

_breeding

values

_including

restrictions for no

_genetic

_change

_among correlated traits

_{(restricted BLUP).}

Restricted BLUP was derived

_{by imposing}

restrictions

directly

on the

multiple

trait mixed model

_{equations. Consequently,}

the restricted BLUP

_{procedure requires}

solutions of

_high

order simultaneous

_equations,

_particularly

when a

_large

number

of animals are evaluated for many traits. For this reason,

computational techniques

have been studied for

computing

restricted BLUP. Lin

_[10]

showed how restricted

BLUP of

_breeding

values can be estimated not

_only

for zero

change

but also for

proportional change

in restricted traits. It _was,

_however,

assumed that the variance-covariance matrix _among

_predicted

_breeding

values was the same as that among

true

_breeding

values. This

_approach

adds bias when estimates of

_genetic

variances and covariances are used instead of the true

_parameters.

The

_{assumption ignores}

the effect of

differing

accuracies of

prediction

of individual

_breeding

values for each

animal, particularly

when animal models are fitted to

_large

unbalanced field data

sets

_!15).

Itoh and Iwaisaki

_[7]

found that a canonical transformation of the traits to new

_independent

variables was

_{possible and,}

_{consequently, only}

mixed model

equations

of

_relatively

smaller order for each trait need to be solved.

_However,

this

is

_{applicable only}

to an animal model with identical models for all traits and no

partially missing

observations.

The

_objectives

of the

present

paper are to show a

_{simple procedure}

for

_computing

restricted BLUP of

_breeding

values and to discuss its

application.

2. THEORETICAL APPROACH

2.1. Theoretical

_background

An additive

_genetic

mixed animal model for _{q traits} is assumed. The model for

the ith trait is written as:

vector of unknown random additive

_genetic

_effects,

Z

i

is a known incidence matrix

relating

elements of ui to yi, and ei is a vector of random errors.

_{Let n}

_j

be the

number of records on the

jth animal; j

=

1, 2, ... , n and

0 x n

j

x

q. The model

for all traits is written as: -- - _ _

where records are ordered

_by

animals within traits. It is assumed that u and e are multivariates

_normally

distributed with

_E(u)

=

0,

E(e)

=

v0,

var(u)

=

G,

var(e)

=

R,

and

cov(u, e’)

=

_0;

G =

Go

_Q9

A,

where

Go

is

a q x

q additive genetic

variance-covariance matrix for

_the

_{q traits,}

A is the additive

_relationship

matrix for the n

_animals,

and R is an n x

_n(n

=

En

i

)

error variance-covariance matrix for the q traits for the n animals.

Let the set of restrictions on u be C’u. If the same constraints are

_imposed

on the additive

_genetic

values of all

animals,

C =

o

C

_Q9

I

m

where

C

o

is a q x r matrix with full column

_rank;

m is the number of animals

_represented

in u. The number of columns of

C

o

,

_r,

depends

on the number and

_type

of

constraints

_imposed:

no

_change

and/or

_{proportional change.}

This will be illustrated

subsequently.

Kempthorne

and

_Nordskog

_[9]

defined

_C

_o

for no

_change

constraints.

For

_example,

if the restriction is no

change

for the first two

_{traits, C}

_o

_might

be

L !

Here r is the number of traits constrained. For the case of

_proportional

con-straints

_(involving

_{2 ! p <}

q

traits),

define:

Then let

_C’

₌

_{[C’ 0<p-i> x <q-p>)}

where ci is a

_{predetermined}

_{proportional change}

for traits

_{1, 2,}

_{..., p}

[8, 11!.

Note

that r =

p - 1.

Furthermore,

if constraints include no

change

and

proportional

change,

C

o

is z columns whose elements are

_unity

or zero in addition to

_Cp,

and then r = _{z + p -} 1. For

example,

if we want no

_change

in trait 1 but

_proportional

change

in traits

_2,

3 and 4 is desired based on

proportional

constraints in the ratio

2:3:4, C

o

is

_expressed

as

The restricted BULP of _u,

_u,

is obtained

_{by solving}

the

_following

_equations

_!13!:

2.2. Restriction for

_general

case

Premultiplying

the second

_equation

in

₍₃₎

_by

C’G

and then

_subtracting

from

this

_product

the third

_{equation gives}

This

implies

that some us are null

_and/or

there are

_simple

linear

_dependencies

among them. These can be

exploited

to reduce the size of the

_problem.

Now we

consider

_imposing

the same restrictions on the

_predicted

_breeding

values of all

animals, however,

it is

_possible

to relax the situation. When constraints are

_imposed

on some

_traits,

model

(1)

can be rewritten as

where

_{subscripts Z,}

R and N

_correspond

to z characters with no

change,

_p characters

with

_proportional

constraints and _q-z-p characters without

_{constraint, respectively.}

From

_(4),

Then,

and from

₍₂₎

and

_(5),

where

Using (6)

and

_(7),

where

_Ko

=

[ 0

0

’

0 and

fi =

(u

P

,

uN!’.

Substituting

(8)

into

₍₃₎

and

There are fewer us in

(9)

than us in

(3).

_Equations

(9)

show that

_computation

for

_formulating

BLUP

_equations

is

_relatively

_simpler

than

_(3).

2.3. Constraints with

_change

in some traits restricted to zero

If constraints

_only

include no

_change

in some

_traits,

then

_Kg

is

_simpler

and to

form

₍₉₎

one

_just

deletes rows and columns from

(3)

corresponding

to

_u

_Z

_.

K

o

is

q x

(q - z)

can then be taken as

and

2.4. Constraints for desired

_changes

If constraints are for

_{proportional changes}

(predetermined

relative

changes)

for

all

_traits,

then

and

The size of restricted BLUP

_equations

_{corresponding}

to random additive effects is reduced to that of

_single

trait BLUP.

2.5. Animal model without

_repeated

records

If we denote the

_equations

₍₉₎

as

Wh

=

h,

an

_equivalent

set of

_equations

for B and u is

where

where t is the number of the columns of X. Then because

_(I

_r

Q9

A-

1

)C’G

=

C’G

where

_p’

=

CoG

o

_Q9

1

m

and s =

(I

r

_Q9

A!! )W

(see

Quaas

and Henderson

[13]).

If

C =

Co

_Q9

1

m

,

then _we have ZZ’ = I and

eliminating

s from

(10),

we obtain

where S =

Z’R-

1

Z -

Z’R-!ZP(P’Z’R-!ZP) - P’Z’R !Z

[13]

and Z is _rows of I

corresponding

to

_missing

records are deleted if there are

_missing

records. Matrix S

has

simple

forms and the calculation of their elements is easy. For

example,

S has

the form

-

-1 -1

’1’1-where each

_D

_ij

is a m x m

_diagonal

matrix.

_Suppose

that

_d

_ijk

is the kth element of

_D

_ij

_,

then

_d

_ijk

is the

_ijth

element of

_S

_k

_,

that

_is,

a matrix

_peculiar

to the kth

animal,

and

where

H!

is a _q x _q matrix

_peculiar

to the kth animal. As shown

_by

Quaas

and Henderson

_(13!,

it is

_computed

as follows:

1)

if the animal has no records

_missing,

H!

=

Ro

1

,

where

R

o

is

the

q x q error variance-covariance

_matrix;

2)

if the animal has all records

_{missing, H}

k

=

_0;

3)

otherwise,

find the inverse of the elements of

_R

_o

_pertaining

to records that

are

_present

and fill out the

_remaining

elements with zeros for the other elements of

H

k

.

3. NUMERICAL EXAMPLE

A numerical

_example

obtained from the

_study

of Henderson and

_Quaas

_[6]

is

used to illustrate the method. Data on five animals for birth

weight

(BW),

weaning

The

_genetic

and error variance-covariance matrices are

and

respectively.

The fixed effects in the model are a common mean for

_BW,

and season

of birth for WW and FG.

_{Consequently,}

and

Because all animals are assumed to have all records for the three

_traits,

then

The additive

_{genetic relationship}

matrix is

Suppose

that the restrictions are for no

_genetic

change

in BW and for desired

changes

in WW and FG which are one

_genetic

standard deviation

_unit,

_namely

23.79:0.1661. Then

First,

the direct solution of restricted BLUP will be shown. Matrices

X’R-

1

X,

X’R-1

Z,

X’R-

1

ZGC,

Z’R-

1

Z

+

,

1

G-

Z’R-

1

ZGC

and

C’GZ’R-

1

ZGC

of

Vectors

_X’R-

_l

_y,

Z’R-

1

Z

and

_C’GZ’R-

_l

_y

of

_equations

₍₃₎

are

X’R-’y = [6.7868

0.4663 0.6401 161.6266

_22.2007]’,

Z

/

R-l

y

= [

1

.

1309

1.3997 1.3764 1.5732 1.3068 0.2254 0.2409 0.2032 0.2383 0.1986 79.1971 82.4295 72.2491 80.3907

_69.5610]’,

C’GZ’R-l

y

=

[143.7809

137.3768 124.4704 112.4962 - 18.6050 - 18.7419 - 17.7570 - 14.9453 -

13.9514]’.

Because rank

_{(X’ZSZ’X)}

=

2,

solving

the

equations requires finding

a

general-ized inverse of

₍₄₎

_{which may be obtained}

by setting appropriate

columns of Z’X

to 0 to remove all linear

_dependencies

and then

_inverting

the non-null

_portion.

For

this

example, X

l

and

X

2

are assumed to be 0. From these

_equations,

solutions are as shown in table I.

Next,

the

technique

developed

here will be shown. From

_(9),

then matrices

_X’R-’ZK,

K’Z’R-

1

ZK

+

K’G-

I

K

and

K’Z’R-

1

ZGC

of

The solutions for FG

_give

the

_{predicted breeding}

values which are identical to

table I. From

₍₆₎

and

_(7),

u of BW = ₀ and u of WW = 143.227 _x u of

_FG.,

which

are identical to table 1.

In this

_{example, equations}

_[11]

are useful.

then,

the matrix on the left hand side of

_equations

(11)

_presented

in

(14)

were

obtained

_by

_removing

rows and columns of all linear

_{dependencies.}

We also obtain the

_{expected breeding}

values which are identical to

₍₁₃₎

from the

solutions

_using

_(14).

4. DISCUSSION

Henderson and

Quaas

[6]

derived BLUP of

_breeding

values for

_multiple

traits

using

records on a

_large

number of relatives. Restricted BLUP was derived

_by

imposing

restrictions on

_multiple

trait BLUP

_!13).

_Hence,

in restricted

_BLUP,

the

computing

load to obtain estimates of

_breeding

values can be

huge.

Itoh and Iwaisaki

_[7]

showed that a canonical transformation

technique

was

an animal model.

_However,

the method has a limitation in that models must

be identical for all traits and there must be no

_partially

_missing

observations.

Hence,

the canonical transformation

_technique

can be used

only

if models and

data structure conform to the above conditions.

Various restricted selection index theories have been

_presented

since

_Kempthorne

and

_Nordskog

_!9!.

_However,

_only

two

_types

of constraints have been used

_practically

for

selection,

i.e. zero

changes

in one or a few traits and

_{proportional changes}

for all traits

_(e.g.

_Hagger

_[2]).

Now if all constraints are

_zero-change

for some traits

and m is several millions even

_removing

a u

_equation

for a set of constraints

_might

be useful. If

_{proportional changes}

are

_imposed

for all

_traits,

the size of

_equations

corresponding

to random additive

_genetic

effects is much reduced. The

_technique

developed

here needs no conditions to be

_applied

and reduced the number of sets of

equations

corresponding

to random additive effects

_from

_{q to}

_q-rank(C

_o

_).

Hence,

if a

large

number of restrictions is

_imposed

in a model such as a constraint of

_achieving

predetermined

relative

_changes

for all traits

_[1,

12,

16],

the size of

_equations

for random additive effects is the same as that of a

_single

trait model.

REFERENCES

[1]

Brascamp E.W.,

Selection indices with

constraints,

Anim. Breed. Abstr. 52

₍₁₉₈₄₎

645-654.

[2]

Hagger C.,

Two

_generations

of selection on restricted best linear unbiased

prediction

breeding

values for income minus feed cost in

_{laying hens,}

J. Anim. Sci. 70

₍₁₉₉₂₎

2045-2052

[3]

Harville

_{D.A., Optimal procedures}

for some constrained selection index

_problems,

J. Am. Stat. Assoc. 69

₍₁₉₇₄₎

446-452.

[4]

Harville

_D.A.,

Index selection with

proportionality constraints,

Biometrics 31

₍₁₉₇₅₎

223-225.

[5]

Henderson

C.R.,

Sire evaluation and

_{genetic trends,}

in:

_Proceedings

of the Animal

Breeding

and Genetics

_Symposium

in honor of Dr JL

_{Lush, Blacksburg, VA, August 1973,}

American

_Society

of Animal

_{Science, Champaign, IL, 1973,}

pp. 10-41.

[6]

Henderson

_{C.R., Quaas R.L., Multiple}

trait evaluation

_using

relatives’

_records,

J. Anim. Sci. 43

₍₁₉₇₆₎

1188-1197.

[7]

Itoh

_Y.,

Iwaisaki

_H.,

Restricted best linear unbiased

prediction using

canonical

transformation,

Genet. Sel. Evol. 22

₍₁₉₉₀₎

339-347.

[8]

Itoh

Y.,

Yamada

Y., Comparisons

of selection indices

achieving predetermined

proportional gains,

Genet Sel. Evol. 19

₍₁₉₈₇₎

69-82.

[9]

Kempthorne 0., Nordskog A.W.,

Restricted selection

_indices,

Biometrics 15

₍₁₉₅₉₎

10-19.

[10]

Lin

_C.Y.,

A unified

_procedure

of

_computing

restricted best linear unbiased

prediction

and restricted selection

_index,

J. Anim. Breed. Genet. 107

₍₁₉₉₀₎

1-316.

[11]

Mallard

J.,

The

theory

and

computation

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critical

_synthesis,

Biometrics 28

₍₁₉₇₂₎

713-735.

[13]

(auaas R.L.,

Henderson

_C.R.,

Restricted best linear unbiased

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_breeding

values, Mimeo,

Cornell

_{Univ., Ithaca, NY, 1976,}

_{pp. 1-14.}

[14]

Quaas R.L.,

Henderson

_C.R.,

Selection criteria for

_altering

the

_growth

curve,

J. Anim. Sci.

_(abstr)

43

₍₁₉₇₆₎

221.

[15]

Schneeberger M.,

Barwick

S.A.,

Crow

G.H.,

Hammond

K.,

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using

breeding

values

_{predicted by BLUP,}

J. Anim. Breed. Genet. 109

₍₁₉₉₂₎

180-187.