Inference about multiplicative heteroskedastic components of variance in a mixed linear Gaussian model with an application to beef cattle breeding

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Inference about multiplicative heteroskedastic

components of variance in a mixed linear Gaussian

model with an application to beef cattle breeding

M San Cristobal, Jl Foulley, E Manfredi

To cite this version:

Original

article

Inference about

_{multiplicative}

heteroskedastic

_components

of variance

in

a mixed

linear

Gaussian

model with

an

application

to

beef cattle

_breeding

M

San

Cristobal

JL

_Foulley

E Manfredi

1

INRA, Station de Genetique Quantitative et Appliquée,

78352 _{Jouy-en-Josas Cedex;}

2

INRA, Station d’Amelioration G6n6tique des _{Animaux, BP, 27,} 31326 Castanet-Tolosan Cedex, France

(Received

28 _{April 1992 ; accepted} 23 _September

₁₉₉₂₎

Summary - A statistical method for _{identifying meaningful} sources of _{heterogeneity} of residual and genetic variances in mixed linear Gaussian models is presented. The method is

based on a structural linear model for _log variances. Inference about dispersion parameters is based on the

marginal

likelihood after _integrating out location _parameters. A likelihood ratio test _using the

_marginal

likelihood is also _proposed to test for

_hypotheses

about sources of variation involved. A _Bayesian extension of the estimation procedure of the dispersion parameters is _presented which consists of _determining the mode of their

_marginal

_posterior distribution _{using log} inverted _chi-square or Gaussian distributions as _priors. Procedures

presented in the paper are illustrated with the _analysis of muscle development scores

at _weaning of 8575 progeny of 142 sires in the Maine-Anjou breed. In this

analysis,

heteroskedasticity is _found, both for the sire and residual _components of variance.

heteroskedasticity / mixed linear _{model / Bayesian} technique

R.ésumé - Inférence sur une _{hétérogénéité multiplicative} des composantes de la

variance dans un modèle linéaire mixte _{gaussien: application} à la sélection des

bovins à viande. Une méthode statistique est _{présentée, capable d’identifier} les sources

significatives d’hétérogénéité de variances résiduelles et _génétiques dans un modèle linéaire mixte gaussien. La méthode est

_fondée

sur un modèle structurel de décomposition du

logarithme des variances. _{L’inférence} concernant les _paramètres de _dispersion est basée sur la vraisemblance _marginale obtenue _{après intégration} des _paramètres de position. Un

*

Correspondence

and reprints

**

test du rapport des vraisemblances utilisant la vraisemblance _marginale est aussi _proposé

afin de tester des hypothèses sur _différentes sources de variation. Une extension _bayésienne

de la procédure d’estimation des _paramètres de _dispersion est _présentée; elle consiste en la maximisation de leur distribution _marginale a _{posteriori, pour} des distributions a _priori

log x

2

inverse ou _gaussienne. Les _{procédures présentées} dans ce _papier sont illustrées _par

l’analyse de notes de _pointages sur le _{développement} musculaire au _sevrage de 8 575 _jeunes veaux de race _Maine-Anjou, issus de _{142 pères.} Dans cette _analyse, une hétéroscédasticité a été trouvée sur les composantes père et résiduelle de la variance.

hétéroscédasticité _/ modèles linéaires mixtes _{/ techniques bayésiennes}

INTRODUCTION

One of the main concerns of

_{quantitative geneticists}

lies in evaluation of individuals

for selection. The statistical framework to achieve that is

_nowadays

the mixed linear

model

_{(Searle, 1971),}

_usually

under the

_assumptions

of

_normality

and

_homogeneity

of variances. The estimation of the location _parameters is

_performed

with BLUE-BLUP

_(Best

Linear Unbiased

_{Estimation-Prediction),}

_leading

to the well-known Mixed Model

_Equations

_(MME)

of Henderson

_(1973),

and REML

_(acronym

for REstricted -or REsidual- Maximum

_Likelihood)

turns out to be the method of

choice for

_estimating

variance _components

_(Patterson

and

_Thompson,

_1971):

However,

heterogeneous

variances are often encountered in

_practice,

eg for milk

yield

in cattle

_(Hill

et

al,

1983; Meinert et

al,

1988;

Dong

and

Mao, 1990;

Visscher et

_al,

_1991;

_Weigel,

₁₉₉₂₎

for meat traits in swine

_(Tholen,

₁₉₉₀₎

and for

_growth

performance

in beef cattle

_(Garrick

et

_al,

_1989).

This

_{heterogeneity}

of

_variances,

also called

_{heteroskedasticity}

_{(McCullogh, 1985),}

can be due to _many

_factors,

_eg

management

level,

genotype x environment _{interactions,}

_{segregating major}

_genes,

preferential

treatments

_(Visscher

et

al,

1991).

Ignoring heterogeneity

of variance may reduce the

_reliability

of

_ranking

and

selection

_procedures

_although,

in cattle for _instance, dam evaluation is

_likely

to

be more affected than sire evaluation

(Hill,

_1984;

Vinson,

_1987; Winkelman and

Schaeffer,

1988).

To overcome this

problem,

3 main alternatives are

possible.

First,

a

transfor-mation of data can be

performed

in order to match the usual

_assumption

of

ho-mogeneity

of variance. A

_log

transformation was

proposed by

several authors in

quantitative

genetics

(see

eg Everett and

Keown,

1984; De Veer and Van

Vleck,

1987; Short et

al, 1990,

for milk

_production

traits in

_cattle).

However,

while ge-netic variances tend to

_stabilize,

residual variances of

_{log-transformed}

records are

larger

in herds with the lowest

_production

level

_(De

Veer and Van

Vleck,

1987; Boldman and

_Freeman,

_1990; Visscher et

al,

1991).

’

The second alternative is to

_develop

robust methods which are insensitive to

moderate

_{heteroskedasticity}

_{(Brown, 1982).}

The last choice is to take

_{heteroskedasticity}

into account. Factors

_(eg

_region,

herd,

year,

parity,

sex)

to

adjust

for

heterogeneous

variances can be identified. But

such a stratification _generates a _very

_large

number of cells

_{(800 000}

levels of herd x _{year in} the French Holstein

_file)

with obvious

_problems

of

_{estimability.}

Hence,

via a

modelling

(or structural)

_approach

so as to reduce the _{parameter space,}

_by

appropriate

identification and

_testing

of

_meaningful

sources of variation of such

variances.

The model for the variance _components is described in the Model section. Model

fitting

and estimation of _parameters based on

_marginal

likelihood

_procedures

are

presented

in the Estimation

_{of Parameters,}

followed

_by

a test statistic in

_Hypothesis

Testing.

A

_Bayesian

alternative to maximum

_marginal

likelihood estimation is

presented

in A

_Bayesian

_Approach

to a Mixed Model Structure In the Numerical

application

section,

data on French beef cattle are

_analyzed

to illustrate the

procedures

given

in _{the paper.}

_Finally,

some comments on the

_methodology

are

made in the Discussion and Conclusion.

MODEL

Following Foulley

et al

_(1990,

₁₉₉₂₎

and Gianola et al

_(1992),

the

_population

is

assumed to be stratified into I

_{subpopulations,}

or strata

_(indexed

_by

i =

_{1, 2, ... ,}

I)

with an

(n

i

x

₁₎

data vector _yi,

_sampled

from

a

normal distribution

_having

mean

i ii and variance

R.

i

=

a2 ei I&dquo;

i

.

Given ii

i

and

_R

_i

Following

Henderson

_(1973),

the vector _IIi is

_{decomposed according}

to a linear

mixed model structure:

where

i

X

and _Z; are

(n

i

x p)

and

x q

(n

i

i

)

incidence

matrices,

corresponding

to fixed

J3 (p

x 1 )

and random

_u

_i

_(q

_i

_{x 1 )}

effects

_{respectively.}

Fixed effects can be factors or

covariates,

but it is assumed in the

_following

that,

without loss of

_{generality, they}

represent factors.

In the animal

_breeding

_context, ui is the vector of

_genetic

merits

_pertaining

to

breeding

individuals used

_(sires

_{spread by}

artificial

_{insemination)}

or _present

(males

and

_females)

in stratum i. These individuals are related via the so-called numerator

relationship

matrix

_A

_i

_,

which is assumed known and

_positive

definite

_(of

rank

_q

_i

_).

Elements of ui are not

_usually

the same from one stratum to another. A borderline case is the &dquo;animal&dquo; model

_((auaas

and

_Pollak,

₁₉₈₀₎

where animals

with records are

completely

different from one herd to another.

_{Nevertheless,}

such individuals are

genetically

related across herds.

Therefore,

model

[3]

has to

be refined to take into account _{covariances among} elements of different

_u!s.

As

proposed by

Gianola et al

_(1992),

this can be

accomplished by relating

Ui to a

with A

_being

the overall

_relationship

matrix of rank q,

relating the

q breeding

I

animals involved in the whole

_population,

with _{q x}

_L

qj.

i=l

Thus,

S

i

is an incidence matrix with 0 and 1 elements

_{relating the q}

_i

levels of

u

* _present in the ith

_{subpopulation}

to the whole vector

_(q

x

₁₎

of u elements. For

instance, if stratification is made

_by

herd

level,

the matrices

S

i

and

_S

_i’

_{(i !}

_i’)

do not share _any non-zero elements in their

columns,

since animals

_usually

have

records

_only

in one herd. On the _contrary, in a sire

_model,

a

_given

sire k _may have

progeny in 2 different herds

(i,

i’)

thus

resulting

in ones in both kth columns of

S

i

and

_Si.

Notice that in this

model,

any genotype x stratum interaction is due

_entirely

to

scaling

(Gianola

et

_al,

_1992).

Formulae

_[2],

_(3!,

_[4]

and

_[5]

define the model for means; a further _step consists

in

_modelling

variance _components

_{{!e! !i=1,...1}

_and

_{{Q!. },!=1,...t}

in a similar _way,

ie

_using

a structural model.

’ ’

The

_approach

taken here comes from the

_theory

of

generalized

linear models

involving

the use of a link function so as to _express the transformed _parameters with a linear

_predictor

_(McCullagh

and

_Nelder,

_1989).

For variances, a common

and convenient choice is the

_log

link function

_(Aitkin,

1987;

Box and

Meyer, 1986;

Leonard,

1975;

Nair and

_Pregibon,

_1988):

where

_wey

and

_{w’ .}

are incidence row vectors of size

k

e

and

_k

_u

_,

_{respectively,}

corresponding

to

_dispersion

_{parameters fg} and _!u. These incidence vectors can

be a subset of the factors for the mean in

(2!,

but _exogeneous information is also

allowed.

_Equations

_[6]

and

_[7]

define the variance component models. These models can be rewritten in a more _compact form as follows.

Let

y =

(y!,..., y!,...,

y’)’

be the n x 1 vector of data for the whole

_population,

I

with n

= ! ni,

i=l

IIi

xil

3

+

0&dquo;&dquo;izisiu*

11

=

(II!,... ,11:,... ,

ll

’)’

be the mean vector of _y,

I

R

_{= ®}

_Ri be the variance-covariance matrix of _y, with ?

representing

the

i=l

direct sum

_{(Searle, 1982).}

Equation

[1]

can then be rewritten as:

In the same _way,

[2]

becomes:

X!

the

_(n

_i

x

p)

incidence matrix defined in

_!2J;

Z

_{= (Z1 ,...,ZZ ,...,ZI ) ,}

Z!

=

o,,,iZ

i

S

i

the

(n

i

x

_q)

&dquo;incidence&dquo; matrix

_pertaining

to

u

*

,

T = (X, Z

*

)

and

0 =

Q3’,

U

*

’ I’.

The vector 0 includes _p + q location parameters. The matrix T can be viewed as an &dquo;incidence&dquo;

_matrix,

but which

_depends

here on the

_dispersion

_{parameters T}

u

through

the variances

_Q

_ua.

Both variance models can also be

compactly

written as:

The _ke+ ku

dispersion

parameters !e and y! can be concatenated into a vector

(

T

=

(T!, T!)’

with

_{corresponding}

incidence matrix W =

W

e

EÐ W

u’

The

dispersion

model then reduces to:

where

a

2

=

_{(CF e 2&dquo; cF! 2’ )’}

and

1n a

2

is

a

_symbolic

notation for

_{(In a;}

₁

_’

...

Inaejl 2

In a!1 ’ , .. , In a![)’.

ESTIMATION OF PARAMETERS

In

_{sampling theory,}

a _{way to} eliminate nuisance _parameters is to use the

_marginal

likelihood

_{(Kalbfleisch, 1986). &dquo;Roughly}

_speaking,

the

_suggestion

is to break the data in two _{parts, one, part} whose distribution

_{depends only}

on the _parameter of

interest,

and another _part whose distribution may well

depend

on the _parameter

of interest but which

_will,

in

_{addition, depend}

on the nuisance _parameter.

_!...!

This second _part

_will,

in

_general,

contain information about the _parameter of

interest,

but in such a _way that this information is

inextricably

mixed _up with

the nuisance

_{parameter&dquo;}

_{(Barnard, 1970).}

Patterson and

_Thompson

₍₁₉₇₁₎

used

this

_approach

for

_estimating

variance components in mixed linear Gaussian models. Their derivations were based on error contrasts. The

corresponding

estimator

_(the

so-called

_REML)

takes into account the loss in

_degrees

of freedom due to the

Alternatively,

Harville

₍₁₉₇₄₎

_proved

that REML can be obtained

_using

the

non-informative

_{Bayesian paradigm.}

_According

to the definition of

_{marginalization}

in

Bayesian

inference

_(Box

and

_{Tiao, 1973;}

_Robert,

_1992),

nuisance _parameters are

eliminated

_by

_integrating

them out of the

_{joint posterior}

_density.

Keeping

in mind that the

_sampling

and the non-informative

_{Bayesian approaches}

give

rise to the same estimation

_equations,

we have chosen the

_{Bayesian techniques}

for reasons of coherence and

_simplicity.

The _parameters of interest are here the

dispersion

_{parameters r,} and the location

parameters 6 appear to be nuisance _parameters. Inference is hence based on the

log marginal

likelihood

_L(

_T

_{; y)}

_{of r:}

An

_{estimator y}

of T is

given

by

the mode of

L(

T

;

y):

where r is a _{compact part} of

R

k

e

+k

u.

This maximization can be

_{performed using}

a result

by

_Foulley

et al

_{(1990, 1992)}

which avoids the

_integration

in

_[13].

Details can be found in the

_A

_PP

_endix.

This

procedure

results in an iterative

_{algorithm. Numerically,}

let

_[t]

denote the iteration

t; the current estimate

9

[Hl]

_{of r}

is

_computed

from the

_following

_system:

where

i

lt]

is the current estimate at iteration _t, W the incidence matrix defined in

_!12!,

Q

M

is the

_weight

matrix

_depending

on

0

and on

ê

[t]

,

which are the solution and

the inverse coefficient matrix

_respectively

of the current _system in 0

_(this

_system is described

_next),

z!

is the score vector

_depending

on

6

and

C!.

Elements of

Q

l

’

l

and

i

lt

)

are

_given

in the

Appendix.

The second _system is:

where

i

[t]

is the &dquo;incidence&dquo; matrix T defined in

_[9]

and evaluated at T =

y

[t]

;

ft- 1

1

’]

is the

_weight

matrix evaluated _{at T}

= y[

t]

,

with R defined as in

[8];

E-C

0

0

1

)

and takes into account the

_prior

distribution of u* in

_!5!.

Regarding

computations

involved in

_!15!,

2 _types of

_algorithms

can be considered as in San Cristobal

_(1992).

A second order

_algorithm

_{(Newton-Raphson type)}

converges

rapidly

and

gives

estimates of standard errors of

y,

but

_computing

time

can be excessive with the

_large

data sets

_typical

of animal

_{breeding problems.}

As shown in

_Foulley

et al

_(1990),

a first order

_algorithm

can be

_easily

obtained

_by

approximating

the

_(a

matrix in

_[15]

_by

its _{expectation component}

_(Qa!,E

in the

appendix

notations).

This EM

_{(Expectation-Maximization;}

_Dempster

et

al,

1977)

algorithm

converges more

_slowly,

but needs fewer calculations at each iteration

_and,

on the

_whole,

less total CPU time for

_large

data sets.

HYPOTHESIS TESTING

An

_{adequate modelling}

of

_{heteroskedasticity}

in variance _components

_requires

a

procedure

for

_hypothesis

_testing.

Let

H

o

:

H !

= 0 be the null

hypothesis

with

H

_being

a full

(row)

rank matrix with row size

equal

to the number of

_linearly

independent

estimable functions of _T

_defining

H

o

,

and

H

1

its alternative. For

example,

one can be interested in

_testing

the

_hypothesis

of

_homogeneity

of residual

variances

_H

_o

_:

_{u2 e}

_i

=

exp

(-y,,)

= Const for all i.

Letting

Ye =

f

7

R

,

&dquo;fe2-’7R. - - -, Yel

&dquo;f

R

}

f

with _&dquo;fR

_being

the

_dispersion

_parameter for the residual variance in the first

stratum taken as reference.

_H

_o

can be

_expressed

as

_e

_e

_r

_H

= 0, _or

(H

e

, 0

h

= 0

with

_He

=

(O(

I

-I

)x

l

,,

I

-1

).

Let

_M

_o

and

_Nft

be the models

_{corresponding}

to

_H

_o

and

_H

₁

_,

_{respectively.}

Since

P(

YIT

)

=

e u

the

marginal

likelihood _can be

interpreted

_{as a} likelihood

of error contrasts

_{(Harville, 1974),}

hence the likelihood ratio test based on the

marginal

likelihood can be

applied:

Under

H

o

, A

is

_{asymptotically}

distributed

_according

to

_a

_X

₂

with

_degrees

of freedom

_equal

to the rank of H. In the normal _case,

_explicit

calculation of

_L(

_T

_;

_y)

is

_analytically

feasible:

A BAYESIAN APPROACH TO A MIXED MODEL STRUCTURE

One can be interested to

_generalise

Henderson’s BLUP for subclass means

₍

₁₁

=

T9)

to

_dispersion

_parameters

_{(ln a}

₂

=

W7 )

ie

_proceed

as

if

T

had a mixed model

structure

_(Garrick

and Van

_Vleck,

_1987).

To overcome the

difficulty

of a realistic

interpretation

of fixed and random effects for

_{conceptual populations}

of variances

from a

_frequentist

(sampling)

_perspective,

one can

alternatively

use

Bayesian

procedures.

It is then _{necessary to}

_place

suitable

_prior

distributions on

dispersion

In linear Gaussian

_methodology,

theoretical considerations

_{regarding conjugate}

priors

or fiducial _arguments lead to the use of the inverted _gamma distribution as a

_prior

for a variance

a

2

_(Cox

and

_Hinkley,

_1974;

_Robert,

_1992).

Such a

density

depends

on

hyperparameters

₇₇ and

s

2

.

The former _conveys the so-called

degrees

of

belief,

and the latter is a location _parameter. The ideas

_{briefly exposed}

in the

following

are similar to those described in

_Foulley

et al

_(1992).

Hence,

a

_prior

_{density for y}

=

ln

Q

2

can

be obtained _{as a}

log

inverted

gamma

density.

As a matter of

_fact,

it is more

_interesting

to consider the

_prior

distribution of v =

&dquo;y —

T

°,

with

q°

= In

s

2

,

ie

where

_r(.)

refers to the gamma function.

Let us consider a K-dimensional &dquo;random&dquo; factor v such that

Vk

1

77k

(k

= _{1, ...}

K)

is distributed as a

log

inverted _gamma

’

)

1]k

(

l

InG-

Since the levels of each random

factor are

_{usually exchangeable,}

it is _{assumed that 1]k = 1] for every} k in

_{{1, ... K}:}

For v

k

in

_[20]

small

_enough,

the kernel of the

_product

of

_independent

distributions

having

densities as in

_[19]

can be

_{approximated (using}

a

Taylor expansion

of

[19]

about v

_equal

to

₀₎

_by

a Gaussian

_kernel,

_leading

to the

_following

_prior

for v:

As

_{explained by Foulley}

et al

_(1992),

this

_{parametrization}

allows

_expression

of the _T vector of

_dispersion

parameters under a mixed model _type form.

Briefly,

from

_[19]

one

has

1

=

1

°

_{+ v}

or

1 =

P

’oS

_{+ v} if _one writes the location

parameter

-to

= In

S

2

as a linear function of some vector 8 of

_explanatory

variables

_(p’

_being

a row incidence vector

_{of coefficients).}

_Extending

this

_writing

to several classifications

in v leads to the

_{following general}

expression:

where P and

Q

are incidence matrices

corresponding

to fixed effects E and random effects v,

respectively,

with

[20]

or

[21]

as

_prior

distribution for v.

Regarding dispersion

parameters T, it is then

possible

to

_proceed

as Henderson

(1973)

did for location _{parameters 11, ie} describe them with a mixed model

structure.

_Again,

as illustrated

_by

formula

[22],

the statistical treatment of this

model can be

_{conveniently implemented}

via the

Bayesian

paradigm.

In

_fact,

_equations

_[22]

define a model on residual variances:

where

_{Pe, Pu, Qe, Qu}

are incidence matrices

_{corresponding respectively}

to fixed effects

5

e

, <!

and random effects _ve =

(v!, Ve2, ... ,

v!,...)’,

Vu =

(v!, V

u2,

’

’ ’ ,

vu,

;

...)!

with,

for the

_jth

and kth random classification in ve and vu

respectively,

Let 11 =

(11!, 11!)’

with _11e =

{77ej}

and

1 1

u =

{77Uk}

be the vectors of

hyperparameters

introduced in the variance _component models

_{[23], [24],}

_[25]

and

[26].

An

_{empirical Bayes procedure}

is chosen to estimate the _parameters. The

hyperparameters,

11

(or § =

(!e, !u)’)

are estimated

by

the mode of the

marginal

likelihood of these

_{hyperparameters}

_(Berger,

_{1985; Robert,}

_1992):

Then,

the

_dispersion

_parameters are obtained

by

the mode of the

posterior

density of

T

given

the

_{hyperparameters equal}

to their estimates:

or

_similarly

_for

t.

Maximization in

_[27]

and

_[28]

can be

performed

with a

Newton-Raphson

or an EM

_{algorithm, following}

ideas in the Estimation

_of

_parameters,

_{Unfortunately,}

the

_algorithm

derived from

_[27]

is

_{computationally demanding,}

since it involves

digamma

and

_trigamma

functions. On the other

hand,

an EM

_algorithm

derived

from

_[28]

has the same form as the EM-REML

_algorithm

for variance components.

It

_just

involves the solution and the inverse coefficient matrix of the system in T

at iteration

_(t).

This latter _system is similar to

_(15),

but it takes into account the

informative _prior on the

_dispersion

_parameters. In the case of a Gaussian

_prior,

this

system can be written as

where

r

is the matrix

I- (!) = ( 0

i

.

)

evaluated

at the current estimate

I

of

!,

tanking

into account the

_priors

via

_{A(!) =}

Var

_{(v’, v’)}

=

A

e

?

A!

with

A,

=

0!1!,

and

A!, _

₍₁₎

_I

_K.,,.

i ’ k

NUMERICAL APPLICATION

Sires of French beef breeds are

routinely

evaluated for muscular

development

(MD)

based on

_{phenotypic performance}

of their male and female _progeny.

_Qualified

personnel subjectively classify

the calves at about 8 months of _age, with MD

scores

_ranging

from 0 to 100. Variance _components and sire

_genetic

values are

then estimated

_by

_applying

classical

_procedures,

ie REML and BLUP

_(Henderson,

1973;

Thompson,

1979),

to a mixed model

_including

the random sire effect and a

set of fixed effects described in table I. The second factor listed in table

_I,

condition

score

(&dquo;Condsc&dquo;),

accounts for the

_previous

environmental conditions

_{( eg}

nutrition via

_fatness)

in which calves have been raised.

Some _{factors among those} described in table I _may induce

_{heterogeneous}

variances. In

_particular,

different classifiers are

_expected

to _generate not

_only

different MD means, but different MD variances as well.

Thus,

the usual sire model

with

_assumption

of

_homogeneous

variances _may be

_inadequate.

This

_hypothesis

was

tested on the

Maine-Anjou

breed. After elimination of twins and further

editing

described in table

_I,

the

_Maine-Anjou

file included

_performance

records on 8 575

progeny out of 142 sires

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

recorded in 5

_regions

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

and 7 _years

(&dquo;Year&dquo;).

Other factors taken into account were: sex of calves

_{(&dquo;Sex&dquo;),}

_age at

scoring

(&dquo;Age&dquo;),

claving

parity

(&dquo;Parity&dquo;),

month of birth

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

and classifier

( &dquo;Classi&dquo; ).

In most strata defined as combinations of levels of the

_{previous factors,}

only

one observation was _present.

Preliminary analysis

A

_histogram

of the MD variable can be found in

_figure

1. The distribution of MD

seems close to

_normality,

with a fair

_PP-plot

_(although

the use of this

_procedure

is somewhat

_{controversial),}

and skewness and kurtosis coefficients were estimated as

null

_hypothesis,

while others did not

_reject

_it,

_namely

_Geary’s

u, Pearson’s tests for

skewness and kurtosis

_{(Morice, 1972)}

at the 1% level.

Bartlett’s test for

_homogeneity

of variances was

computed

for each of the first

8 factors described in table I. Results in table IIa indicate strong evidence for heteroskedastic variances _among subclasses of each factor considered in this data set.

The usual sire model with all factors from table I in the mean

model,

and variance _components

_estimated

_by

_EM-REML,

was

fitted,

leading

to estimates

6d

=

70.1l,a,2,

=

6.91,

and

h

2

=

46fl /(6d

₊

3!)

=

0.36. Note that this model is

equivalent,

in our

_notation,

to the

_homogeneous

model in _{fg and Yu.}

Search for a model for the variances

The

_following

additive mean model

_M

_B

was considered as true

_throughout

the whole

analysis

A forward selection of factors _strategy was chosen to find a

good

variance model

My

but in 2 _stages; a backward selection _strategy would have been difficult to

implement

because of the

_large

number of models to _compare and the small amount of information in some strata

_{generated by}

those models.

(i)

since

_a2

_represents

> 90% of the total

variation,

it was decided to model that component

first, assuming

the ru- part

homogenous;

’

(ii)

the &dquo;best&dquo;

_T

_u

_-model

was thereafter chosen while

_{keeping unchanged}

the

&dquo;best&dquo;

_T

_e-model.

The different nested models were fitted

_using

the maximum

marginal

likelihood

ratio test

_{(MLRT) A}

described in

_!17J.

During

the first _stage

_(i),

the

_homogeneous

sire variance was

_estimated,

for

computational

ease, with an EM-REML

_algorithm,

and the _{Te parameter} estimates

were calculated as in

_Foulley

et al

_(1992).

This _strategy

_leads,

of course, to the

same results as those obtained with the

_algorithm

described in the Estimation

_of

parameters.

The first _step consisted of

_choosing

the best one-factor variance model from results

_presented

in table lib. The next _steps, ie the choice of an

_adequate

2-factor

model,

and then of a 3-factor

model,

_etc, are summarised in table III.

Finally,

the

The model can also be

simplified

after

_comparing

estimates of factor

levels,

and

then

_collapsing

these levels if there are not

_{significantly}

different. For the

_(ii)

_stage, the &dquo;best&dquo;

_r

_u

_-model

was the model

(see

table

IV):

We were not able to reach convergence of the iterative

procedure

for the models

(Mo,

M.y

e

,

Classi)

and

_(Mo,

_M!(,,

_Region),

_although

some levels of the Classi factor were

_collapsed.

This

_phenomenon

is related to a _strong unbalance of the

_design:

for

instance,

one classifier noted the calves of

_only

4 _sires,

_making

_quite

_impossible

a

coherent estimation of

_{Classi-heterogeneous}

sire variances. The other factors

_(except

Year)

had no

_significant

effect on the variation of the sire variances. Because of

imbalance,

the model

gave

unsatisfactory

results _eg

_heritability

estimates _greater than one.

Results

Estimates of the

_dispersion

_parameters for the selected model

_designated

here

as

_{(MB, M!.!, M!.!)}

are shown in table Va. As

_expected,

the

_T

_,-estimates

of the

(Mo, M&dquo;

f

c’

homogeneity)

model,

ie of the best

_r

_e

_-model

with

_only

one

_genetic

variance,

are

quite

similar to the

_-estimates

_e

_T

of the

_(Mo,

_M

_7e

_,

_My&dquo;)

model

_(table

Va).

In _contrast,

_T

_e

_-estimates

of the

_{(Mo, M&dquo;}

_f

_c’

_homogeneity)

model,

with:

M!c :

Classi

_(random)

+ Condsc + Year

_(random)

+ Month

_(random)

_[35]

are different for the &dquo;random&dquo; factors

(see

table

Vb).

Estimated

!e,Year

= 0.009 and

!e,Month

= 0.0024

respectively,

or

_{alternatively}

_using

% values of

the coefficient of variation for

_ae,

_(!e

_CV

₂

₎

_Class

_i

_e

_,

_CV

=

14.5%, CV,,

Yar

= 9.5%

and

_CV

_e

_,

_Month

= 4.9%

respectively.

In

fact,

the smaller the cell size

(n

i

),

and the

smaller

CV,

the

greater

the

shrinkage

of the

sample

estimates

(6

f

)

toward the

mean variance

(3 )

since the

_regression

coefficient toward this mean in the

equa-tion

_Q2

_{= õ’2}

+ b(6

i

2

_

_õ’2)

is

_{approximately}

b =

n

d[

n

i

₊

(2/CV!)]

with

!7 =

2/CV

2

:

see also Visscher and Hill

(1992).

The

_genetic

variation in heifers turns out to be less than one half what it is in

bulls even

_though

the

_phenotypic

variance was

_virtually

the same. This may be due

to the fact that classifiers do not score

_exactly

the same trait in males

_(muscling)

as

in females

_{(size and/or fatness).}

It may also suggest that the

_regime

of male calves

is

_supplemented

with concentrate.

Location _parameters are

_compared

in

_figures

2a-d under different

_dispersion

models,

through

scatter

_plots

of estimates of standardized sire merits

_(u

_*

_).

Indexes

based on &dquo;subclass means&dquo;

_(V

_i

=

y

i

,

i = 1, ...

I,

with

homogeneous

variances)

and

those based on the &dquo;sire model&dquo; under the

_homogeneity

of variance

_assumption

are

far away from each other

(see

fig

2a).

Figure

2a is

_just

a reference of

_discrepancy,

which illustrates the

impact

of the BLUP

_methodology.

When

_{heterogeneity}

is

introduced _among residual

_variances,

sires’

_genetic

values do not vary too

much,

as

shown in

_figure

2b.

_Modelling

of the

_genetic

variances has a

_larger

_impact

on the sire

_genetic

values

_(see

_figure

_2c)

than

_modelling

of residual variances.

_Finally,

the

Bayesian

treatment of

_r

_e

_{-parameters by introducing}

random effects in the model

(M

B

,

M&dquo;

YJ

does not have any influence on the sire

_genetic

merits

(fig 3d).

Evaluation of sires can be biased if true

_{heterogeneity}

of variance is not taken into

account. As shown in table

_VI,

sire number 13 went down from the 16th to the 24th

position because

his calves were scored

mostly by

classifier no 1 who uses a

large

scale of notation

_(see

_T

_-estimates

in table

_V).

On the other

hand,

sire 103 _{went up}

from the 25th to the 14th

_place

since the

_{corresponding}

Classi and Condsc levels have low residual variance

_(for

the other factor levels

_represented,

the variances

were at the

_average).

For the same _reason, the sire

_genetic

merits were also affected

by

modelling

In ad.

The difference in

_genetic

merit for sire 56

_{(1.40 vs}

1.74 under the homoskedastic and the residual heteroskedastic models

_{respectively)}

is also

explained by

the fact that the calves of this sire were scored

_{exclusively by}

classifier no 12 and in 1983

(Year

=

1).

Due _to

modelling

Q

u,

this sire _went down

_again

(from

1.74 to 1.63 under the full heteroskedastic

_model)

because all its progeny

are females with a lower

Q

u

component

than in males. Other

_{things being equal,}

a reduction in the

oru

variance results in a

larger

_ratio,

or

equivalently

a smaller

heritability

and

_consequently

in a

_{higher shrinkage}

of the estimated

_breeding

value

toward the mean. In other

words,

if a decrease in

_genetic

variance is

ignored,

sires

above the mean are overevaluated and sires below the mean are underevaluated.

Hypothesis checking

The

_estimated

sire variance was

_7.08.

Variances of the

_Classi,

Year and Month factors are,

After

_modelling

residual

_variances,

the distribution of standardized residuals

became closer to

_normality,

in terms of skewness and

_especially

kurtosis. This

phenomenon

was observed in the whole

_sample

and also in the

_subsamples

defined

by

the levels of the factor considered in _re. On the other

hand,

normality

of the residuals was stable in the

subsamples

defined

_by

the factors absent from the _re

-model.

Normality

of the distribution of the standardized sire values in terms of

kurto-sis and

_PP-plot

was

continuously

damaged

at each _step of the variance

_modelling:

estimated kurtosis was

_0.61,

0.72 and

_0.90,

for the

_{homoskedastic,}

residual

het-eroskedastic and

_fully

heteroskedastic models

_{respectively.}

_Moreover,

skewness for

the 142 sire

_genetic

merits

_improved

_{slightly during}

that process: -0.09, -0.003

and -0.03 for the same models

respectively.

Computational

aspects

Programmes

were written in Fortran 77 on an IBM 3090

by implementing

an

EM

_{algorithm corresponding}

to

[15].

The _convergence was fast: 15-20

_cycles

for heteroskedastic

_{T e-models}

with

_<7J

estimated

_by

EM-R.EML

_{((i) stage),}

and 15-40

cycles

for

_fully

heteroskedastic

_T

_-models

or heteroskedastic

_T

_e

_-models

with random

effects. CPU time was between 2-5 min per model fit

(estimation

of parameters and

computation

of the

_{log marginal}

likelihood.

DISCUSSION AND CONCLUSION

This paper is an extension to u-components of variances of the

_{approach developed}

by Foulley

et al

₍₁₉₉₂₎

to consider

_{heterogeneity}

in residual variances

using

a

structural model to describe

_dispersion

_parameters, in a similar _way as

usually

done on subclass means.

In that _respect, our main concern focuses on _ways to render models as

parsi-monious as

_possible

so as to reduce the number of _parameters needed to assess

heteroskedasticity

of variances. An

_interesting

feature of this

_procedure

is _{to assess,}

through

a kind of

analysis

of variance, the effects of factors

marginally

or

jointly.

For _instance, one can test

_{heterogeneity}

of _{sire variances among} breeds of dams

be related to a

_segregating

_major

_gene)

can be tested while

_taking

into account the

influence of other nuisance factors

_{(season, nutrition...).}

_However,

the _power of the

likelihood ratio test for

_detecting

_{heterogeneity}

of variance can be a real issue in

many

practical

instances.

From the

_{genetic point}

of _view, the

_approach

is

_quite

_general

since it can deal with

heterogeneity

among within and between

_family

_components of

_variances,

or _among

genetic

and environmental variances. Factors involved for u and e _components of variance _may be different or the same,

making

the method

_especially

flexible. Our

modelling

allows one to assume

_(or

even

test)

whether the ratios of variances or

heritabilities are constant over levels of some

_single

factor or

_combination

of factors

(Visscher

and

_Hill,

_1992).

If a constant

_heritability

or ratio of variances a =

or

2

i/

or

2

among strata is

assumed,

the model involves the parameters y and a

only,

and

reduces to

_{In o, ei 2}

=

we!re

with

_{oru 2i}

_replaced

_by

_a;

_j

_0:

in the likelihood function. The

_shrinkage

estimator for the variances

_{proposed by}

_eg, Gianola et al

_(1992),

follows the same idea of the

_Bayesian

estimator described in the

_{Bayesian approach}

section. When a Gaussian

_prior

_density

is

_employed

for the

_dispersion

_parameters

Y

, the

hyperparameter

6 acts

as a shrinker. But the

_{Bayesian approach}

for a direct

shrinkage

of variance _components assumes that

_{heterogeneity}

in such components

(residual

and u

_components)

is due

_only

to one factor. The

_{approach presented}

in this _paper is more

_general

since it can _{cope with} more

complex

structures of stratification which _{may differ} from one _component to the other.

_Moreover,

its mixed model structure allows _great

_flexibility

to

_adjust

variances in relation to the amount of information for factors in the

_model;

eg when data

provide

little information for some factors

_{(or levels)}

or considerable for

_others,

our

procedure

behaves like BLUP

_{(or James-Stein)}

ie shrink estimates of

_dispersion

parameters toward zero if there is little

_information;

_only

with sufficient information

can the estimate deviate. For

instance,

our

methodology provides

a

_simple

and

rational

_procedure

to shrink herd variances

_(whatever

_they

are,

genetic,

residual

or

_phenotypic)

towards different

_population

values

_(eg

_regions,

as

_{proposed by}

Wiggans

and

_VanRaden,

₁₉₉₁₎

due to _{poor accuracy} of within herd or

herd-year

variances

_{(Brotherstone}

and

_Hill,

_1986).

It then suffices to use a hierarchical

_(linear)

mixed model for herd

_{log-variances}

and take the

_population

factor

_{( eg region)}

as

fixed and herd as random within that factor. An illustration of the

_flexibility

and

feasibility

of our

_procedure

was

_recently

_given

_{by Weigel}

₍₁₉₉₂₎

in

_analyzing

sources

of

_{heterogeneous}

variances for milk and fat

_yield

in US Holsteins.

Coming

back to the case of a

_unique

factor of variation for the sire

_variances,

one can think of a

_{simpler model,}

such as _yZ!,! =

m

i + _uij +

_e2!!(i

= 1, ...

I; j

=

1, ... J; k

=

l, ...

n2! ),

where _J.li is the mean effect of environment

_i,

_uij is the

(random)

effect of

_{the jth}

sire in the ith

_environment,

such that ui =

{u2!}! !

N(O,a!,A),

and e2!! is the residual effect

pertaining

to the kth calf of the

_jth

sire in the ith environment.

_Usually

in such hierarchical

_models,

it is assumed that Cov

_(u

_i

_,

_Ui’

₎

= 0 if i

=A

i’. On the _contrary, our

_{modelling procedure}

via the

change

in variables

_u!

=

Qu! u2!

(see

!4) )

takes into account covariances among

the same

_(or

_genetically

_related)

sires used in different

herds,

ie

_{Cov (ui, uj, )}

=

a u , a u

&dquo;

A!i! (Aii!

=

relationship

_matrix

pertaining to

ui and

u

i

, )

and so recovers

the inter-block information. The loss in _power in

_hypothesis

_testing

due to

_ignoring

Although

this

_presentation

is restricted to a

single

random factor

u

*

,

it can be

generalized

to a

multiple

random factor situation. If such factors are

uncorrelated,

the extension is

_{straightforward.}

When covariances

_exist,

one _may

_simply

_assume, as

proposed

by

Quaas

et al

_(1989),

that

_{heterogeneity}

in covariances is due to

scaling.

This _means, for

instance,

that in a sire

_(s

_i

_)-maternal

_grand

sire

_(t!)

model Y

ijk =

X!. k 13 + Si

+t!

+e2!k,

one will model

as

h’

2

a th 2

as

_previously,

and assume that

the _{covariance is Q9t,,} =

pa sha t h

for stratum h. If the model is

parameterized

in

terms of direct ao and maternal am effects as follows

_through

the transformation

s

i =

ao

i/

2

and t

j

=

ao!/4

₊

a

m;

/2,

one can set the

_genetic

correlation pa to a

constant, ie

_a

_ao

=

P

a

a

aoh

O

a

,

n

-

Notice that this condition is not

equivalent

to

the

_previous

one, except if

_a

_aoh

_/!d&dquo;,h

does not

_depend

on h.

Although

the

_methodology

is

_appealing,

attention must be drawn to the

feasi-bility

of the method. The first

_problem

is the inversion of the coefficient matrix in

[16]

required

for the

_computation

of the variance _system

_(15].

In animal

_breeding

applications,

this matrix is

_usually

_very

_large.

This

_limiting

factor is

_already

becom-ing

less

_important

due to _{constant progress} in

_computing

software and hardware.

The

_technique

of

_absorption

is

_usually

used to reduce the size of matrices to invert.

Another

_approach

is to

_approximate

the inverse. One can, for instance, use a

Taylor

series

_expansion

of order N for a _square invertible matrix A

where the square matrix

A

o

is a matrix close to A and _is, of course, easy to invert, and

_{where 11}

-

11

denotes some norm on the space of invertible matrices. Methods

viewed in Boichard et al

₍₁₉₉₂₎

can also

help

to

_{approximate A-}

1

in

_particular

cases such as _sparse matrices, &dquo;animal

model&dquo;,

etc.

Statistical power for likelihood ratio tests was

_investigated

for detection of

heterogeneous

variances in the usual

_designs

of

_{quantitative genetics}

and animal

breeding.

Results

_given

_by

Visscher

₍₁₉₉₂₎

and Shaw

₍₁₉₉₁₎

indicate

_generally

low power values for

detecting heterogeneity

in

_genetic

variance.

_According

to

_Shaw,

a nested

_design

of 900 individuals out of 100 sire families

_provides

a _power of 0.5 for

_genetic

variances

_differing

_by

a factor of 2.5. This

clearly

indicates the

minimum

_requirements

in

_sample

size and

_family

numbers which should be met

before

_carrying

out such an

analysis

and the limits therein. Therefore it seems

unrealistic to model

_genetic

variances in

_practice

_according

to more than 1 or 2

factors,

and it

_might

be wise to consider some of them as random if little information

is

_provided

_by

the data in each level of such factors.

Although

statistical constraints are satisfied for estimates of the _parameters of

the model

_(positive

variance _estimates, intra-class correlations within

[-1, + l]J,

some

_genetic

constraints such as about the

_heritability

estimate

_(h

₂

=

_4a!/(a!+ae)

for a sire

model)

_ranging

within

[0,1]

are not

_{imposed by}

our model. This can be

dealt with

_by

_choosing

_appropriate

_prior distributions on the

dispersion

_parameters

that would take this constraint into _account, but this

_procedure

appears to be

extremely complicated. Fortunately,

the

_{unconstrained}

solutions are the constrained

maximization

_procedure

under constraints must be

_performed

or the

_posterior

distribution under the constraint can be obtained from the unconstrained

_posterior

distribution

_{multiplied by}

a corrective factor

_(Box

and

_Tiao,

_1973).

This

_problem

does not occur with an &dquo;animal

model&dquo;,

but can arise when a &dquo;sire model&dquo; is

_used,

and is not

_specifically

related to

_{heteroskedasticity.}

From a statistical _point of _view, the

procedure

uses the _concept of variance function

_(Davidian

and

_Carroll,

₁₉₈₇₎

as an extension to

_dispersion

_parameters of the link function. Our _presentation focuses on the

_log

link function which is the

most common choice in this field

_(see

for instance San

Cristobal,

1992,

for a review of variance

_models)

&dquo;for

_physical

and numerical reasons&dquo;

_(Nair

and

_Pregibon,

_1988).

Following

Davidian and Carroll

₍₁₉₈₇₎

or

_Duby

et al

_(1975),

the

_question

can be

asked whether or not variances vary

according

to means or location _parameters. In

the

_Maine-Anjou

_{data, however,}

it does not seem to be the case, thus

validating

our choice in

!10!.

It would be

_interesting

to extend our method to a

fully generalized

linear mixed

model on means and on variances with or without common _parameters between

the mean model and the variance model. Numerical

_integration

or Gibbs

sampling

procedures

would then be

_required

_although

_approximate methods of inference

can also be used for such models

_(Breslow

and

_Clayton,

_1992;

_Firth,

_1992).

Statistical

_{problems arising}

with common _parameters are

_{already highlighted by}

van

_Houwelingen

_(1988).

With a

_fully

fixed effect variance

_model,

_techniques

of estimation and

_hypothesis

testing

for

_dispersion

_parameters

_presented

here are those of the classical

_theory

of

likelihood inference

_(likelihood

and likelihood ratio

_test),

_except that the

_marginal

likelihood function

_L(

_Y

_;y)

was

_preferred

to the usual likelihood

_L(13,

_T;

_y),

in the

light

of ideas behind REML estimators of variance _components. This test reduces

to Bartlett’s test

_{(Bartlett, 1937)}

for a one classification model in variances and

under a saturated fixed model on the means

_(ie

_i

_ji

=

_y

_i

_,

i = 1, ...

I).

Unfortunately,

Bartlett’s test is known to be sensitive to

_departure

from

_normality

_{(Box, 1953).}

Simulations are needed to

_study

the robustness of this test and other

_competing

tests. From a

Bayesian

_perspective,

the

Bayes

factor is

usually applied

for

hypothesis

testing

(see

Robert, 1992,

for a

_discussion).

The

_posterior

_Bayes

factor

_(Aitkin,

1991)

could also be used to _compare

_dispersion

_models,

but numerical

_integration

would then be

_required

_(see

the

_expression

of the likelihood in

_!18!).

In this paper, focus was on an

_appropriate

way to model

_{heterogeneous}

vari-ances, but the initial motivation was a best

fitting

of location _parameters

(animal

evaluation for animal

_breeders).

This difficult

_problem

of

feedback,

also related to

the Behrens-Fisher

_problem,

has to be solved in our

_{particular approach.}

Moreover,

a _great

research

_perspective

is _open on the

_important

and

complicated

question of the

_{joint modelling}

of means and variances

(Aitkin,

1987;

Nelder, 1991;