Inferences on homogeneity of between-family components of variance and covariance among environments in balanced cross-classified designs

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Inferences on homogeneity of between-family

components of variance and covariance among

environments in balanced cross-classified designs

Jl Foulley, D Hébert, Rl Quaas

To cite this version:

Original

article

Inferences

on

homogeneity

of

between-family

components

of

variance and

covariance

_among

environments

in

balanced cross-classified

_designs

JL

_Foulley

D

Hébert

2

RL

_Quaas

₃

1

Institut National de la Recherche

_Agronomique,

Station de

_{Génétique Quantitative}

et

Appliquée,

Centre de Recherches de

_{Jouy-en-Josas,}

78352

_{Jouy-en-Josas Cedex;}

2

Domaine

Expérimental

Agronomie d’Auzeville,

Centre de Recherches de

_Toulouse,

BP

_27,

31326 Castanet Tolosan

_{Cedex, h}

_b

_ance;

3

Cornell

_{University, Department of Animal Science, Ithaca,}

NY

14853,

USA

(Received

4

_August

_1993;

_accepted

29 November

₁₉₉₃₎

Summary - Estimation and

_testing

of

homogeneity

of

_{between-family}

_components of variance and covariance among environments are

_investigated

for balanced cross-classified

designs.

The variance-covariance structure of the residuals is assumed to be

_diagonal

and heteroskedastic. The

testing

procedure

for

_homogeneity

of

_family

_components is based on the ratio of maximized

_{log-restricted}

likelihoods for the reduced

_(hypothesis

of

_homogeneity)

and saturated models. An

_{expectation-maximization}

_(EM)

_algorithm

is

_proposed

for

_calculating

restricted maximum likelihood

_(REML)

estimates of the residual and

_{between-family}

_components of variance and covariance. The EM formulae to

implement

this are iterative and use the classical

_analysis

of variance

(ANOVA)

_statistics,

ie the between- and

_{within-family}

sums of _squares and

_{cross-products. They}

can be

_applied

both to the saturated and reduced models and _guarantee the solutions to be in the parameter space. Procedures

presented

in this paper are illustrated with the

analysis

of 5

_vegetative

and

_reproductive

traits recorded in an

_experiment

on 20 full-sib families of

black medic

_(Medicago

_lupulina

_L)

tested in 3 environments.

_Application

_{to pure} maximum likelihood

_procedures,

extension to unbalanced

_designs

and

comparison

with

approaches

relying

on alternative models are also discussed.

genotype X environment interaction

_/

_{heteroskedasticity}

_/

expectation-maxi-mization

_/

restricted maximum likelihood

_/

likelihood ratio test

La

_procédure

de test

_{d’homogénéité}

des composantes

familiales

repose sur le _rapport des vraisemblances restreintes maximisées sous les modèles réduit

_(hypothèse

_{d’homogénéité)}

et saturé. Un

_{algorithme d’espérance-maximisation}

_(EM)

est

proposé

pour calculer les

estimations du maximum de vraisemblance restreinte

_(REML)

des composantes résiduelles et

_familiales

de variance et de covariance. Les

_formules

EM à

appliquer

sont itératives et utilisent les

_statistiques

_classiques

de

_l’analyse

de variance

_(ANOVA),

c’est-à-dire les sommes de carrés et

_coproduits

inter- et

_{intrafamilles.}

Elles

s’appliquent

à la

fois

aux modèles réduit et saturé et garantissent

l’appartenance

des solutions à

_l’espace

des

paramètres.

Les méthodes

_présentées

dans cet article sont illustrées _par

_l’analyse

de 5 caractères

_végétatifs

et

_reproductifs

mesurés lors d’une

_expérience

_portant sur 20

_familles

de

pleins frères

testées dans 3 milieux chez la minette

_(Medicago

_lupulina

_L).

_{L’application}

au maximum de vraisemblance stricto _sensu, la

_{généralisation}

à des

_{dispositifs déséquilibrés}

ainsi _que la

_comparaison

à des

approches

reposant sur d’autres modèles sont

_également

discutées.

interaction _génotype x milieu

_/

hétéroscédasticité

_/

_{espérance-maximisation}

_/

maxi-mum de vraisemblance restreinte

_/

_rapport de vraisemblance

INTRODUCTION

There is a

_great

deal of interest

_today

in

_quantitative

and

applied genetics

in

heterogeneous

variances.

_Ignoring

such

_{heterogeneity,}

as is

_usually

_done,

_may

substantially

affect the

_reliability

of

_genetic

evaluation and thus reduce the

_efficiency

of selection

_(Hill,

_1984;

Visscher and

_Hill,

_1992).

There is concern not

_only

about

_{estimating dispersion}

_parameters

for hetero-skedastic

_models,

but also about

_testing

_hypotheses

for the real

_degree

of

hetero-geneity

which can be

expected

from

_experimental

results. In this

_respect,

Visscher

(1992)

investigated

the statistical _power of the likelihood ratio test in balanced half-sib

_designs

for

_{detecting heterogeneity}

of

_phenotypic

variance and intra-class correlation between environments.

In that

_approach,

the

_(family)

correlation between environments

_(p)

is assumed to be

_equal

to

_1,

and

_{heterogeneity}

of

_{between-family}

_components

of covariance among environments in

only

due to

_scaling

of variances.

The aim of this paper is to extend that

_approach

to the case of true

genotype

by

environment interactions

_{(p #}

_1).

Our attention will be focused on:

i)

cross-classified balanced

_designs;

and

_ii)

the null

_hypothesis

_{involving homogeneity}

of

_{between-family}

_components

of variance and covariance between environments. This variance-covariance structure has been

_widely

used for

_{analyzing family}

data recorded in different

_{environments,}

in

_particular

due to its close link with a

2-factor classification model

_(ie

_family

and

_environment)

with interaction

_(Mallard

et

_{al, 1983; Foulley}

and

_Henderson,

_1989).

_Moreover,

even for balanced

_designs,

the estimation of the 2

_parameters

involved in this

_simple

structure via maximum likelihood

_procedures

has no

_analytical

solution in the

_general

case when no

assumption

is made about the residual variances. This motivated the

_proposal

made in this

_study

to use the

_{expectation-maximization}

_(EM)

_algorithm

_(Dempster

et

THEORY

Generalities

Let us assume that the records from the balanced cross-classified

layout family

(or

genotype)

x environment can be written as:

where _y2!! is the

_performance

of the kth progeny

(or individual) (k

=

1, 2, ... ,

n)

of the

_jth

family

(or

genotype) ( j

=

1, 2, ... ,

s)

evaluated in the ith environment

(i

=

1, 2, ... , p) ; b

ij

is the random effect of the

_{jth family}

in the ith

_environment,

assumed

_normally

distributed,

such that

_Var(b

_ij

₎

=

a

B

i,

Cov(bij,bi

’

j)

=

O’Bii

,,

for

i

_{! i’,}

and

_{Cov(bi!,bi!!!)}

= 0

for j # j’

and

_any

i and

i’;

and

e2!k is a residual

effect

_pertaining

to the kth progeny in the subclass

ij,

assumed

A!77D(0, <7!)

viz,

normally

and

_{independently}

distributed with mean zero and variance

_U2

_wi

Using

vector

_notation,

ie _yjk =

{

Yijk

}, !!

=

Ig

il

,

bj =

{bij}

and ej k =

{e2!k}

for i =

1, 2, ... ,

p, the model

[1]

can

_{alternatively}

be written as:

where

_b

_j

_rv

_N(0,!3)

and _{ejk rv}

_N(0,E!),

with

E

B

=

{a-Bii’

}

’

standing

for the

(p

x

_p)

matrix of

_{between-family}

components

of variance and covariance between environments and

£w

=

Diag{

Q

w

i

} for

the

_(p

x

_p)

_diagonal

matrix of residual

components

of variance.

&dquo;

Actually,

this

approach

consists of

considering

the

_expression

of the trait in different environments

_{(i, i’)}

as that of 2

genetically

related traits with a coefficient

of correlation _pii, =

asi!!/!B!!s!&dquo;

(Falconer, 1952).

In a

_given

environment

_(i),

this

_1-way

linear model

generates

the classical ANOVA

statistics,

ie the

_{between-family}

_(SB!,

_B

_i

₎

and

within-family

(Swi, W

i

)

sums of _squares and mean _squares,

respectively,

whose distributions are

propor-tional to

_chi-squares:

Due to the cross-classified structure of the

_design,

one also has to consider a sum

(S

B

..,)

and a mean

(BZi!)

between-family cross-product

for each

(ii’)

combination

If we let yj_ =

_{lyij. 1,}

_Y.. =

{Yi..},

then the matrix

S

B

=

{BBi&dquo;}

} with

elements from

_[3a]

and

_[4],

such that:

&dquo;&dquo;

has a Wishart

_{distribution,}

denoted

_W(r,

s -

1),

with

_parameters

_{(s &mdash; 1)}

and

r =

Eyv

₊

_nE

_B

_,

thus

generalizing

to a matrix of

_{between-family}

sums of _squares

and

_{cross-products}

the

_{(o,2 w}

_i

_{+naBi)X!s_1!}

distribution

_arising

in

_(3a!.

In the 1-dimensional _case, the set of

_SB!

and

_S

_w

_.

are

independent,

location

invariant sufficient statistics for

_a-

₁

and

_w

_i

_similarly

the matrices

_S

_B

_and

Syv

=

Diag{5w,} have

the _same

property

for

E

B

and

_{Eyv. Hence,}

one can write

the

_density

of

_[5]

as

Similarly,

Using

[6a]

and

_[6b]

in the

_expression

for the

_log

_likelihood,

where Ct is a constant. This leads to:

with W =

Diag{W

d

=

Syv/s(n - 1)

and

tr(.)

= the trace

_operator.

Notice that maximization of

_[7]

_yields

REML estimators of

_E

_B

and

_Eyj,

because the

_marginally

sufficient statistics

_S

_B

and

_Syir

are used in the

_{log-likelihood}

function. Under the saturated model

_{(a wi 2 7!}

₀

_{,2 Wi,}

and

_UBij

_{, 0}

_!8.!!.!!!

_{for any}

i, i’, i&dquo;

and

_i&dquo;’),

the

_partial

derivatives

with respect

to

_g

_o

_B

_and

_0’

₂

_wi

of minus twice the

_log

likelihood

_(-2L)

are:

8F/8aB;;,

is a

_(p

x

_p)

matrix

_{having n}

as the

_{(i, i’)}

element and 0

_elsewhere,

so that the

_equation

[8a]

= 0

_gives

f

= B.

Similarly,

8£w /

8

a

/

j

has 1 as the ith

diagonal

_element

and 0 elsewhere. Given that

E

w

and W are

diagonal

matrices and that

f

=

B,

the solutions to

equations

[8a]

= 0 and

[8b]

= 0 _are

provided

that B &mdash; W is

_positive

definite. The maximum of the

_{log-likelihood}

function is then

_(apart

from a

constant):

Otherwise,

REML estimates of

E

B

and

E

W

are no

_longer

identical to ANOVA

estimates and

_require

the use of another

algorithm

for their calculation

(see

Appendix

A).

The null

_hypothesis

consists of

_assuming

the

_homogeneity

of the

_{between-family}

components

of variance

_(’di,

_or

₂

₌

_CT

₂

₎

and covariance

_{(Vi #}

_i’,

a-Bii’ =

C

B

)

_as

postulated

in many

analyses

of genotype

by

environment

_experiments

_(Dickerson,

1962; Yamada, 1962;

Mallard et

_al,

_1983).

The

_{approach presented}

in this paper allows us to test this

_simplified

structure of

_E

_B

_against

Falconer’s saturated model for any structure of the residual variances. The nulle

hypothesis

(H

o

)

considered here can be written as:

where

_Ip

=

identity

matrix of order

p and

Jp

=

(p

x

p)

matrix of ones.

Under

H

o

,

REML estimation of

_E

_B

and

_Eyv

becomes much more

_complex.

Here

8F/8a

§

=

nIp

and ar/aC

B

=

n(J

P

-

Ip)

result in the

following equations:

were

lp = (p x 1)

vector of ones.

Since

_r-

₁

_-

_I

_-

_F-’BF-

₁

_,

the REML solution for the residual

components

([8b])

is no

_longer

Êw

= W and the

system

of

_equations

_{[8b] (see}

also

_{(B11!), [12a]}

and

[12b]

has no

analytical

solutions in the

general

case. This was the reason

_motivating

our search for another

approach

for

computing

REML solutions to

E

B

and

£

w

under

_Ho.

An EM

_{diagonalization approach}

The

_{expectation-maximization approach}

is a _very efficient

_concept

in maximum

(auaas,

1992).

The basic

_principle

is to treat the unobservable random variables

_b

_ij

and _e2!! as

_missing

data.

_Actually,

the EM

_algorithm

will not be

_applied

_directly

to the model described in

_[1]

and

_[2]

but after a

_{spectral decomposition}

of

E

B

according

to its

_eigenvalues

and

_vectors,

ie:

In this

_formula,

A =

Diagf6

il

is the

(p

x

_p)

matrix of

_{eigenvalues 6}

_i

_,

_{with 6}

_i

repeated

as _many times as its

_multiplicity

_order,

and U =

(U

1

,

U

2

, ... ,

Ui, ... ,

Up)

is the

_(p

x

_p)

matrix of the

corresponding

p normed

eigenvectors

U

i

of

_E

_B

_(U’U

=

I

P

).

Under the

_special

form shown in

_!11!,

_E

_B

has

_only

2 distinct

_eigenvalues:

with

_multiplicity

orders 1 and

_{(p &mdash; 1)}

_{respectively.}

_Moreover,

the matrix U of

eigenvectors

does not

_depend

on the values of

6

1

and

₆

₂

_,

U’

_being

the Helmert

matrix of order _p, see for

_{example Searle,}

1982

_(p

71 and

₃₂₂₎

for more details about such matrices. For

_instance,

for p =

_3,

Due to the invariance

_property

of

_(RE)ML

_estimators,

the one-to-one transfor-mation in

_[14a]

and

_[14b]

allows us to

_change

the

_{parameterization}

from

_(<

_T1

_,C

_B

₎

to

₍₆

₁

_,b

₂

_),

or more

_conveniently

to

_(<!i,T)

where T =

6

1

₊

(p &mdash; 1)6

2

,

the back transformation is:

From the

_{spectral decomposition}

of

_E

_$

_,

the model in

_[2]

can be written as:

where U is

_defined

as before and

the vector

_f

_j

=

{ fi!

is

such

that

_f

_j

N

N(0,A).

Using

the

_Dempster

et al

₍₁₉₇₇₎

_terminology,

a

_complete

data set x can be

constructed from /-1,

f

j

,

jke

for j

=

1,2,...,s

and k =

1, 2, ... ,

n,

whereas the

incomplete

data set is the vector y of

observations.

’

Let us first consider the case of

E

B

.

If the

_f

_j

_’s

were

known,

sufficient statistics

REML would then be obtained

by equating

the

_expectation

of these sufficient

statistics,

ie:

to their calculated values

_{(M step).}

_Actually,

these sufficient statistics are not

directly

observable and the EM

_{algorithm proceeds}

first

_{by estimating}

them

_by

taking

their conditional

_expectation

_given

the observed data set

_{(E step).}

Since such an estimation

_depends

on the value of the unknown

_parameters,

the

procedure

is iterative and consists of

_implementing

the 2 usual

_steps:

E

_step:

at iteration

_(t!,

calculate

M

step: compute

6!&dquo;’I

and

T[

t+

1

]

from the

_following

equations:

As shown in

_{Appendix A,}

the

_(p

x

_p)

matrix

A’

l

can be

expressed

as:

where

_U,

B and r are defined as above

(see !13!, [5a]

and

[5b] respectively)

and

C!t!

is the matrix of variance of

_prediction

errors of

[

t]

=

E(f

j

)

y,

8!t] ,

r

[t]

,

_S!),

the best

predictor

of

_f

_j

at iteration

_[t]

such that:

Similarly,

sufficient statistics for

E

w

under the

complete

data set x are:

and the E and

_{M ’steps}

are

as follows:

For the E

_step,

at iteration

_[t],

calculate:

using

the

_following

formula based on the same

_reasoning

as

_previously

_(see

Ap-pendix

A):

sums of _squares and

cross-products

(y..

being

defined as

For the M

_step

compute

the next value of

_Eyv

from:

Formulae

_[25]

and

_[24a]-[24b]

define the E and M

_steps,

_{respectively,}

of an EM

parameters.

Notice that in this scheme

_{tr(P)/p (average}

_diagonal

element of

_P)

and

((1’P1) -

_{tr(P!!/p(p -}

1)

(average

off-diagonal

element of

_P)

behave as sufficient

statistics for _aB and

_C

_B

with

respect

to the

complete

data set.

Formulae for the residual

_components

are

_unchanged

with

UC

[t]

U’

=

:E

W

1 E!y!

and

M[

t

]

₌

_{n£ §}

_I

_(r

_iti

1

. For

the saturated

_model,

the formulae to

_apply

are the same for

_E

_W

_{and, simply,}

&dquo;

+

t

f

E

=

Il

ltl

ls

for

E

B

.

Testing

procedures

Hypotheses

of interest concern the vector 0 of

_parameters

involved in the matrices of

_{between-family}

_(E

_B

₎

and

_{within-family}

_(Ey!)

_components

of variance and covariance between environments. The

_theory

of the

_generalized

likelihood ratio

can be

applied

to that purpose, as

already proposed by Foulley

et al

_(1990,

_1992),

Shaw

₍₁₉₉₁₎

and Visscher

₍₁₉₉₂₎

_among others.

Let

_o

_:

_H

0 E

8

0

be the null

_hypothesis

and

_:

_H

_I

0 6 8 -

8

0

its

_alternative,

where 8 refers to the

_complete

_parameter

space and

O

o

,

a subset of it

_pertaining

to

_H

_o

_.

Under

H

o

,

the statistic

where

_L(0;y)

is defined as in

!7!,

has an

_asymptotic

chi-square

distribution with r

degrees

of

_freedom,

r

_being

the difference in the numbers of estimable

_parameters

involved in e and

O

o

(Mood

et

al,

1974).

Here e contains

_p(p

+

_3)/2

parameters

corresponding

to _p residual

_components

of variance and

_p(p

+

_2)/2

_{between-family}

_components

of variance and covariance

between environments whilst

Oo

has

_p+2

_{2 parameters}

_only

_(p

residual

_components,

<T1 and

C

B

),

so that r =

!p(p+ 1)/2! -

2.

In the

_{Neyman-Pearson approach}

of

_hypothesis

_{testing, H}

_o

is

_rejected

at the

a level if the calculated value of

_A(y)

exceeds a critical value

À

c

such that

Pr(xr >

À

c

)

= _a.

_However,

the likelihood ratio statistic

.!(y)

in

[24]

_can also

be

_interpreted

as the difference in

_degree

of fit via maximum likelihood

_procedures

by

2 models: a reduced

_model(R)

with

parameter

vector 0 E

_O

_o

and a full model

(F)

with 0 E O

_encompassing

both the null

_hypothesis

and its alternative. In the

theory

of

_significance

_testing

_(Kempthorne

and

_Folks,

_1971),

this statistic is also used as a measure of

_strength

of evidence

_against

the reduced model or the null

hypothesis.

The lower the

_probability

under

_H

_o

of

_exceeding

this statistic evaluated

from the data

_(also

referred to as the P-value or

_significance

level or size of the

test),

the

stronger

the evidence

_against

H

o

.

Example

Data used here to illustrate the

procedures

are from an

_experiment

carried out in

Montpellier

(south

west of

_France)

on 20 full-sib families of black medic

(Medicago

lupulina

L)

tested in 3 different environments

_(control,

_harvesting

and

competition

treatments).

The

_{experimental design}

was described in detail

_by

H6bert

(1991).

There were

replicate. Thirty-six

traits were recorded and the variable used was the mean of the 5

_plants

cultivated in each

_replicate

so that _p =

_3,

_s = 20 and _n = 2.

Basic ANOVA statistics for the

_{between-family}

and

_{within-family}

sums of squares and

cross-products

are

_given

in table I for a subset of 5 traits.

Firstly,

the null

_assumption

that the

diagonal

terms of

_E

_w

were

_equal

was tested

via a Bartlett’s test based on ANOVA mean _squares statistics. P values were

_0.007,

0.08, 1.4

x

10-

7

,

8 x

10’!

and 0.04 so that this

_assumption

can be

_{reasonably rejected}

(except

perhaps

for trait

_2).

Test statistics about

E

B

and estimates of

E

B

and

£w

under both the reduced and saturated models are

_given

in table II. P-values for

_vegetative

yield

_traits,

represented

here

_{by dry}

matter

_weight

_(trait

No

₃₎

and

_dry

matter

_weight/max

plant

size diameter

_(trait

No

_4),

were _very

low,

indicating

a

large heterogeneity

in

genetic

variation between evironments with full-sib variances

_{substantially}

reduced in the

_harvesting

_(i

=

1)

and

competition

(i

=

3)

environments

compared

with the control

_(i

=

2).

In

_contrast,

the

harvesting

and

_competition

environments do not

generate

a

meaningful

level of stress

_compared

with the control for the

_expression

of

_genetic

variation of

_days

to 1st

_{ripe pod}

_(trait

No

₂₎

and relative

_{pod weight}

(trait

No

_5).

These 3 environments then behave as

_{’exchangeable’,}

as statisticians would _say. In this

_example,

_genetic

correlations between environments were rather

high

and it would have been

_interesting

to test for some traits

_(eg

No 1 and

₅₎

using

the

_assumption

that these correlations are

equal

to

_unity

_by

Visscher’s

₍₁₉₉₂₎

DISCUSSION

This _paper describes a further contribution to the solution of the

_problem

of

testing

homogeneity

of

_{between-family}

components

of variance and covariance between environments in the case of balanced cross-classified

designs.

The

_testing

procedure

is based on the likelihood ratio test as

_already

advocated

_by

Shaw

(1987)

in

_{quantitative genetics.}

This

_study

extends that of Visscher

_(1992),

which was

restricted to the case of _pure

_scaling

effects between environments

(ie

all

_genetic

correlations between environments

_equal

to

_one).

The choice of an EM

_algorithm

for

_computing

REML estimates of

E

B

and

E

w

under the null

_hypothesis

allows us to make

_explicit

the

_equations

of the iterative process to

_implement

via formulae based on the usual ANOVA statistics. This

algorithm

does not

_require

_any constraint on the value of these ANOVA statistics

( eg

B - W can be

_non-positive

_definite)

_provided

the

_starting

values for

_E

_B

are within

the

_parameter

space. A

simple

reason for this is that the E

phase

under the restricted

model involves the conditional

expectation

(given

the

_data)

of sums of _{squares, eg}

£ f£

and

_e

_{e !!}

as estimators of variance. Because E

(L

f

l

f

ly,

A = 8!t! )

is

j 7 k j

,

always positive

definite

_(Foulley

et

_al,

_1987),

this

_property

of the EM

_algorithm

is also true under the saturated

model;

it can then be used to

_provide

REML estimates of

_E

_B

and

_E

_w

when ANOVA estimators are not

_permissible

_(eg

for traits

_1,

_3,

4 and 5 in the

_example).

Some authors such as Anderson

(1984)

and Shaw

(1987)

advocate the use of

ML rather than REML

_procedures

to test

_hypotheses

about variance covariance matrices. Our EM

_algorithm

can be

easily adapted

to obtain such ML estimates of

E

B

and

_E

_w

_.

It suffices to

_replace

_(s-1)B+r

in formulae

_[19]

and

_!21!

for A and Q

respectively by

(s-1)B

corresponding

to the

change

in the conditional

_expectation

(given

the

_data)

_{of n (y}

_j

_{. -}

_{p) (yj_ - !)’}

_according

to

_{whether g}

is considered as

j

a

_parameter

of interest

_(ML)

or a nuisance factor to be

_integrated

out

_(REML).

This

_algorithm

can also retrieve the usual ANOVA estimates for

_E

_B

and

_Ew

using

the

_2-way

crossed-mixed model:

involving

fixed environmental effects

_(h

_i

_),

random

_family

effects

_[s

_j

_{- NIID (0, a/ ) ] ,}

random interactions

_[hs

_ij

_-

_{NIID(0,afl!)]}

and residuals

_[e

_ijk

_{°° NIID(0,a}

_£

_)].

In

_{fact, Foulley}

and Henderson

₍₁₉₈₉₎

showed that this is an

equivalent

model

for a

_simplified

version of the

_multiple

trait model in

_[1]

restricted to

_E

_B

=

(

U2

B -

C

B

)Ip

+ C

B

Jp

and

_E

_w

=

Q!2,yIP

with

0,2 B

=

Q

s

+

or2!’, h

C

B

= or

.

and

_ow

₂

=

O

re.

2

Notice however that this

_{simplified multiple}

trait model differs from that considered

throughout

this

_study

_(see

_[11])

not

_{only by}

the

_assumption

of homoskedastic

residual variances but also

_by

its restriction to a

_positive

covariance

(C

B

)

between

environments.

For

_instance,

for trait 1

_(days

to

_flowering),

EM-REML estimates of variance and covariance

_components

obtained from the

_algorithm

described in

_[17]

to

_[22]

with

values can

easily

be checked with ANOVA

_estimatps:

a2

=

79.89;

&2,

=

0.97;

and

a

2

= 26.39. Here

again,

the EM

algorithm provides

estimates within the

parameter

space, which is not

_{always the}

case with ANOVA estimators as shown for instance

with trait 5 :

_{&1 =}

_{79.50, C}

_B

= 79.43 and

a2

= 23.23 with EM

versus as

=

79.65,

!hs

= -1.02 and

Qe

=24.07 with ANOVA.

The EM

_algorithm

is a first-order

_procedure

and therefore has close

_{relationships}

with a maximization

procedure

based on zeroed first derivatives. As shown in

Appendix

B in the case of

_E

_B

_,

the difference between the formulae to

_implement

in the 2 iterative

_procedures

consists of

_replacing

_{(s - 1)B}

+

r!t!

in

_[19]

_by

sB in the 1st derivative

_algorithm

_(B

_being

a sufficient statistic for r in the saturated

model).

Again,

the use of EM

_guarantees

_staying

in the

_parameter

space whereas there is no obvious

_proof

of that for the 1st derivative

_algorithm.

_Moreover,

the EM

approach

turns out to be easier to understand and to use than the other one, which relies

_mainly

on

_algebraic

tricks. As far as REML estimates of

_£w

are

_concerned,

a functional iteration

algorithm

based on first derivatives was also

proposed

in

Appendix

B due to the lack of an obvious

_analogue

of the EM formulae.

Finally,

this EM

_reasoning

can be extended to an unbalanced structure of data and to additional nuisance fixed effects cross-classified with

_family

effects,

ie to an

extended version of model

_[1]

such as

where

₁₃

is a vector of fixed effects

_including

the ith classification for environment and effects of other factors to be

_adjusted

for,

and

_x’ij

_k

is the

_{corresponding}

incidence

_(row)

vector. Under that

model,

the E and M

steps

defined in

_[17]

and

[18a]

and

_[18b]

for

_E

_B

and in

_[21]

and

_[23]

for

E

w

are still valid.

_However,

the

E-statistics in

_{!17!, [21]}

and

_[24b]

are not evaluated as functions of ANOVA statistics but

_directly

from the numerical values of the BLUP and the variance of

_prediction

errors of the

_f

_j

or

_b!’s.

In this

_{situation, also,}

the EM

_algorithm

should be

_applied

systematically

to both the reduced and saturated model for

Ey!.

Another

approach

would be to write

_[28]

under its

_equivalent

form

_(for

_C

_B

>

_0):

where

_{hi, sj}

and

_hsjj

are defined as in

_[25]

and _{e2!! N}

_NID(O,

_o,2w.).

Under such

a mixed-model

_structure,

one can then use the methods

developed

_{by Foulley}

et al

_{(1990, 1992)}

and San Cristobal et al

₍₁₉₉₃₎

for

_calculating

REML estimates of variances in the _presence of

_{heterogeneous}

residual

_components.

_However,

the

procedure

derived in this _{paper remains}

_definitively

more

_general,

for

_instance,

it

can also be

easily applied

to a

non-diagonal

structure of

E

w

using

formulae

_[17]

to

[22]

unchanged

and

_[23]

_slightly

modified into

E!+l]

=

S2!t!/ns.

This _paper deals with a null

_hypothesis

of constant

_{between-family}

variance and covariance. In some

_instances,

a more

_appropriate

null

_hypothesis

would be a

constant

_{between-family}

correlation

_(p)

between environments

_(a-

_B

_ii

_’

=

/9<?’B,<!B,/)

and/or

of constant intraclass correlation

_[a-1i

₌

_t(a-1i

₊

_or2

_wi

_)].

_Testing

procedures

for these

_assumptions

will be

_reported

in a

separate

article.

ACKNOWLEDGMENTS

The authors are

_grateful

to I Olivieri

_{(INRA, Montpellier)}

for

_stimulating

discussions which motivated this

_study,

to J Ruane for the

_English

revision of the

_manuscript

and to the _anonymous referees for their valuable comments.

_Special

thanks are

_expressed

to C Robert and M San Cristobal who read the

_{manuscript thoroughly}

and checked the numerical

_example.

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Anderson TW

₍₁₉₈₄₎

An Introduction to Multivariate Statistical

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J

Wiley

and

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Dempster

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Laird

_NM,

Rubin DB

₍₁₉₇₇₎

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Falconer DS

₍₁₉₅₂₎

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JL,

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K

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JA

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_{A, Graybill}

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(auaas

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₍₁₉₉₃₎

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heteroskedastic

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of variance in a mixed

_linear

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Searle SR

₍₁₉₈₂₎

Matrix

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_Useful

to Statistics. J

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₍₁₉₈₇₎

Maximum likelihood

_{approaches applied}

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_{quantitative genetics}

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Hill WG

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321-329

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₍₁₉₆₂₎

_{Genotype by}

environment interaction and

_genetic

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_Jap

J Genet

_37,

498-509

APPENDIX

An EM

_algorithm

for REML estimations

_{of E}

_B

and

_E

_w

_{(part A)}

E

B

and

_Eyv

under

_H

_o

An

_explicit

formula for A can be obtained

_{by successively conditioning}

and

deconditioning

the

_expression

in

_[17]

with

_respect

to the mean vector _{p, ie:}

where 0 stands for the vector of

parameters

involved in the matrices of

between-family

(E

B

)

and

_{within-family}

_(E

_w

₎

_components

of variance and covariance between environments and

0

1

’

is the current estimate of 0 at iteration

_!t!.

s

Now, conditionally

on _{y, p and 0} =

0’

l

,

the

expectation

of

Ef

j

f

can

be j= 1

decomposed

into:

This

_{decomposition}

is

_{especially helpful}

because it

_allows

us to introduce the usual statistics of Gaussian

_models,

ie the conditional mean

f

j

=

E(f!

ly,

_1-1,

e)

and its

_prediction

error variance

_C

_j

=

The next

_step

is to

_specify

the

_expression

for:

with

respect

to the distribution of

₄₁

_y,

e. On account of

assumptions

made in

_!1),

the distribution of this random variable is

_{N[y.., (E}

_B/

_s)

+

_(Ew/ns)),

so that:

The formula for the variance of

_prediction

error in

[A3b]

does not

_depend

on _p, so that the

_expression

of

_(Al]

reduces to:

where

C!t!

is defined as in

[A3b]

_using

0!

as a current estimate of 0 in

Ew

and

E

B

,

the matrices B and U

_being

constant over rounds of iteration.

The next values of the unknown

_components

of

_E

_B

_(ie

_B

_a

and

C

B

or

6

1

and

T

)

are

_computed

at the M

_step

from the

_diagonal

elements of

A’

l

_(!18a!

and

_!18b!).

The

_diagonal

terms of the matrix

E

w

under the

_complete

data set x

since:

Because this statistic is not

_observable,

it is

_{replaced by}

its conditional expec-tation

_given

the data y and e =

9!.

As for

_{B-components,}

this

_expectation

is calculated after

_conditioning

and

_{deconditioning}

with

_respect

to the mean vector !, ie:

where

&dquo;

0

and C are as before.

From

_[A9a],

the

_quadratic

_{L êjkêjk}

can be written as

jk

with M =

nUCU’E-1

The next

_step

is to take the

_expectation

of each element in the

_right-hand

side with

_respect

to the distribution of

_gly,

0. Then

where T

_designates

the matrix of total mean _squares and mean

cross-products,

Similarly,

Placing

[All], [A12]

and

_[A13]

in

_[A8]

and

_noting

that the conditional variance in

_[A9b]

does not

_depend

on _!,

_gives

In

_{fact, ft}

is evaluated

_{conditionally}

on 0 =

0!,

ie

by taking

F =

F

l

’

]

,

C =

Cl’l

and M =

M’

l

in

[A. 14].

The _next value of

or

2

E

B

and

_Eyjr

under the saturated model

Actually,

the EM

_algorithm

described

_previously

can be

_easily

accomodated to deal with the saturated model. This is

_especially

_helpful

when the ANOVA estimates of

E

B

fall outside the

parameter

space.

Nothing

is

_changed

with

_respect

to

_Eyv,

which has the same

_diagonal

structure with p different elements in both situations. As far as

E

B

is

concerned,

a sufficient

statistic under the

_complete

data set is now the

(p

x

_p)

matrix

_bjb

_’

_.

₃

j

U ( ! f! f! ) U’.

However,

for a

_{given U,}

the

general

_expression

of the conditional j

expectation

of

_L

_:

_{fj f)}

_given

the data was

_already

derived

(see

[19]

and

[A5],

so that

j

the E

_step

remains the same.

Because all the elements of A are now

_required,

the

changes

to

_implement

at the M

_step

are the

_following:

_compute

the next value

I:!+1]

of

_E

_B

_(saturated

_model)

by

where

A’

l

is obtained from

_[A5]

with

U

M

being

the matrix of normed

_eigenvectors

of

E

W

.

Notice that here U is

_updated

at each iteration from the

_equation

E[t]U

lt]

=

U[t]![t].

Algorithms

based on first derivatives

_(part

_B)

Between

_family

_{components E}

B

Using

the

spectral

decomposition

in

_!13!,

ie r = nU’!U ₊

E

w

,

we obtain

where

_Uip

_Ui

₂

_{>&dquo;&dquo;}

_{Uie, ... ,}

_Ui

_r-

are

the ri

normed

_eigenvectors

of

E

B

correspond-ing

to the

_{eigenvalue 6}

_i

with

multiplicity

order ri. Remember that under the reduced

model,

rl = 1 and _r2 =

p &mdash; 1 for

6

1

and

6

2

defined in

[14a]

and

[14b],

respectively.

Substituting

8F /861

by

its

_expression

_[B1]

in the

equation

<9(-2L)/<9<*’t

= 0 leads

to:

Let L =

vn:U ! 1/2

with L

partitioned

in the same _way as

U,

ie

_Lie

=

vn

-

6iUi,,

where

_(A)

_i£ie

stands for the

_i

_f

_th

_diagonal

element of the A matrix. Now: and with

Then,

from and Furthermore: or so that and

Using

[B7]

and

_!B8!,

the

_system

_[B3]

reduces to

The EM

_procedure

described in

_(18a!,

_[18b]

and

_[19]

can be

_{alternatively}

written

The

_parallel

between

_[B9]

and

_[B10]

is

_{straightforward.}

Here B

_replaces

((s - 1)B

+

_r!/s,

since these 2

_quantities

have the same

_expectation

F under the

saturated model.

Residual

_components

_Ew

<9F

OEW

_{(OEW ! OEW}

_{..!v!. !}

Here,

ar - a!2 , C a!2 J -

1

and 2

= 0 for _any i

_{! i’.}

_Then,

the

a!W -

0,W!

a

B!!.A.’

equation

[8b]

= 0 _can be written _as

Equation

!B11!

defines a non-linear

_system

that can be solved

_iteratively

_using,

for