Genetic variation of traits measured in several environments. II. Inference on between-environment homogeneity of intra-class correlations

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Genetic variation of traits measured in several

environments. II. Inference on between-environment

homogeneity of intra-class correlations

C Robert, Jl Foulley, V Ducrocq

To cite this version:

Original

article

Genetic

variation

of traits measured

in

several

environments.

II.

Inference

on

between-environment

homogeneity

of

intra-class

correlations

C Robert

JL

_Foulley

V

_Ducrocq

Institut national de la recherche agronomique, station de

g6n6tique quantitative

et

_appliquee,

centre de recherche de

_{Jouy-en-Josas,}

78352

_{Jouy-en-Josas cedex,}

Rance

(Received

28

_{April 1994; accepted}

26

_September

₁₉₉₄₎

Summary - This paper describes a further contribution to the

problem

of

_testing

homo-geneity

of intra-class correlations among environments in the case of univariate linear

models,

without

_making

any

assumption

about the

genetic

correlation between

environ-ments. An iterative

_{generalized expectation-maximization}

_(EM)

_algorithm,

as described in

_Foulley

and

_Quaas

_(1994),

is

_presented

for

computing

restricted maximum likelihood

(REML)

estimates of the residual and

_{between-family}

_components of variance and

co-variance. Three different

_{parameterizations}

_(cartesian,

_polar

and

_spherical

_coordinates)

are

_proposed

to _compute EM-REML estimators under the reduced

_(constant

intra-class correlation between

_{environments)}

model. This

_procedure

is illustrated with the

_analysis

of simulated data.

heteroskedasticity

/

parameterization

/

intra-class correlation

_/

expectation-maximization

_/

restricted maximum likelihood

Résumé - Variation _génétique de caractères mesurés dans _plusieurs milieux. II. Infé-rence relative à des corrélations intra-classe constantes entre milieux. Cet article décrit

une

_approche

_permettant d’estimer les composantes de variance-covariance entre milieux dans le cas de corrélation intra-classe

_homogènes

entre

_{milieux, sans faire d’hypothèse}

sur

les corrélations

_génétiques

entre milieux pris 2 à 2. Un

_{algorithme itératif}

d’espérance-maximisation

_(EM),

_comparable

à celui décrit _par

_Foulley

et

_Quaas

_(1994),

est

_proposé

pour calculer les estimations du maximum de vraisemblance restreinte

_(REML)

des com-posantes résiduelles

_{et familiales}

de variance covariance. Trois

_{paramétrisations différentes}

(coordonnées

cartésiennes, polaires

et

_sphériques)

sont

_proposées

pour calculer les esti-mateurs EM-REML sous le modèle réduit

(les

corrélations intra-classe sont

_supposées

toutes

_égales

à une même

constante).

Cette

_procédure

est illustrée par

l’analyse

de données simulées.

hétéroscédasticité

_/

_{paramétrisation}

_/

corrélation intra-classe

_/

INTRODUCTION

Statistical

_procedures

based on the

_theory

of the

_generalized

likelihood

_ratio,

previously proposed by Foulley

et al

_(1994),

Shaw

₍₁₉₉₁₎

and Visscher

_(1992),

have been

_applied

to test the

_homogeneity

of

_genetic

and

_phenotypic

_parameters

against

Falconer’s

₍₁₉₅₂₎

saturated model. In

_particular,

Robert et al

₍₁₉₉₅₎

have described a

procedure

for

_estimating

_components

of variance and covariance

between environments and for

_testing

the

_homogeneity

of the

_following

_parameters:

(a)

a constant

_genetic

correlation between

_{environments;}

and

_(b)

constant

_genetic

and intra-class correlations between environments.

The

_objective

of this article is to

_present

a

procedure

for

dealing

with

homo-geneous intra-class correlations among environments without

making

any

as-sumption

about the

_genetic

correlations between environments. The method is based on restricted maximum likelihood estimators

(REML)

and on a

general-ized

_{expectation-maximization}

_(EM)

_algorithms

as

_proposed

initially by Foulley

and

_Quaas

₍₁₉₉₄₎

for heteroskedastic univariate linear models. Three

parameteri-zations of variance-covariance

_components

are

_suggested

for

_solving

this

_problem.

A simulated

_example

is

_presented

to illustrate this

_procedure.

THEORY

A model often used to deal with

_genotypic

variation in different environments is the

_2-way

crossed

genotype

(random)

x environment

_(fixed)

linear model with interaction. In

_particular,

this model has been

_proposed

as an alternative to a

multiple-trait

approach

when variance and covariance

_components

are

_homogeneous

and

_genetic

correlations between environments are

_positive

(Foulley

and

Henderson,

1989).

It has also been

_{employed by}

Visscher

₍₁₉₉₂₎

to

_study

the _power of likelihood ratio tests for

_{heterogeneity}

of intra-class correlations between environments when

genetic

correlations among them are assumed

_equal

to

_unity.

The aim of this paper

is _{to go} one

_step

further in

_addressing

the same

_problem

with the same model but with a

_{heterogeneous}

structure of variance-covariance

components.

The full model

Let us assume that records are

_generated

from a cross-classified

_layout.

The model is defined as follows:

where It

is the

_{mean, h}

_i

is the fixed effect of the ith environment:

_{a Si sj is}

the random

_{family j}

contribution such that

_{s! !}

_NID(0,1)

and

_Q

_s

_v

is the

_family

variance for records in the ith

_environment;

_0’!;!!,

is the random

_family

x

Model

_[1]

can be written more

generally

_using

matrix notation as:

where _Yiis a

(n

2

x

₁₎

vector of observations in environment

_i;

₁₃

is a

_(p

x

₁₎

vector of fixed effects with incidence matrix

_X

_i

_;

_ui

=

(s) )

and

u2

=

{h,s !

} are

2

_independent

random normal

_components

of the model with incidence matrices for standardized effects

_Zit

and

Z

2i

respectively;

_{cr! !}

and

_Q

_u

₂

_,.

are the

corresponding

_components

of

_{variance, pertaining}

to stratum i and ei is

the

vector of residuals for stratum i assumed

_{N( 0 , a}

_f

_{l, In, ) .}

The reduced model

The null

_hypothesis

_(H

_o

₎

consists of

_assuming

_homogeneous

intra-class correlations between environments

_{(ie, d i, ti}

=

(a;i +a!8i) / (!9!+!hsi+!e!)

= t). The

variance-covariance structure of the residual is assumed to be

_diagonal

and heteroskedastic. Under model

_[I],

this

_hypothesis

is tantamount to

_assuming

a constant ratio of variances between environments: V

_i,

_afl

_/

_(as.

+

a!8i)

=

8

2

,

where 8 is a constant. Under this

_hypothesis,

3 different

parameterizations

will be considered to solve this

_problem.

Cartesian coordinates

where 6 is a

_positive

real number. Polar coordinates

where pi and 6 are

_positive

real numbers.

Spherical

coordinates

where

!2

is a

_positive

real number. Under this

_{parameterization}

6’

=

tan’

a.

An EM-REML

_algorithm

A

_{generalized expectation-maximization}

_(EM)

_algorithm

to

_compute

REML

where y is the set of estimable

parameters

for each of the 3 models

_(under

each

parameterization

considered).

Ei

l

[.]

represents

the conditional

_expectation

taken with

_respect

to the distribution of fixed and random effects

_given

the data vector and

y =

y[

t

].

Ei

l

(.!

can be

_expressed

as a function of bilinear forms and a trace of

parts

of the inverse coefficient matrix of the mixed-model

_equations

_(as

described in

_Foulley

and

_Quaas,

_1994).

So,

for each

_{parameterization,}

we derive function

_[3]

with

respect

to each

parameter

of y

and we solve the

resulting

_system

8Q(Yly[t])

/

9

y

= 0. After

some

_algebra

and

_using

the method of

’cyclic

ascent’

(Zangwill, 1969),

we obtain

the 3

_{following algorithms.}

For model

_[2]

and

_using

cartesian

_coordinates,

the

_algorithm

at iteration

_[t,

I

_+1]

can be summarized as follows. Let

8

2ft

,

l]

,

0

,[t,l]

and

Q!t2!!.

be the values at iteration

[t, 1].

The next iterates are obtained as:

0

![tlc+i1

is the

only

_positive

root of the

following

cubic

_equation:

with

with

For model

_[2]

and

_{polar coordinates,}

the

_algorithm

at iteration

_!t,

I +

_1]

can be

summarized as follows. Let

8

2

[

t

,

1

],

ll

p

,

ft

and

0&dquo; !

be the values at iteration

_{[t, I].}

The next iterates are obtained as:

v

.

p!t,l+11

_{is the}

_{only positive}

_{root of the}

_{following quadratic}

_equation:

with:

.

0i’!!U

is the solution of _the

_equation

7-!!

=

tan(!’!!/2)

where

Z

f

t

,

t+11

is

the

_only

_positive

root of the

_{quartic equation:}

For model

_[2]

and

_spherical

coordinates,

the

_algorithm

at iteration

_[t,

l +

_1]

can

be summarized as follows. Let

1/1l

t

,

l]

,

o

pi

’

and

al!,4

the values at iteration

[t, l!.

The

next iterates are obtained as:

9

1/

1l

t

,

l+1]

is the

only

positive

root of the

following

quadratic

equation:

with:

with:

.

a!t,!+1!

_{is the} solution of the

_equation

,!!t’t+1!

=

tan!(a!-’+!/2)

where

xi’!!U

is the

only positive

root of the cubic

_equation:

The convergence of the EM-REML

procedure

is measured as the norm of the

vector of

_changes

in variance-covariance

_components

between iterations. In our

simulation and for the 3

_{parameterizations,}

_{convergence is assumed when the} norm

is less than

10-

6

.

In

_practice,

the number of inner iterations is reduced to

_only

one in the method of

_’cyclic

ascent’. The

_algebraic

solution of

_quadratic,

cubic or

quartic equations, using

the discriminant

_method,

demonstrates that each time

_only

one root is

_possible

in the

_parameter

_space. In the simulated

_example,

the

_polar

parameterization

converged

the fastest.

Testing

procedure

Let

_L(y;

_y)

be the

_{log-restricted}

_likelihood,

F be the

_complete

_parameter

_space and

_r

_o

a subset of it

_pertaining

to the null

_{hypothesis H}

_o

_.

_H

_o

is

_rejected

at the level a if the statistic

((y) =

_2Max

_r

_L(y;

_{y) - 2Maxr}

_o

_L(y;

_y)

exceeds

_(o

where

₍

₀

corresponds

to

_{Pr[X2 r , > (}

_o]

= _a

(

X2

is

the

_chi-square

distribution with r

_degrees

of freedom

_given

_by

difference between the number of

_parameters

estimated under the full and the reduced

_models).

Formulae to evaluate

_{-2MaxL(y; y)}

can

_easily

be made

_explicit:

where B is the coefficient matrix of the mixed-model

_equations.

NUMERICAL EXAMPLE

This

_procedure

is illustrated from a

_hypothetical

data set

_{corresponding}

to a

balanced,

crossed

_design

with 3

_{environments,}

20 families per environment and

50

_replicates

_per

_family

_(p

=

_3,

_s = 20 and n =

50).

The 20 families were

randomized within each environment. Basic ANOVA statistics for the

between-family

and

_{within-family}

sums of _squares and

cross-products

are

_given

in table I.

Table II

_presents

the estimation of

_genetic

and residual

parameters

under the full and reduced

_(hypothesis

of a constant intra-class correlation between

_{environments)}

models

_{respectively,}

and the likelihood ratio test of the reduced model

_against

the full model. The P values in table II indicate that there are no

_significant

differences

*

1,2,3

3 = the 3 environments.

8

Sums of

cross-products

between

_{families: n !(y2 j. -}

_{!/t..)(yt’?. !}

_Yi

_’

_..)

8 n j=1 8 n

Sums of _squares within

families: L L(Yijk -

Yijf

2

j=1 k=1

DISCUSSION AND CONCLUSION

In this _paper, estimation and

_testing

of

_homogeneity

of intra-class correlations among environments have been studied with heteroskedastic univariate linear models. Another

_{possible approach}

to account for

_’genotype

x environment’ effects

would be to consider the

_{multiple-trait}

linear

_approach,

defined

_by

Falconer

_(1952).

As described

_hereafter,

these 2

_approaches

_{may or} may not be

equivalent.

In this

discussion,

the conditions

required

to have

_equivalence

between the

_{multiple-trait}

and the univariate linear models will be established.

In Falconer’s

_approach,

_expressions

of the trait in different environments

_{(i, i’)}

are those of 2

_genetically

correlated

_traits,

with a coefficient of correlation

d(i,

i’),

Pii

’ =

!s!!,

/

aBaB.,.

The model is defined as follows:

where _lJ2!k

_is

the

_performance

of the kth individual

_(k

=

_{1, 2, ... ,}

n)

of the

jth family

(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

1

i,

Cov(b

ij

,

bi!!)

= a

Biil for i

7! i’

and

Cov(bi!, bi.!!)

= 0

for j #

j’

and _{any i} and

_i’;

_ljk is a residual effect

_pertaining

to the kth individual in the subclass

_ij,

assumed

_normally

and

_{independently}

distributed with mean zero and

variance

_{o,2 wi}

Under the

_hypothesis

of

_homogeneity

of intra-class correlations between

a

Likelihood ratio _test;

_b

_degrees

of freedom =

_2;

* same EM-REML estimates under the

multiple

trait

_approach.

number of

parameters.

Model

_[1]

has

_[2p

+

_1]

genetic

and residual

_parameters

and model

_[4]

has

_[(p(p

+

_1)/2)

+

_1]

parameters.

For p =

_3,

whatever the

hypotheses considered,

even

_though

these 2 models have

the same number of estimable

parameters,

the

parameter

spaces are not

_exactly

the same. Two conditions must be added to

_satisfy

the

equivalence

between the

multiple-trait

and the univariate linear models. The univariate linear model does

not allow the estimation of a

_{negative genetic}

correlation between

_{environments,}

since it is a ratio of variances.

_Thus,

we have the

_following

condition:

Then we have:

and

By definition,

_or

₂

_Si

and

_a!8i

are

_positive

_parameters,

so the

_following

relation must

be satisfied:

&dquo; &dquo;

It is worth

_noticing

that the condition in

_[6]

means that the

_partial

_genetic

correlation between any

pair

( j, k)

of environments for environments i fixed is also

positive.

The

problem

of

_testing

_homogeneity

of intra-class correlations between

environ-ments was

_finally

solved under 3 different

_assumptions

about the

_genetic

correla-tions between environments:

_equal

to one

_{(Visscher, 1992);}

constant and

_positive

(Robert

et

_al,

_1995);

and

_{just positive}

_{(this work).}

For more than 3

_traits,

model

[1]

is no

_{longer equivalent}

to the

_multiple

trait

approach

of Falconer. As a matter of

fact,

it

_generates

fewer

parameters

than

!4!,

2p

vs

p(p

+

_1)!2

for

_[1]

and

_[4]

_{respectively.}

This

parsimony

might

be an

_{interesting feature,}

because the difference in numbers of

_parameters

increases with the number of traits considered

_(eg,

10 vs

15

_parameters

for 5

_traits).

_Comparison

of

_approaches

on real

genetic

evaluation

problems

such as sire evaluation of

dairy

cattle in several countries would be of

great

interest.

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Falconer DS

₍₁₉₅₂₎

The

problem

of environment and selection. Am Nat

_86,

293-298

Foulley JL,

Henderson CR

₍₁₉₈₉₎

A

simple

model to deal with sire

_by

treatment

interactions when sires are related. J

Dairy

Sci _72, 167-172

Foulley JL, Quaas

RL

₍₁₉₉₄₎

Statistical

analysis

of

heterogeneous

variances in Gaussian linear mixed models. Proc 5th World

_Congress

Genet

_Appl

Livest

Prod,

Univ

_Guelph,

Guelph, ON, Canada,

18, 341-348

Foulley JL,

Hébert

D, Quaas

RL

₍₁₉₉₄₎

Inference on

_homogeneity

of

_{between-family}

components of variance and covariance among environments in balanced cross-classified

designs.

Genet Sel

_{Evol 26,}

117-136

Lawley DN,

Maxwell AE

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_{C, Foulley JL, Ducrocq}

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₍₁₉₉₅₎

Genetic variation of traits measured in several environments. I. Estimation and

testing

of

_homogeneous

and intra-class correlations between environments. Genet Sel

Evol 27,

111-123

Shaw RG

₍₁₉₉₁₎

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comparison

of

_{quantitative genetic}

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Evolution

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143-151

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₍₁₉₉₂₎

On the power of likelihood ratio tests for

detecting heterogeneity

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