A link function approach to heterogeneous variance components

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A link function approach to heterogeneous variance

components

Jean-Louis Foulley, Richard L. Quaas, Caroline Thaon d’Arnoldi

To cite this version:

Original

article

A link

function

_approach

to

_{heterogeneous}

variance

_components

Jean-Louis

_Foulley

Richard

L.

_Quaas

_b

Thaon

d’Arnoldi

a

Station de

_{génétique quantitative}

et

_appliquée,

Institut national de la recherche

agronomique,

CR de

_Jouy,

78352

_{Jouy-en-Josas Cedex,}

France b

Department

of Animal

Science,

Cornell

_{University, Ithaca,}

NY 14853, USA

(Received

20 June

1997; accepted

4 November

₁₉₉₇₎

Abstract - This _{paper presents}

_techniques

of parameter estimation in heteroskedastic mixed models

_having

_i)

_{heterogeneous log}

residual variances which are described

_by

a

linear model of

_explanatory

covariates and

_ii)

_log

residual and

_log

u-components

linearly

related. This makes the intraclass correlation a monotonic function of the residual variance. Cases of a

_homogeneous

variance ratio and of a

_homogeneous

_u-component of variance are also included in this _{parameterization.} Estimation and

_{testing procedures}

of the

_{corresponding dispersion}

_parameters are based on restricted maximum likelihood

procedures. Estimating equations

are derived

using

the standard and

gradient

EM. The

analysis

of a small

_example

is outlined to illustrate the

_{theory. ©}

_{Inra/Elsevier,}

Paris

heteroskedasticity

/

mixed model

_/

maximum likelihood

_/

EM _algorithm

Résumé - Une _approche des _composantes de variance _{hétérogènes} par les fonctions de lien. Cet article

_présente

des

_techniques

d’estimation des

_paramètres

intervenant dans des modèles mixtes caractérisés

_i)

par des

logvariances

résiduelles décrites par un modèle linéaire de covariables

_explicatives

et

_ii)

par des composantes u et e liées _par une fonction affine. Cela conduit à un coefficient de corrélation intraclasse

_qui

varie comme une fonction monotone de la variance résiduelle. Le cas d’une corrélation constante et celui d’une

composante u constante sont

également

inclus dans cette

_{paramétrisation.}

L’estimation et les tests relatifs aux

_paramètres

de

dispersion correspondants

sont basés sur les méthodes du maximum de vraisemblance restreint

_(REML).

Les

équations

à résoudre pour obtenir

ces estimations sont établies à

_partir

de

_{l’algorithme}

EM standard et

_gradient.

La théorie est illustrée par

l’analyse numérique

d’un petit

exemple. ©

Inra/Elsevier,

Paris

hétéroscédasticité

_/

modèle mixte

_/

maximum de vraisemblance

_/

_algorithme EM

*

1. INTRODUCTION

A

_previous

_paper of this series

_[4],

_presented

an EM-REML

_{(or ML)}

_approach

to

_{estimating dispersion}

_parameters

for heteroskedastic mixed models. We assumed

i)

a linear model on

log

residual

(or e)

_variances,

_and/or

_ii)

constant u to e variance

ratios.

There are different _ways to relax this last

_assumption.

The first one is to

_proceed

as with residual

_variances,

i.e.

_hypothesize

that the variation in

_log

_u-components

or of the u to e-ratio

_depends

on

explanatory

covariates observed in the

_experiment,

e.g.

region,

herd, parity,

management

conditions,

etc. This is the so-called structural

approach

described

_by

San Cristobal et al.

_[23],

and

_{applied by Weigel}

et al.

_[28]

and De Stefano

_[2]

to milk traits of

_dairy

cattle.

Another

_procedure

consists in

_assuming

that the residual and

_u-components

are

directly

linked via a

relationship

which is less restrictive than a constant ratio. A

basic motivation for this is that the

_assumption

of

_homogeneous

variance ratios or intra class correlations

_(e.g.

_heritability

for animal

_breeders)

_might

be unrealistic

[19]

although

very convenient to set _up for theoretical and

_{computational}

reasons

(see

the

_procedure

_by

Meuwissen et al.

_[16]).

As a matter of

_fact,

the power of

statistical tests for

detecting

such

_{heterogeneous}

heritabilities is

_expected

to be low

[25]

which may also

explain why

homogeneity

is

_preferred.

The _purpose of this second _paper is an

_attempt

to describe a

_procedure

of this

_type

which we will call a link function

_approach

_referring

to its close connection with the

_{parameterization}

used in GLM

_theory

_{[3, 14].}

The _paper will be

_{organized along}

similar lines as the

_previous

_paper

[4]

_including

i)

an initial section on

_theory,

with a brief summary of the models and a

_presentation

of the

_{estimating equations}

and

_{testing procedures,}

and

_ii)

a numerical

_application

based on a small data set with the same structure as the one used in the

_previous

paper

[4].

2. THEORY

2.1. Statistical model

It is assumed that the data set can be stratified into several strata indexed

_by

(i

=

_{1, 2, ... ,}

I)

_representing

_a

_potential

_{source of}

heteroskedasticity.

For the sake of

_simplicity,

we will consider a standardized _one-way random

_(e.g.

_sire)

model as in

_Foulley

_[4]

and

_Foulley

and

_Quaas

_[5].

where yi is the

(n

i

x

1)

data vector for stratum

_i;

_j3

is a

(p

x

1)

vector of unknown fixed effects with incidence matrix

_X

_i

_,

and ei is the

(n

2

x

1)

vector of residuals. The contribution of the

_systematic

random

_part

is

_represented

_by

_O

_&dquo;uiZ

_iU

_*

where u

*

is a

(q

x

₁₎

vector of standardized

_deviations,

_Z

_i

is the

_{corresponding}

incidence

matrix _{and <7u,} is the _square root of the

_u-component

of

_variance,

the value of which

The influence of factors

_causing

the

_{heteroskedasticity}

of residual variances is modelled

_along

the lines

presented

in Leonard

_[13]

and

_Foulley

et al.

_{[6, 7]}

via a

linear

_regression

on

log-variances:

where 5 is an unknown

(r

x

1)

real-valued vector of

_parameters

and

_p’

is the

corresponding

(1

x

_r)

row incidence vector of

qualitative

or continuous covariates.

Residual and

u-component parameters

are linked via a functional

_relationship

or

_equivalently

where the constant T

equals

exp(a).

The differential

_equation

_pertaining

to

_[3ab],

i.e.

_(d

_C7u

_jC

_7uJ

_-

_b(dC7ejC7eJ

= 0 is a scale-free

_relationship

which shows

clearly

that the

_parameter

of interest in this

model is b. Notice the close connection between the

parameterization

in

_equations

[2]

and

_[3ab]

with that used in the

_approach

of the

’composite

link function’

proposed

by

Thompson

and Baker

_[24]

whose

_steps

can be summarized as follows:

i

) (C7ui,C7eJ’

=

f(a,b,C7e

J

;

ii)

C7ei =

exp(?

7

i),

and qi

=

(112)p’

6

.

As

compared

to

Thompson

and

Barker,

the

_only

difference is that the function

_f

in

_i)

is not linear and involves extra

_parameters,

i.e. a and b.

The intraclass correlation

_{(proportional}

to

_heritability

for animal

_breeders)

is an

_increasing

function of the variance

_{ratio p}

_i

=

ou

i

/!e..

In turn _pi increases or decreases with

_u,

₂

_depending

on b > 1 or b <

1,

respectively,

or remains constant

(b

=

1)

since

dpi/p

i

=

2(b - l)do’e!/o’e!.

Consequently

the intraclass correlation

increases or decreases with the residual variance or remains constant

_(b

=

1).

For

2.2. EM-REML estimation

The basic EM-REML

_procedure

_{[1, 18]}

proposed

by

Foulley

and

Quaas

(1995)

for

_{heterogeneous}

variances is

_applied

here.

Letting

_g

/ ’ ’ y’ )’ e=(e’ e’ e’ .. e’ )’

_{i 1 1 2 i 1} > and

_{’y = (6’, T, b)’,}

1

> > >

the EM

_algorithm

is based on a

_complete

data set defined

_by

x =

(p ,

u

*

,

e’)’

and

its

_{loglikelihood}

_{L(y; x).}

The iterative process takes

place

as in the

_following.

The

_E-step

is defined as

usual,

i.e. at iteration

_[t],

calculate the conditional

expectation

of

_{L(y; x)}

_given

the data _{y and y =}

_y’

_l

as shown in

Foulley

and

Quaas

[5],

reduces to

where

E1

t

] (.)

is a condensed notation for a conditional

_expectation

taken with

respect

to the distribution of x in

_{Q given}

the data vector _y and y =

1

’[

t]

.

Given the current estimate

₁

_’[

_t]

of y, the

M-step

consists in

_calculating

the next

value

₁

_’[

_tH

_]

_by

_maximizing

₎

_t]

_1’[

_I

_’

₁

_Q(

in

_equation

₍₄₎

with

_respect

to the elements of the vector _y of unknowns. This can be

_accomplished

efficiently

via the

Newton-Raphson algorithm.

The

_system

of

_equations

to solve

_iteratively

can be written in

matrix form as:

where

_P(

_rxl

_{) _}

_{(P1!P2,...,Pi,...,P1)i}

_Vó

_[

_Ix1

_]

=

f a!la!n!e!J! vT -

fi9QIa-rl,

v

b =

{8Q/ab!;

W

a

p

=

åQ/åaå/3’,

for _a and

j3

being

_components

of y =

(5’,

T

, bl’.

Note that for this

_algorithm

to be a true

EM,

one would have to iterate the NR

_algorithm

in

_equation

₍₅₎

within an inner

cycle

(index

£)

until _convergence to

the conditional maximizer

_y

_[t+1]

_{_}

_y

_l

_’,’

_]

at each M

_step.

In

_practice

it _may be

advantageous

to reduce the number of inner

_iterations,

even _up to

_only

one. This is an

application

of the so called

’gradient

EM’

algorithm

the convergence

properties

of which are almost identical to standard EM

_[12].

The

_algebra

for the first and second derivatives is

_given

in the

_Appendix.

These derivatives are functions of the current estimates of the

parameters

y =

’

Yl

’l,

and

of the

components

of

E!t](eiei)

defined at the

_E-step.

with a _cap for their conditional

_{expectations,}

_e.g.

These last

_quantities

are

_just

functions of the sums

X’y

i

,

Z’yi,

the sums of squares

y§y

i

within

strata,

and the GLS-BLUP solutions of the Henderson mixed

model

_equations

and of their accuracy

[11],

i.e.

where ’ 1

Thus,

deleting

[t]

for the sake of

_simplicity,

one has:

r <&dquo;* f i where

_j3

and u* are solutions of the mixed model

equations,

and C _

[

CO

,

3

CUO

C

O

.

Cuu

_Ju

1

is the

_partitioned

inverse of the coefficient matrix in

_equation

_(7).

For

_grouped

data

(n

i

observations in subclass i with the same incidence matrices

_X

_i

=

l

ni

x’

and

Zi

=

1,,iz’),

formulae

(8)

reduce to:

where

2.3.

_{Hypothesis testing}

Tests of

_hypotheses

about

_dispersion

parameters

y =

(Õ’,

7

, b)’

can be carried

out via the likelihood ratio statistic

_(LRS)

as

_proposed

_by

Foulley

et al.

_{[6, 7].}

Let

_H

_o

_:

_y E

.f2

o

be the null

_hypothesis,

and

H

1

:

y E ,f2 -

_,f2

_o

its alternative where

,f2

o

and Q refer to the restricted and unrestricted

parameter

spaces,

respectively,

where

_y

_and

_y

_are

the REML estimators of y under the restricted

(Ho)

and unrestricted

_(H

_o

U

_H

₁

₎

models. Under standard conditions for

_H

_o

_(excluding

hypotheses allowing

the true

_parameter

to be on the

_boundary

of the

_parameter

space as addressed

by

Robert et al.

_{[22], A}

has an

_asymptotic

_chi-square

distribution with r = dim ,f2 - dim

_S

_-2

₀

degrees

of freedom. Under model

_(1),

an

_expression

_{of -2L(y; y)}

is:

The theoretical and

practical

aspects

of the

_hypotheses

to be tested about the structural

component

5 have been

_already

discussed

_by

Foulley

et al.

_{[6, 7],}

San

Cristobal et al.

_[23]

and

_Foulley

_[4].

As far as the functional

_relationship

between the residual and

_u-components

is

concerned,

interest focuses

_primarily

on the

_hypotheses

i)

a constant variance ratio

(b

=

1),

and

ii)

_{a constant}

u-component

of variance

_(b

=

0) [2,

_{16, 22,}

28].

Note that the structural functional model can be tested

_against

the double

struc-tural model:

_{fn o, ei}

₂

_{= pi b}

_e

_,

and

_{fn o, u 2i}

=

p§ 5

u

with the _{same covariates. The reason}

for that is as follows. Let P =

[11P’],

5

e

=

,

e

[ry

6*]

and

&eth;

u

=

(r!!, 6*]

_pertaining

_to a traditional

_{parameterization}

_involving

_intercepts

_{qe and ?7u} for

_describing

the residual and

u-components

of

_variance,

_{respectively,}

of a reference

_population.

The structural functional model reduces to the null

hypothesis

6*

=

2bbe,

thus

result-ing

in an

_asymptotic

_chi-square

distribution of the LRS

_contrasting

the two models with

_Rank(P)-2

_degrees

of freedom.

2.4. Numerical

_example

For readers interested in a test

_example,

a numerical illustration is

presented

based on a small data set obtained

_by

simulation. For

_pedagogical

reasons, this

example

has the same structure as that

presented

in

_Foulley

_[4],

i.e. it includes two

crossclassified fixed factors

_(A

and

_B)

and one random factor

_(sire).

The model used to

_generate

records is:

where a

is a

_general

_{mean, ai,}

₁

₃

_j

are the fixed effects of factors

_{A(i = 1, 2)}

and

B(j

=

1,2,3),

sk

the standardized contribution of male k _{as a} sire and

1/2se )

that of male as a maternal

_grand

sire.

Except

for T= 0.001016 and b =

1.75,

the values chosen for the

_parameters

_are

Table

_{II presents}

-2L

_values,

LR statistics and P-values

_contrasting

the

_following

different models:

1)

additive for both

_log

_Q

_e

and

_{log or}

₂

_;

2)

additive for

_{log u2}

_and

_{log as}

= _{a +} b

log a,;

3)

constant variance ratio

_(b

=

1);

4)

constant sire variance

_{(b = 0).}

In this

_example,

models

₍₃₎

and

₍₄₎

were

_rejected

as

expected

whatever the

alternatives,

i.e. models

₍₁₎

or

(2).

Model

(2)

was

_acceptable

when

compared

to

₍₁₎

thus

illustrating

that there is room between the

complete

structural

approach

and

the constant variance ratio model.

The

_{corresponding}

estimates of

_parameters

are shown in table IIL Estimates

of the functional

relationship

are T = 0.001143 and b =

_3.0121,

this last value

being higher

than the true one, but - not

_surprisingly

in this small

_{sample -}

not

significantly

different

_(A

= 1.5364 and P-value =

0.215).

3. DISCUSSION AND CONCLUSION

This has been an issue for _{many years in} the animal

breeding community.

For

instance for milk

_yield,

the

_assumption

of a constant

_heritability

across levels of

environmental factors

_(e.g.

countries, regions, herds,

years,

management

conditions)

has

_generated

considerable

controversy:

see Garrick and Van Vleck

[8],

_Wiggans

and VanRaden

_[29];

Visscher and Hill

_[26],

_Weigel

et al.

_[28]

and DeStefano

_[2].

Maximum likelihood

computations

are

_{based, here,}

on the EM

algorithm

and

different

_simplified

versions of it

_(gradient

EM,

ECM).

This is a

powerful

tool

for

_addressing

problems

of variance

_component

_estimation,

in

_particular

those of

_{heterogeneous}

variances

_[4,

_5,

_{7, 20,}

_21].

It is not

_only

an _easy

_procedure

to

implement

but also a flexible one. For

_instance,

ML rather than REML estimators can be obtained after a

slight

modification of the

E-step

resulting

for

grouped

data in

where

_M

_uu

is the u x u block of the coefficient matrix of the Henderson mixed

model

_equations.

Moreover the

_procedure

can be extended to models with

_several(k

_{= 1, 2, ... ,}

_K)

uncorrelated u random

factors,

_e.g.

Such an extension will be _easy to make via the ECM

_(expectation

conditional

maximization)

algorithm

[15]

in its standard or

_gradient

version

_along

the same

lines as those described in

Foulley

[4].

However caution should be exercised in

applying

the

_{gradient ECM,}

for this

_algorithm

no

_longer

_guarantees

_{convergence in}

likelihood values. Other alternatives

_might

be considered as well such as the _average information-REML

_procedure

_{[10, 17].}

In

_conclusion,

the likelihood framework

_provides

a

_powerful

tool both for

estimation and

_hypothesis

_testing

of different

_competing

models

_regarding

those

problems. However,

additional research work is still needed to

_study

some

_properties

of these

_procedures

_especially

from a

practical

_point

of

_view,

for

example

the _power

of

_testing

such

_assumptions

as b = 1.

ACKNOWLEDGEMENTS

This work was conducted while Caroline Thaon was on a

_’stage

de fin d’6tudes’ at the Station de

g6n6tique quantitative

et

_appliqu6e

_(SG(aA),

_Inra,

_{Jouy-en-Josas}

as a student from the Ecole

_{Polytechnique,}

Palaiseau. She

greatly acknowledges

the

support

of both institutions in

_making

this

_stay

feasible. Thanks are

expressed

to

Elinor

Thompson

(Inra-Jouy

en

_Josas)

for the

_English

revision of the

_manuscript.

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[1]

Dempster

A.P.,

Laird

_N.M.,

Rubin

_D.B.,

Maximum likelihood from

incomplete

data

via the EM

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₍₁₉₇₇₎

1-38.

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and

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Al. APPENDIX: Derivatives for the EM

_algorithm

The

_Q

function to be maximized is

_(in

condensed

_notation)

with

and,

Al.l. Derivative with

_respect

to 5

_(log

residual

_parameters)

According

to the chain

_rule,

one has

Now

That is

and

I

Letting

v5 =

8Q /8£n ufl

so that

!! =

L

_v5ipi =

P’

V

,5,

then

! °! i = 1 Let us define

and,

the same

_symbols

with a hat for their conditional

expectations,

i.e.

an alternative

_expression

for

computing

(A4)

is

already

reported by Foulley

et al.

_[6]

and

_Foulley

_[4]

for models with a

homogeneous

u-component

of

_variance,

and a constant u to e variance

_ratio,

_{respectively.}

Al.2. Derivative with

_respect

to T

with

and

or, more

_explicitly

Al.3. Derivative with

_respect

to b

and

After

_developing

and

_rearranging,

one obtains

Letting

b = 0 and b = 1 in

(All)

leads _to

and

Again

these are the same

_expressions

as those

_{given by Foulley}

et al.

_[6]

and

Foulley

[4]

for a constant

_u-component

of variance and a constant variance

ratio,

respectively.

Al.5. b - T derivatives

where

Now

Al.6. 5 - b derivatives where Now and so that A1.7. T - T derivatives

Differentiating

(A7)

once

_again

with

_respect

to T leads to

Al.8. T - b derivatives

From

_(A7),

one has

Finally,

the

_{Newton-Raphson algorithm}

to

_implement

for the

M-step

of the EM

algorithm

can be written in condensed form as:

where at iteration

_[n],

A6

[n]

=

b!’!! -

1]

,

-

n

¿;[

and

!T!!! = T

[

n

] - T

[

n

-

1

]

and

Abl&dquo;

I

=

b[

n

] - b

[

n

-1]

.