Variance estimation from integrated likelihoods (VEIL)

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Variance estimation from integrated likelihoods (VEIL)

D Gianola, Jl Foulley

To cite this version:

Original

article

Variance estimation

from

_integrated

likelihoods

_(VEIL)

D

Gianola

1

JL

_Foulley

1

University of Illinois, Department of Anima1 Sciences, Urbana, IL 61801 _USA;

2 _{Institut National de la Recherche}

Agronomique,

b’tation de (Uénétique Quantitative et Appliquée,

Centre de Recherches de _{Jouy-en-Josas,} 78352 _{Jouy-en-Josas Cedex,} France

(Received

17 _{January 1990; accepted} 22

_{August 1990)}

Summary - A method of variance _component estimation in univariate mixed linear mod-els based on the exact or _{approximate posterior} distributions of the variance components

is _presented. From these distributions, posterior means, modes and variances can be

cal-culated _exactly, via numerical methods, or _{approximately, using} inverted

_X

₂

distributions.

Although

particular attention is given to a _{Bayesian analysis} with $at priors, informa-tive _prior distributions can be used without _great additional _{difficulty. Implementation} of

the exact _analysis can be _taxing from a numerical _point of _view, but the _{computational} requirements of the _approximate method are not greater than those of REML.

variance _{components / maximum likelihood / Bayesian} methods

Résumé - Estimation des _composantes de la variance à partir de vraisemblances

intégrées. Cet article présente une méthode d’estimation des composantes de la _variance,

pour des modèles linéaires mixtes _{univariés, qui} est _fondée sur des _expressions exactes et _approchées de la distribution a _posteriori de ces _composantes. On peut calculer les espérances, modes et variances de ces _{distributions,} de _façon exacte par des méthodes numériques ou

_{de façon}

_approchée en utilisant des lois

de

X

2

inverses. Bien _que l’attention

soit centrée sur une _{analyse bayésienne faisant} intervenir des a _priori

_uniformes,

le recours

à des lois a _priori

_informatives

est

_envisageable

sans

_difficulté

notoire additionnelle. La mise en œuvre de _l’analyse exacte peut être _{contraignante} d’un _point de vue _numérique,

mais la méthode _approchée ne demande pas plus de calculs que la résolution du REML.

composantes de la variance _/ maximum de vraisemblance _/ méthodes bayésiennes.

INTRODUCTION

Harville and Callanan

₍₁₉₉₀₎

noted that likelihood-based methods of variance

com-ponent estimation have

_gained

favor among

quantitative

geneticists.

In

particu-lar,

the

_procedure

known as "restricted maximum

likelihood",

or REML for short

(Thompson,

1962;

Patterson and

_Thompson,

_1971),

is now

widely

_regarded

in

respect to the variances

_only

the _part of the likelihood function

_(normality

is

as-sumed)

that does not

_{depend on}

fixed effects. In so

_doing,

Patterson and

_Thompson

(1971)

obtained an estimator that &dquo;accounts for the

_degrees

of freedom lost in the estimation of fixed effects&dquo;

_{which, according}

to their

_reasoning,

is not

_accomplished

by

full maximum likelihood

_(ML’.

With balanced

_data,

the REML

_estimating

equa-tions reduce to those used in estimation

_{by analysis}

of variance

_(ANOVA)

so that

if the ANOVA estimates are within the parameter space, these are REML as well.

The basic idea

_underlying

REML is

_obtaining

a likelihood-based estimator

(thus

retaining

the usual

_asymptotic

_properties)

while

_reducing

the bias of ML.

_However,

REML estimators are biased as well and because these are constructed from a

partial likelihood,

one should _expect

larger

variance of estimates than with ML. A more reasonable

_comparison

is in terms of a loss function such as

_mean-squared

error, but Monte Carlo studies

(for

example,

Swallow and

Monahan,

1984)

have

given ambiguous

results. In

_fact,

in a

purely

fixed model with a

_single

location

parameter and 1 variance _component, the ML estimator has smaller

_mean-squared

error than REML

_throughout

the whole _{parameter space,}

_ie,

REML is inadmissible under

_quadratic

loss for this model.

_Lacking

evidence to the contrary, a similar

sampling

behavior can be

_expected

at least in some other models. Searle

(1988)

puts it

_{concisely :}

&dquo;It is

_difficult

to be

_anything

but inconclusive about which

_of

ML and REML is the

_pneferred

method&dquo;.

The 2

_procedures

have the same

_asymptotic

_properties

(although

their

asymp-totic variance is

_different)

and unknown small

_sample

_{distributions,}

a feature shared

by

all

_{sampling theory}

estimators of variance _components.

Although

ML and REML

estimates are defined inside the

appropriate

parameter space, their

asymptotic

dis-tribution

_(normal)

can _generate interval estimates that include

negative

values. This

_{potentially embarrassing phenomenon}

is often overlooked in discussions of

likelihood based methods.

Harville

₍₁₉₇₄₎

showed that REML

_corresponds

to the mode of the

_marginal

posterior

density

of the variance _components in a

_Bayes

_analysis

with flat

_priors

for fixed effects and variance _components. The fixed effects

₍₀₎

are viewed in REML as nuisance

parameters

and,

therefore are eliminated from the full likelihood

by

integration

so that the restricted likelihood is a

_marginal

likelihood

(Dawid, 1980),

seemingly

a more

appropriate

term. This likelihood does take into account the error incurred in

_{estimating ,0}

because it can be viewed as a

_weighted

average of

likelihoods of variance components evaluated at all

_possible

values of

_,Q,

the

_weight

function

_being

the

_posterior

_density

of the fixed effects

_{(proportional}

to the

_marginal

likelihood of

,0

in this

_context).

Gianola et al

₍₁₉₈₆₎

gave a

_Bayesian

_{justification}

of REML when

_{estimating breeding}

values with unknown variance _components. The same,

by

virtue of the symmetry of the

Bayesian analysis, applies

to estimation of fixed effects.

There

are 2 _aspects of REML that have not received sufficient discussion.

First,

the method

_produces

_joint

modes rather than

_marginal

modes. If the loss function is

quadratic,

the

_optimum

_Bayes

estimator is the

posterior

mean, and

marginal

modes

provide

better

_{approximations}

to this than

_joint

modes

_{(O’Hagan, 1976).}

_Second,

in some

_problems

not all variance parameters have

_equal

importance.

For

example,

suppose that there is interest in

making

inferences about the amount of additive

requires

a model

_that,

in addition to fixed

effects,

includes random

herd, additive,

dominance,

permanent environmental and temporary effects. In this

_situation,

the

marginal

or restricted likelihood of the variance components involves 5

_dimensions,

1 for each of the

_variances,

_yet all components other than the additive one should be

regarded

as nuisance _parameters.

Carrying

the

logic

of Patterson and

_Thompson

(1971)

1 _step

_further,

REML would not take into account the error incurred in

estimating

the nuisance variances

_{and, therefore, only}

the part of the likelihood that is a function of the additive

_genetic

variance should be maximized. Construction of

this likelihood

_employing

classical statistical _arguments seems

_impossible.

The

_objective

of this note is to _present a new method of variance _component estimation in univariate mixed linear models based on the

principle

of

_finding

the

_marginal

_posterior

_density

of each of the variances. From

_this,

means,

modes,

variances and

_higher

order moments of the distribution of variance _components can

be calculated

_exactly,

_using

numerical

_methods,

or

_analytically

via

_{approximations.}

Particular attention is

_given

to the

_Bayesian

_analysis

with flat

_priors

for fixed effects and variance _components.

MODEL AND DEFINITIONS

Model

Consider the mixed linear model:

where :

y: data vector;

_{X, Z}

_i

_:

known incidence matrices. For i = _c,

Z

i

=

I,

an

identity

matrix of

_appropriate

_order;

₍

_3:

_p x 1 vector of

_uniquely

defined fixed effects

_(X

has full-column

_rank);

_{ui: qi} x 1 random vector distributed as

N(O,

Ai!2),

where

A

i

is a known _square,

_{positive-definite}

matrix and

_Q2

_(i

=

_{1, 2, ... ,}

_c)

_{is a variance}

component. It is assumed that all ui and Uj are

_mutually

_independent.

For i = _c,

A!

is often taken to be an

identity

matrix of

appropriate order;

this will be assumed

in the _present paper.

The

_marginal

distribution:

gives

rise to the &dquo;full&dquo; likelihood

_function,

from which ML estimators

_{of P}

and of

each of the variance _components can be obtained.

Bayesian

elements

and the normal distributions

_{assigned independently}

to the

u

i

’s

are viewed as

prior

probability

distributions as well. To

_complete

the

model,

_assignment

of

_prior

probability

distributions to the variance _{components is necessary. In this paper,}

and for the sole _purpose of

_showing

how a

_{higher degree}

of

_{marginalization}

can be achieved than in

_REML,

_independent

&dquo;flat&dquo;

_prior

distributions are

_assigned

to each

of the variances

_!2.

With this

_formulation,

the

_joint

_posterior

_density

_{of P}

and of

variances becomes:

so that there is no information about fixed effects and variance parameters

beyond

that contained in the likelihood function. If

₍₄₎

is maximized

_jointly

with respect to

(3, u!, !z ! ! ! ! ! u§

one obtains the maximum likelihood estimates of all parameters. If

(4)

is

_integrated

with respect to

_p,

the

_marginal

_(restricted)

likelihood function of the variance components is obtained

_{(Harville, 1974).}

This contains no information about

_!3

and maximization of this

_marginal

likelihood with _respect to the variances

gives

restricted maximum likelihood estimates of these parameters.

REPRESENTATION OF THE MARGINAL

_(RESTRICTED)

LIKELIHOOD

As shown

_by

Gianola et al

_{(1990a, 1990b),}

_integration

of

₍₄₎

with _respect

_to

_/

₃

_gives:

where:

and

is the solution to Henderson’s mixed model

_equations.

_Although

₍₅₎

is a

_posterior

density,

it is

_{strictly proportional}

to the

_marginal

likelihood of the variance

POSTERIOR DENSITY OF THE RESIDUAL VARIANCE

AND OF RATIOS OF VARIANCE COMPONENTS Consider the one-to-one

reparameterization

from

_o,

₂

_,0-

₂₂

_,...,0,2

to

and

Because maximum likelihood estimates are invariant under one-to-one

reparam-eterizations,

maximization of

₍₅₎

with _respect to the 0-2 or to the _a-parameters

gives

the same estimates.

_However,

and

_arguing

from a likelihood

_veiwpoint,

it is

unclear how

₍₅₎

can be factorized into _{separate components} for each of the old or new _parameters.

_Hence,

estimates of a

_particular

uf

or _ai from

₍₅₎

are obtained in the _presence of the

_{remaining or}

2

1

or a’s. To the extent that these are viewed as nuisance parameters, REML would not take into account the

_{&dquo;degrees}

of freedom

lost&dquo; in the estimation of such _parameters.

In the

_{Bayesian approach}

_parameters are viewed as random

_variables,

so a

_change

from a a2 to an

_{a-parameterization}

must be made

_{using theory}

of transformation of random variables. The absolute value of the determinant of the Jacobian matrix

of the transformations indicated in

_(6a)

and

_(6b)

is:

Using

this in

₍₅₎

_gives

as

_{joint posterior}

_density

of the a’s:

The _range of the a’s

_depends

on the model in

_question.

The a! parameter is

always larger

than 0. In an &dquo;animal&dquo; model with 2 variance _components, the ratio

between the environmental and the additive

_genetic

variances can vary between 0

and

_infinity.

On the other

_hand,

in a &dquo;sire&dquo;

_model,

the

_{corresponding}

ratio varies

between 3 and

_infinity.

POSTERIOR DENSITY OF THE RATIOS OF VARIANCE COMPONENTS

With the above

parameterization,

a! can be

integrated

out

analytically

using

Although

the

_joint

posterior

density

(8)

is not _{in any}

_{obviously recognizable form,}

the _components of the modal vector, or a-values

_maximizing

₍₈₎

can be calculated. Let the

_logarithm

of

₍₈₎

be

_L,

so that:

where:

The mixed model &dquo;residual&dquo;

_sum

of squares

B(.)

depends

on the variance ratios

_through

the solution vector

b.

_Following

Macedo and Gianola

₍₁₉₈₇₎

and

suppressing

the

_dependence

of

_B(.)

on the a’s in the

_notation,

we have for

i = 1,2,...,c- 1:

Setting

(10)

to zero and

_rearranging:

with:

Above:

is a

_{block-diagonal}

_matrix,

and

where

_C

_ii

is defined below.

_Using

_(12a)

and

_(12b)

in

₍₁₁₎

and

_{rearranging gives,}

for i =

_{1,2,...,c - 1:}

Above,

and

Ci

i

is the

_partition

of

_(W’W+E)-

₁

_{corresponding}

to the ui effects. Note that

q

= > 4 and n >

_!+2c

are

_required.

The

_expression

in

₍₁₃₎

can be

_interpreted

as the ratio between an &dquo;estimator&dquo; of

o,2

_and

an &dquo;estimator&dquo; of

!2.

Because

(13)

is not

explicit

in the

_a’s,

the

_expression

must be iterated upon, thus

generating

a sequence

of successive

approximations.

For

_{example, starting}

with

a!O]

_(i

=

1, 2, ... ,

c—1)

_one

can solve the mixed model

_equations,

evaluate the

_right

hand-side of

_(13),

obtain

a new set of a’s and _repeat the calculations _{until convergence.}

In addition to first order

_algorithms

such as the one

_{suggested above, computing}

strategies

based on second derivatives are

_possible.

These are tedious and are not

presented

here. The

_{important point}

is that the components of the mode of the

joint posterior

density

of al, a2, ... , a!_1 can be obtained without _great

_conceptual

di!culty. Hereafter,

these _components will be denoted as

5i,Q’2,5e-i. Contrary

to

REML,

account is

_being

taken here of the

_&dquo;degree

of freedom&dquo; lost in

_estimating

a!

(a! in

the

original

parameterization).

EXACT MARGINAL DISTRIBUTION OF EACH

OF THE VARIANCE COMPONENTS Residual _component

Consider

_density

₍₇₎

and write it as:

The first conditional

_density

in

₍₁₄₎

is obtained from

₍₇₎

_{by regarding}

ac as a

variable and the

_remaining

a’s as constants. Hence:

This is

_exactly

the kernel of the

_density

of an inverted

x

2

random variable with

parameters:

as before. The

_{complete density}

_{(Zellner, 1971)}

is:

and the mean, mode and variance of this distribution can be shown

by

standard

techniques

to be:

’

provided

that n — _{p —} 2c > 4. From

_(14),

the

_{marginal density}

of a, is:

This is a

_weighted

_{average of} an infinite number of inverted

x

2

densities as in

(16),

the

_weight

function

_being

the

_{posterior density}

_(8).

Whereas the

_{marginal density}

of ac cannot be written

explicitly,

its mean and variance can be calculated

_by

numerical means. For

_example,

note that:

because the

_expectation

is calculated with respect to the distribution al, a2, ..., -;

a!-1 ! y.

This is the

_Bayes

estimator

_{minimizing expected}

_posterior

_quadratic

loss.

_Evaluating

the

_expectation

in

₍₁₉₎

_requires

_solving

Henderson’s mixed model

equations

a

_large,

_technically

an

infinite,

number of times. Monte-Carlo

(Kloek

and van

_Dijk,

₁₉₇₈₎

or numerical

(Smith

et

_al,

₁₉₈₇₎

_{integration procedures}

are needed

to

_complete

the calculation of

_(19).

Other

_components

Now consider the

_{parameterization:}

i different from c

The Jacobian matrix of the transformation from a

_{parameterization}

in terms of

is ai. The

joint posterior

density

of

_uf

_and

_{al, a2, ... ,} a!_1 is then from

(7):

As in the

_preceding

_{section, given}

the ratios of variance _components, and

provided

that each of the variances

_!2

_takes

values between 0 and _oo, one obtains:

where:

Again,

(22)

is the

_density

of an

inverted

X

2

random variable with the features:

Using

the _argument of

_(18),

the

_marginal

_density

of

_a?

_is

a

_weighted

_average of an infinite number of inverted

X

2

_densities,

and the

_{posterior density}

of the ratios

of variance _components,

_(8),

is

the

_weight

function. The

_marginal

_posterior

_density

cannot be written in

_{explicit form,}

but its moments can be evaluated

_numerically.

For

_example:

where the

_expectation

is taken with _respect to the distribution c!l, a2, ... ,

a!_1

[ y.

As discussed

_{earlier, carrying}

out this calculation

_requires

numerical

_integration

or Monte Carlo

_techniques.

APPROXIMATE MARGINAL DISTRIBUTION OF EACH OF THE VARIANCE COMPONENTS

Because of the numerical difficulties

_arising

in the

_{marginal analysis}

of the variance

analytical

treatment of the

_resulting

distributions. If the

posterior density

of the ratios of variance components is

_peaked

or

symmetrical

about its

mode,

one can resort to the well known

_{approximations}

_(Box

and

Tiao, 1973;

Gianola and

Fernando,

1986;

Gianola et

_al,

_1986):

and

where the a’s are the components of the mode of the

posterior

distribution of the variance

ratios,

with

_density

as in

_(8).

The

_marginal

densities of the variance

components can be

_{approximated using}

densities

₍₁₆₎

and

_(22),

but with the

parameters now

being:

where

0

is the solution to Henderson’s mixed model

_equations

evaluated at

S

i

,S

2

)&dquo;-)

S

c

-i-

Because the densities are in inverted

x

2

form,

one can

readily

obtain their moments and construct

_point

estimates of the variance _components in

a Bayesian

decision-theoretical context. For

_example,

for a

_quadratic

loss

function,

the estimator that minimizes

_{expected quadratic}

loss is the

posterior

mean, or:

and

Estimators

_minimizing

&dquo;all or none&dquo;

_posterior

loss are:

and

The

_posterior

variances can be

_approximated

_{by evaluating}

_expressions

_S

_’

_in

distributions can be

_depicted

_{graphically by plotting}

densities

₍₁₆₎

and

_(22),

evaluated at the a

_values,

thus

_giving

a full

_{(approximate)}

solution to the

_problem

of inferences

_regarding

variance _components in a univariate mixed linear model.

DISCUSSION

Harville

₍₁₉₉₀₎

stated: &dquo;A more extensive use

_{of Bayesian}

ideas

_by

animal breeders and other

_{practitioners}

is desirable and is more

feasible from

a

_{computational}

standpoint

than

_{commonly thought...}

The

_{Bayesian approach}

can be used to devise

prediction

procedures

that are more sensible -

_from

both a

_Bayesian

and

_frequentist

perspective -

than those in current use&dquo;.

In this paper we have used the

_Bayesian

mechanism of

_integrating

nuisance

parameters to construct

_marginal

likelihoods from which

_marginal

inferences about each of the variance _components can be made. In this _respect, the method advanced in this paper goes

beyond

REML in

_degree

of

_{marginalization.}

In

_REML,

the fixed effects are viewed as nuisance _parameters and are

_integrated

out of the likelihood

(Harville, 1974)

so inferences about variance _components are carried out

_jointly.

In the present paper fixed effects and other variance components also

_regarded

as nuisances are

integrated

out so that inferences about individual variances of interest can be

completed

after

_taking

into _account,

exactly

or

approximately,

the error due to not

_knowing

all nuisance _parameters.

Hence,

practitioners

that _accept

R.EML in terms of the _argument advanced

_by

Patterson and

_Thompson

_(1971),

_ie,

taking

into account

_degrees

of freedom

&dquo;lost&dquo;,

should also feel comfortable with our

_approach

because additional

_degrees

of freedom

_(stemming

from the nuisance variance

_components)

are also taken into account.

When fixed effects are viewed as nuisance _parameters, it is fortunate that the

likelihood can be

_separated

into a _component that

_depends

on the fixed effects

_plus

an &dquo;error contrast&dquo; _part

_(Patterson

and

_Thompson,

_1971).

_However,

in likelihood

inference it seems

_impossible

to eliminate nuisance _parameters other than

_by

ad hoc methods or without recourse to

_Bayesian

ideas. A

_possible

method for

_dealing

with nuisance variances would be

_treating

all random effects other than that of interest as

fixed,

and then

_estimating

the

_appropriate

_component

_by

some translation invariant

procedure

such as REML. This

_suggests

that the distribution of the

_resulting

statistic would not

_depend

on _parameters of the distribution of the nuisance random

effects. On the other

_hand,

the variance of interest would be estimated

_jointly

with

the residual

_variance;

in most instances this should

cause

little

_difficulty

because of the

_large

amount of information available about the residual _component. It seems

intriguing

to examine the

_sampling

_properties

of this method.

An

_interesting

_question

is the extent to which the

_proposed

method differs from

REML. It is known that ML and REML can differ

appreciably

when the number of

fixed effects is

_large

relative to the number of observations.

Likewise,

the difference between estimates obtained with the new method and REML should be a function of

the number of variance _components in the model and of the amount of information about such _components contained in the

_sample.

For the sake of

_brevity,

we will refer

to the new method as &dquo;VEIL&dquo;

_(standing

for &dquo;variance estimation from

_integrated

The finite

_sampling

_properties

such as

_bias,

variance and mean

squared

error of

VEIL are

_unknown,

but the same holds for REML and

ML,

except for some trivial

models. Differences among methods

depend

on data structures and _parameter

values,

and can

_only

be assessed via simulation

_experiments.

When the

_approximate

analysis

described in _{this paper is}

_used,

the

_goodness

of VEIL will

_depend

on

the extent to which the

_density

_{Q’i,a2)-&dquo;)Q’c-i ! y}

is

_peaked;

this

_assumption

surely

does not hold when there is limited information in the data about the

variance ratios. In this case, the exact

_approach

is

_strongly

recommended over the

approximate

one. With _respect to

_{large sample}

_properties,

the work of Cox

₍₁₉₇₅₎

on

_partial

likelihoods

_suggests

that the

_properties

of maximum likelihood estimates

apply

to maximum

_marginal

likelihood estimates as well. A

_good

illustration is

precisely

REML relative to ML.

Hence,

the

_{large sample properties}

of estimates obtained from the

_{highly marginalized}

likelihoods

_presented

here should be the same as those of REML

_although,

of course, the parameters of the

asymptotic

distribution must differ. For

_example,

the

_asymptotic

variance-covariance matrix of

ML estimates of variance _components is different from that of REML estimates. From a

Bayesian

_viewpoint,

the method described in this _paper is a marked

improvement

over ML and REML

provided

the

_objective

of the

analysis

is

_making

marginal

inferences about individual _components of

_variance,

as the other param-eters are treated as nuisances. As stated

_earlier,

the

_marginal

densities of the vari-ance _components are

_{approximately}

inverted

X

2

.

This allows to _generate a

_complete

posterior

distribution from which moments can be calculated as

_needed,

and

prob-ability

statements can be obtained at least

_by

numerical

_procedures.

The inverted

x

2

distribution is defined

_only

over the

_positive

_part of the real line so the

embar-rassing

situations caused

_by

_{encountering point}

estimates that fall outside of the

parameter space, or

_negative

values of the variances

_yielding

a

_positive

likelihood

(Harville

and

_Callanan,

₁₉₉₀₎

are avoided.

It is

_interesting

to observe that the

_{(approximate)}

_marginal

_posterior

densities of the variance _components appear in the inverted

x2

form.

In a fixed

model,

the

marginal

posterior

distribution of the residual variance is

_exactly

inverted

_X

_Z

and its

modal value is the usual unbiased estimator

_(identical

to REML in this

_case).

This

implies

that this distribution is suitable for

_{specifying marginal}

_prior

_knowledge

about variance _parameters as

_done,

for

_{example, by Lindley}

and Smith

(1972)

and

H6schele et al

_(1987).

For

instance,

suppose that the

prior

distribution

of

Q2

is taken

as

_inverted

x2 with

parameters

vo

and so.

Then the

marginal

posterior

density

of

Q2

can be written as:

Taking

into account that the

_integrated

likelihood is

_{approximately proportional}

to

where:

and

Hence,

the

_marginal

_posterior

_density

of _component

_a?

remains in the inverted

_X

₂

form. This

_{&dquo;conjugate&dquo;}

_property of the inverted

_{x2 density}

can be used to

_advantage

when

_computing

the

_{marginal density}

of a variance _component from a

_large

data

set. If the data can be

partitioned

into

_{independent pieces, repeated}

use of

₍₃₀₎

with

parameters

updated

as in

_(31a)

and

_(31b)

will lead to the

_{(approximate)}

_marginal

posterior

density

of the variance _components. Each of these

_pieces

of data should have a

_relatively

_large

amount of information about the variance ratios so that

replacing

the variance ratios _ai

_by

the modal values

_ix

_i

can be

_justified.

Note in

(31a)

and

_(31b)

that while the

_&dquo;degree

of belief&dquo; _parameter accumulates

_additively,

the

_{parameter s}

₂

is

_updated

in the form of a

_weighted

_average which is a common

feature in

_Bayesian

_posterior

_analysis.

The

_computations

_required

for

_completing

the

_approximate

_analysis

are similar

to those of REML. As illustrated in

_(13),

an iteration constructed on the basis of successive

_{approximations requires}

at each round the solution vector and the inverse of the coefficient matrix of Henderson’s

equations.

These are also the

_computing

requirements arising

in first and second order

_algorithms

for REML

_(Harville

and

Callanan,

1990).

Derivative free

_algorithms

should be

_explored

as well. On the other

hand, carrying

out the exact

_analysis

_requires

numerical

_integration

and the mixed model

_equations

would need to be solved at each of the _steps of

_integration.

This

may not be feasible in a

_large

class of models. In summary, the exact method can be

_applied

_only

in data sets of moderate

_size,

but it is

_precisely

in this situation

where it would be most useful.

As stated

by

Searle

_(1988),

the

_sampling

distributions of

_analysis

of variance and

likelihood based estimates of variance _components are unknown and will

probably

never be because of

_{analytical intractability.}

It would be

_intriguing

to

_study

the

extent to which the inverted

_x2

distributions

with densitites as in

₍₁₆₎

and

₍₂₂₎

can

provide

a

description

of the

variability

encountered over

repeated sampling

_among estimates of variance _components obtained with the _present method. In such a

_study

the

_QZ

_’s

would be estimates obtained in different

_{(independent)}

_samples

and

v and

s!

would

be _parameters of the

_sampling

distribution. In other

_{words, perhaps}

the small

_sample

distribution of

_point

estimates obtained with VEIL can be

_suitably

fitted

_by

an inverted

X

2

distribution.

ACKNOWLEDGMENTS

This work was carried out while the senior author was at

_INRA,

and on sabbatical

leave from the

_University

of Illinois. The support of these 2 institutions is

_gratefully

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