ECM approaches to heteroskedastic mixed models with constant variance ratios

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ECM approaches to heteroskedastic mixed models with

constant variance ratios

Jl Foulley

To cite this version:

Original

article

ECM

_approaches

to

heteroskedastic

mixed models with

constant

variance ratios

JL

_Foulley

Station de

_{génétique quantitative}

et

_appliquée,

Institut national de la recherche

_agronomique,

78352

_{Jouy-en-Josas cedex,}

France

(Received

6

_February

_1997;

_accepted

28

_May

₁₉₉₇₎

Summary - This paper presents

techniques

of parameter estimation in heteroskedastic

mixed models

_having

constant variance ratios and

_{heterogeneous log}

residual variances

that are described

_by

a linear model. Estimation of

_dispersion

parameters is

_by

standard

(ML)

and residual

_(REML)

maximum likelihood.

_{Estimating equations}

are derived

_using

the

_{expectation-conditional}

maximization

_(ECM)

_algorithm

and

_simplified

versions of it

(gradient ECM).

Direct and indirect

_approaches

are

_proposed

with the latter

_allowing

hypothesis testing

about the variance ratios. The

analysis

of a small

example

is outlined to illustrate the

_theory.

heteroskedasticity

/

mixed

_{model / maximum}

likelihood

_/

EM _algorithm

Résumé - Approches ECM des modèles mixtes hétéroscédastiques à _rapports de variances constants. Cet article

présente

des

techniques

d’estimation des

paramètres

intervenant dans des modèles mixtes _ayant des _rapports de variance constants et des variances résiduelles décrites par un modèle linéaire de leurs

logarithmes.

Les

_paramètres

de

_dispersion

sont estimés _par le maximum de vraisemblance

classique

(ML)

et restreint

(REML).

Les

équations

à résoudre pour obtenir ces estimations sont établies à _partir de

l’algorithme d’espérance-maximisation

conditionnelle

_(ECM)

et d’une version

_simplifiée

dite du

_gradient

ECM. Des

_approches

directe et indirecte sont

_proposées,

cette dernière conduisant à un test

_{d’hypothèse}

sur le _rapport de variances. La théorie est illustrée _par

l’analyse numérique

d’un petit

exemple.

hétéroscédasticité

_/

modèle mixte

_/

maximum de vraisemblance / algorithme EM

INTRODUCTION

Heteroskedasticity

has

_recently

_generated

much interest in

_{quantitative genetics}

and animal

_breeding.

To

_{begin with,}

there is now a

_large

amount of

_experimental

evidence of

_{heterogeneous}

variances for most

_important

livestock

_production

traits

of

_{heterogeneous}

variances

_arising

in univariate mixed models

_(Foulley

et

_{al, 1990;}

Gianola et

_{al, 1992; Weigel}

et

_al,

_1993;

_DeStefano,

_1994;

_Foulley

and

_Quaas,

_1995).

For _many reasons

_(accuracy

of

_estimation,

ease of

_{handling large}

data

_sets),

a

major

objective

in this area lies in

_making

models as

_parsimonious

as

_possible.

This can be

_accomplished

in at least two _ways:

_i)

_{by modelling}

variances in the

case of

_potentially

numerous sources of

_{heteroskedasticity,}

and

_ii)

_by

_assuming

that

some functions of those

_parameters

(eg,

intra-class correlation or

heritability)

are

constant. The first

aspect

corresponds

to the so-called structural

approach

in which the

_{heterogeneity}

of the

_log

_components

of variances is described via a linear model

structure similar to that used for means

(Foulley

et

_{al; 1990,}

_1992;

San

_Cristobal,

1993).

Restrictions as in

ii)

were considered

by

Meuwissen et al

₍₁₉₉₆₎

and Robert et al

_(1995a,b).

Meuwissen et al

₍₁₉₉₆₎

introduced a

_{multiplicative}

mixed model to estimate

_breeding

values and

_{heteroskedasticity}

factors

_assuming

_heritability

_(h

₂

₎

constant across

_herd-years.

Robert et al

_(1995a,b)

_developed

estimation and

_testing

procedures

for

_homogeneity

of

_heritability

within

_and/or

_genetic

correlations across

environments. But Meuwissen’s

_study

postulates

known

h

2

and

Robert’s research

applies

to

_only

a

_single

classification of

_{heteroskedasticity.}

The _purpose of this _{paper is} to _propose a

_complete

inference

_approach

for

parameters

having

both features

_i)

and

_ii),

ie,

for continuous data described

by

mixed models with constant variance ratios and

_{heteroskedasticity analyzed}

via

a structural

_approach.

For

_simplicity,

the

_theory

will be

_{presented using}

a

one-way random mixed model for data and afterwards it will be

generalized

to several

u-components.

Inference is based on likelihood

_procedures

(REML

and

ML)

and

estimating equations

derived from the

_{expectation-maximization}

_(EM)

_theory,

more

precisely

the

expectation/conditional

maximization

(ECM)

algorithm

_recently

introduced

_{by Meng}

and Rubin

_(1993).

THEORY

Statistical model

As

_usual,

it is assumed that the

population

can be structured into strata

_(i

=

1, 2, ....,1)

corresponding

to

_potential

factors of

_{heterogeneity.}

Let the _one-way random model be written as:

where _yi is the

_(n

₂

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

i

x

₁₎

vector of residuals. The contribution of random effects is

_expressed

as in

_Foulley

and

_Quaas

₍₁₉₉₅₎

as

_{O&dquo;uiZiU’}

where u* is a

_(q

x

₁₎

vector of standardized

deviations, Z

i

is the

corresponding

incidence matrix and au, is the square root of the

u-component

of variance the value of which

_depends

on stratum i. Classical

_assumptions

are

made

_for

the distributions of u* and ei,

ie,

Nu*

_{N(0, A),}

eiN

N(0, ae.In! ),

and The notation in

_[1]

is unusual as

compared

to that used in the statistical literature

an

_expression

of the random

_part

_especially

in animal

_breeding.

For instance the between sire variance _{may vary}

_according

to the environment in which the progeny

of the sires are raised. Note also that (_JUi can be viewed as a

_regression

coefficient of

any element of yi on the

corresponding

element of

Z

i

u

*

.

Thus,

in animal

_breeding,

a

u

, acts as a

scaling

factor of a vector u* of standardized sire values on

which,

for

instance,

selection can be based.

A structure is

_hypothesized

on the residual variance so as to model the influence of factors

_{causing heteroskedasticity.}

This is carried out

_along

the lines

presented

in

_Foulley

et al

_{(1990, 1992)}

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.

Furthermore,

the

_assumption

of a constant intra-class correlation

_{(or heritability)}

implies

setting

EM-REML estimation

Use is made here of the EM

_algorithm

of

_Dempster

et al

₍₁₉₇₇₎

to

_compute

REML estimates of

_parameters

involved in variance

components

(Patterson

and

Thompson, 1971;

Searle et

_al,

_1992).

The basic

_procedure

_{proposed by Foulley}

and

Quaas

(1995)

is

_applied

here after some

_adjustment

of the

_M-step

_taking

_advantage

of the ECM

_algorithm

of

_Meng

and Rubin

_(1993).

-the ECM

_algorithm

is based on a

_complete

data set defined

_by

x =

(0’, u

*

’,

e’)’

and its

_{log-likelihood}

_{L(y; x).}

The iterative process takes

place

as follows.

The

E-step

is defined as

usual,

_ie,

at iteration

_[t],

calculate the conditional

expectation

of

_{L(y; x)}

_given

the data _y and _y =

y!t!

which,

as shown in

_Foulley

and

_Quaas

(1995),

reduces to

where

E!t]

_(.)

is a condensed notation for a conditional

_expectation

taken with

respect

to the distribution of

_{x!y, y}

=

-yf

t

l.

Since the

_parameters

to be estimated are

_{heterogeneous,}

the

_{estimating equations}

are derived at the maximization

_stage

from a

_slightly

different version of the EM

_algorithm,

the so-called ECM

_algorithm.

As

explained

in detail in

_Meng

and Rubin

_(1993),

a CM

_stage

_replaces

the

_M-step

_by

a _sequence of several conditional

ascent maximization

_procedure

_(Zangwill,

_1969).

We

_suggest

here the

_following

procedure:

-

maximize

_Q

over _y to

_get

6

[tH]

with T set at its last value

_T

_[t]

_,

ie

-

then,

maximize

_Q

over Tto obtain

T!’+’l

with 5 in _y of

1’[

])

t

Q(

I

’

1

set to

5!!!,

ie,

Thus,

the maximization

_step

consists of two

_CM-steps

within the same

_E-step

in order to reduce the need to

_compute

the conditional

_expectation

of

_eie

_i

_,

and its

components

more than once. The

algebra

of differentiation is

given

in

Appendix

A.

The iterative

_system

for

_computing

formulae 5 can be written as

with the elements of the

_right-hand

side

_being

Note that for this

_algorithm

to be a true

_ECM,

one would have to iterate the NR

algorithm

in

_[7]

within an inner

_cycle

_{(index £)}

until _convergence to the conditional

maximizer

_y[

_tH]

=

yl’,’]

_at each

M-step

[t].

In

practice

it may be

advantageous

to

reduce the number of inner

_iterations,

even _up to

_only

_one,

_ie,

_{by solving}

_just

once

However,

caution should be exercised when

_applying

such a

_{hybrid algorithm}

that no

_longer

_guarantees

the monotonic convergence in likelihood values

(Lange,

The formula to

_update

Treduces to

mimicking

the form of a scaled

_regression

coefficient

_pooled

over strata.

The elements to

_compute

at the

E-step

can be

expressed

as functions of the sums

X’yi, Z’yi,

the sums of _squares

yiyi

within

_strata,

and GLS-BLUP solutions of

Henderson’s mixed model

_equations

and of their _accuracy

_{(Henderson, 1984),}

ie

Thus,

deleting

[t]

for the sake of

_simplicity,

one has:

where

_{(3 and}

u* are mixed model

_equations

for

₁₃

and

u

*

,

and C - _

[Cf

3f

3

_Cuf3

C

f

3

u

]

Cuu

J

is the

_partitioned

inverse of the coefficient matrix.

Expressions

in

_[12a-c]

can

_easily

accommodate

_grouped

data

(see

Appendix

B).

The close connection between the

_system

of

_equations

_[7]

for residual

_parameters

and formula

_[12]

_given

in

_Foulley

et al

₍₁₉₉₀₎

can be observed. There is also a

remarkable

_similarity

between formula

_[9]

for the ratio and formula

_[7]

in

_Foulley

and

_Quaas

_(1995).

This means that the

_computations

can be

_implemented

with

very little

change

in the code used

previously.

True or

_gradient

EM could also have

Extension to several

_u-components

Formulae

_{(7!, [8ab]}

and

_[9]

can

_easily

be

_generalized

to a mixed model

including

several

_(k

=

1,

2,...,

K)

independent

u-components

with Tk =

a

Uik

/a

ei

_{constant over strata} i.

Letting

y =

(b’,

T

’)’

as

_previously

but now with T=

I

Tk

being

a vector of ratios

of standard

deviations,

the

_Q

function to be maximized has the same form as in

[4]

with ei

expressed

from

!13!.

One can

_perform

the

CM-steps using

either

i)

the

sequence

6, ’r

l

, T

2

....

I

Tk

, - - - ,

, KT

ie,

each Tk one

_by

_one, the

_remaining

ones

_being

held

constant,

or

ii)

the _sequence

/5,

and T as a whole with all the Tks maximized

jointly.

In both cases, the

algorithm

for

computing

5 is

formally

the same as in

[7]

with

_only

a

_slight

_change

in the definition of the elements of

W

bb

,

vb

being

unchanged

If the conditional maximization of

the T

k

s

takes

place

one

_by

one

_{(case i),}

formula

[9]

still

_applies

for each of them. Otherwise

_{(case ii),}

one has to solve the

following

system:

An indirect

_approach

The

_original

model with a constant T ratio

specified

in

[1-3]

can be viewed as a

special

case of a more

general

model

with,

as

_previously,

fno, 2 -

p§5,

but also with a linear structure on

_log-ratios

Letting

y =

(6’, 71’)’

_here,

the

sequence of the

CM-steps

are

The

_algorithm

for S is the same as in

_[7].

The

_algebra

for A is shown in the

Appendix,

and leads to a

_system

that can be written under a similar form as that

of 6

1--J For

_practical

_reasons, one _may also wish to limit the number of inner iterations

(index

£)

even to

_only

one in order to reduce the volume of

_computation

but the

application

of this ECM

_{gradient algorithm}

should be

performed

carefully.

Further

empirical

simplifications

for the elements of

_[22]

can be

_proposed

_along

the same

lines as in

_Foulley

et al

_(1990).

Again,

these results can be extended to a model with several random

_independent

factors

_(k

=

1, 2,...,

K)

by

_setting

Actually,

if the

_CM-steps

are

_performed

for each vector

₇₁

_k

_separately,

the same

formulae as in

[20], [21]

and

_[22]

_{apply: just replace}

_Ti,

_,

_i

_Z

u*

_by

_Ti,k,

_Zi

_k

_,

_{uk and}

ML estimation

It _may be

_interesting

in some instances to use ML rather than REML for

_estimating

variance

_components

_{(see Discussion).}

The ECM

_{procedure developed}

in this paper

can be

_easily

_adapted

to obtain ML

_parameter

estimates.

₁₃

is now

_part

of the

respect

to the distribution of u*

_given

_{y, y} =

y!t!,

and

13

=

13

[

’

I

.

Maximization with

respect

to

₁₃

can be based on the

_equation

<9Q/<9j3

=

0,

ie

One can

proceed

as

_previously,

_ie,

run two

_CM-steps

for the

_dispersion

parameters

based on the same

_E-step

so as to obtain

6!t+

and

T

]

t+1

]

_(or

_!ft+1]),

and then

perform

an additional

_CM-step

for

_computing

¡3

[t+l]

based on

!23!,

ie

l &dquo;&dquo; ’ -!J

Alternatively,

it _may be

_advantageous

to

_perform

the

_CM-step

for

_j3

and the

next

_E-step

_{jointly by solving}

Henderson’s mixed model

_equations

in

I3

[Hl]

and

u*[

t

+

i

]

=

E!u*!y,

6

1

tH

], rrl

t

H

]

)

based _on

6[

Hl

]

and

T

f

c

+

1

1.

Formulae for the two

_CM-steps

do not

_change.

The

_only

additional modification results from

taking

the conditional

_expectation

of

components

of

_e!e,

given

y, y =

y[

t

],13

=

l3

[t]

instead of

y, y =

y

[t]

.

Formulae in

[12]

reduce to

where

_M

_uu

is the u

_by

u block of the coefficient matrix

!11!.

Note that the trace terms inside those formulae have

disappeared

or have been

greatly simplified owing

to

_conditioning

with

respect

to

₍₃

=

l3

[t]

.

More

generally,

for models

_[13]

_involving

several

_{u-components,}

_[25c]

becomes

where

_{(M§) )}

_k£

is the block

_pertaining

to random factors k and in the inverse of the random

_part

of the coefficient matrix.

Numerical

_example

The

_{procedures presented}

in this paper are illustrated with a small data set obtained from simulation. Data were

_generated

according

to a cross-classified model

having

two

_{(environmental)}

fixed factors

_{(A =}

_{2 levels;}

B = 3

levels)

and one

(genetic)

random factor

_(S

= 9

levels).

The

_genetic

contribution consists of sire and maternal

grand

sire

_effects,

the latter

_being

assumed to have half the value of the first one.

where _p is a

_general

_{mean, ai} the effect

of

environmental factor A

(i

=

1, 2), b!

the effect of environmental factor B

_(j

=

1, 2, 3), s* the

standardized contribution of male k as a

_sire,

and

1/2se

the standardized contribution of male as a maternal

grand

sire,

and _eZ!w&dquo;, the residual term.

Values chosen for the fixed effects were

(using

a full-rank

parameterization):

¡.

.t

+al

+b1

=

_100;

_az-on =

_20;

_b2

_-

_b

_,

=

_-10;

_b3

_-

_bl

= -20. The vector s* =

fs

*

kl

}

of sire effects is assumed to be

_{N(0, A)}

with elements of the

relationship

matrix A shown at the bottom of table I.

Residual variances were obtained from

with a base line value

_(]&dquo;!11

=

exp(p

*

+ai +bl)

=

400,

and

multiplicative adjustment

factors:

_{exp(a2 - a*)}

=

2;

exp(b2 - bi)

=

1/2

and

exp(b3 - b*)

=

3/2.

The ratio

T =

(

]

&dquo; 8ij

/ (]&dquo;

eij

of the square root of the sire to the residual variance was taken as

constant over A x B cells and set to

8.75-

1/2

_{(heritability}

_equal

to

_0.41).

There were 267 observations distributed among 18 different AB x sire x maternal

grand

sire subclasses. The data structure is

_displayed

in table I as well as cell size

Tests of

_hypotheses

about the location

parameters

{3,

the residual

_dispersion

parameters

5 and the ratios

_r

_ij

were carried out via the likelihood ratio statistic as

described in

_previous

studies

_(Foulley

et

_al,

1990

_1992;

San Cristobal et

_{al, 1993;}

Meyer

et

_{al, 1993; Foulley}

and

_Quaas,

_1995).

Formulae

by

Quaas

(1992)

were used

to

_compute

maximized likelihood functions

_(Ln,

_aX

_).

Results can be

arranged

as an

analysis

of variance

(or

deviance)

table: see

table II for

hypothesis testing

about

_{3,

and table III for residual

_(b)

and ratio

(A)

parameters.

Note also that the test statistic for

₁₃

relies on

-2L

n

,aX

evaluated

from the ML estimates of all

parameters,

whereas a maximized residual likelihood can be better

_employed

for 5 and 7!.

Interaction effects on location

_parameters

are

constantly rejected

under different

assumptions

for the other

_parameters.

The

_hypothesis

of residual variance

homo-geneity

is

_strongly

_rejected

as well as

single

factor

descriptions

of

heterogenity.

The

assumption

of a constant ratio Tturns out to be a reasonable one. The test results

eventually

agree with the simulation

model; they

support

the

practical

conclusion that the _{p +} A + B model is the most

_appropriate

to account for variation both in

location and in

_log-residual

variances,

the ratio T

_being

constant.

The estimation

_procedure

for l5 and T

_{(or J!)}

is illustrated in table IV for this model and an alternative one

_using

both standard and residual maximum likelihood

methods of estimation. ML and REML estimates of residual variances do not differ

very

much;

on the

_contrary,

the ML estimates of the ratio T turns out to

be,

as

expected,

lower than the REML ones, the values of the latter

being

close to the

true value.

DISCUSSION AND CONCLUSION

The main _purpose of this _paper was to extend the

_general

structural

_approach

to

heteroskedasticity

in mixed models

proposed by Foulley

et al

_{(1990, 1992)}

to the

case of

homogeneous

ratios of u to e variance

_components.

In a sire

by

environment

interaction,

this is

equivalent

to

postulating

homo-geneous intra-class correlations or heritabilities. This seems to be a reasonable

assumption

in

_practice,

or at least serves as a suitable

compromise

between the existence of

_{heteroskedasticity}

and

_parsimony

of models. Less restrictive assump-tions

_might

also be

_investigated

_(Quaas,

1995,

pers

comm).

This paper also

provides

a

_{generalization}

of LR tests of this

_assumption

to unbalanced data and

_complex

model structures: see the

previous

work of Visscher

(1992)

on a _one-way random

balanced

_design,

and that

by

Robert et al

_(1995a,b)

for

_{heterogeneous}

variances

due to a

_single

classification.

The EM

_algorithm

turns out to be a convenient and

powerful

tool for

solving

variance

_component

estimation

_problems.

The ECM

_algorithm

allows us to

_simplify

the

estimating equations,

in

_particular

the ECM

_gradient

version. The

_advantage

of this

_algorithm

was

especially

clear here in the case of the indirect

approach.

A few

_examples

of this for the mixed model have been

_already

mentioned

_(Meng

and

Rubin,

1993

_{example 1; Walker,}

_1996).

It offers

_great

_flexibility

in

_defining

the

sequence of the conditional maximization

steps,

all the alternatives of which have

generated by

the EM

_algorithm

are

_strikingly

natural

(see

Appendix

B)

thus

_giving

a flavour of

_simplicity

to the whole

_procedure.

It also makes it

_possible

to switch from REML to ML or vice versa with _very little

change

in

_implementing

_computations

_(Foulley

et

_al,

_1994).

Some authors such as

Leonard

_(1975),

Denis

₍₁₉₈₃₎

and Anderson

₍₁₉₈₄₎

in statistics and Shaw

₍₁₉₈₇₎

in

quantitative genetics

have advocated the use of ML rather than REML

_procedures

to estimate variance

_components.

_Although

the interest of ML versus REML in

that case remains

_{questionable,}

ML estimates remain

_mandatory

for

_hypothesis

testing

about

13

via the LR statistic

_(see

the numerical

_example).

Bayesian point

estimators can also be envisioned via EM

_(Foulley

et

_{al, 1992;}

Gianola et

_{al, 1992;}

San Cristobal et

_al,

_1993;

_Weigel

and

_Gianola,

_1993;

_Foulley

and

_Quaas,

_1995).

In

_addition,

as

already pointed

out

_by

Denis et al

_(1996),

a LR test about

₍₃

requires

5 and T

_{(or 71)}

_being

the same for the null

_hypothesis

and its

_alternative;

the same rule holds in

_{hypothesis testing}

about 5

_{(or 71)}

by keeping

the other

and difficult

_problem

of

_joint

_modelling

of means and

_variances,

which is related to such issues as the Behrens-Fisher

_problem

and

_{multi-stage hypothesis testing,}

and which needs further consideration.

Another area that deserves caution and further

development

is that of

estima-bility.

Difficulties are

expected

when all the cells

_contributing

to an

element j

of

5 or A

(or

to a linear combination of

them)

have a

_{weight tending}

towards zero.

This _may arise due to

_i)

purely overparameterization problems,

or due to

_ii)

pa-rameter values

_becoming

extreme

_(eg,

ratios _Ti

_tending

to zero

_implying

elements

of 71

_being

_infinite).

This last

phenomenon

is similar to what

happens

in the

anal-ysis

of

_binary

and ordinal data with latent variable models

_(Misztal

et

_{al, 1989;}

Fahrmeir and

_Tutz,

_1994).

Such difficulties can be avoided

_{by reparameterization}

(i),

or

_by

_setting

lower bounds to the

diagonal

elements of the

_system

of

_equations

to solve

_(ii).

Finally,

asymptotic

accuracy can also be worked out

_numerically

within the EM framework via the so-called SEM

_algorithm

_(Meng

and

_Rubin,

_1991).

_Although

it

only requires

the code of the

complete

data variance covariance matrix and of the EM or ECM

_outputs,

the burden of calculations is then

_heavier,

so that we

_suggest

that it is restricted to the

_simplest

models. Other

_computing

alternatives should also be

_considered,

_eg, the _average information-restricted maximum likelihood

_algorithm

(AI-REML)

of Gilmour et al

_(1995).

ACKNOWLEDGMENTS

The author is

_grateful

to Elinor

_Thompson

_{(INRA, Jouy-en-Josas)}

and Dr Max Rothschild

_(ISU,

_Ames)

for the

_English

revision of the

manuscript

and to the two anonymous referees for their valuable comments

_especially

about some technicalities

of the EM

_theory.

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Non Linear

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A

_{Unified Approach.}

Prentice Hall,

Engle-wood Cliffs

APPENDIX A:

_Algebra

for the

_{estimating equations}

The

_Q

function to be maximized is

_(in

condensed

_notation)

Letting

!=<9(2Q)/<9 In

(y2 ei

Second derivatives: from

_!A4a!,

one has

or,

alternatively

Finally,

the non-linear

_system

to solve reduces to

Derivatives with

_respect

to T

_(ratio)

Additional derivatives for the exact EM

From the second

expression

in

_[A7]

it follows

immediately

One has also to _express

Now,

from

_[A7]

where

W r8

is a

(I x 1)

defined

_by

The

_system

for true EM

_(or

_gradient

EM without inner

_iteractions)

is then

Extension to K

_independent

random factors

Q

in

_[Al]

remains

_{formally unchanged}

with ei in

[A3]

being

now

Based on this

_expression

of the

residual,

formulae

[A4ab]

still hold.

or,

alternatively

As far as Tis

_concerned,

_[A6]

becomes

leading

to the

_following

_system

Derivatives with

_respect

to 7!

_(parameters

of the

_log-ratio)

Q

and the model for

_{In (}

_T

_e

_,

₂

are the same as in

[Al]

and

[A2]

but the vector of residuals is defined

as

o , I I

or, after some

_algebra

Furthermore,

This can be

_{easily generalized}

to several

_independent

random factors

if conditional maximization is

performed

factor

by

factor. The

_system

_[AI8]

_applies

to each factor k with

APPENDIX B: Formulae

_[12]

for

_grouped

data

In some instances

_(see,

_{eg, the}

_example

in table

_I)

data can be

_grouped

so that the

n

2 observations within a stratum i share the same covariates

Substituting

X

i

by

its

_expression

in

_[Bl]

_gives

where yzj is the

jth

element of yi, and