Inference for threshold models with variance components from the generalized linear mixed model perspective

HAL Id: hal-00894076

https://hal.archives-ouvertes.fr/hal-00894076

Submitted on 1 Jan 1995

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Inference for threshold models with variance components

from the generalized linear mixed model perspective

B Engel, W Buist, A Visscher

To cite this version:

Original

article

Inference for threshold models

with

variance

_components

from the

_generalized

linear mixed model

_perspective

B Engel

W

Buist

A Visscher

1

DLO

_Agricultural

Mathematics

_Group

_(GLW-DLO),

PO Box

_100,

6700 AC

_Wageningen;

2

DLO

Institute

for

Animal Science and Health

_(ID-DLO),

PO Bo!

_501,

3700 AM

Zeist,

The Netherlands

(Received

10 December

_{1993; accepted}

5 October

₁₉₉₄₎

Summary - The

_analysis

of threshold models with fixed and random effects and associated variance components is discussed from the

_perspective

of

_generalized

linear mixed models

(GLMMs).

Parameters are estimated

_by

an interative

_procedure,

referred to as iterated

re-weighted

REML

_(IRREML).

This

_procedure

is an extension of the iterative

_re-weighted

least squares

algorithm

for

_generalized

linear models. An

_advantage

of this

_approach

is that

it

_immediately

_suggests how to extend

_ordinary

mixed-model

_methodology

to GLMMs. This is illustrated for

_{lambing difficulty}

data. IRREML can be

implemented

with standard

software available for

ordinary

normal data mixed models. The connection with other estimation

procedures,

eg, the maximum a

po8teriori

(MAP)

approach,

is discussed. A

comparison by

simulation with a related

_approach

shows a distinct pattern of the bias of MAP and IRREML for

_{heritability.}

When the number of fixed effects is

_reduced,

while the total number of observations is

_kept

about the same, bias decreases from a

_{large positive}

to a

_{large negative value, seemingly independently}

of the sizes of the fixed effects.

binomial data

_/

threshold

_{model / variance}

components

/

generalized linear model

_/

restricted maximum likelihood

Résumé - Inférence sur les composantes de variance des modèles à seuil dans une perspective de modèle linéaire mixte généralisé.

L’analyse

des modèles à seuils avec

effets fixes

et aléatoires et des composantes de variance

_{correspondantes}

est ici

_placée

dans la perspective des modèles linéaires mixtes

_{généralisés}

_(GLMMs!.

Les

paramètres

sont estimés par une

_{procédure itérative, appelée}

maximum de vraisemblance restreinte

re-pondéré

obtenu _par itération

_(IRREML).

Cette

_procédure

est une extension de

_{l’algorithme}

itératif

des moindres carrés

_repondérés

pour les modèles linéaires

généralisés.

Elle a

l’avantage

de

suggérer

immédiatement une manière d’étendre la

méthodologie

habituelle du modèle mixte aux GLMMs. Une

_application

à des données de

_{difficultés d’agnelage}

est

présentée.

IRREML peut être mis en ceuvre avec les

_logiciels

standard

disponibles

pour les

modèles linéaires mixtes normaux habituels. Le lien avec d’autres

procédures d’estimation,

par simulation avec une méthode voisine montre un biais

_{caractéristique}

du MAP et de l’IRREML pour l’héritabilité.

Quand

le nombre des

effets fixés

est

diminué,

à nombre

to-tal d’observations constant, le biais passe d’une valeur

fortement

positive à une valeur

fortement négative,

apparemment

indépendantes

de

l’importance

des

_{effets fixés.}

distribution binomiale

_/

modèle à seuil

_/

composante de variance

_/

modèle linéaire

généralisé

/

maximum de vraisemblance restreinte

INTRODUCTION

In his paper on sire evaluation

Thompson

(1979)

already pointed

out the

_potential

interest for binomial data in

_modifying

the

_generalized

linear model

_(GLM)

esti-mating equations

to allow for random effects. He

_conjectured

that if modification is

feasible,

generalization

towards other distributions such as the Poisson or gamma

distribution should be _easy. The iterated

_re-weighted

restricted maximum

likeli-hood

_(IRREML)

_procedure

_(Schall,

_1991;

_Engel

and

_Keen,

₁₉₉₄₎

for

_generalized

linear mixed models

_(GLMM)

proves to be

exactly

such a modification. IRREML

is motivated

_by

the fact that in GLMMs the

_{adjusted dependent}

variate in the

iterated

_re-weighted

least squares

(IRLS)

algorithm

(McCullagh

and

Nelder,

1989,

§ 2.5)

approximately

follows an

_ordinary

mixed-model structure with

_weights

for the residual errors

_and,

in the absence of under- or

_{overdispersion,}

residual error variance fixed at a constant value

_{(typically 1).}

IRREML is

_quite

flexible and not

only

covers a

_variety

of

_underlying

distributions for the threshold model but also

easily

extends to other

_types

of data such as count

_data,

for

example,

litter size. This entails

_simple

_changes

in the

_algorithm

with

_respect

to link and variance func-tion

_employed.

When the residual error variance for the

_{adjusted dependent}

variate is not

_fixed,

it

_represents

an additional under- or

overdispersion

_parameter

which is a useful

_feature,

for

_example,

under- or

_{overdispersed}

Poisson counts. Calculations

in this paper are

performed

with REML

(Patterson

and

Thompson,

1971)

facilities for

_ordinary

mixed models in Genstat 5

_(1993).

Software for animal models such as DFREML

_{(Meyer, 1989),}

after some

modification,

can be used for IRREML as well.

Methods for inference in

_ordinary

normal data mixed

_models,

_eg, the Wald test

(Cox

and

_{Hinkley, 1974,}

p

323)

for fixed

effects,

are also

_potentially

useful for

GLMMs,

as will be illustrated for the

_{lambing difficulty}

data. Simulation results for the Wald test in a GLMM for

_{(overdispersed)}

binomial data were

_presented

in

Engel

and Buist

_(1995).

For threshold models with normal

_underlying

distributions and known

compo-nents of

_variance,

Gianola and

_Foulley

₍₁₉₈₃₎

observe that their

_Bayesian

maximum a

_posteriori

(MAP)

_{approach produces estimating}

_equations

for fixed and random effects such as those

_anticipated

_{by Thompson.}

Under

_normality

_assumptions,

for

fixed

_components

of

_variance,

IRREML will be shown to be

_equivalent

to MAP. IRREML therefore offers an

alternative, non-Bayesian,

derivation of MAP. The

MAP

_approach

was also

presented

in Harville and Mee

_(1984),

_including

estima-tion of variance

_components.

Their

_updates

of the

_components

of variance are akin

to those of the estimation maximization

_(EM)

_algorithm

_(Searle

et

_al,

_1992,

_§

_8.3)

paper, is related to Fisher

_scoring.

Both

_algorithms

solve the same final

_estimating

equations,

but the latter is

_considerably

faster than the former.

Gilmour et al

₍₁₉₈₅₎

_presented

an iterative

_procedure

for threshold models with

normal

_underlying

distributions,

which also uses an

_{adjusted dependent}

variate and

residual

_weights.

This

_approach,

which will be referred to as

_GAR,

is different from

MAP and IRREML. In the

_terminology

of

_Zeger

et al

₍₁₉₈₈₎

MAP and IRREML are

closely

related to the

_{subject-specific}

nature of the

_GLMM,

while GAR is of a

population-averaged

nature,

as will be

_explained

in more detail in this _paper.

A number of

_authors,

_eg, Preisler

_(1988),

Im and Gianola

_(1988),

and Jansen

(1992),

have discussed maximum likelihood estimation for threshold models.

_Apart

from the fact that

_{straightforward}

maximum likelihood estimation does not correct

for loss of

_degrees

of freedom due to estimation of fixed

_effects,

as REML does in the conventional mixed

model,

it is also

handicapped

by

the need for

high-dimensional numerical

_integration.

Maximum likelihood estimation for models with several

_components

of

_variance,

_especially

with crossed random

_effects,

is

_practically

impossible.

IRREML is more akin to

_{quasi-likelihood}

estimation

_(McCullagh

and

Nelder, 1989, chap 9;

McCullagh,

1991):

conditional upon the random effects

only;

the

_relationship

between the first 2 moments is

_employed

while no full distributional

assumptions

are needed

beyond

existence of the first 4 moments.

Since

_practical

differences between various methods

_proposed

_pertain

_mainly

to

their

_{subject-specific}

or

population-averaged

_nature,

we will

_give

some attention

to a

_comparison

between GAR and IRREML. Simulation studies were

_reported

in Gilmour et al

_(1985),

Breslow and

_Clayton

_(1993),

Hoeschele and Gianola

_(1989),

and

_Engel

and Buist

_(1995).

Conclusions from the Hoeschele and Gianola

_study

differ from conclusions from the other studies with

_respect

to bias of

_MAP/IRREML

and GAR. Since the Hoeschele and Gianola

_study

was rather modest in

_size,

it was decided to

repeat

it here in more

detail,

ie under a

_variety

of

parameter

configurations

and for

_larger

numbers of simulations.

GLMMs and threshold models The GLMM model

Suppose

that random effects are collected in a random vector _u, with zero means

and

_dispersion

matrix

_G,

eg, for a sire model G =

Ao, 8 2,

where A is the additive

relationship

matrix and

₀

_&dquo;;

the

sire

_component

of variance. Conditional _{upon u, eg,}

for

_given

_sires,

observations _{y are} assumed

_independent,

with variances

_proportional

to known functions V of the means p:

For

_binary

_data,

_y = 1 may denote a difficult birth

_and

_{y =} 0 a normal birth. The mean _f.1, is the

probability

of a difficult birth for

offspring

of a

particular

sire. The conditional variance is

_Var(y!u)

=

V(p)

=

/l

(1-

p)

and 0 equals

1. For

proportions

Parameter 0

may be included to allow for under- or

_{overdispersion}

relative

to binomial variation

_(McCullagh

and

_Nelder,

1989,

§

4.5).

Observe that

_[2]

is

inappropriate

when n is

_{predominantly}

small or

_large:

for n = 1 no

_{overdispersion}

is

_possible

_{and 0}

should

equal

1 and for n -! _oo,

[2]

vanishes to 0 while

extra-binomial variation should remain. More

complicated

variances

_(Williams,

₁₉₈₂₎

_may be obtained

_{by replacing 0}

in

_[1]

_by

_{1

+

_{(n - 1)!0{}

or

_by

_{1

+

_{(n -}

1)u5

f

.1,(1- f.1,n.

In both

_expressions

_ao

is a variance

_{corresponding}

to a source of

_{overdispersion}

(for

a discussion of

_{underdispersion}

see

_Engel

and Te

_Brake,

1993).

Limits for n -> oo of the variances are

o,

2

t

L

(1 -

0

f

.1,)

and

o, 0 2 /-t

2

(1_M)2

respectively.

Both can be accomodated in a GLMM for continuous

_proportions,

eg,

motility

of

_spermatozoa,

and are covered

_by

IRREML.

The mean f.1, is related to a linear

_predictor

_by

means of a known link function _g:

=

g(f.1,).

The linear

predictor

is a combination of fixed and random effects: ! =

x

l

l3

₊

z’u,

where _x and z are

design

_vectors for fixed and random effects

collected in vectors

₁₃

and u,

respectively.

For

_difficulty

of

_birth,

for

_instance,

₇₇

may include main effects for

parity

of the dam and a covariable for

_birthweight

as fixed effects and the

_genetic

contribution of the sire as a random effect.

Popular

link functions for

_binary

or binomial data are the

_logit

and

_probit

link functions:

logit(p)

=

log(f.1,/(l-¡.¡,))

=

qand

probit (p) =

!-1(p.)

=

17

, where

!-1

is the inverse

of the cumulative

_density

function

_(cdf)

of the standard normal distribution.

The threshold model

Suppose

that r is the

_{’liability’,}

an

_underlying

random variable such that _y = 1 when

r exceeds a threshold value 0

and

_{y =} 0 otherwise. Without loss of

generality

it _may be assumed that 0 = 0. Let ₇₇ be the mean of r, conditional upon u.

_Furthermore,

let the cdf of the residual e =

(r — 7!),

say

F,

be

independent

of u. Then

where

F-

1

is the inverse of F. It follows that the threshold model is a GLMM with link function

_g(

_f

_{.1,) =}

_-F-

₁

_{(1 -}

_f.1,),

which

_simplifies

to

_g(f.1,)

=

F-

1

(f.1,)

when e is

_{symmetrically}

distributed. Residual e may

represent

variation due to Mendelian

sampling

and environment. Probabilities _p do not

_change

when r is

_multiplied

_by

an

_arbitrary

_positive

constant and the variance of e can be fixed at _any convenient

constant

_value,

say

or

2

.

When F is the cdf L of the standard

logistic distribution,

ie

_F(e)

=

L(e)

=

1/(1

₊

exp(-e)), g

is the

logit

link and

a

2

=

!r2/3.

When

F is the cdf 4) of the standard normal

_{distribution,}

_{g will}

be the

_probit

link and

Q

Z

= 1.

Although

the

logistic

distribution has

relatively

longer

tails than the normal

distribution,

to a close

_{approximation}

_(Jonhson

and

Kotz, 1970,

p

6):

where c =

(15/16)!/!.

Results of

_analyses

with _a

probit

_or

_logit

link _are

usually

virtually equivalent,

apart

from the

_scaling

factor c for the effects and

C

2

for the

components

of variance.

_Heritability

may be defined on the

_liability

scale,

eg, for a sire model:

h

2

=

4a;/(a;+a2).

As a

function

of

₀

_,2/or

₂

_heritability

does not

_depend

Conditional and

_marginal

effects

In a

_GLMM,

effects are introduced in the link-transformed conditional means, ie in

the linear

_{predictor q}

=

g(p).

Consequently,

effects refer to

subjects

or individuals.

The GLMM and the threshold model are both

_{subject-specific models,}

using

the

terminology

of

_Zeger

et al

_(1988).

This is in contrast with a

_{population-averaged}

model where effects are introduced in the link-transformed

marginal

means

g(E(f.1,))

and refer to the

_population

as a whole. In animal

breeding,

where sources of variation have a direct

_physical

_{interpretation}

and are of

_primary

_interest,

a

subject-specific

model,

which

_explicitly

introduces these sources of variation

_through

random

effects,

seems the natural choice. For fixed effects

however, presentation

in terms _{of averages} over the

_population

is often more

appropriate.

In the threshold

model,

there is no information in the data about the

phenotypic

variance of the

liability,

allowing

QZ

to be fixed at an

_arbitrary

value.

_Intuitively

one

would expect

the

expressions

for

_marginal

effects to involve some form of

scaling by

the

underlying

phenotypic

standard deviation. For

_normally

distributed random

effects

and

_probit

link this is indeed so. From r -

N(x’j3,

z’Gz +

₁₎

the

_{marginal probability,}

say p,

follows

_directly:

Hence,

the

probit

link also holds for

_{marginal probabilities,}

but the effects are shrunken

_by

a factor

_Ap

=

(z’Gz

₊

1)-°!5.

For a sire model

_Ap

=

(u

£

+

1) -

0

.

5

.

That

the same link

_applies

for both conditional means _f.1, and

_marginal

means p is rather

exceptional.

For the

_{logit link,}

the exact

_integral

_expression

for p cannot be reduced

to any

simple

form

_(Aitchison

and

_Shen,

_1980).

_However,

from

_[3]

it follows that the

logit

link holds

_{approximately for p,}

with

_shrinkage

factor

_A

_L

=

((z’Gz/c

2

) + 1 )-

0

.

5

.

Without full distributional

assumptions,

for

_relatively

small

components

of

_variance,

marginal

moments may also be obtained

_by

a

_Taylor

series

_expansion

(see

Engel

and

_Keen,

_1994).

Binary

observations y

i

and y

j

corresponding

to,

for

_instance,

the same sire will be correlated. For the

_probit

link the covariance follows from:

Here

_V2

_(a,

_b;

_p)

is the cdf of the bivariate normal distribution with zero _means, unit variances and correlation coefficient p, p2! is the correlation on the

underlying

scale,

eg, in a

_simple

sire _{model Pij} =

or 8 2/(

0

,2

+

1).

For the

_logit

_link,

using

!3!,

Ap

should be

_{replaced by}

_!L/c,

while the value of the

_correlation,

_expressed

in terms

of the

_components

of variance in the

_logit

_model,

is about the same. The double

integral

in

_!2

_may

_effectively

be reduced to a

single integral

(Sowden

and

Ashford,

1969),

which can be evaluated

by

Gauss

quadrature

(Abramowitz

and

Stegun,

1965,

and

_Stegun,

1965, 26.3.29,

p

940)

may be used:

where

T

(

t

)

is the tth derivative of the

_probability

_density

function

_(pdf)

T of the

standard normal distribution. For a sire

_model,

under

_normality

_assumptions,

the first-order

_{approximation}

appears to be

satisfactory,

except

for extreme incidence

rates _p

_(Gilmour

et

_al,

_1985).

_{By grouping}

of n

_binary

observations

_pertaining

to

the same fixed and random

effect,

moments for binomial

_proportions

_{y immediately}

follow from

_!4),

eg:

where p is the intra-class correlation on the

_liability

scale.

_Expression

_[6]

can be

simplified by

using

!5!.

Results for the

_logit

link follow from

_!3!.

Estimation of

_parameters

The

_algorithm

for IRREML

The

_algorithm

will be described

_briefly.

For details see

_Engel

and Keen

₍₁₉₉₄₎

and

Engel

and Buist

_(1993a).

_Suppose

that

_[3

_o

and uo are

_starting

values obtained from an

ordinary

GLM fit

with,

for

example,

random effects treated as if

they

were fixed or with random effects

ignored,

ie uo = 0. After the initial GLM has been fitted

by

IRLS,

the

_{adjusted dependent variate}

_and

iterative

_weights

w

(McCullagh

and

Nelder, 1989,

§

2.5)

are saved:

where

_g’

is the derivative of the link function with

_respect

to f.1&dquo; eg, for the

probit

link: w =

/{f.1,

0

(1 -

2

)

o

nr(ry

f

.1,0)},

4

approximately

follows an

_ordinary

mixed-model

structure with

weights

w for the residual errors and residual variance

0.

Now a minimum norm

_quadratic

unbiased estimation

_{(MINQUE) (Rao,}

_1973,

_§

_4j)

is

applied

to

_l&dquo;

_employing

the Fisher

_scoring

_algorithm

for REML

₍₁

_step

of this

algorithm corresponds

to

_MINQUE).

From the

_mixed-model

_equations

_(MMEs)

(Henderson,

1963;

Searle et

_{al, 1992,}

_§

_7.6)

new

values [3

and

6

for the fixed and

random effects are solved:

Here,

X and Z are the

_design

matrices for the fixed and random effects

distributional

_assumptions

_beyond

the existence of the first 4 moments and may be

presented

as a

_{weighted least-squares}

method

_(Searle

et

_{al, 1992,}

Ch

_12).

Some

_properties

of IRREML

When the MMEs are

_expressed

in terms of the

_original

observations y, it is

readily

shown that _{at convergence} the

_following

_equations

are solved:

where *

denotes a direct elementwise

_(Hadamard)

_product.

These

_equations

are similar to the GLM

_equations

for fixed u

_(with

_appropriate

side

_conditions)

_except

for the

_{term 0}

G-

1

u

on the

_right-hand

side.

_Equations

_[8]

_may also be obtained

by

setting

first derivatives with

_respect

to elements of

_j3

and u of D +

u’G-

l

u

equal

to zero, where D is the

_(quasi)

deviance

_(see

_McCullagh

and

_Nelder,

_1989,

§

2.3 and

_§

_9.2.2.)

conditional upon u. The

_assumption

of randomness for u

_imposes

a

_{’penalty’}

on values which are ’too far’ from 0. When the

_pdf

of observations _y conditional _upon

_normally

distributed random effects u is in the GLM

_exponential

family,

eg, a binomial or Poisson

_{distribution,}

maximization of

D

+

u’G-

l

u

is

_easily

shown to be

_equivalent

to maximization of the

_joint

_pdf

of y and u.

Suppose

that we have a sire model

with

q sires

and sire variance

_component

_or

₂

The IRREML

_{estimating equations}

for

_o,2

_{s and 0}

_(see,

for

_{example, Engel,}

₁₉₉₀₎

are:

Here

Z

o

=

I,

Z

I

is the

design

matrix for the

_{sires, A}

_o

=

W-

1

,

A

l

=

A,

P =

SZ-

1

_

[21 X(X’[2-1 X)-l X’[2-

I

and S2 =

ZGZ’

₊

cpW-

1

.

The difference with

ordinary

REML

_equations

is

_that

_depends

on the

_parameter

values as well. The

MINQUE/Fisher

scoring

update

of IRREML can be recovered from

_!9!,

_by

_using

P = P[2P

8

AZ’P

₊

cpPW-

1p

_on the left-hand side:

where

_0’5

_{= ø}

and

_at

=

as.

When 0

is fixed _at value

_1,

the

_equation

for

k’

= 0 is

dropped

from

_(10!.

Alternative

_updates

related to the EM

_algorithm

_may also be obtained from

_[9]

_(see,

for

_example,

_Engel,

_1990),

and will be of interest when other estimation

_procedures

are discussed:

Here

_{T / ø}

is the

_part

of the inverse of the MME coefficient matrix

_{corresponding}

to u.

With

_{quasi-likelihood}

_(QL)

for

_{independent data,}

it is

_suggested

_(McCullagh

and

_{Nelder, 1989,}

_§

4.5 and

_chap

₉₎

that one can estimate

₀

from Pearson’s

(generalized)

chi-square

statistic. From

_[9]

it _may be shown

_{that !}

=

variances and d = N -

rank(X) -

{q -

trace (A -

I

T/â;)}

is an associated ’number

of

_degrees

of freedom’.

Application

to birth difficulties in

sheep

The data are

_part

of a

_study

into the scope for a Texel

_{sheep breeding}

program

in the Netherlands

_employing

artificial insemination.

_{Lambing difficulty}

will be

analyzed

as a

binary

variable: 0 for a normal birth and 1 for a difficult birth. There are 43

_{herd-year-season}

_(HYS)

effects. Herds are nested within

_regions

and

_regions

are nested within years. There are 2 years, 3

_regions

_{per year,} about 4 herds per

region,

and 3 seasons. The 33 sires are nested within

regions.

The 433 dams are

nested within herds with about 20 dams per herd. Observations are available from

674

_offspring

of the sires and dams.

Variability

on the

liability

scale _may

depend

on litter size.

Therefore,

observations

corresponding

to a litter size of 1 and litter sizes of 2 or more are

_{analyzed separately.}

Corresponding

data sets are referred to as the S-set

_(single;

191

_{observations)}

and

M-set

_(multiple;

483

_{observations).}

The M-set is

_reproduced

in

_Engel

and Buist

(1993)

and is available from the authors. Some _summary statistics are shown in

table I.

Table II shows some results for

_components

of

_variance,

for models fitted to the

S- and M-sets. Dam effects are absorbed. To stabilize convergence, the occurrence of extreme

_weights

was

_prevented

_by

_limiting

fitted values on the

probit

scale to the

range

[-3.5, 3.5!.

In addition to fixed HYS

_effects,

factors for age and

parity

of the dam

_(P),

sex of the lamb

(S),

and for the M-set a covariate for litter size

_(L

= litter

size -

₂₎

and included. Levels for factor P consist of the

_following

6 combinations

of _{age and}

_parity:

_{(1;1), (2;1), (2;2),}

_{(3; !}

_2),

_{(4; ! 3)}

and

_(>

_{5; !}

_4).

In models

_3,

4 and 5 a factor D for

_pelvic

dimension of the dam

_(’wide’,

’normal’ or

_{’narrow’),}

and in models 4 and 5 a covariate W =

birthweight -

average

birthweight

of the

lamb is also

_included,

with

_separate

_averages of 4.27 and 3.63 for the S- and M-sets

respectively.

Fixed effects _may be screened

_{by applying}

the Wald test to the values of

Standard errors are in

_{parentheses. S,}

P and D denote main

_effects,

W and L are

covariables,

S x L and D x W are interactions. When an estimate is

_negative

_(-),

the

component is assumed to be

_negligible

and set to 0.

table III. In all cases, test statistics are calculated for the values of the variance

components

obtained for the

_{corresponding}

full

model,

ie model 1-5.

_Variability

due

to estimation of the variance

_components

is

_ignored.

For each line in the table the

corresponding

test statistic accounts for effects above that

line,

but

ignores

effects below the line.

_Referring

to a

_chi-square

_{distribution,}

in model 1 seasonal effects seem to be

unimportant

and are excluded from the

_subsequently

fitted models.

In model 3 for the

_M-set,

the

_following

contrasts for

_pelvic

_opening

_(D)

are found: 0.520

_(0.231),

2.019

_(0.547)

and 1.499

_(0.531),

for ’normal’ versus

_’wide’,

’narrow’ versus ’wide’ and ’narrow’ versus

_{’normal’,}

_{respectively.}

Pairwise

_comparison,

with a normal

_{approximation,}

shows that any 2 levels are

significantly

(P

<

_0.05)

different.

The effects refer to the

_probits

of the conditional

_{probabilities.}

For the

_probits

of

the

_marginal

probabilities,

effects have to be

_multiplied

_by

_{(1+(J&dquo;!+(J&dquo;5)-0.5 =}

0.769

(from

table

_II).

The difference between ’narrow’ and

’wide’,

for

_example,

becomes 1.55

_(0.43).

In model 5 for the

_M-set,

_separate

coefficients for

_birthweight

are fitted for the 3 levels of

_pelvic

_opening.

The estimated coefficient for

_birthweight

for a dam with a narrow

_pelvic

_opening

is 0.72

(0.61);

this is about 0.47

(0.63)

higher

than the estimates for the other 2

_levels,

which are about the same.

_Although

a

larger

coefficient is to be

_expected

for a narrow

pelvic opening,

the difference found is far from

_significant.

Fitting

a common

coefficient,

ie

dropping

the interaction D x W between

_{pelvic opening}

and

_birthweight

in model

_{4, gives}

an estimated

coefficient for

_birthweight

of 0.28

_(0.15),

which becomes 0.21

_(0.12)

after

_shrinkage.

By

comparison,

the coefficient for the

S-set, after

shrinkage,

is 0.92

_(0.30).

The reduced effect of

_birthweight

for the M-set agrees with the

negligible

values found

Relation to other methods

We will

mainly

concentrate on differences between GAR and IRREML.

GAR is based on

QL

for the

marginal

moments with a

_probit

link and

nor-mally

distributed random effects

_(see

also

Foulley

et

_al,

_1990).

_{QL-estimating}

equations

for

_dependent

data are

_(McCullagh

and

_{Nelder, 1989,}

_§

9.3):

D’Var(y)-1

(y - p)

=

_0,

where the matrix of derivatives D =

(d

ij

),

dij =

((

3

pt/<9/3,t

j

),

follows from _p

_{= <I>(X(3}

_*

_),

and

₍₃

_*

=

where R =

diag({p(l - p) -

pr

2

(ÀpXj3)}/n),

B =

diag(r(ÀpXj3)),

C =

A2G

and

p

represents

the

_appropriate

intraclass correlation.

_Var(y)

will be

_{replaced by}

_[12],

ignoring

terms of order

_p

₂

and

_higher.

The

_QL

_equations

_may be solved

_by

iterated

generalized

least _squares on an

_{adjusted dependent variate,}

_say

₍

_GAR

_:

where

B

is B evaluated at

j3,.

_By

_comparison,

the

_{adjusted dependent}

variate from

[7]

is:

The latter variate relates to a first-order

_{approximation}

of the conditional means _p, while the former relates to a

_first-order

_{approximation}

_of

the

_marginal

means _p.

Approximately

Var((

GAR

)

=

1

Var(y)B-

l

B-

=

B-

I

RB-

1

₊

ZCZ’ =

w

G1

R +ZCZ’,

and Gilmour et al

₍₁₉₈₅₎

solve MMEs in terms of

_!GAR

and

WGAR!

For

_example,

in a sire model

with

q sires,

predictions

from these MMEs for

_Apu,

say

Û

GAR

,

are used to

_update

the intraclass correlation _p.

_Analogous

to

_!11!

with

<!=1:

Both GAR and IRREML are based on an

_approximate

mixed-model structure for an

_{adjusted dependent}

variate. Note however that in

_GAR,

in contrast to

_IRREML,

predictions

of the random effects are

_only

used to

_update

the

components

of variance and do not enter

_W

_GAR

and

₍

_GAR

_directly.

At the end of this section we shall see

that this may be an

_advantage

for GAR over IRREML in situations where there is

little information in the data about individual random

_effects,

eg, in a sire scheme

with small

_family

sizes or extreme incidence. The

_{marginal quasi-likelihood}

method

(MQL)

presented

in Breslow and

_Clayton

₍₁₉₉₃₎

can be

_regarded

as a

_{generalization}

of GAR.

_However,

in its more

_general

_setting,

_MQL

does not have the benefit of some of the exact results for binomial data and

_underlying

normal distributions

used to derive GAR.

In

_MAP,

_assuming

a _vague

_prior

for

₍₃

and a normal

_prior

for _u, the

_posterior

mean for

(0’,

u’)’

is

_{approximated by}

the

_posterior

mode.

_Hence,

(13 , û’)’

maximize

the

_{joint pdf}

of y and u. Under

normality

this is

_equivalent

to maximization of a

penalized

deviance.

_Hence,

the

_{estimating equations}

for

13

and u for MAP and

IRREML are the same and

_given

_by

_[8].

As shown in

_Foulley

et al

₍₁₉₈₇₎

an estimator

_for,

for

_instance,

a sire

_component

_QS

_,

_may be solved from:

_Eo[

_8/8

_u

_£

_s

log (f (u

I

or 2))]

= 0.

Here,

f(.[.)

denotes a

_{(conditional)}

_pdf

for the variables

indicated,

and

_{expectation E}

_o

is with

respect

to

_{f (u!y,}

_as).

The latter

_pdf

_may be

approximated

by

a normal

_density

with mean u and

_dispersion

T. In an iterative scheme this leads to the

EM-type update

!11!.

Consequently,

MAP and IRREML also solve the same final

_{estimating equations}

with

_respect

to the

_components

of

variance,

although

the

_algorithms

used are different.

The

_{penalized quasi-likelihood}

method

_(PQL)

_presented

in Breslow and

_Clayton

(1993)

is based on

_Laplace

_integration.

Random effects are assumed to be

_normally

1987)

can be

used,

see

Engel

and Buist

(1993).

In the

resulting

integral expression,

the

_logarithm

of the

_integrand

in terms of u is

_{approximated by}

a

_quadratic

function around its

optimum

in

_u,

_employing

_expected

second-order derivatives. Now random effects are

_{easily integrated}

out. Some further

_{approximations,}

eg,

approximating

conditional deviance residuals

_by

Pearson

_residuals,

result in a

normal

_log

likelihood for the

_{adjusted dependent variate}

_{( from}

_[7].

An REML

type

adjustment

for loss of

_degrees

of freedom due to estimation of fixed effects

finally yields

an REML

log likelihood,

which is maximized

_by

Fisher

_{scoring. Hence,}

although

motivated

_quite

_differently,

PQL

is

_equivalent

to IRREML

_(for

details see

Engel

and

_Buist,

_1993).

A

_comparison

between GAR and

_{MAP/P(!L/IRREML}

_by

simulation was

pre-sented in

Gilmour

et al

_(1985).

They

consider a

_simple

_one-way

_model,

_eg, a sire model with _q unrelated

_sires,

with n

_offspring

per

sire,

and with a

_binary

observation per

offspring.

An overall mean is the

only

fixed effect. Gilmour et

al

₍₁₉₈₅₎

observe that there is a

tendency

for

MAP/P(!L/IRREML

to under-estimate

h

2

for

small

_family

size n or extreme incidence _p. For extreme

inci-dence _p, GAR tends to overestimate

h’.

As also noted

_{by Thompson}

_(1990),

for

this

_simple

set-up,

closed-form

_expressions

can be derived from

_[8]

and

_!11!,

_viz,

P

GAR

=

{MS

B

-

y

(1 -

y

)}/{(n -

1)T(!-1(y)z}.

Here

MS

B

is the mean sum of

squares between sires for the

binary

observations

and y

is the overall mean. The

adjusted dependent

variable and

_weights

are:

₍

_GAR

=

!-1(y)

₊

(y -

y)/T(!’-1(y))

and

_W

_GAR

=

T(!-1(y))z/{!(1 -

y) -

/&dquo;’(!!07))!}-

As

long

as the first-order

ap-proximation

from

_[5]

_{holds, and}

_{q is}

not too

_small,

_P

_GAR

will be

_nearly

unbiased.

Actually,

for this

_simple

scheme,

GAR _may be shown to be

_equivalent

to Williams’ method of moments for

_estimating

_{overdispersion}

_(Williams,

_1982).

_Starting

from

u

o = 0 and

?7o =

!-1 (y),

the

adjusted dependent

for IRREML is the same as for

GAR,

but the

_weights

w =

T(!-1(y))z/{y(1 -

V)l

are smaller!

_{Consequently,}

the first estimate

_a

₂

₌

_{MS

_B

_{- y(l - y)}/{nr(<p-}

_l

_(y))

₂

_}

underestimates

_U

_s

approx-imately by

a factor

_{(n -}

_1)/n.

Simulation results in Gilmour et al

₍₁₉₈₅₎

_suggest

that with further iteration underestimation

_by

this factor

_persists.

For extreme

incidence,

say close to

1,

IRREML

weights

are

approximately T[r(i¡) ,

as follows from Abramowitz and

_Stegun

_(1965,

_26.2.12,

_p

_932).

Most of the information is in

the

_{negative-valued}

random

_effects,

and

_{over-shrinkage}

of random effects will

_yield

smaller

_weights

in the next

iteration,

ie ‘residual variances’ will be too

_high.

There-fore,

underestimation

_by

IRREML may be

expected

to be more serious for extreme

incidence,

even when the first-order

_{approximation}

from

_[5]

still holds. Lack of

in-formation about individual random

_effects,

because of small

_family

size or extreme

incidence,

seems a serious

_problem

with

MAP/P(aL/IRREML.

The smaller

weights

in IRREML are a _consequence of ’residual variances’

being

derived from

predicted

values of conditional variances. Alternative

_weights

wo =

E(g’(f.1,)

2

V(f.1,))-

1

for

IR-REML are

_suggested

in

_Engel

and Keen

_(1994).

For the

_logit

link evaluation of alter-native

_weights

is

_{straightforward:}

wo

= {2

+

_2exp(

_QZ/

₂₎

_{cosh(x’I3)}-}

_B

but for the

probit

link it is

problematic.

It is

_suggested

that we use wo =

{g’ (ji)2 E(V (f.1,))) -1

instead. Note that in the first

step

for the

_simple

sire scheme this

_{weight equals}

WGAR

effects.

_{MAP/PQL/IRREML}

has smaller bias and smaller

_mean-squared

error than

GAR. For both methods an

_upward

bias in the order of

20%

_(based

on 45

simula-tions)

of the true

h

2

= 0.25 is observed. An

_upward

bias is also

_apparent

for some of the models simulated in Hoeschele et al

_(1987).

Simulation results for

overdis-persed

binomial data in

_Engel

and Buist

₍₁₉₉₅₎

_only

indicate a serious

upward

bias for variance

_components

estimated

_by

IRREML when the true

_component

value is small and the

_{non-negativity}

constraints are active.

Simulation results

Data are

_generated

as described in Hoeschele and Gianola

_{(1989) (referred}

to as

HG).

The 135 HYS effects are

_generated

from a

N(0,

U2

H y

s

)

distribution. For each HYS

_class,

a sire has

_0,

1 or 2

_offspring

with

_{probabilities}

1 - pHyS,

PHYS/

2

and

_p

_HY

_s/2

_{respectively.}

The HYS effects and

design

for sires and

_offspring

are

generated only

once, after which the same HYS values and the same

design

are used in all

_subsequent

simulations. HYS effects are included in the model as fixed effects. Other fixed effects are sire group effects: 4 groups with effects

-0.40,

-0.15

(the

value -0.10 in HG was assumed to be a

_typing

_error),

0.15 and 0.40. The numbers of sires in the groups are

_{12, 14,}

13 and 11. The 50

_independent

sire

effects are

_generated

from an

N(0,

<7

g)

_{distribution,}

where

or

2

₌

h

2

/(4 - h

2

).

The residuals for the

_liability

are from an

_{N(0, 1)}

distribution. The overall constant on the

probit

scale is

_-!-1(po)

₍₁

+

_Q

_H

_YS

+

a!)0.5,

where po determines the overall

incidence.

_Suppose

that

_aH

_YS

_corresponds

to a

_proportion

_HYS

_/

of 1 +

_o

_h

_{ys 2}

+

_e

_r

_g,

then

_QHYS

=

(4f

HYS/

(1 -

,I

HYS

))/(4 -

).

h

2

In

HG, h

2

=

_0.25,

f

HYS

=

0.30,

po = 0.9 and

PHYS = _0.2. The

_expected

total number of records is

_{2 025,}

with

about 40

_offspring

_per sire. This is

_{configuration}

_10,

which is

_presented

with some of the other

_{configurations}

of

_parameter

values studied in table IV.

For each

_{configuration,}

either 1 or 2 series of 200 simulations are

_performed.

In a

_series,

HYS

_effects,

_{sire-offspring configuration}

and data are all

_generated

from a _sequence of random numbers from the same seed. The first series

_corresponds

to

the same seed for all

_{configurations.}

Therefore,

for the first

_series,

for

_example

for

configurations

3 and

_8,

the

_design

is the same. Seeds of the second series are all different. In table IV bias

_(%bias)

and root mean _square error

(8

l

4Q)

are both

expressed

as a

_percentage

of the true value of

h

2

.

In contrast with the results

in Gilmour et al

₍₁₉₈₅₎

and in

_agreement

with Hoeschele and Gianola

_(1989),

with one

_exception,

_{MAP/PGZL/IRREML}

shows a

_positive

bias. The

_exception

is

_{configuration}

13 where HYS effects are not

_included,

neither in the

_generation

of the data nor in the model fitted. The

_negative

value for this

_{configuration}

indicates

that, although

the bias seems

_{fairly independent}

of the size of the fixed

_effects,

it

may

depend

on the

(relative)

number of fixed effects in the model.

Table V shows that this is indeed so.

_Starting

from the

_original

HG

_scheme,

_using

new seeds for

_generating

the

_data,

the number of fixed effects is reduced

_by

factors

1/3, 1/2, 2/3

and

_3/4.

A reduction of

_1/3,

for

_instance,

is effected

_by

combination of HYS effects

_{(2, 3),}

_(5,

_{6), (8,}

₉₎

and so

forth,

replacing

the

original

values for

Standard errors in

parentheses;

h

2

=

heritability;

_Po = incidence _{rate; PHYS} =

probability

for a sire for progeny in a HYS

class; f

_HYS

₌

_proportion

of variance

explained by

the

The

_negative

_{bias agrees with the} calculations in the

_preceding

section for the

simple

sire model and with the results in Gilmour et al

_(1985),

where an overall

constant was the

_only

fixed effect in the model. Data for

configurations

10 and 16 was also

_generated

with random HYS

_effects,

with a new set of HYS effects for each

simulation,

and

_analysed

with HYS as a random factor in the model. Estimated bias was -35.8

_(2.7)%

for

_{configuration}

10 and -11.8

_(1.7)%

for

_{configuration}

16.

Corresponding

root mean _square errors were 52.0 and

27.3%,

respectively.

Bias and root mean square error are reduced when the progeny size is

increased,

see,

for

_{example, configurations}

18,

21 and 22. Estimators are also less biased when incidence is less

extreme,

eg,

configurations

3 versus 8 and 1 versus 7. Differences

between series within

_{configurations}

are sometimes

_greater

than would be

expected

on the basis of the standard errors

_{involved, showing}

that the

_{configuration}

of sires

and

_offspring

is also of

_importance.

Bias and root mean square error for estimated

heritability

are

expressed

as a _percentage

of the true value of

h

2

.

Standard errors in

_parentheses.

GAR was

_programmed,

similar to

_{MAP/PQL/IRREML,}

in terms of Fisher

scoring

for the

_updates

of the sire variance. Estimates are the same as for the

original

method

_proposed,

which uses EM

_steps,

but

speed

of _convergence is often increased

_by

at least a factor 10. Some

_{results, using}

the same seeds as

for

_{MAP/P(aL/IRREML,}

are

presented

in tables IV and V. For

high

incidence

(po

=

0.9)

results of

MAP/PQL/IRREML

and GAR are

comparable.

For

mod-erate incidence

_(p

_o

=

0.6),

the bias for GAR is

clearly

less than for

MAP/P(aL/

mean

square error is

_{considerably larger}

than the

_{corresponding}

bias. Plots of the es-timated

h

2

values of

_{MAP/P(aL/IRREML}

_against

GAR

_suggest

a

_strong,

fairly

lin-ear

_{relationship.}

Correlations between sire

_predictions

from

_{MAP/P(aL/IRREML}

and GAR are

_high,

_eg, over 0.99 for the first series of

_{configurations}

10 and 16.

Table V also includes results obtained for IRREML with the alternative

_weights

Wo =

T

(

1]?/

{p(l- p) -

pT(Àp1])2}

from the

_previous

section. There is a distinct

change

in the bias due to the

_change

of

_weights.

_However,

_although

bias is

considerably

improved

for the

top

lines in the

_table,

underestimation for the bottom

lines has become more serious.

In table

_{VI, approximate}

standard errors of

_heritability

estimates from

_MAP/

_/

PQL/IRREML

are

compared

with standard errors estimated from the series of 200 simulations

_(referred

to as

_empirical

values).

The

_approximate

standard _errors, based on Fisher information

assuming

normality

for the

_{adjusted dependent}

variate,

perform quite well,

as was also observed in

_Engel

and Buist

(1995).

These standard errors are standard

_output

in Genstat 5.

Mean,

10 and _{90% percentage}

_points

of the

_approximate

standard errors are

_presented

as percentages of the

empirical

value.

The

_{approximately}

linear

_relationship

between IRREML and GAR estimates within a

configuration

of

_parameter

values,

and the similar standard _errors,

_suggest

that it should be

_possible

to correct for bias in

_IRREML,

at the least in those cases

where

GAR

_performs

well. For incidence 0.90 and the full set of 135 HYS

effects,

shrinkage

towards the

origin.

Possibly,

the results of Cordeiro and

_McCullagh

(1991),

involving

an extra iteration

_step

for

_adjusted

response

variables,

_may

be

extended to minimization of the

_penalized

deviance,

thus

improving

estimates /3

and

predictions

u. This would

_imply

modification of the

_{adjusted dependent}

variate

_(.

Users of

_{MAP/P(aL/IRREML}

should be aware of the

_problems

involved when

family

sizes are

_small,

or in the context of an animal

model,

when many animals are

only weakly

related. With extreme

_incidence,

_say over

_0.90,

and a sizeable number of HYS

_effects,

say in the order of

6%

of the number of

binary

observations,

MAP,

PQL,

IRREML and GAR are liable to

_seriously

overestimate

_{heritability,}

and actual selection response may be

considerably

less than

expected

on the basis of model calculations.

ACKNOWLEDGMENTS

The assistence of A Keen with the

_analysis

of the

_{lambing difficulty}

data and the use

of his Genstat

_procedure

IRREML is

_{greatly appreciated.}

We also thank J de

_Bree,

PW Goedhart and an _{anonymous reviewer} for

helpful

comments on earlier versions of the

manuscript.

REFERENCES

Aitchison

_J,

Shen SM

₍₁₉₈₀₎

_{Logistic-normal}

distributions: some

_properties

and uses.

Biometrika

67,

261-272

Abramowitz

_{M, Stegun}

I

₍₁₉₆₅₎

Handbook

_of

Mathematical Functions.

_{Dover, NY,}

USA Breslow

_NE,

_Clayton

DG

₍₁₉₉₃₎

_Approximate

inference in

_generalized

linear mixed

models. J Am Stat Assoc 88, 9-25

Cordeiro

_{GM, McCullagh}

P

₍₁₉₉₁₎

Bias correction in

_generalized

linear models. J R Stat Soc B

_53,

629-643

Cox

_{DR, Hinkley}

DV

₍₁₉₇₄₎

Theoretical Statistics.

Chapman

and

Hall, London,

UK

Engel

B

₍₁₉₉₀₎

The

analysis

of unbalanced linear models with variance _components.

Statistica Neerlandica

_44,

195-219

Engel B,

Te Brake J

₍₁₉₉₃₎

_Analysis

of

_{embryonic development}

with a model for under-or

_{overdispersion}

relative to binomial variation. Biometrics

_49,

269-279

Engel B,

Buist W

₍₁₉₉₃₎

Iterated

_Re-weighted

REML

Applied

to Threshold Models with

Components of

Variance

for Binary

Data.

_Agricultural

Mathematics

_Group.

Research

report

LWA-93-16,

available from first author

Engel B,

Buist W

₍₁₉₉₅₎

_Analysis

of a

_generalized

linear mixed model: a case

_study

and

simulation results. Biometr J

_{(in press)}

Engel B,

Keen A

₍₁₉₉₄₎

A

_{simple approach}

for the

_analysis

of

_generalized

linear mixed models. Statistica Neerlandica

_48,

1-22

Foulley JL,

Im

_S,

Gianola

_D,

Hoeschele I

₍₁₉₈₇₎

_Empirical

estimation of parameters for

n

_{polygenic binary}

traits. Genet Sel Evol _19, 197-224

Foulley JL,

Gianola

D,

Im S

₍₁₉₉₀₎

Genetic evaluation for discrete

_polygenic

traits

in animal

breeding.

In: Advances in Statistical Methods

_for

Genetic

Improvement of

Livestock

_(D

Gianola,

K

_Hammond,

_eds).

_{Springer Verlag, Berlin, Germany,}

361-409 Genstat 5 committee

₍₁₉₉₃₎

Censtat 5 Release 3

_Reference

Manual.

_(RW

_Payne

Gianola

D, Foulley

JL

₍₁₉₈₃₎

Sire evaluation for ordered

categorical

data with a threshold model. Genet Sel

Evol 15,

201-223

Gilmour

_AR,

Anderson

_RD,

Rae AL

₍₁₉₈₅₎

The

_analysis

of binomial data

_by

a

generalized

linear mixed model. Biometrika

_72,

593-599

Harville

_D,

Mee RW

₍₁₉₈₄₎

A mixed-model

_procedure

for

_analyzing

ordered

_categorical

data. Biometrics

_40,

393-408

Henderson CR

₍₁₉₆₃₎

Selection index and

_{expected genetic}

advance. NAS-NRC 982 Hoeschele

I,

Gianola D

₍₁₉₈₉₎

_Bayesian

versus maximum

quasi-likelihood

methods for sire

evaluation with

categorical

data. J

Dairy

Sci

_72,

1569-1577

Hoeschele

_I,

Gianola

_{D, Foulley}

JL

₍₁₉₈₇₎

Estimation of variance components with

quasi-continuous data

_{using Bayesian}

methods. J An Breed Genet

_104,

334-349

Im

_S,

Gianola D

₍₁₉₈₈₎

Mixed models for binomial data with an

_application

to lamb

mortality Appl

Stat

_2,

196-204

Jansen J

₍₁₉₉₂₎

Statistical

analysis

of threshold data from

experiments

with nested errors.

Comp

Stat Data

_{A!,al 13,}

319-330

Johnson

NL,

Kotz S

₍₁₉₇₀₎

Continuous univariate distributions. -2.

_{Houghton Miffiin,}

Boston,

USA

McCullagh

P

₍₁₉₉₁₎

_{Quasi-likelihood}

and

estimating

functions. In: Statistical

_Theory

and

Modelling

(DV

Hinkley,

N

_Reid,

EJ

Snell,

eds)

Chapman

and

Hall, London, UK,

265-286

McCullagh P,

Nelder JA

₍₁₉₈₉₎

Generalized Linear Models.

Chapman

and

Hall, London,

UK,

2nd edn

Meyer

K

₍₁₉₈₉₎

Restricted maximum likelihood to estimate variance _components for animal models with several random effects

_using

a derivative-free

algorithm.

Genet

Sel Evol

_31,

317-340

Nelder

_{J, Pregibon}

D

₍₁₉₈₇₎

An extended

_{quasi-likelihood}

function. Biometrika

_74,

221-232

Patterson

HD,

Thompson

R

₍₁₉₇₁₎

_Recovery

of inter-block information when block sizes

are

_unequal.

Biometrika

_58,

545-554

Pearson K

₍₁₉₀₁₎

Mathematical contributions to the

_theory

of evolution. VII. On the correlation of characters not

_{quantitatively}

measureable. Phil Trans R Soc

_(Lond)

A

195,

1-47

Preisler HK

₍₁₉₈₈₎

Maximum likelihood estimates for

binary

data with random effects. Biometr J

3,

339-350

Rao CR

₍₁₉₇₃₎

Linear Statistical

_Inference

and its

_{Applications.}

John

Wiley

&

_Sons,

New

York, USA,

2nd edn

Schall R

₍₁₉₉₁₎

Estimation in

_generalized

linear models with random effects. Bio!n,etrika

78,

719-728

Searle

_SR,

Casella

G,

McCulloch CE

₍₁₉₉₂₎

Variance

Components.

John

_Wiley

& Sons. New

_York,

USA

Sowden

RR,

Ashford JR

₍₁₉₆₉₎

_Computation

of the bi-variate normal

_{integral. Appl}

Stat

18, 169-180

Thompson

R

₍₁₉₇₉₎

Sire evaluation. Biometrics

_35,

339-353

Thompson

R

₍₁₉₉₀₎

Generalized linear models and

_applications

to animal

_breeding.

In: Advarcces in Statistical Methods

for

Genetic

Improvement of

Livestock

_(D

Gianola,

K

_Hammond,

_eds)

_{Springer Verlag, Berlin, Germany,}

312-328

Williams DA

₍₁₉₈₂₎

Extra binomial variation in

_logistic

linear models.

_Appl

Stat

_31,

144-148

Zeger SL, Liang KY,

Albert PS

₍₁₉₈₈₎

Models for

_longitudinal

data: a

_generalized