On biased inferences about variance components in the binary threshold model

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On biased inferences about variance components in the

binary threshold model

C Moreno, D Sorensen, La García-Cortés, L Varona, J Altarriba

To cite this version:

Original

article

On biased inferences about variance

components

in the

_binary

threshold

model

C Moreno

1

D

Sorensen

LA García-Cortés

1

L Varona

J

Altarriba

1

Unidad de Genética Cuantitativa _y

_{Mejora Animal,}

Facultad de

Veterinaria,

Universidad de

Zaragoza, Miguel

Servet

_177,

50013

_{Zaragoza, Spain;}

2

Danish

Institute

_of

AnimaL

_Science,

Research Center

Foulum,

PO Box

_39,

DK-8830

Tjele,

Denmark

(Received

2

_September

_1996;

_accepted

6

February

1997)

Summary - A simulation

_study

was conducted to

study frequentist

properties of three estimators of the variance _component in a mixed effect

_binary

threshold model. The three estimators were: the mode of a normal _{approximation} to the

marginal posterior

distribution of the _component, which is denoted in the literature as

_marginal

maximum likelihood

_(MML);

the mean of the

marginal

posterior distribution of the _component,

_using

the Gibbs

_sampler

to

_perform

the

_{marginalisations}

_(GSR);

and

_third,

the mode of the

_joint

posterior

distributions of location and the variance _parameter, used in

_conjunction

with the iterative

_bootstrap

bias correction

_(MJP-IBC).

The latter was

_{recently proposed}

in the literature as a method to obtain

_nearly

unbiased estimators. The results of this

_study

confirm that MML can

_yield

biased inferences about the variance _component, and that the

sign

of the bias

depends

on the amount of information associated with either fixed effects or with random effects. GSR can

_{produce positively}

biased inferences when the amount

of data per fixed effect is small. When fixed effects are

_{poorly estimated,}

the bias

_persists,

despite

the fact that

posterior

distributions are

_guaranteed

to be _proper, and that the amount of information about the variance _component is

_large.

In this case, the

marginal

posterior

distribution of the variance _component is

_{highly peaked}

and

symmetric,

but it shows a shift towards the

_right

with _respect to the true

_(simulated)

value. This bias can be reduced

_{by assigning}

a Gaussian

_{probability density}

function to the

_prior

distribution of the fixed

_effects,

but this strategy does not work with very sparse data structures. The method based on MJP-IBC

yielded

unbiased inferences about the variance _component in all the cases studied. This estimator is

_{computationally simple,}

but _contrary to GSR with normal

_priors

for the fixed

_effects,

can lead to estimates that fall outside the _parameter space.

binary traits

_/

Gibbs _sampling

_/

biased estimators

_/

bootstrap

*

Résumé - À propos des biais d’estimation de composantes de variance dans le modèle binaire à seuil. Une étude de simulation a été conduite _pour étudier les

propriétés

fréquentistes

de trois estimateurs de la _composante de variance dans un modèle mixte à seuils pour variable binaire. Les trois estimateurs ont été : le mode de

l’approximation

normale de la distribution

marginale

a

_posteriori

de la composante

qui

est

_appelée

dans la littérature « maximum de vraisemblance

marginale

» (MML),

l’espérance

de distribution

marginale

a _posteriori de la composante, en utilisant

l’échantillonnage

de Gibbs

_(GSR)

et

_finalement,

le mode de la distribution

conjointe

a _posteriori des

paramètres

de position

et de variance, utilisé avec une méthode itérative de correction des biais _par

_bootstrap

(MJP-IBC).

Cette dernière a été récemment

_proposée

dans la littérature comme une

méthode d’obtention d’estimateurs approximativement non biaisés. Les résultats de l’étude

confirment

que le MML peut donner des

_inférences

biaisées à _propos de la composante

de variance _{et que} le

_signe

du biais

dépend

de la

_quantité

_{d’information}

attachée aux

effets fixes

ou aux

_effets

aléatoires. Le GSR peut

produire

des estimées à biais

_positif

quand il

y a _peu

d’information

_par niveau

d’effet fixe.

Dans ce _cas, le biais _persiste même

quand l’information

sur la composante de variance est importante et

_quand

la distribution

postérieure

est _{propre. La} distribution

_marginale

a

_posteriori

est hautement concentrée et

symétrique

mais montre un

_décalage

à droite de la vraie valeur

_(simulée).

Ce biais peut être réduit en considérant _que la distribution a _priori des

_{effets fixes}

est de type

gaussien

mais cette

_stratégie

n’est _pas

_e,!cace

_pour les structures de données où

_{l’information}

est très

dispersée.

La méthode MJB-IBC a

_produit

des estimées non biaisées de la _composante de variance dans tous les cas étudiés. Cet estimateur est

_simple

à calculer mais contrairement à GSR avec des

effets fixes

à distribution

_gaussienne,

elle _peut mener à des estimées situées en dehors de

l’espace

des

paramètres.

caractères binaires

_/

échantillonnage de Gibbs

_/

estimateurs biaisés / bootstrap

INTRODUCTION

Ordered

_categorical

traits are often

_analysed

_by

means of the threshold

_liability

model,

first used

_{by Wright}

₍₁₉₃₄₎

in studies of the number of

_digits

in

_guinea

pigs,

and

_by

Bliss

₍₁₉₃₅₎

in

_{toxicology experiments.}

In the threshold

_model,

it is

postulated

that there exists a latent or

_underlying

variable

_(liability)

which has

a continuous distribution. A _{response in} a

_{given category}

is observed if the actual

value of

_liability

falls between the thresholds

_defining

the

_appropriate

_category.

Here

we will consider the situation where the

_liability

is modelled as a linear function of

fixed effects and

_possibly

correlated random effects. In a

_typical

likelihood

_setting,

one _may be interested in

making

_joint

inferences about estimable functions of fixed

effects and on variance

_components

associated with the random effects. Within a

_{Bayesian framework,}

one _{may concentrate} inferences on the

_marginal

_posterior

distributions of contrasts of ’fixed’ effects and of the variance

components,

and

possibly

on the

_{marginal posterior}

distributions of the random effects. In a

_genetic

context,

the latter involve

_typically

additive

_genetic

values and

_knowledge

about these is

_important

to carry out selection decisions and to describe response to selection. Other inferences of interest may

require computing probabilities

that observations fall in a

_given

_category

for

_given

individuals or combinations of

parameters.

All these inferential

_problems,

whether in a likelihood or in a

_Bayesian

_setting,

require

marginalisations.

For

_example,

_computation

of the likelihood or of the

_joint

with

_respect

to the random effects.

_Although

these

_integrals

do not have a closed

form

_solution,

numerical

_{integration using quadrature}

is feasible when the random effects are uncorrelated

(Anderson

and

Aitkin,

1985).

When random effects are

correlated,

as is the case with animal

_models,

these numerical

_integration

_techniques

cannot be used. To circumvent this

_problem,

a

large

number of

_{approximations}

and different

_approaches

has been

_suggested

in the literature

_(see,

for

_example,

McCullagh

and Nelder

₍₁₉₈₉₎

for a discussion of some of

these,

and

Foulley

et al

(1990)

and

_Foulley

and Manfredi

₍₁₉₉₁₎

for a review of the

_analysis

of

_categorical

traits in an animal

breeding

context).

The introduction of Markov chain Monte Carlo methods allows

_computation

of multidimensional

_integrals

and

_analytic

_{approximations}

can therefore be avoided.

The use of the Gibbs

_sampler

to

_analyse

ordered

_categorical

traits has been

_reported

by Zeger

and Karim

₍₁₉₉₁₎

and

_by

Albert and Chib

_(1993),

_using

a

_Bayesian

perspective.

McCulloch

₍₁₉₉₄₎

_applies

the Gibbs

_sampler

to estimate variance

components

for

_binary

data in a likelihood

_setting.

In an animal

_breeding

_context,

the Gibbs

_sampler

was

_applied

to the

_analysis

of

_categorical

traits

_by

Moreno

₍₁₉₉₃₎

and

by

Sorensen et al

_(1995),

both from a

Bayesian

_perspective.

Zeger

et al

₍₁₉₈₈₎

studied non-linear models for the

_analysis

of

_longitudinal

_data,

and drew attention to the

_dependence

between the

_marginal

_expectation

of the data and the variance of the random effects. In the context of a

_{generalized estimating}

equations approach,

they

showed that estimators of the fixed effects and of the variance

_component

associated with the random effects are not even

asymptotically

orthogonal,

as

_they

are in the Gaussian model. A similar

_dependency

in the case

of the

_logistic

mixed model was shown

_by

Drum and

_McCullagh

₍₁₉₉₃₎

in the context of restricted maximum

likelihood,

where the latter was constructed

_using

error contrasts. This

_approach

to

_constructing

the restricted likelihood

_generates

an

equation

that is not a function of the fixed effects in the case of the Gaussian

_model,

but not in the case of non-linear models. The association between the fixed effects

and the variance

component

indicates that inference about the latter is

dependent

upon the amount of information in the data with which fixed effects are estimated.

The

_study

of Moreno

₍₁₉₉₃₎

_reported

_frequentist

_properties

of the mean of

the

_marginal

_posterior

distribution of the variance

_component

in a

_binary

mixed

threshold model. In this

_study,

a sire model was assumed. Moreno

(1993)

noted

that when there is little information about fixed

_effects,

the mean of the

marginal

posterior

distribution of the variance

_component

is biased

upwards.

This is so even

in cases where there are _very

_large

numbers of sires and

_offspring

_per

_sire,

and

consequently

this

_marginal

_posterior

distribution is

_peaked

and

_symmetric.

It is well known that when the data associated with a

_particular

fixed effect fall all

within the same

_category

(known

as the extreme

_category

_problem

_(ECP),

the likelihood is underidentified

_(eg,

Misztal et

_al,

_1989;

Gelman et

_al,

_1995).

In this

situation,

in a

_Bayesian

_setting,

the

_assignment

of

_{improper prior}

distributions to

the ’fixed effects’ can lead to

_improper

_posterior

distributions.

_However,

we show

that the bias related to inference about the variance

_component

when there is little information about fixed effects

persists,

when all the

parameters

of the model are

This bias

problem

has been the

subject

of an extensive simulation

study by

Hoeschele and Tier

_(1995).

These

_authors,

who used the Gibbs

_sampler

with data

augmentation

(Tanner

and

_Wong,

_1987),

confirm the results of Moreno

_(1993),

and indicate that

_part

of the

_problem

can be alleviated

_by

_assuming

that ’fixed effects’

follow a

_priori

a Gaussian distribution. As we show

below,

treating ’fixed

effects’ as

if

_they

were ’random’ can reduce the bias associated with the mean of the

marginal

posterior

distribution of the variance

_component,

but this is not a

_strategy

that

always

works.

An alternative

_approach

to inferences in the mixed threshold model was

pre-sented

_by

Tempelman

(1994).

The method is

quite

general

and is based on an

approximation

to the

_marginal

_posterior

distribution of the variance

component,

where the

_{marginalisation}

is obtained

_{using Laplacian integration.}

The

computa-tional burden of this method amounts to

_obtaining

_repeatedly

the determinant of the Hessian of the

_{joint posterior distribution,}

conditional on the current value of

the variance

component.

The dimension of the Hessian is

_equal

to the number of ’fixed’ and random effects in the model. In a small simulation

_study,

Tempelman

(1994)

showed that the method

_performed

well in a sire

_model,

but

_yielded

biased estimates of the variance

component

in an animal model.

Breslow and Lin

₍₁₉₉₅₎

have

_{recently developed}

methods to

_adjust

for the bias in

_approximate

estimators of the variance

_component

and

regression

coefficients for

generalised

linear mixed models

_having

one source of random variation. The bias

correction

_applies

to

_{approximations}

based on

_penalised

_{quasilikelihood}

and holds

for

_generalised

linear mixed models

_having

canonical link

_functions,

the latter

_being

a condition that is not satisfied in the case of the

_probit

mixed model studied in

this work.

Kuk

₍₁₉₉₅₎

_proposed

an alternative iterative method

using

the

_bootstrap

to obtain

_{asymptotically}

unbiased estimators. The

_appealing

aspect

of this

approach

is that in

_principle

it can be

_applied

to _any estimator.

_{Though computationally}

intensive,

the method can be

_implemented

in a

straightforward

manner.

The

_objective

of this work is to

_study

_frequentist

_properties

of three

_approaches

to

_making

inferences about variance

_components

in mixed

_binary

threshold models. One of these

approaches,

which does not

_rely

on

analytic

_{approximations,}

is based

on the mean of the

_{marginal posterior}

distribution of the variance

_components.

We

study

its

_{properties using}

different data structures and

_assuming

either proper or

improper prior

distributions for the ’fixed’ effects and for the variance

_component.

We

_report

also

_frequentist

_properties

of a

_{computationally simple}

estimator based

on the mode of the

_{joint posterior}

distribution of location and scale

_parameters,

used in

_conjunction

with the iterative bias correction

_using

the

_{bootstrap proposed}

by

Kuk

_(1995).

MATERIAL AND METHODS

Model for the

_analysis

of

_binary

responses

A Bernoulli random variable _yg is recorded for observation

_j(j

=

1,...,!,)

in

subclass

_i(i

=

1, ... ,

s),

and takes the value y2! = 1 if

!i!

_> t _or

y2! = 0

otherwise,

Conditionally

on fixed effects

(herds) (b,

of order

p)

and on random

(sire)

effects

(u,

of order

_q),

the

_!z!

are

_{independently}

and

_{normally distributed,}

_ie,

N(x’b

+

z!u, 1),

and the _Yij are

_independent

with

_Pr(yg

=

O!b, u) _

4l(t -

xib - z!u).

The

_symbol

_4l(.)

denotes the standard normal cumulative distribution function and

_x!(z!)

denotes the ith row of the known

_design

matrix

_X(Z).

The subclass i is defined as the herd-sire combination i. We will denote the

_prior

distribution for

the fixed effects as

p(b).

_Invoking

an infinitesimal

_model,

the vector of sire effects

is multivariate

_normally

distributed:

_{ulA, a 2 -}

_N(O,

_A

_U2

_),

where A is a known q

by

q matrix

af additive

_{genetic relationships}

_among

_sires,

and

_Q

_u

_is

the unknown variance between sires. The

_{prior probability density}

for

_Q

_u

_will

be denoted

_{p(o, u 2).}

Under the

model,

the conditional distribution of the data y,

given

b and u is:

where nil is the number of observations in class i with _!2! = 0 and _n22is the number

of observations in class i with _y2! = 1. The

_{joint posterior}

distribution is

_given

by:

Unless otherwise

_stated,

_p(b)

is assumed uniform in the interval

_[t

-

5,

t +

_5!,

and

p(

Q

u)

is assumed uniform in the interval

_!0,1/3!.

_{Consequently,}

in these cases,

posterior

distributions are _proper.

As is well established

_(Cox

and

_Snell,

_1989),

in the

binary

threshold model neither the residual variance nor the threshold can be estimated. In this

work,

the

simulated data are

_analysed

_using

a model in which the residual variance is set

equal

to

_1,

and the value of the threshold is

_arbitrarily

set

_equal

to the simulated value.

Methods of inference

In this section we

_briefly

describe the three methods of inference studied in this

work. These _are,

_first,

_marginal

maximum likelihood

_(MML)

_(Foulley

et

_al,

_1987).

The second method is based on an estimator of the mean of the

marginal

_posterior

distribution of the sire variance.

Here,

marginalisations

were obtained

using

the Gibbs

_{sampler applying}

an

_{adaptive rejection sampling algorithm}

due to Gilks and

Wild

_{(1992) (GSR).}

The third method is based on the mode of the

_{joint posterior}

distribution

_!2!,

used in

_conjunction

with the iterative

_bootstrap

bias correction

(MJP-IBC).

The method based on MML was chosen because it is one of the most common

methods used in the

_analysis

of

_categorical

traits. Method GSR does not

involve

analytic

approximations

and therefore it is

_interesting

_{to compare} it to MML under

a

_variety

of data structures. As we show

below,

both MML and GSR in the form

implemented

in this

_work,

can lead to biased

_inferences,

even in cases with a

_large

Marginal

maximum likelihood

_(MML)

This method is based on

_locating

the modal value of an

_{approximation}

to In

_p(QuIY)!

the

_logarithm

of the

_marginal

_posterior

distribution of the variance

_component.

As shown in

_Foulley

et al

₍₁₉₈₇₎

and

_applying

their result for the case

Q

u

u 2 _

Uniform [0, 1/3]:

The

_expression

on the

_right

hand side of

[3]

involves

_finding

the

expected

value of

u’A-l

u,

where this

_expectation

is taken with

respect

to the distribution of

_uly,

_{afl .}

That is:

In the Gaussian

_model,

_{uJy, (7!}

_is

normal,

and the

_expectation

in

_[4]

involves the solution of Henderson’s

₍₁₉₇₃₎

mixed model

equations.

This

_yields

_{E(uly, (7!),}

and the inverse of the coefficient matrix of the mixed model

_equations,

_yields

Var(uly,

(7!).

In the context of the

_present

_binary

_{model, the}

form of the distribution of

_{uly, (7!}

_is

not known. MML consists of

_{approximating}

_{E(uly, (7!)}

by

the mode of

_{p(b, uJy, (7!)}

and

_{Var(uly, (7!)}

_by

the inverse of minus the

_expected

value of the second derivatives of

_{p(b, uly, (7!)}

with

_respect

to b and u.

Setting

[3]

equal

to zero and

_solving

for the

_unknown,

leads to an iterative scheme which involves

the solution of a non-linear

_system

reminiscent of Henderson’s

(1973)

mixed model

equations.

The

_equation

for

_{(7! requires}

the

_computation

of:

where

_C

_uu

is the

_part

of the

_complete

inverse of the coefficient matrix

_{corresponding}

to the u

by

u block.

This method is known to

_yield

biased inference in small

_samples

with

_respect

to the variance

_component

_(Hoeschele

et

_{al, 1987; Dellaportas}

and

_Smith,

_1993;

_Engel

et

_al,

_1995).

The

_sign

of the bias

_depends

on the data structure

(Hoeschele

and

Tier,

1995;

Engel

et

_al,

_1995).

With a

large

amount of information on fixed effects

and little information on random

_effects,

the bias is

_negative.

When the amount of

information on the random effects is

_large,

but fixed effects are

_poorly

_estimated,

the bias is

_positive.

This behaviour of the estimator can lead to situations where

no bias is detected.

Mean of the

_{marginal posterior}

distribution of sire

variance,

using

the

Gibbs

_sampler

_(GSR)

of all the

_parameters

of the model. The literature on Markov chain Monte Carlo

methods _{is very}

_large

and _many theoretical and

_applied

issues can be found in the recent

_publication

of Gilks et al

_(1996).

In _many

_{applications,}

the

_fully

conditional

_posterior

distributions are

_known,

and the

_{implementation}

of the Gibbs

_sampler

is

_{straightforward.}

This is the case of

the

binary

threshold mixed

_model,

when the Gibbs

_sampler

is used in

_conjuction

with data

_{augmentation,}

as in Albert and Chib

_(1993).

If the

_technique

of data

augmentation

is not

_used,

the

_present

model leads to

_fully

conditional

_posterior

distributions of the location

_parameters

that are not of standard form. In this case, other

sampling

schemes can be

implemented,

such as the

adaptive rejection

sampling

proposed

by

Gilks and Wild

_(1992).

This is the

_approach

chosen in this paper. Details of the

fully

conditional

posterior

distributions can be found in the

Appendix.

We follow

_closely

the

_{adaptive rejection}

_sampling

_algorithm

as described

in

_Dellaportas

and Smith

_(1993).

The

_{implementation}

of GSR

_requires

draws from the

_fully

conditional

_posterior

distributions of the

parameters.

One of these is the

_fully

conditional

_posterior

distribution of the sire

_variance,

_{p (}

₀

_,

₂

_{u 1 b,}

_u,

_y),

which is in the form of a scaled

inverted

_chi-square.

The

_{heritability,}

h

2

=

40&dquo;!

u / (0,2

+

1) ,

was constructed for each value drawn from

_{p(a u}

₂

_lb,

_{u, y).}

The Monte Carlo estimate of the mean of the

marginal

posterior

distribution of

_heritability

was

_computed

as the mean

h

2

over the Gibbs

_samples.

In the

_present

_study,

differences between mean, mode and median of the

_marginal

_posterior

distribution of

_heritability

were

_negligible

and results based

on the mean

_only

are

_reported.

The Gibbs

_sampler

was

_implemented

_using

several

chains,

and all the

_samples

in each were

_kept.

Burn in

periods

were determined on a

_pragmatic

basis. The criterion of Gelman and Rubin

(1992)

was

_adopted

to determine _convergence.

The Gibbs

_sampler

allows the

_drawing

of inferences

_using

a

_{fully Bayesian}

approach,

without

_resorting

to

_analytical

_{approximations.}

_{Asymptotically,}

_posterior

distributions are

normal,

with mean

_equal

to the maximum likelihood estimator. It seems relevant to

_study

the behaviour of this

_approach

under different data

structures,

in view of the fact that

analytic approximations

are not involved.

Mode of

the

_{joint posterior}

distribution followed

_by

the iterative boost-rap bias correction

(MJP-IBC)

The first

_step

of this method is to find the mode of the

_{joint posterior}

distribution of location and

dispersion

parameters

!2!,

under the

_assumption

that the

_prior

of

p(

0

,2)

follows a scaled inverted

_chi-square

distribution. The latter is of the form:

where

_q

_*

and

_u*

₂

are

_parameters

of the

_prior

distribution. In the

_present

_study,

_q

_*

the

_resulting

joint posterior

with

respect

to

_{o, u 2,}

_setting

to zero and

solving

for

_{or2 U )}

produces

the iterative scheme:

as shown

_by

Hoeschele et al

_(1987),

where

u

is the mode of

_p(b,

_ula!,y)

as in

MML. Notice that in contrast to

_{!5!, [7]}

does not

_require

_knowledge

of the inverse of any matrix and is very easy to

_compute.

Use of a _proper

_prior

distribution for

o

l2

is

important,

because with an

_improper

uniform

_prior,

this method is known to

yield

zero estimates for

_Q

_u

_(Lindley

and

Smith,

1972).

The second

_step,

consists of

_applying

the iterative bias correction that was

proposed by

Kuk

_(1995).

This

_procedure

can be described as follows. Let 0 be

the

parameter

of

_interest,

and let

e

represent

an initial estimate obtained as a

solution to a

_system

of

_{estimating equations}

!(0!y),

where _y are the data that are

assumed to follow a distribution

_f(y!0).

The

system

of

equations

!(9!y)

could be

the score

function,

for

_example,

or some

_{approximation}

to it. In the

_present

case,

ljJ(ely)

is the

_system

of

_equations

that results from the

_joint

maximization of the

posterior

distribution with

respect

to location and

dispersion

parameters.

Let

_A

_*

_,

the

_expected

value of the estimator of

_e,

_satisfy:

If the estimator of 0 is

_biased,

the

_asymptotic

bias is

_given

_by:

where

_g(e)

=

e

*

.

Kuk

(1995)

_proposes the

following

correction for this bias. Let

(3!°!

be an initial estimate of the bias of

0.

Then define an

_updated

estimate of the bias as:

... I- 1 .1 I- . 11

and define an

updated

bias-corrected estimate of 0 as:

Kuk

₍₁₉₉₅₎

shows that this estimator is

_{asymptotically}

consistent. Since the function

_g(8)

is not

_{usually known,}

Kuk

₍₁₉₉₅₎

_proposes to

_approximate

_g(8)

by

g

M

(8)

!

which is the average of

e

over simulated

samples.

That is:

The simulated values _{yi,..., ym} are obtained from

_{f (y!0),}

where in the kth round of

_iteration,

e is set to 8 -

(3!).

The

_{implementation}

of the above

_{procedure using}

_method

_(MJP-IBC)

was as

follows.

_First,

estimates of location and scale

_parameters,

_e,

are obtained from the

_joint

maximization of the

_posterior

distribution

_!2!,

conditional on the

_original

data.

_Second,

simulated

_samples

_{are generated}

from

_f(ylO),

where e is set to the value of these initial estimates

6.

_Third,

_6(y

_i

₎

are obtained from the

_joint

maximization of the

_posterior

distribution

_[2],

conditional on the ith simulated

sample

(i

=

1, ... ,

m).

Fourth,

_an

average of

8(

Yi

)

over the m

_samples

is

_computed,

which is denoted

_by

_g

_M

_(9),

and the estimate of the bias is obtained from

_[13].

Finally,

6

is corrected

_using

_!14!,

which is used to

_generate

a new set of simulated

samples.

This iterative scheme is

_repeated

until _convergence is achieved.

In the _presence of the extreme

_category

_problem

_(ECP),

it was

_required

that

the

_proportion

of ECPs In the simulated

_samples

was the same as in the

_original

data. This was achieved

_{by adjusting}

the values of the fixed effects associated with ECPs. Another

_point

to

_highlight

in the

_application

of MJP-IBC in models with

ECPs is that the correction in

_expression

_[14]

involves the

_component

of variance and the non-ECP fixed effects

_only.

_Moreover,

if in a simulated

sample

yi, an ECP

is

_generated

for a

_fixed

effect that was a non-ECP fixed effect in the

_{original data,}

then the element in

6(y

i

)

associated with this fixed

effect,

is set to the value that ’

was used to simulate this fixed effect in the

_sample

_yi, and not to its estimated value. This ad hoc

_strategy

was arrived at on a trial and error

_basis,

and it was

followed because it avoided

_problems

_{of lack of convergence. Other}

_strategies

could be

_envisaged,

one such

being

based on

_assigning

_mildly

informative

_priors

to the ’fixed’ effects.

Simulation

_study

A Monte Carlo simulation

_study

was carried out in order to evaluate the three

methods of inference under a

_variety

of data structures. The criteria used to

assess the

performance

of the methods were bias and mean _square error. The data were simulated as follows.

_Liability

was drawn from a normal

_{distribution,}

with

mean

_equal

to the fixed effect

_{corresponding}

to the individual in

_question

and variance

_{given by}

₍₇₂

₊

_o,,2,

where

_U2

_is

the sire variance and

_U2

_was set

_equal

to 1.

Only

one set of fixed effects were simulated

_(herds),

and these were drawn from

uniform distributions within the interval

_{(-2, 2).}

Sires were drawn from a normal

distribution,

with mean

_(1/2)v,P

and variance

_(3/4)

_Q

_u,

_{where up} is the realized

value of the sire of the individual. Base

_population

sires were drawn

_assuming

a mean of zero and variance

u U. 2

Three

_{non-overlapping}

_generations

of data

_(with

no

The data at the

phenotypic

scale were

generated

as follows:

where the value of the known threshold

_t,

was set

_depending

on the desired level of

incidence.

The simulations of the

_original

data were carried out such that the

_resulting

structures led to cases with different amounts of information associated with the

parameters

of the model. The simulation results were all based on 50

_replicates.

RESULTS

Estimates of

_heritability

using

MML and GSR

_(mean

of the

_marginal

_posterior

distribution of

_{heritability)}

for three data structures are shown in table I. The

analyses

based on GSR assumed the _proper uniform

_priors

for the ’fixed’ effect and

the variance

component,

as described in Model

for

the

analysis

of binary

_responses.

The structure Al is one in which there is

_adequate

information in the data to

make inferences about all the

_parameters

of the model. There is no ECP in this structure and the incidence is set at

60%.

Both MML and GSR

_yield

estimates that on _average are in

_good

_agreement

with the true value. In structure A2 there is

adequate

information related to fixed

effects,

but there is

_relatively

less information

associated with the random effects. In A2 there are no ECPs. In structure A3 there

are

_ECPs,

but there is a

_large

amount of data associated with the non-ECP fixed

effects and there is little information about the random effects. As

_expected,

the MML

_yields

estimates of

_heritability

that are biased downwards. GSR does not show

signs

of bias. Structures A4 and A5

_represent

cases where there is little information

about fixed effects. The

_percentage

of ECPs is

_higher

under structure A5

_(65%

in

A5 versus

25%

in

_A4).

Both methods

_{produce positively}

biased

_results;

the bias

Table II shows estimates of the mean of the

_{marginal posterior}

distribution of the

_heritability

_using

GSR under a

_variety

of data

_structures,

and

_performing

the

Bayesian analysis

on the basis of proper and

improper posterior

distributions. The

point

of the results in the table is to draw attention to the

_type

of data structures that can cause GSR to

_yield

biased inferences about the variance

_component

and to illustrate that the bias is not

_necessarily

related to the

_impropriety

of the

_posterior

distribution. In cases Bl and

_B3,

the data

_comprise

unrelated sire families and

an

_improper

uniform

_prior

was

assigned

to the fixed effects and to the variance

component

in such a _way that the conditions

_required

to

_guarantee

that the

posterior

distribution is _proper

_(Natarajan

and

_McCullogh,

₁₉₉₅₎

are not met.

Although

the ECP is

_present

at a level of

27%

in

_Bl,

with a

_large

number of

families and a

_large

number of observations _per

_family

and per fixed

effect,

the estimate of the mean of the

_marginal

_posterior

distribution of

_heritability

over the 50 Monte Carlo

_replicates

shows

_good

_agreement

with the true value. This result is

_disturbing

since the

_sampler

retrieved a

_meaningful

estimate from a

_posterior

distribution which is not

guaranteed

to be

_proper!

This

_point

is discussed at

_length

in Casella

_(1996).

In cases B2 and B3 there is a limited amount of information per random effect and _per fixed effect. The ECP is not

_present.

The difference between these two cases

is that under

_B2,

GSR was

_implemented

with _proper

_priors

for all the

_parameters,

which leads to a _proper

_{joint posterior}

distribution.

_However,

the estimates

_using

GSR are biased in both cases and the size of the bias is similar. Thus the _presence

of bias does not seem to be

_necessarily

related to the fact that the

_posterior

distribution is

_improper.

In

_B4,

the

_analysis

was carried out

assigning

proper

priors

to all the

_parameters

and there are no ECPs

(the

_marginal

_posterior

distribution of

_heritability

is

guaranteed

to be

_proper).

In common with cases B2 and

_B3,

there are

_relatively

few observations _per fixed and random effect. Relative to cases B2 and

_B3,

case

of the

heritability

is almost three times smaller than in cases B2 and B3.

_Here,

the

_marginal

_posterior

distribution of

_heritability

is

_{highly peaked}

and

_symmetric,

but is shifted to the

_right

relative to the true value of 0.5. One of the 50

replicates

was

arbitrarily

chosen and this distribution is shown in

figure

1.

Despite

the

large

number of sire

families,

the absence of ECPs and the fact that the

marginal

posterior

distribution is proper, the bias

using

GSR

persists.

Table III shows

_comparisons

of GSR and MJP-IBC. Method GSR was

imple-mented

_assigning

_normally

distributed

_priors

for the fixed effects and the proper uniform distribution in the interval

_{!0, 1/3!}

for the

_prior

sire variance.

Cases Cl and C2 have the same data structure as B2 in table

II,

where there are

relatively

few observations per ’fixed’ effect but no ECPs.

_Using

GSR and

_assigning

a normal

_prior

to the ’fixed’ effects is effective in

_removing

the bias. Data structures C3 and C4 are identical to A4 and A5. These are characterised

_by

few observations

per ’fixed’

effect,

but with ECP.

Again,

assigning

a normal

prior

to the ’fixed’ effects

produces

estimates of

heritability

in better

_agreement

with the true value. In all these four cases, MJP-IBC

yields

estimates of

heritability

in

good

agreement

with the true value. The mean _square errors of estimates based on GSR and on MJP-IBC

are of similar

magnitude.

In data structure

_C5,

the

_frequency

of ECP is

80%

and the incidence is

90%.

The

_strategy

of

_using

GSR

_treating

fixed effects as random

_yields

estimates that are

DISCUSSION

The purpose of this work has been to draw attention to the poor

frequentist

properties

of the estimator of the mean of the

_{marginal posterior}

distribution of

variance

_components

in mixed threshold models when there is little information associated with the fixed effects. The simulation results indicate that this bias is not

_necessarily

related to the

_impropriety

of

_posterior

distributions.

_Indeed,

we have

shown that biased inferences result in cases when the

_posterior

is

guaranteed

to be

proper, and

agreement

with the true value was obtained when the

_propriety

of the

posterior

distributions cannot be

_guaranteed.

_Further,

this is seen as a

_problem,

because the bias

_persists

in situations when this

_marginal

_posterior

distribution is

highly peaked,

symmetric

and

_guaranteed

to be _proper. In other

_words,

this is not the bias to be

expected

from a non-linear estimator in a small

_{sample setting.}

The

_type

of data structures that can lead to this

_problem

is

_ubiquitous

in animal

breeding.

This is characterised

_by

models with a

_large

number of fixed

_effects,

with

relatively

few observations _per fixed effect. As we have

shown,

the biased inferences

associated with the variance

_component

are not

_necessarily

related to the extreme

category

problem, although

this can be conducive to then.

The

_inability

to obtain a closed form solution to the

marginal

_posterior

distri-bution of the variance

_component

has

_precluded

us from

_{fully understanding}

the

cause of the bias. The lack of

_{asymptotic independence}

between the estimator of the

fixed effect and of the variance

_component,

either in the context of the

_generalized

estimating equation

studied

_by

Zeger

et al

₍₁₉₈₈₎

or in the context of the likelihood of error contrasts

_reported

_by

Drum and

_McCullagh

_(1993),

does not

_necessarily

cast

_light

on the

_present

_problem.

In a

_{Bayesian setting,}

one is

_{marginalising}

with

respect

to the fixed

effects,

and

provided

that one does not stumble into

problems

of

_impropriety

of

_{distributions,}

it is not clear to us how assertions that are valid

A

_strategy

that can be

_interesting

to

_explore

is to run the Gibbs

sampler

after

_{having integrated}

the fixed effects out

_analytically

from the

_{joint posterior}

distribution. The relevance of this is based on the observation that

_running

the

sampler

conditioning

on the true values of the fixed

_{effects, yields}

correct inferences

about

_heritability

_{(Moreno, unpublished).}

Using

GSR but

_treating

fixed effects as

_random,

removes much of the bias in

inference about variance

_components,

as shown

_by

Hoeschele and Tier

_(1995).

However,

we have illustrated that this alternative does not

always

work. In such

cases, it is useful for the researcher to have a choice of other methods to solve the

problem.

The iterative

_bootstrap

bias correction is an

alternative,

computationally

simple method,

that shows

_good

_frequentist

behaviour. In this

_work,

we chose to

use it in

_conjunction

with the mode of the

_{joint posterior distribution,}

due to its

computational

simplicity.

We

_stress,

_however,

that the iterative bias correction can

in

_principle

be

_applied

_{to any}

_method,

but if it is

_used,

for

_{example, together}

with

the Gibbs

_sampler,

_computing

time can become a

limiting

factor. An

important

point

to stress is that the

_price

_{to pay} for unbiasedness is that estimates can fall

outside the

parameter

space. This will never be the case

_using

GSR

_treating

fixed

effects as

random,

or

_{using approximations}

to the

marginal posterior

distribution

of the variance

_component.

It would

_certainly

be

_interesting

to extend the

_study

of

_Tempelman

₍₁₉₉₄₎

and to

_explore

the behaviour of alternative

_{computational}

forms of the

_{Laplacian approximation}

further.

The

present

simulation

_study

was based on the sire model where each sire

effect is

_{represented by}

several observations. A more relevant model in an animal

breeding

context is the individual animal

_model,

where each additive

_genetic

value is associated at the most with one observation. In the case of the animal

model,

the bias

_problems

discussed in this _paper are

_accentuated,

as illustrated in the

simulation

_study

of Hoeschele and Tier

_(1995).

The

_study

of

_Tempelman

₍₁₉₉₄₎

also showed poor behaviour of the

Laplacian approximation

in the case of the

animal model. It is

_precisely

in such a situation that methods which control this

bias such as the MJP-IBC

implemented

here _may be useful.

ACKNOWLEDGEMENTS

The authors are _very

_grateful

to J

Ocana,

who

_suggested

the

bootstrap

to

_study

the

problem

of bias on inferences about variance components in threshold models. We

gratefully acknowledge

the support of the

Diputaci6n

General de

_Arag6n

_{(Grant 2-2-94)}

and of

Caja

de Ahorros de la Inmaculada

_(Grant

CA

_396).

Three anonymous reviewers made useful

_suggestions.

REFERENCES

Albert

_JH,

Chib S

₍₁₉₉₃₎

_{Bayesian analysis}

of

_binary

and

_{polychotomous}

_response data. J Am Stat Assoc

_88,

669-679

Anderson

_DA,

Aitkin M

₍₁₉₈₅₎

Variance _component models with

_binary

_response: inter-viewer

_variability.

J R Stat Soc B

_47,

203-210

Bliss CI

₍₁₉₃₅₎

The calculation of the

dosage-mortality

curve. Ann

_A

_P

_pd

Biol 22, 134-167 Breslow

NE,

Lin X

₍₁₉₉₅₎

Bias correction in

_generalised

linear mixed models with a

single

Casella G

₍₁₉₉₆₎

Statistical inference and Monte-Carlo

algorithms.

Test _5, 1-30 Cox

_DR,

Snell EJ

₍₁₉₈₉₎

_{Analysis of Binary}

Data.

_Chapman

and

_Hall,

London

Dellaportas P,

Smith AFM

₍₁₉₉₃₎

Bayesian

inference for

_generalized

linear and propor-tional hazards models via Gibbs

_{sampling. Appl}

Stat

42,

443-459

Drum

_{ML, McCullagh}

P

₍₁₉₉₃₎

REML estimation with exact covariance in the

_logistic

mixed model. Biometrics

49,

677-689

Engel B,

Buist

_W,

Visscher A

₍₁₉₉₅₎

Inference for threshold models with variance

components from the

_generalized

linear mixed model

_perspective.

Genet Sel Evol

_27,

15-32

Foulley JL,

Im

_S,

Gianola

_D,

Hoeschele I

₍₁₉₈₇₎

_{Empirical Bayes}

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,

New

York,

361-409

Foulley JL,

Manfredi E

₍₁₉₉₁₎

_{Approches statistiques}

de 1’evaluation

_g6n6tique

des

reproducteurs

pour des caract6res binaires a seuils. Genet Sel Evol 23, 309-338 Gelman

_A,

Rubin DB

₍₁₉₉₂₎

A

_single

series from the Gibbs

_{sampler provides}

a false

sense of

_security.

In:

_Bayesian

Statistics IV

_(JM

_Bernardo,

JO

_Berger,

AP

_Dawid,

AFM

Smith,

eds),

Oxford

_{University Press, Oxford,}

625-631

Gelman

A,

Carlin

JB,

Stern

HS,

Rubin DB

₍₁₉₉₅₎

Bayesian

Data

_{Analysis. Chapman}

and

Hall, London,

1st ed

Gilks

_WR,

Wild P

₍₁₉₉₂₎

_{Adaptive rejection sampling}

for Gibbs

_{sampling. Appl}

Stat 41,

337-348

Gilks

_WR,

Richardson

_{S, Spiegelhalter}

DJ

₍₁₉₉₆₎

Markow Chain Monte-Carlo in Practice.

Chapman

and

_{Hall, London,}

1st ed

Henderson CR

₍₁₉₇₃₎

Sire evaluation and

_genetic

trends. In:

_{Proceedings of}

the Animal

Breeding

and Genetics

_Symposium

in Honor

of

Dr DL

_{Lush, Blacksburg, Virginia,}

10-41

Hoeschele I,

Foulley

JL, Gianola D

₍₁₉₈₇₎

Estimation of variance _components with

quasi-continuous data

using Bayesian

methods. J Anim Breed Genet

104,

334-349

Hoeschele

_I,

Tier B

₍₁₉₉₅₎

Estimation of variance components of threshold characters

_by

marginal posterior

modes and means via Gibbs

sampling.

Genet Sel Evol

_27,

519-540

Kuk AYC

₍₁₉₉₅₎

_{Asymptotically}

unbiased estimation in

_generalized

linear models with random effects. J R Stat Soc B

_57,

395-407

Lindley DV,

Smith AFM

₍₁₉₇₂₎

Bayes

estimates for the linear model. J

_Roy

Stat Soc

B 34, 1-41

McCullagh P,

Nelder JA

₍₁₉₈₉₎

Generalized Linear Models.

_Chapman

and

_{Hall, London,}

2nd ed

McCulloch CE

₍₁₉₉₄₎

Maximun likelihood variance _components estimation for

binary

data. J Am Stat Assoc 89, 330-335

Misztal

_I,

Gianola

_{D, Foulley}

JL

₍₁₉₈₉₎

_Computing

_aspects of a nonlinear method of sire

evaluation for

categorical

data. J

_Dairy

Sci 72, 1557-1568

Moreno C

₍₁₉₉₃₎

Estudio de los _componentes

_gen6ticos

del modelo umbral. Tesis doctoral. Universidad de

_{Zaragoza, Spain}

Natarajan

R,

McCullogh

CE

₍₁₉₉₅₎

A note on the existence of the

_posterior

distribution for a class of mixed models for binomial responses. Bio!rrzetrika 82, 639-643

Sorensen D, Andersen

_S,

Gianola

_{D, Korsgaard}

I

₍₁₉₉₅₎

_Bayesian

inference in threshold models

using

Gibbs

_sampling.

Genet Sel

Evol 27,

229-249

Tempelman

RJ

₍₁₉₉₄₎

Evaluation of derivative-free

_marginal

maximum likelihood esti-mation of variance _components in threshold mixed models. In: Proc

of

the 5th World

Cong

Genet

_Appl

to Livestock

_{Prod, University}

of

_{Guelph, Guelph,}

22, 15-18

Wright

S

₍₁₉₃₄₎

An

_analysis

of

variability

in number of

_digits

in an inbred strain of

_guinea

pigs.

Genetics

19,

506-536

Zeger SL, Liang K,

Albert PS

₍₁₉₈₈₎

Models for

_longitudinal

data: a

_{generalized estimating}

equation approach.

Biometrics

_44,

1049-1060

Zeger SL,

Karim MR

₍₁₉₉₁₎

Generalized linear models with random effects: a Gibbs

sampling approach.

J Am Stat Assoc 86, 79-86

APPENDIX

This

_appendix

shows the

_{log-conditional}

_posterior

distributional forms for the fixed and random effects and the variance

_component

of the

_binary

threshold model. The

adaptive

rejection sampling

algorithm

was

_{implemented using}

these distributional

forms.

The

_log-fully

conditional

posterior

distributional form for the xth fixed effect is:

where ci

= ! _

1

if i (j) = x

and 0

_represents

the rest of

parameters.

0

0

_{if i(i) 7! x}

Log-conditional

distributional form for the xth random effect is:

Log-conditional

distributional form for the variance

_component

is: