Estimation of variance components of threshold characters by marginal posterior modes and means via Gibbs sampling

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Estimation of variance components of threshold

characters by marginal posterior modes and means via

Gibbs sampling

I Hoeschele, B Tier

To cite this version:

Original

article

Estimation of variance

_components

of threshold

characters

_by

_marginal

posterior

modes and

means

via

Gibbs

_sampling

I

Hoeschele

B Tier

Virginia Polytechnic

Institute and State

_University,

Department of Dairy Science, Blacksburg,

VA

_24061-0315,

USA

(Received

19 October 1994;

accepted

24

_August

₁₉₉₅₎

Summary - A Gibbs

_sampling

scheme for

_{Bayesian analysis}

of

_binary

threshold data

was derived. A simulation

_study

was conducted to evaluate the _accuracy of 3 variance

component estimators, deterministic

approximate marginal

maximum likelihood

_(AMML),

Monte-Carlo

_{marginal posterior}

mode

_(MCMML),

and Monte-Carlo

_{marginal posterior}

mean

(MCMPM).

Several

_designs

with different numbers of

_genetic

_groups,

herd-year-seasons

(HYS),

sires and progeny per sire were simulated. HYS were

_generated

as

_fixed,

normally

distributed or drawn from a _proper uniform distribution. The downward bias

of the AMML estimator for small

_family

sizes

₍₅₀

sires,

average of 40

progeny)

was

eliminated with the MCMML estimator. For

_designs

with many

HYS,

0.9

incidence,

50 sires and 40 progeny on _average, the

marginal posterior

distribution of

_heritability

was

non-normal;

MCMML and MCMPM

significantly

overestimated

heritability

under

the sire

mode,

while under the animal model the Gibbs

sampler

did not converge. For

designs

with 100 sires and 200 progeny per

sire,

the

_{marginal posterior}

distribution of

heritability

became more normal and the

_discrepancy

_among MCMML and MCMPM

estimates vanished.

_Heritability

estimates under the animal model were less accurate than those under the sire model. For the smaller

_designs,

the MCMML estimates were _very

close to the true value when

using

a normal

prior

for HYS

effects, irrespective

of the true state of nature of the HYS effects. For extreme

_incidence,

small data sets and _many

_HYS,

observations within an HYS will

_frequently

fall into the same _category of _response. With flat

_priors

for the HYS

_effects,

the

posterior density

is

_{likely improper, supported by}

an

analytical proof

for a

_simplified

model and

_analyses

from Gibbs _output. In

_analyses

of

limited

_binary

data with extreme

_incidence,

effects of a factor with _many levels should be

given

a normal

prior.

Assigning

a _proper uniform

_prior

or

_fixing

values of such levels was

*

On leave from Animal Genetics and

_{Breeding Unit, University}

of New

_England,

not useful. Most accurate estimation of

_genetic

_parameters

_requires

very

large

data sets.

Further work is needed on

diagnosis

of

_improperness

and on alternative _proper

_priors.

Bayesian estimation

_/

Gibbs sampling

/

categorical data

_/

marginal maximum likelihood

_/

variance component estimation

Résumé - Estimation des composantes de variance de caractères à seuil par les modes et les moyennes marginales a _posteriori à l’aide de _{l’échantillonnage} de Gibbs.

Un

plan d’échantillonnage

de Gibbs _pour

_{l’analyse bayésienne}

de caractères binaires à seuil a été établi. Par

_simulation,

on a _{pu comparer} la

_précision

de 3 estimateurs

de _composante de variance, un maximum de vraisemblance

_marginale

déterministe et

approximatif

(MVMA),

le mode

_marginal

a

_posteriori

de Monte Carlo

(MVMMC)

et la

moyenne

marginale

a

_posteriori

de Monte Carlo

(MMPMC).

Plusieurs

_{plans d’expérience}

avec des nombres

différents

de _groupes

_{génétiques,}

de cellules

troupeau-année-saison

(TAS),

de

pères

et de descendants par

père

ont été simulés. Les TAS ont été établis

comme des

effets fixes,

ou tirés d’une distribution

normale,

ou tirés d’une distribution

uniforme.

L’erreur par

défaut

du MVMA pour de petites tailles de

_famille

₍₅₀

_pères,

40

descendants par

père)

a été éliminée _par le MVMMC. Pour des

dispositifs

incluant de

nombreux

_TAS,

avec une incidence de

_0,9,

50

_{pères et}

₄

₀

descendants _par

_père

en _moyenne,

la distribution

marginale

a

_posteriori

de l’héritabilité n’est pas

normale ;

MVMMC et

MMPMC surestiment

_{significativement}

l’héritabilité dans un modèle

_paternel,

alors

_qu’avec

le modèle individuel la

_procédure

de Gibbs ne _{converge pas.} Avec 100

pères

et 200 descendants par

père,

la distribution

marginale

a

_posteriori

de l’héritabilité se

rapproche

de la normale et les discordances entre les estimées MVMMC et MMPMC

disparaissent.

Les estimées d’héritabilités avec le modèle individuel sont moins

_{précises qu’avec}

le modèle

paternel.

Pour les _petits

_dispositifs,

les estimées MVMMC sont très

_proches

de leur vraie

valeur

_quand

la distribution a

_priori

des

effets

TAS est

_{normale, quelle}

que soit la réalité

des TAS. Pour des incidences

_extrêmes,

de petits échantillons et un

_grand

nombre de

TAS,

les observations à l’intérieur d’une cellule TAS tombent souvent dans la même

_catégorie

de

_réponse.

Avec des a _priori

uniformes

_pour les

effets TAS,

la densité a _posteriori est

probablement

impropre, comme tend à

l’indiquer l’analyse

des résultats d’une

procédure

Gibbs

_appliquée

à un modèle

_simplifié.

Dans

_l’analyse

de données binaires en nombre limité

et avec une incidence _extrême, une distribution normale a

_priori

devrait être

_assignée

aux

effets

des

facteurs

ayant de nombreux niveaux,

plutôt qu’une

distribution

_uniforme

ou

des valeurs

_fixées.

Des estimations

_précises

des

_{paramètres génétiques requièrent}

dans ce cas de très

_grands

ensembles de données. Il reste encore à étudier la manière de déceler

l’impropriété

des distributions a

_priori

et de choisir de meilleurs a

_priori.

estimation _{bayésienne / échantillonnage} de Gibbs

_/

données catégorielles

/

maximum de vraisemblance _{marginale /} composante de variance

INTRODUCTION

Bayesian

analysis

of

_binary

or

_{polychotomous}

threshold traits via Gibbs

_sampling

has

_recently

been described

_by

Albert and Chib

_(1993),

Sorensen et al

₍₁₉₉₅₎

and Jensen

_(1994).

In all 3 _papers, the Gibbs

_sampler

was

_implemented

in

combina-tion with data

_{augmentation,}

ie

_parameters

and

_missing

data were

sampled

from

effects and variance

_components,

while the

_missing

data consisted of the liabilities

or latent continuous variables in the threshold model. In contrast with the afore-mentioned _papers,

_Zeger

and Karim

₍₁₉₉₁₎

_{implemented Bayesian analysis}

with a

Gibbs

_sampler

based on the

_posterior

_density

of the

_parameters

rather than the

joint posterior

of

_parameters

and

_missing

data.

_Zeger

and Karim

₍₁₉₉₁₎

also used a different

_prior

for the

_dispersion

_parameters.

McCulloch

₍₁₉₉₄₎

derived maximum

likelihood rather than

_Bayesian

estimation via an

_{expectation-maximization}

_(EM)

algorithm

with Gibbs

_sampling

of the liabilities within each

_{E step.}

In this

_{contribution,}

a Gibbs

sampling

scheme

_applied

to

_parameters

and

liabil-ities was

implemented

for

_{Bayesian analysis}

of a

_binary

trait. A simulation

_study

was conducted to evaluate the _accuracy of 3 estimators of variance

_components,

deterministic

_approximate

_marginal

maximum likelihood

_(AMML)

_(Foulley

et

_al,

1987;

Hoeschele et

_al,

_1987),

the Monte-Carlo evaluated

_marginal

_posterior

mode

(MCMML),

and the Monte-Carlo evaluated

_marginal

_posterior

mean

(MCMPM).

The

_analysis

was restricted to a threshold model with one variance

_component

and was

_performed

under sire and animal models. In

_MML,

the

_marginal

poste-rior

_density

of the variance parameters is maximized with

_respect

to these

parame-ters. ’Fixed’ effects and variance parameters have

_improper,

flat

_prior

distributions.

Thus,

the

_{marginal posterior}

_density

of the variance

_components

is

proportional

to

their

_marginal

likelihood. In

_Bayesian

_analysis

_implemented

via Gibbs

_sampling

applied

to all

_parameters

_(or

all

_parameters

and

_missing

_data),

the

parameter

sam-ples provide

inferences about any

parameter

from its

_{marginal posterior}

_density.

For a

single

variance

_component,

the

_marginal

_posterior

mode is

_equivalent

to the MML estimator.

_Therefore,

the Monte-Carlo evaluated

_marginal

_posterior

mode

was termed the MCMML estimator.

With conventional deterministic

_algorithms,

MML estimates are

_{computed using}

a normal

_{approximation,}

and severe biases of variance and covariance estimates have been

_reported

_(Gilmour

et

_al,

_1985;

Hoeschele et

_{al, 1987;}

Hoeschele and

Gianola, 1989;

Simianer and

_Schaeffer,

_1989).

While Gilmour et al

₍₁₉₈₅₎

observed an underestimation

_{of heritability}

with small

_family

_sizes,

Hoeschele et al

₍₁₉₈₇₎

and Hoeschele and Gianola

₍₁₉₈₉₎

found an overstimation of

_heritability

for a

_binary

trait with extreme incidence and in the _presence of a fixed factor with _many levels. The

_objective

of this

_study

was to _compare the 3 variance

_component

estimators

AMML, MCMML,

and MCMPM in terms of their

_{frequentist properties, and,}

in

particular,

to

_investigate

whether the biases observed for the AMML estimator can

be eliminated

_by

_computing

exact MML estimates via MCMC

_algorithms.

A related side

objective

was to

_{investigate potential}

causes of the biases.

MATERIALS AND METHODS

Methodology

The

_{Bayesian analysis}

described below is identical to that of Sorensen et al

_(1995),

when

_applied

to a

_binary

_trait,

but differs from Albert and Chib

₍₁₉₉₃₎

in the

sampling

of the variance

_components.

The Gibbs

_sampling

scheme differs from the

sampler

of Sorensen et al

₍₁₉₉₅₎

_only

when run under an animal model where

Let _y

_represent

the observed dichotomous variable and w the

_liability

variable. In the threshold model for 2

_categories

of _{response, yi} = ₁ if

w

i > 0.0 and

y

2 = 0 otherwise.

Conditionally

on the fixed

_(!3)

and random effects

_(u),

the

w

i are

independent

N(x!fJ

+

_{z!u, 1),}

and the _yi are

independent

Bernouilli with

Prob(y

i

=

1)

=

4

i(x

i

#

₊

ziu),

where

4

i(.)

denotes the standard normal cumulative

distribution function. Matrices X and Z are the usual incidence matrices with row

i denoted

_by

_x!

_(zD.

The

_parameter

vector

₍₀₎

includes

_#,

u =

[u! ..., u!,...,

u9!’,

and the

_af

_j

_{(I, j}

=

1, ... ,

q),

with

Cov(u

i

,

u) )

=

A

ij

a

ij

and the A matrices known.

The

_{joint posterior}

_density

of e and w =

{w

i

}

is

where c is a

_constant,

₀₍

_p,;

a

2

)

is the

_density

function of

_N(!.;

_a

₂

_),

and

_I(

_XE

_S)

is the indicator function

equal

to 1 if variable x is contained in set S and zero

otherwise. For

_independent

_u

_j

_s’,

the

_prior

densities of the _uj and the _<7,!

_simplify

to

_products

of the

_f(u

_j

_la

_J

_),

the

_density

of the MVN

_(0;

_{A!Q! ),}

and

_products

of the

f (a?) 3

or

_prior

densities of the

aJ.

Samples

from the

_{joint posterior}

distribution can

be obtained

_{by sampling}

in turn from

_{f(Blw, y) f}

_{(0 1 w)}

and

_{f (w 10, y).}

These 2

conditional distributions are of standard

_forms,

and the

_fully

conditional

_parameter

densities,

derived from

_/(!w),

are identical to those in the standard mixed linear

model

_(eg,

Gelfand et

_{al, 1990;}

_Wang

et

_al,

_1993).

_Then,

from standard mixed linear model results

_(eg,

Gianola et

_al,

_1990),

and with q =

_1,

Uj = _u and

a

§

=

Q!,

where

(!Z

is element i of vector

_f

_{l, (}

_i

_3-

is this vector with element i

_omitted,

xi is

column i of matrix _x, x-i is x with column

_{i deleted, a}

ii

is element

(i, i)

of the inverse additive

_{genetic relationship}

matrix

_A-

₁

_,

and

_Ai

_is

row i of

A-

1

without element i.

Under the animal

model,

breeding

values

_(u)

of a sire and his

_offspring

were

sampled jointly by sampling

a sire’s u from its

_marginal

normal distribution while

sampling

the u of each

_offspring

from its full conditional distribution in

_(3!.

Mean and variance of the

_marginal

distribution were obtained as the BLUP of the sire’s

u and its

_prediction

error variance after

_absorbing

the

_offspring

u into the sire’s u

in MME for this sire and his

_offspring.

For _q = 1 and

f(u u 2)

=

_constant,

where the inverse

_chi-square

distribution has n - 2

_df

with n

_equal

to the number

reported problems

in

_estimating

variance

_components

due to the Gibbs

_sampler

&dquo;being

trapped

at zero&dquo;. Their

_prior

resulted from an inverse

chi-square

(or

inverse

Wishart)

distribution with zero

_{prior degrees}

of

_freedom,

_yielding

the

_{prior density}

f(afl) =

(afl)!!

and

_changing

the

_df

in

_[4]

to n. This

problem

vanishes when

using

the flat

_prior,

as also observed

_by

other researchers

(D

Sorensen,

personal

communication).

Note also that with an

_{improper prior}

distribution the

_resulting

posterior

distribution can be _proper or

improper

(eg,

Berger

and

Bernardo, 1992a,

b;

Hobert and

_Casella,

_1994).

Hober and Casella

₍₁₉₉₄₎

established necessary

and sufficient conditions for the

_{joint posterior}

_density

of the fixed and random

effects and the variance

_components

in a

hierarchical

Bayes

LMM to be _proper,

ie

_integrable.

One condition was that for _any variance

_component

_j,

besides the

residual,

with

_prior

_{f (!! )}

=

(!! )-!a!+1>,

we must _{have aj} < 0.

_Setting

_{aj =} -1

yields

the flat

_prior

_{f ( ?)}

= 1 used in this _paper, while _aj = 0

produces

the

prior

used

_by

_Zeger

and Karim

_(1991).

More

_{specifically,}

in this

_study,

a bounded flat

prior

for

_afl

_{was used under} a sire model as in Sorensen et al

(1995),

while an

improper

flat

_prior

was used under the animal model.

The

_marginal

_posterior

mode of

_heritability

was

computed by

a

grid

search of the Rao-Blackwell estimate

_(eg,

Gelfand et

_al,

₁₉₉₀₎

of the

_marginal

_{posterior density}

of

_a

₂

_and

a

_change

of variable to

_h

₂

_,

or

where 6 = 4 or 6 = 1 for the sire or animal

_{model, respectively,}

k denotes the Gibbs

sample,

and J is the Jacobian of the transformation

_{afl -}

h

2

,

Conditional on the

_parameters,

the latent data were

sampled

from truncated normal

_{distributions,}

or

with

To

_implement

the Gibbs

_{sampler, starting}

values for the

_parameters

were

obtained

_by

_computing

the maximum a

_priori

(MAP)

estimates of

/

3

and u

_(eg,

Gianola and

_Foulley,

₁₉₈₃₎

evaluated at the

_approximate

MML estimates

_(Foulley

et

al,

1987)

of the

_0’!.

Given initial

_parameter

_values,

a first

_sample

of the latent

data was drawn from

!5!,

followed

_by

the

_sampling

of new

_parameter

values from

!2!, [3]

and

_!4!,

and further Gibbs

_cycles.

Because the

_prior

distributions for the fixed effects and for the variance

compo-nent under the animal model are

_improper,

the

_posterior

distribution

might

also

posterior

distribution which

_only

hold for hierarchical linear mixed models.

Fur-thermore,

a reviewer

_pointed

out that for a

simple

fixed model with one factor and the

_logit

link

_function,

the

_{joint posterior}

distribution of the fixed effects is

im-proper, if in at least one factor level all observations fall into the same

_category

of

response. An

analytical investigation

of whether

f(0)y)

is proper for the nonlinear

mixed model considered

here, however,

is _very difficult or intractable.

_Therefore,

it is desirable to be able to detect an

_{improper posterior}

from Gibbs

_output.

A

candidate

approach

is the MC evaluation of the

_marginal

likelihood

_{f (y).}

With B

representing

the

_{parameter vector,}

the

_marginal

likelihood is

The

reciprocal

of

_{f (y)}

is a

_normalizing

constant

_ensuring

that the

_{posterior density}

f (9!y)

integrates

to one, if _{it is proper.} If it is _not,

_[7]

is infinite.

While the

_computation

of

_Bayesian

_point

estimates and related inferences

_(eg,

marginal posterior density plots,

highest

posterior

density

regions)

from Gibbs

output

is

_{straightforward,}

_computation

of

_marginal

likelihoods or

_Bayes

factors has

_proved

to be _very

_challenging

_(eg,

_{Chib, 1994;}

Newton and

_Raftery,

_1994).

Because all conditional distributions

_{(ie !2!, [3]}

and

_!4!)

are

_standard,

the

_approach

of Chib

₍₁₉₉₄₎

for

_marginal

likelihood estimation from Gibbs

_output

was

adopted

here. The

_marginal

likelihood can be written as

and

is estimated

_by

_evaluating

_[8]

at a

_{particular point,}

_eg, the

_posterior

mean

8 =

.E(<!y).

With data

_{augmentation,}

the denominator of

[8]

_can be written _as

where w is the vector of

_missing

data. The Monte-Carlo estimate of

_[9]

is

where N is Gibbs

_sample

size and the

w

i

are

_samples

from

f (B, w!y)

and hence are

also

_samples

from

_f(wly).

To evaluate the densities in the

_right-hand

side of

_!10!,

let B =

_[0’,

_u’,

_u

Then,

one _{way of}

_factoring

the

_{joint posterior density}

evaluated at the vector of

_posterior

means of the

_parameters

is

Note that with the parameter vector

_partitioned

into 3

_subvectors,

_[12a,

_b,

_c]

represents

one of 6

_possible

factorizations of the

_posterior

_density.

Furthermore

note that

_density

_[12a]

can be evaluated

_immediately

at the termination of the Gibbs

_chain,

while the evaluation of

_[12b]

_{requires output}

from a second Gibbs chain with

_{afl fixed}

at its

_posterior

mean estimate obtained from the first

_chain,

and evaluation of

_[12c]

is from a third Gibbs chain with u and

_Q

_u

_fixed

at their

posterior

mean estimates from the first Gibbs chain. Simulated data and

_analyses

Data were simulated on a

_binary

threshold trait.

_Eight

different

_designs

were

investigated,

which

_firstly

differed in the numbers of

_genetic

groups,

herd-year-season

(HYS)

_effects,

_sires,

and _{progeny per sire.}

_Secondly,

_designs

different in the

way HYS effects were simulated: as fixed

(generated

once from a normal distribution and held constant in all

_replicates),

as random and

_normally

_distributed,

or as

random and drawn from a _proper uniform distribution. Genetic _groups were

_always

fixed and sires or animals were random. The

_designs

are defined in table I.

Genetic _group means for

_liability

were

_{-0.4, -0.15, 0.15,}

and 0.4. Sire or animal effects on

liability

were

_generated

from

N(0,

0D

.

for a

_heritability

of

h

2

= 0.25

(or2= h 21(6 -h 2)

for HYS fixed and

₇₂

=

h

2

(1

+

_{(}!Ys)/(8 - h}

₂

₎

for HYS

_random,

with 6 = 4

(6

=

1)

under the _sire

generated

from

_{N(0, 1).}

HYS effects were

generated

from

N(0,

0’ Hys ! 2

0.46)

or from

U [a, b],

where a =

&mdash;0.5(12cr!Ys)!! !

-b _{was set} such that HYS effects had the

same variance under both distributions. The truncation

_point

used to dichotomize the liabilities was

!-1

(p)*(1 +a!Ys+a;)O.5,

where _p was the desired incidence

_equal

to 0.9 for all

_designs

_except

VI-F with _p = 0.5. With 135

HYS,

the

probabilities

of

having

0,

1,

or 2

_offspring

in any HYS were

_{0.8, 0.1,}

and

_0.1,

_{respectively,}

for each

sire;

with 32

_HYS,

sires had

_0,

6 or 7

_offspring

_per HYS with the same

_{probabilities;}

and with 320

HYS,

sires had

_0,

2 or 4 _{progeny per} HYS with

_{probabilities 0.8,}

0.09,

and

_0.11,

_{yielding approximately}

the average progeny group sizes

given

in table I. The numbers of sires in the 4

_genetic

_groups were

_12,

_14,

_13,

and 11.

_Designs

with

average progeny group size of 40

(200)

were

replicated

40

(20)

times.

All data sets were

_analyzed

with the sire

_model,

and data sets

_II-F,

V-F and VI-F were also

_analyzed

with the animal model. Note that for these

_designs

and if a linear mixed model were

used,

sire and

animal,

models would be

(linearly)

equivalent

(Henderson, 1985),

ie

_yield

the same estimates of fixed and sire effects.

HYS effects were treated as fixed

( f )

in the

_analysis,

ie had

_improper

uniform

_priors,

for all

_designs

where HYS effects were fixed

(F).

Additionally,

HYS were treated as random with normal

_prior

(n)

in the

_analyses

of data sets for

_designs

_II-F,

II-N

and II-U where HYS were

_fixed,

random and

_normally

_distributed,

and random and

drawn from the proper uniform

distribution,

respectively. Treating

HYS as random with normal

_{prior required}

to also estimate HYS variance

_a

_H

_y

_s

_.

For some of the

_designs,

data sets contained HYS levels with the so-called

extreme

_category

_problem

_(ECP)

_(Misztal

and

_Gianola,

_1989).

Extreme

_categories

are the first and last

_category,

hence for a

_binary

trait both

_categories

are extreme.

In an HYS

exhibiting

the

_ECP,

all records are in the same extreme

_category.

On

average,

frequency

of HYS with the ECP was

45%

for

designs

I-F and

II-F/N/U,

22%

for

_design

_III-F,

17%

for

_design

IV-F,

and

0%

for all other

_designs.

Solutions

for such HYS classes do not converge but tend toward

_infinity

in deterministic AMML

_algorithms.

Therefore,

in AMML solutions for these HYS were fixed at

I 10

_(Misztal

and

_Gianola,

_1989).

For the Gibbs

_{sampler, options}

were considered for the treatment of HYS in the _presence of the ECP:

_(i)

to use a _proper normal

_prior

in the

analysis;

(ii)

to use a _proper uniform distribution as

prior,

_eg,

U(&mdash;10., 10.);

and

_(iii)

to fix HYS effects with the ECP at ± 10 or at ± 3 in all Gibbs

cycles,

denoted

_by

f10 and

_{f3, respectively.}

RESULTS AND DISCUSSION

Estimates of

_heritability

_(h

₂

₎

were obtained under the sire model for all

designs

and under the animal model for

_designs

_{II-F, V-F,}

and VI-F. For

_designs

II-F and

_IV-F,

HYS variance was also estimated when HYS effects had a

_prior

normal distribution

in the

_analysis.

Estimators were

deterministic, AMML,

MCMML and MCMPM. MC estimates were

_computed

from 20 000 consecutive Gibbs

_samples

for the

sire model and 200 000

samples

for the animal

model,

with a burn-in

period

of an additional 2000

cycles.

The burn-in

period

was determined based on

plots

of

sample

values for

_heritability

versus

_cycle

number

_{showing consistently}

that 2 000

cycles

were more than needed.

_{Typical plots}

for

_{design-analysis}

combinations II-F-f

due to the use of

_{good starting}

values for both location and variance

_parameters,

which were the estimates obtained from the AMML

_procedure.

The number of Gibbs

cycles

of 20 000

_{(200 000)}

was determined as sufficient

based on the autocovariance structure of the

_sample

of heritabilities. This entailed

calculating

an effective

sample

size as the ratio of variance of

samples

to the variance of the

sample

mean

computed

from the estimated autocorrelations

(Sorensen

et

al,

1995).

Autocovariance for

_lag

t was estimated as

Variance of

_sample

mean

_given

the estimated autocovariances was estimated with

where m is the

_largest

_integer

satisfying

the condition

and effective

_sample

size ESS was

For the sire model with 20 000

_cycles,

the minimum ESS was 580 and was found in one of the

design

I-F data sets

analyzed

with HYS treated as fixed

( f ).

For a

_typical

replicate

with ESS =

1435,

autocorrelations at

lags

1, 10,

20 and 50 were

_0.66,

0.34,

0.13 and

_0.007,

_{respectively.}

For most data sets of this

design,

ESS exceeded 1000. For other

_designs

_(II-VI),

ESS was in the 2 000 to 6 000 range,

and,

for

example,

autocorrelations at the above

_lags

were

_0.39,

_{0.14, 0.03,}

and 0.006 with ESS = 3 705

for a

_design

IV-F data set with HYS treated as fixed. For the animal model and

Mean estimates and

_empirical

SE across 40

_replicates

are

_presented

in table II. The first column of table II

_specifies

the

design

(eg, I-F)

in combination with the

type

of

_analysis

_{(eg, I-F-f)}

_according

to the treatment of HYS effects.

_Progeny

group sizes

_{typically ranged}

from 25 to 55 for average progeny group size of 40 and from

120 to 280 for an _average of 200.

All estimators

_strongly

overestimated

heritability

for the

_{design-analysis}

com-binations I-F-f and II-F-f under the sire model. The

_upward

bias was

largest

for

MCMPM,

followed

_by

MCMML and AMML. The

_designs

were characterized

_by

true value than the mean. A

plot

of the

marginal posterior density

of

h

2

based on

[5]

for II-F-f can be found in

figure

3.

For

_{design-analysis}

combination

_III-F-f,

with the

_{design containing}

the same

number of sires and records as, but a smaller number of HYS levels

_than,

I-F

and

_II-F,

the

_upward

biases of the MCMPM and MCMML estimators decreased while the AMML estimator tended to underestimate

h

2

.

Overestimation

_(Hoeschele

et

_al,

_1987;

Hoeschele and

_Gianola,

_1989;

Simianer and

_Schaeffer,

₁₉₈₉₎

and underestimation

_(Gilmour

et

_al,

_1985;

_Thompson,

₁₉₉₀₎

with AMML are in

_good

agreement

with the literature. It _appears that the AMML estimates are closest to

the true value of

h

2

.

_However,

a

_comparison

between the results for I-F-f and II-F-f

versus III-F-f

_strongly

_suggests

that this

_finding

is due to a

_{partial counterbalancing}

of

_upward

and downward biases of the AMML estimator for data sets with a small number of records and progeny per sire.

For the

_{larger designs}

with 100 sires and average progeny group size of

200,

biases of and

_{discrepancies}

_among estimators were

_strongly

reduced

_{(design IV-F)}

and

negligible

for

_design

VI-F with the mean as the

only

fixed effect and an incidence

levels were increased

_by

a factor of 10 relative to III-F-f. As a

_result,

the

_AMML,

estimator no

longer

underestimated

h

2

,

while the overstimations obtained with and

the

_discrepancy

between the MCMPM and MCMML estimators

_strongly

decreased. A

_plot

of the

_{marginal posterior}

distribution of

h

2

for

_{design-analysis}

combination

IV-F-f is

_presented

in

_figure

_4,

which shows a more

_symmetric

distribution with

smaller variance relative to

_figure

3.

It _may be concluded that the

_{discrepancies}

found between the true

_h

₂

_,

the MCMPM and the MCMML estimates are due to a lack of information in the data

causing

the

_posterior

distribution to be non-normal and its mean to be a biased estimator of

h

2

.

Another

_explanation

_might

be that the

joint posterior

distribution of the

parameters

is

_improper.

As mentioned

earlier,

for the

logit

link and a

simple

fixed

model, improperness

can be shown

_analytically

when levels of the fixed factor

exhibit the ECP.

_{Consequently,}

use of a _proper

_prior

distribution for the fixed

effects, resulting

in a _proper

_posterior,

should eliminate this

problem. Therefore,

2 proper uniform distributions were

_employed

as

_priors,

U(&mdash;3, 3)

and

U(&mdash;10, 10)

of

sample

value versus Gibbs

cycle

for an HYS with the ECP. In this

analysis,

an

improper

uniform

_prior

was

used,

and

sample

values drifted toward

extremely

small numbers far below the lower limits of -3 and -10 in the proper uniform

priors.

Next,

values of HYS effects with the ECP

_(all

records

_equal

to

₀₎

were fixed at -3 or at -10 across all Gibbs

_cycles

while the other HYS effects were

_sampled

and had an

_{improper prior.}

Results from these

analyses

are in table II in the rows for

_{design-analysis}

combinations II-F-f3 and 11-F-flO.

Upward

biases were still

substantial and

_{only slightly}

smaller than those for II-F-f. The

_posterior

distribution for the

_parameters

sampled

should have been proper

and,

if so, the results indicate

that biases were

(mostly)

due to limited information in the data.

Because a _proper uniform

_prior

could not be used for technical reasons, a normal

prior

distribution was

_postulated

for the HYS. The

designs analyzed

with a normal

prior

were

_II-F,

_II-N,

and

II-U,

where the true state of nature of the HYS effects

was

_fixed,

_normally

distributed and drawn from a _proper

_{uniform, respectively.}

The results for all 3

designs

were _very similar. The AMML estimator

significantly

The

_empirical

SE of the MC estimators tended to be

_{slightly higher}

than the SE of the deterministic AMML estimator for the smaller

_{designs. Doubling}

the number of Gibbs

_cycles

had

_virtually

no

_{effect, suggesting}

that the cause was not

the Monte-Carlo error.

_Possibly

the AMML estimator had smaller variance in cases

where it exhibited a

_large

downward bias.

When an animal model was used for

_{design-analysis}

combination

_II-F-f,

the Gibbs

_sampler

’blew

_up’

in each of several

_replicates

_analyzed,

ie the additive

genetic

variance continued to increase in

_subsequent

Gibbs

_cycles,

soon

_reaching

unreasonably high

values. When

_design

V-F with

_only

one fixed effect was

_analyzed

under the animal

_model,

the Gibbs

_{sampler ’converged’,}

but estimates had

larger

biases and were more variable

_compared

to those obtained under the sire model.

’Convergence’

of the Gibbs

_sampler

refers to the

_sampler

_reaching

_{stationarity,}

where

sample

values fluctuate around a constant

_(see

_figure

6 for _presence of and

figure

5 for lack of

_{’convergence’;}

these

_figures

are discussed further

below).

To

verify

this result and ensure the correctness of the software under the animal model

option,

the

_{larger design}

VI-F was also

analyzed

under the animal model.

Then,

the _average estimates were much closer to the true values and almost identical to

The results discussed so far

_strongly

_suggest

that the cause of the

_upward

bias

in the MCMPM and MCMML estimators is a

_{highly non-symmetrical marginal}

posterior

distribution of

_heritability

in situations where the data contain little information. For the

_larger

_designs

_{(IV-F, VI-F),}

the biases

_strongly

decreased.

However,

it _appears to be

_likely

that the

_{posterior density}

of the

_parameters

is

also

_improper

in the presence of HYS levels with the ECP. As mentioned

earlier,

improperness

can be shown

_analytically

for a

_simple

fixed model and the

_logit

link for

_binary

data when at least one level of the fixed factor exhibits the ECP. The

fact that the Gibbs

_sampler

’blew

_up’

for the smaller

_designs

under the animal model also

supports

the

_hypothesis

of an

_{improper posterior. Moreover,}

when a

joint posterior

density

of the

parameters

is

_improper,

_’marginal

_posterior

mean or

mode’ estimates or

_{’marginal posterior density plots’}

from Gibbs

_output

_may look

’reasonable’,

although

these were not obtained from a Markov chain with known

stationarity

properties

(Hobert

and

Casella,

1994).

In the

_logit

link

_example,

effects of fixed levels not

_showing

ECP are still

_{’reasonably}

well estimated’ in the _presence

of other levels with the ECP

_causing

the

_posterior

to be

_improper.

Because an

_analytical

_proof

of an

_improper

_posterior

under the nonlinear mixed

model for

binary

data

employed

in this

investigation appeared

to be _very difficult or

intractable,

the

_marginal

likelihood in

!7J,

or the

reciprocal

of the

_integration

constant of the

_posterior

_density,

was considered as a

_potential

criterion for

detecting

improperness

from Gibbs

_output.

It was estimated from several modified

design

II-F data sets

_{using expression}

_[11]

and

required

3 consecutive Gibbs chains which were run at

_lengths

of 10 000 and 20 000

_(after

_burn-in)

_cycles.

Two situations

were considered in which the

_posterior

density

was

probably

_improper:

in the

presence of the ECP and when

using

the

_prior

_1/

₀

_,

_for

the variance

_component.

To examine the

_ECP,

4 data sets were created. Data set 1 had 70 HYS with the

ECP,

data set 2 had

_only

1 HYS with the ECP and data set 3 had none. Data sets

2 and 3 were obtained

by

reducing

the incidence of the trait from 0.9 to 0.5 and

by reducing

the size of the differences _among HYS differences. Data set 4 was from

design

II-N with random HYS effects.

Table III contains the estimated

_normalizing

constants

_(reciprocals

of

_marginal

likelihoods)

and the MCMPM estimates of

_heritability

for each of the 5 data sets

and the 2

_lengths

of the

sampler.

The estimates of the

_integration

constant for data sets 1 and 2 of table III differed

_strongly

between

_lengths

of 10 000 and 20 000

cycles,

ie did not

_{’converge’.}

The estimate of the

_integration

_constant,

_however,

did _converge for data set 3 not

_containing

HYS with ECP and for data set 4

treating

HYS as random. For all 4 data

_sets,

the estimate of

_heritability

_appeared

to have

_{’converged’,}

because

virtually

the same means were obtained after 10 000 and 20 000

_{cycles. ’Convergence’}

was also indicated

by

the

plot

of

_sample

value for

heritability

versus Gibbs

_cycle

in

_figure

7

_although

it showed more

_variability

than a

corresponding plot

in

figure

8 for data set 3 not

_containing

_any HYS with ECP.

Note that the

_numbering

of the Gibbs

_cycles

in the

_figures

_begins

after burn-in.

Expectedly,

a

_plot

of HYS

_sample

value versus Gibbs

_cycle

(fig 5)

showed

non-convergence for an ECP HYS in data set 1 of table III. As a

control,

figure

6

presents

the

_plot

of

_sample

value versus Gibbs

_cycle

for an HYS without

_ECP,

which demonstrated _convergence. Plots of the

_marginal

posterior

density

for an

To

_verify

the effect of the

particular

factorization of the

_posterior

_density

evaluated at the vector of

_posterior

means of the

_parameters

in

_[11],

the

_analysis

of data set 1 in table II was

repeated

for the other 5

factorizations,

which all indicated

non-convergence of the

integration

constant

_(results

not

_presented).

To examine the second case of an

improper posterior

caused

by

the

1/<T!

_prior,

improper posteriors

in the linear model

_(Hobert

and

_Casella,

_1994).

Use of this

prior

in estimation under

_design

V-F

_{occasionally yielded}

zero

h

2

estimates

or underestimated

h

2

.

_{Although heritability}

was underestimated

_(h

₂

=

_0.09),

the Monte-Carlo estimates of the

_integration

constant were almost

unchanged

after 10 000 and 20000 Gibbs

_{cycles indicating}

that this form of

_{improperness,}

if it

exists,

was not detected

by

Monte-Carlo estimation of the

integration

constant

based on

!11).

CONCLUSIONS

This

_study

confirmed that the AMML estimator of the

_heritability

of a threshold

character has a downward bias if

_family

sizes are small

(Gilmour

et

_al,

_1985),

in

this case if sire _{progeny group} size is small for a

_binary

trait with

_high

_(or

_low)

incidence. This bias is due to the

_{approximation}

in the AMML estimator

_(Foulley

et

_al,

_1987;

Hoeschele et

_al,

_1987),

because it is eliminated

_by

_computing

exact

MML estimates via Gibbs

_sampling.

For small data sets with extreme incidence and _many fixed effects

_{( eg,}

_design

II-F),

ie with little information about the

_{heritability,}

the

_{marginal posterior density}

of

heritability

is

_{highly non-normal,}

its mean and mode differ and overestimate

h

2

2

effect and the incidence was 0.5. The

_phenomenon

of

_obtaining

biased estimates when random effects are

_incorporated

into a

_generalized

linear model is

_quite

well

_known,

and an iterative bias correction method for

_parametric

models

_using

the

_bootstrap

has been

_developed

_(Kuk,

_1995).

Moreno

_{(personal communication)}

obtained

_promising

results when

_applying

Kuk’s method to a

_binary

threshold

_trait,

however,

he noted that the

_bootstrap

is

_{computationally}

_extremely

_demanding.

Without bias

_correction,

accurate estimation of

_genetic

_parameters

for

_binary

threshold traits

_requires

a

_large

amount of data in absolute terms and relative to the number of fixed effects. This is

_likely

even more

_important

for

_genetic

correlations than for heritabilities with the former known to be

_poorly

estimated from

_binary

data

_(Simianer

and

_Schaeffer,

_1989).

An

_{encouraging result, however,}

is that for situations where

_only

a small data set

_containing

_many HYS levels is available for

analysis,

the MCMML estimator of

_heritability

_{appears to} be unbiased if a normal

prior

is used for the HYS

_effects,

_irrespective

of the true state of nature of the HYS

effects

_(fixed,

_normally

or bounded

_uniformly

_{distributed).}

As

_categorical

traits are

usually

not under intense

_selection,

a non-random association of sires and herds seems

_unlikely

which

_{justifies treating}

HYS effects as random in

_practice.

Other

approaches

to

_improving

_heritability

estimation for such

_data,

_eg,

_fixing

values of

prior,

were not successful. The use of alternative

_prior

distributions

( eg,

Berger

and

Bernardo, 1992a,

b)

for fixed factors with many levels

exhibiting

the ECP should

be

_investigated

further.

The _accuracy of

_heritability

estimation was also found to differ among sire and

animal models. For data with extreme

_incidence,

a limited number of sires

(50)

and small progeny group size

(40),

estimates obtained with the animal model were less

accurate than those from the sire model. When the number of HYS effects

_(treated

as

fixed)

was

_{additionally large,}

the Gibbs

_sampler

did not

_{’converge’.}

_Therefore,

the animal model should be used

_only

when there is sufficient information in the

data;

otherwise the

_apparently

more robust sire model should be

preferred.

ACKNOWLEDGMENTS

Financial support for this work was

_{provided by}

the National Science Foundation _grant

BIR-94-07862,

the Australian Meat Research

_Corporation,

and the Australian

_Department

of

_{Industry, Science,}

and

_Technology.

This research was conducted

_using

the resources

of the center’s

_Corporate

Research Institute. The authors

acknowledge

discussions with MA

_Tanner,

D Sorensen and D Gianola. We also thank MA Tanner for

providing

us with

technical reports on the Monte-Carlo estimation of

Bayes

factors and

marginal

likelihoods. Two anonymous reviewers are thanked for valuable

_suggestions

and for

_improving

the

clarity

of this paper.

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