Bayesian inference in threshold models using Gibbs sampling

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Bayesian inference in threshold models using Gibbs

sampling

Da Sorensen, S Andersen, D Gianola, I Korsgaard

To cite this version:

Original

article

Bayesian

inference

in

threshold models

using

Gibbs

_sampling

DA Sorensen

S Andersen

2

D

Gianola

I

_Korsgaard

1

National Institute

_of

Animal

Science,

Research Centre

Foulum,

PO Box

_39,

DK-8830

_Tjele;

2

National Committee

_{for Pig Breeding,}

Health and

Prodv,ction,

Axeltorv

_3,

_{Copenhagen V, Denmark;}

3

University of Wisconsin-Madison, Department of

Meat and Animal

_Sciences,

Madison, WI53706-1284,

USA

(Received

17 June

1994; accepted

21 December

₁₉₉₄₎

Summary - A

Bayesian analysis

of a threshold model with

multiple

ordered

categories

is

_{presented. Marginalizations}

are achieved

by

means of the Gibbs

sampler.

It is shown

that use of data

_augmentation

leads to conditional

_posterior

distributions which are _easy to

sample

from. The conditional

_posterior

distributions of thresholds and liabilities are

independent

uniforms and

independent

truncated

normals, respectively.

The

_remaining

parameters of the model have conditional

posterior

distributions which are identical to

those in the Gaussian linear model. The

methodology

is illustrated

_using

a sire

model,

with an

_analysis

of

hip dysplasia

in

dogs,

and the results are

_compared

with those obtained

in a

_{previous study,}

based on

_approximate

maximum likelihood. Two

_independent

Gibbs chains of

length

620 000 each were run, and the Monte-Carlo

sampling

error of moments

of

_posterior

densities were assessed

using

time series methods. Differences between results

obtained from both chains were within the range of the Monte-Carlo

sampling

error.

With the

_exception

of the sire variance and

_{heritability, marginal posterior}

distributions seemed normal. Hence inferences

using

the _present method were in

_good

_agreement with those based on

_approximate

maximum likelihood. Threshold estimates were

_strongly

autocorrelated in the Gibbs sequence, but this can be alleviated

using

an alternative

parameterization.

threshold

_{model / Bayesian}

analysis

/

Gibbs _sampling

_/

_dog

Résumé - Inférence bayésienne dans les modèles à seuil avec _{échantillonnage} de

Gibbs. Une

_{analyse bayésienne}

du modèle à seuil avec des

catégories multiples

ordonnées

est

présentée

ici. Les

_{marginalisations}

nécessaires sont obtenues par

échantillonnage

de

Gibbs. On montre que l’utilisation de données

_{augmentées -}

la variable continue sous-jacente non observée étant alors considérée comme une inconnue dans le modèle -

con-duit à des distributions conditionnelles a

_posteriori

faciles

à échantillonner. Celles-ci sont

tronquées indépendantes

pour les sensibilités

(les

variables

sous-jacentes).

Les

paramètres

restants du modèle ont des distributions conditionneLles a

_{posteriori identiques}

à celles

qu’on

trouve en modèle linéaire

_gaussien.

La

méthodologie

est illustrée sur un modèle

paternel appliquée

à une

_dysplasie

de la hanche chez le

_chien,

et les résultats sont

com-parés

à ceux d’une étude

_précédente

basée sur un maximum de vraisemblance

approché.

Deux

séquences

de Gibbs

_{indépendantes, longues}

chacune de 620 000

_{échantillons,}

ont été réalisées. Les erreurs

d’échantillonnage

de _type Monte Carlo des moments des densités a

posteriori ont été obtenues par des méthodes de séries

temporelles.

Les résultats obtenus

avec les 2

_{séquences indépendantes}

sont dans la limite des erreurs

_{d’échantillonnage}

de Monte-Carlo.

À

l’exception

de la variance

_paternelle

et de

l’héritabilité,

les

distribu-tions

_marginales

a _posteriori semblent normales. De ce

fait,

les

_inférences

basées sur la

présente

méthode sont en bon accord avec celles du maximum de vraisemblance

_approché.

Pour l’estimation des

_seuils,

les

_séquences

de Gibbs révèlent de

_{fortes autocorrélations,}

auxquelles

il est

_{cependant possible}

de remédier en utilisant un autre

_{paramétrage.}

modèle à seuil

_/

_{analyse bayésienne}

_/

_{échantillonnage} de Gibbs

_/

chien

INTRODUCTION

Many

traits in animal and

_plant

_breeding

that are

_postulated

to be

_continuously

inherited are

_{categorically}

scored,

such as survival and conformation _scores,

_degree

of

_{calving difficulty,}

number of

_piglets

born dead and resistance to disease. An

appealing

model for

_genetic

_analysis

of

_categorical

data is based on the threshold

liability

concept,

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.

The

probability

distribution of _{responses in} a

_{given population depends}

on the

position

of its mean

_liability

with

_respect

to the fixed thresholds.

_Applications

of this model in animal

_breeding

can be found in Robertson and Lerner

(1949),

_Dempster

and Lerner

₍₁₉₅₀₎

and Gianola

_(1982),

and in Falconer

_(1965),

Morton and McLean

(1974)

and Curnow and Smith

_(1975),

in human

_genetics

and

_{susceptibility}

to

disease.

_Important

issues in

_{quantitative genetics}

and animal

_breeding

include

drawing

inferences about

_(i)

_genetic

and environmental variances and covariances in

populations;

(ii)

liability

values of groups of individuals and candidates for

genetic

selection;

and

_(iii)

_prediction

and evaluation of _response to selection. Gianola and

Foulley

(1983)

used

_Bayesian

methods to derive

_{estimating equations}

for

_(ii)

_above,

assuming

known variances. Harville and Mee

₍₁₉₈₄₎

_proposed

an

_approximate

method for variance

_component

_estimation,

and

_{generalizations}

to several

_polygenic

binary

traits

_having

a

_joint

distribution were

_{presented by Foulley}

et al

_(1987).

In these methods inferences about

_dispersion

_parameters

were based on the mode of their

_{joint posterior distribution,}

after

_integration

of location

_parameters.

This involved the use of a normal

_{approximation which,}

_seemingly,

does not behave well _{in sparse}

_contingency

tables

_(H6schele

et

_al,

_1987).

These authors found that

combination of fixed and random levels in the model was smaller than

_2,

and

suggested

that this _may be caused

by inadequacy

of the normal

_{approximation.}

This

_problem

can render the method less useful for situations where the number of rows in a

_contingency

table is

_equal

to the number of individuals. A data

structure such as this often arises in animal

_breeding,

and is referred to as the ’animal model’

_(Quaas

and

Pollak,

1980).

Anderson and Aitkin

₍₁₉₈₅₎

_proposed

a

maximum likelihood estimator of variance

_component

for a

_binary

threshold model. In order to construct the

likelihood, integration

of the random effects was achieved

using

univariate Gaussian

_quadrature.

This

_procedure

cannot be used when the random effects are

_correlated,

such as in

_{genetics. Here, multiple}

integrals

of

high

dimension would need to be

_calculated,

which is unfeasible even in data sets with

only

50

_genetically

related individuals. In animal

_breeding,

a data set _may contain thousands of individuals that are correlated to different

_degrees,

and some of these

may be inbred.

Recent reviews of statistical issues

_arising

in the

_analysis

of discrete data in

animal

_breeding

can be found in

Foulley

et al

₍₁₉₉₀₎

and

_Foulley

and Manfredi

(1991).

Foulley

(1993)

gave

approximate

formulae for

one-generation predictions

of response to selection

_by

truncation for

_binary

traits based on a

simple

threshold model.

_However,

there are no methods described in the literature for

_drawing

inferences about

_genetic

_change

due to selection for

_categorical

traits in the context

of threshold models.

_Phenotypic

trends due to selection can be

reported

in terms of

changes

in the

_frequency

of affected individuals.

_{Unfortunately,}

due to the nonlinear

relationship

between

_phenotype

and

_genotype,

_{phenotypic changes}

do not translate

directly

into additive

_{genetic changes,}

_{or, in} other

_words,

to _{response to} selection. Here we

_point

out that inferences about realized selection _response for

_categorical

traits can be drawn

_{by extending}

results for the linear model described in Sorensen

et al

_(1994).

With the advent of Monte-Carlo methods for numerical

integration

such as Gibbs

sampling

(Geman

and

_{Geman, 1984;}

Gelfand et

_al,

_1990),

_analytical

_{approximations}

to

_posterior

distributions can be

_avoided,

and a simulation-based

_approach

to

Bayesian

inference about

quantitative genetic

parameters

is now

possible.

In animal

breeding, Bayesian

methods

_using

the Gibbs

_sampler

were

_applied

in Gaussian models

_{by Wang}

et al

_{(1993, 1994a)}

and Jensen et al

₍₁₉₉₄₎

for

_(co)variance

component

estimation and

_by

Sorensen et al

₍₁₉₉₄₎

and

_Wang

et al

_(1994b)

for

assessing

response to selection.

_Recently,

a Gibbs

_sampler

was

_implemented

for

binary

data

_(Zeger

and

Karim,

1991)

and an

analysis

of

multiple

threshold models was described

_by

Albert and Chib

(1993).

_Zeger

and Karim

(1991)

constructed the Gibbs

_{sampler using rejection}

_{sampling techniques}

_{(Ripley, 1987),}

while Albert and Chib

₍₁₉₉₃₎

used it in

_conjunction

with data

_{augmentation,}

which leads to

a

_{computationally simpler}

_strategy.

The _purpose of this _paper is to describe a Gibbs

_sample

for inferences in threshold models in a

_quantitative

_genetic

context.

First,

the

_Bayesian

threshold model is

_presented,

and all conditional

_posterior

distributions needed for

_running

the Gibbs

sampler

are

_given

in closed form.

Secondly,

a

_{quantitative genetic}

_analysis

of

_hip

_dysplasia

in German

_shepherds

is

presented

as an

illustration,

and 2 different

_{parameterizations}

of the model

leading

MODEL FOR BINARY RESPONSES

At the

_phenotypic

_level,

a Bernoulli random variable

_Y

_i

is observed for each individual

_{i (i}

=

1, 2, ... , n)

_taking

values y

i

=

_{1 or y}

₂

= 0

(eg,

alive _or

dead).

The variable

_Y

is the

_expression

of an

_underlying

continuous random variable

_U

_i

_,

the

_liability

of individual i. When

_U

_Z

exceeds an unknown fixed threshold

_t,

then

Y

=

_{1, and Y}

= 0 otherwise. We _assume that

liability

is

_normally

_distributed,

with the mean value indexed

_by

a

_parameter

_0,

_and,

without loss of

generality,

that it has unit variance

_(Curnow

and

_Smith,

_1975).

Hence:

where

0’ =

_{(b’, a’)}

is a vector of

parameters

with _p fixed effects

_(b)

_and

_{q random}

additive

_genetic

values

_(a),

and

_w’

is a row incidence vector

_linking

e to the ith observation.

It is

_important

to note that

_{conditionally}

on

_0,

the

_U

_i

are

independent,

so for the vector U =

{U

i

}

given 0,

we have as

_joint

_density:

where

_!U(.)

is a normal

_density

with

_parameters

as indicated in the

_argument.

In

!2!,

put

WO = Xb ₊

Za,

where X and Z are known incidence matrices of order n

by

p and n

by

q,

respectively,

and,

without loss of

generality,

X is assumed to have full column rank. Given the

_model,

we have:

where

_<p(.)

is the cumulative distribution function of a standardized normal variate. Without loss of

_generality,

and

_provided

that there is a constant term in the

_model,

t can be set to

_0,

and

_[3]

reduces to

Conditionally

on both 0 and

_{on Y}

_i

=

y 2

,

U

i

follows a truncated normal distribution. That

is,

for _Yi = 1:

where

_I(X

E

_A)

is the indicator function that takes the value 1 if the random variable X is contained in the set

_A,

and 0 otherwise. For _Yi =

0,

the

density

is

§

u

; (w§ e, I) /V(-w§ e) I (U

i

x 0) .

Invoking

an infinitesimal model

_{(Bulmer, 1971),}

the conditional distribution of additive

_genetic

values

_given

the additive

_genetic

variance in the

_conceptual

base

where A is a _q

_by

_q matrix of additive

_{genetic relationships.}

Note that a can include animals without

_phenotypic

scores.

We discuss next the

_{Bayesian inputs}

of the model. The vector of fixed effects b will be assumed to follow a

_priori

the

_improper

uniform distribution:

For a

_description

of

_uncertainty

about the additive

_genetic

_variance,

or a 2,

an inverted gamma distribution can be

_invoked,

with

_density:

where v and

S’

are

_parameters.

When v = -2 and

S

Z

=

0,

[8]

reduces _to the

improper

uniform

_prior

distribution. A proper uniform

prior

distribution for

_Q

_a

_is:

where k

a

is a constant and

_{a a 2m!’. is}

the maximum value which

_J£

_can take a

_priori.

To facilitate the

_development

of the Gibbs

_sampler,

the unobserved

_liability

U

is included as an unknown

_parameter

in the model. This

approach,

known as data

augmentation

(Tanner

and

_Wong,

_1987;

Gelfand et

al, 1992;

Albert and

Chib, 1993;

Smith and

_Roberts,

₁₉₉₃₎

leads to identifiable conditional

_{posterior distributions,}

as shown in the next section.

Bayes

theorem

_gives

as

_{joint posterior}

distribution of the

_parameters:

The last term is the conditional distribution of the data

_given

the

parameters.

We notice

_that,

for

_Y

_i

=

_1,

say, we have

For

_Y

_i

₌

_0,

we have:

The

_{joint posterior}

distribution

_[10]

can then be written as:

where the

_conditioning

on

hyperparameters

v and

S’

is

_replaced

by

0

a

ma

.

when the uniform

prior

[9]

for the additive

_genetic

variance is

_employed.

Conditional

_posterior

distributions

In order to

_implement

the Gibbs

sampler,

all conditional

_posterior

distributions of the

parameters

of the model are needed. The

_{starting point}

is the full

posterior

distribution

_!13!.

_Among

the 4 terms in

_(13!,

the third is the

only

one that is a function of b and we therefore have for the fixed effects:

which is

_proportional

to

_{!U(Xb+Za, I).}

As shown in

_Wang

et al

_(1994a),

the scalar form of the Gibbs

_sampler

for the ith fixed effect consists of

sampling

from:

where _xi is the ith column of the matrix

X, and b

i

satisfies:

In

_!16!,

_X-

_i

is the matrix X with the column associated with

i deleted,

and

b_

i

is b with the ith element deleted. The conditional

posterior

distribution of the

vector of

breeding

values is

_proportional

to the

_product

of the second and third

terms in

_!13!:

which has the form

_{!(0,Acr!)!(u!b,a).}

Wang

et al

_(1994a)

showed that the scalar Gibbs

sampler

draws

samples

from:

where zi is the ith column of

Z,

cii is the element in the ith row and column of

A-1 , /B B =

(Qa)-1,

and a

i

satisfies:

In

_[19],

_ci,-i is the row of

A-

1

corresponding

to the ith individual with the ith element excluded. We notice from

_[14]

and

_[17],

that

_augmenting

with the

For the variance

_component,

we have from

!13!:

Assuming

that the

_prior

for

_o,2is

the inverted _gamma

_given

in

_!8!,

this becomes:

and

_assuming

the uniform

_prior

_!9!,

it becomes:

Expression

!21a!

is in the form of a scaled inverted gamma

density,

and

[21b]

in

the form of a truncated scaled inverted _gamma

density.

The conditional

_posterior

distribution of the

_underlying

variable

U

i

is

propor-tional to the last 2 terms in

_!13!.

This can be seen to be a truncated normal

dis-tribution,

on the left if

Y

i

= 1 and _on the

right

otherwise. The

density

function of

this truncated normal distribution is

_given

in

_!5!.

_Thus,

_depending

on the observed

Yi,

we have:

or

Sampling

from the truncated distribution can be done

by generating

from the untruncated distribution and

_retaining

those values which fall in the constraint

region.

Alternatively

and more

efficiently,

_suppose that U is truncated and defined in the interval

_{!i, j]}

_only,

where

_{i and j}

are the lower and _upper

bounds,

respectively.

Let the distribution function of U be

F,

and let v be a uniform

[0, 1]

variate. Then U =

F-1 !F(i)

₊

v(F(j) &mdash;

F(i))!

is _a

_drawing

from the truncated random variable

(Devroye,

1986).

Albert and Chib

₍₁₉₉₃₎

also constructed the mixed model in terms of a hier-archical

_model,

but

_proposed

a block

_sampling

_strategy

for the

_parameters

in the

underlying

scale,

instead.

_{Essentially, they}

_suggest

_sampling

from the distributions

(

Q

a,

a,

blU)

as

(a!IU)(a,

blU,

J£

),

instead of from the full conditional

posterior

distributions

!15!,

!18!

and

_!21!,

and

_they

assumed a uniform

_prior

for

log(a2)

in a finite interval. To facilitate

_sampling

from

p(or2lU),

_they

use an

_{approximation}

which consists of

_placing

all

_{prior probabilities}

on a

grid

of

or2

values,

thus

making

the

_prior

and the

_posterior

discrete. The need for this

_{approximation}

is

question-able,

since the full conditional

_posterior

distribution of

_{(T has}

a

_simple

form as noted in

_[21]

above. In

_addition,

in animal

_breeding,

the distribution

_{(a, b[U,}

_{a a 2)}

is a

_high

dimensional multivariate normal and it would not be

_{simple computationally}

MULTIPLE ORDERED CATEGORIES

Suppose

now that the observed random variable Y can take values in one of C

mutually

exclusive ordered

_categories

delimited

_by

C + 1 thresholds. Let

to

=

- oo,

t

c

= _+oo, with the

remaining

thresholds

satisfying

t1 ! t

2

... <

_t

_C

_-

₁

-Generalizing

[3]:

Conditionally

on

_A,

_Y

_i

=

j,

t

j

-1

and

_t

_j

_,

the

_underlying

variable associated with the ith observation follows a truncated normal distribution with

density:

Assuming

that

_{o, a, 2}

b and t are

_{independently}

distributed a

_priori,

the

_joint

posterior

density

is written as:

where

_p(Ulb,

_a,

_t)

=

p(Ulb, a).

Generalizing

[12],

the last term in

_[25]

can be

expressed

as

(Albert

and

_Chib,

_1993):

All the conditional

_posterior

distributions needed to

_implement

the Gibbs

sam-pler

can be derived from

!25!.

It is clear that the conditional

_posterior

distributions of

b

i

,

ai and

u2

_{are the} same as for the

_binary

_response model and

_given

in

_(15!,

[18]

and

_!21!.

For the

_underlying

variable associated with the ith observation we have from

_!25!:

This is a truncated

normal,

with

_density

function as in

!24!.

The thresholds t =

(t

l

,

_t

₂

_{, ... ,}

tC-1)

are

_clearly

_dependent

a

_priori,

since the model

postulates

that these are distributed as order statistics from a uniform distribution in the interval

_[t

_mini

_t

_max]

_.

_However,

the full conditional

_posterior

distributions of the thresholds are

independent.

That

_is,

p(t

j

It-

j

, b, a, U, o

l

a

2

,

Y

) =

p(t

where T =

{(h, t2,&dquo;’, tc-dltmin !

_{/ x t2 ! ... ! t}

₁

_v

_-

_C

t

max

)

(Mood

et

a

l

,

1974).

Note that the thresholds enter

_only

in

_defining

the

support

of

_p(t).

The conditional

_posterior

distribution

_{of t}

_j

is

_{given by:}

which has the same form as

!26!.

_Regarded

as a function

_{of t,}

[26]

shows

_{that, given}

U and _y, the _upper bound of

_{threshold t}

_j

is

_{min (U I Y}

_{= j}

₊₁₎

and the lower bound

is

_max(UIY

=

j).

The _a

_priori

condition _{t E} T is

automatically

fulfilled,

and the bounds are unaffected

_by

_knowledge

of the

_remaining

thresholds. Thus

_t

_j

has a uniform distribution in this interval

_given

_by:

This

_argument

assumes that there are no

_categories

with

_missing

observations. To accommodate for the

possibility

of

_missing

observations in 1 or more

_categories,

Albert and Chib

₍₁₉₉₃₎

define the _upper and lower bounds of threshold

_j,

as

minfmin(UIY

= j

+

_{1), t}

_j+d

and as

max{max(U!Y

=

j),t

j

-’

11

,

respectively.

In this case, the thresholds are not

_{conditionally independent.}

The Gibbs

_sampler

is

implemented

by sampling repeatedly

from

_{!15!, !18!, !21!, [24]}

and

_(28!.

Alternative

_{parameterization}

of the

_multiple

threshold model

The

multiple

threshold model can also be

parameterized

such that the conditional distribution of the

_underlying

variable

_{U, given 0,}

has unknown variance

_0’

_instead

of unit variance. The

_equivalence

of the 2

_{parameterizations}

is shown in the

Appendix.

This

_{parameterization requires}

that records fall in at least 3

_mutually

exclusive ordered

_categories;

for C

_categories,

_only

C-3 thresholds are identifiable.

In this new

_{parameterization,}

one must

_sample

from the conditional

_posterior

distribution

_{of o, e 2.}

Under the

priors

[8]

or

_(9!,

the conditional

posterior

distribution of

_{Jfl can}

be shown to be in the form of a scaled inverted _gamma. The

_parameters

of this distribution

_depend

on the

_prior

used for

ae

2

If this is in the form

(8!,

then

where,

SSE =

(U - Xb - Za)’ (U - Xb - Za),

and _v, and

S

e

are

_parameters

of the

prior

distribution. If a uniform

_prior

of the form

[9]

is assumed to describe the

_prior

uncertainty

about

_{u . 2,}

the conditional

_posterior

distribution is a truncated version of

_[29]

_(ie

_[21b]),

_{with v}

_e

= -2

and S,2

= 0. With

exactly

3

categories,

the Gibbs

EXAMPLE

We illustrate the

_methodology

with an

_analysis

of data on

hip

dysplasia

in German

shepherd dogs.

Results of an

_early

_analysis

and a full

_description

of the data can be found in Andersen et al

_(1988).

_Briefly,

the records consisted of

_radiographs

of 2 674

_offspring

from 82 sires. These

_radiographs

had been classified

_according

to

guidelines approved by

FCI

_(Federation

_Cynologique

Internationale,

1983),

each

offspring

record was allocated to 1 of 7

_mutually

exclusive ordered

_categories.

The model for the

_underlying

variable was:

.

where ai is the effect of sire

i (i

=

_{1, 2, ... , 82; j}

=

1, 2,... , n

i

).

The

prior

distribution of /! was as in

_[7]

and sire effects were assumed to follow the normal distribution:

The

_prior

distribution of the sire variance

₍

_Q

_a)

was in the form

_given

in

!8!,

with v = 1 and

S

2

= 0.05. The

prior

for

_t

₂

_{, ... , t6}

was chosen to be uniform on the ordered subset of

_[f,

=

-1.365,

f

7

=

+00!5

for which

ti

_<

t

2

_< ... <

t

7

,

where

f

l

was the value at which

t

1

was

_set,

and

f

7

is the value of the 7th threshold. The value for

_f

_l

was obtained from Andersen et al

_(1988),

in order to facilitate

comparisons

with the

_present

_analysis.

The

analysis

was also carried out under the

parameterization

where the conditional distribution of U

_given

0 has variance

_{a 2}

Here,

Q

e

was assumed to follow a

_prior

of the form of

!8!,

with v = 1 and

S

Z

= 0.05

and

_t

₆

was set to 0.429. Results of the 2

_analyses

were

similar,

so

only

those from the second

parameterization

are

_presented

here.

Gibbs

_sampler

and

_post

Gibbs

_analysis

The Gibbs

sampler

was run as a

_single

chain. Two

_independent

chains of

length

620 000 each were _run, and in both _cases, the first 20 000

samples

were discarded.

Thereafter, samples

were saved _every 20

_iterations,

so that the total number of

samples kept

was 30 000 from each chain. Start values for the

_parameters

_were, for the case of chain

_1,

o,2=

_2.0,

Q

a

=

_0.5,

_t

₂

=

_{-0.8, t}

₃

₌

_-0.5,

_t

₄

=

-0.2,

_t

₅

= 0.1. For

chain

2,

estimates from Andersen et al

₍₁₉₈₈₎

were

_used,

and these were

J

fl

=

_1.0,

or =

0.1, t

2

=

_-1.05,

_t

₃

=

₄

_{-0.92, t}

=

_{-0.62, t}

₅

= -0.34. In both _runs,

starting

values for sire effects were set to zero.

Two

_important

issues are the assessment of convergence of the Gibbs

sampler,

and the Monte-Carlo error of estimates of features of

_posterior

distributions. Both issues are related to the

_question

of whether the

chain,

or

chains,

have been run

long enough.

This is an area of active research in which some

_guidelines

based on theoretical work

(Roberts,

1992; Besag

and

Green, 1993;

Smith and

Roberts,

1993;

Roberts and

_Polson,

₁₉₉₄₎

and on

_practical

considerations

(Gelfand

et

al,

1990;

Gelman and

Rubin, 1992; Geweke,

1992;

Raftery

and

_Lewis,

₁₉₉₂₎

have been

and Lewis

₍₁₉₉₂₎

_proposed

a method based on 2-state Markov chains to calculate the number of iterations needed to estimate

_{posterior quantiles.}

Let

_X

_l

_{, ... , X&dquo;,}

be elements of a

_single

Gibbs chain of

_length

m. The m

sampled

values,

which are

_generally

_correlated,

can be used to

_compute

features of the

posterior

distribution. In our

model,

the Xs could

_represent

_samples

from the

posterior

distribution of a sire

_value,

or of the sire

_variance,

or of functions of these.

Invoking

the

_{ergodic theorem,}

for

_large

_m, the

_posterior

mean can be estimated

_by

and the variance of the

_posterior

distribution can be estimated

_by

(Cox

and

_{Miller, 1965; Geyer,}

_1992).

_Similarly,

a

_{marginal posterior}

distribution

function,

F(Z)

=

P(X

_<

Z),

_can be estimated

by

which means that

_F(Z)

is estimated

_by

the

_empirical

distribution function. All these estimators are

subject

to Monte-Carlo

_sampling

_error, which is reduced

_by

prolongation

of the chain.

Consider the _sequence

_g(X

₁

_),...,

_g(Xm),

where

_g(.)

is a suitable

_function,

_eg,

g(X)

= X _or

g(X)

=

l!X!Z!,

and let

_{E(g(X)) =}

_p. The

_lag-time

auto-covariance of the sequence is estimated as:

where

is the

_sample

mean for a chain of

_length

_m, and t is the

_lag.

The auto-correlation is then estimated as

Tm(!)/!m(0).

The variance of the

_sample

mean of the chain is

given by

which will exceed

_y(0)/m

if

_q(t)

> 0 for all

t,

as is usual. Several estimators of the variance of the

_sample

mean have been

_proposed

(Priestley, 1981),

but we chose one

Let

_F

_m

_(t)

=

$

m(2t)

₊

(2t + 1),

im

f

t =

0, 1, ....

The estimator can then be written

as

where t is chosen such that it is the

_{largest integer}

_satisfying

fm (I)

>

_0,

i =

_1,

1 ...

t.

The

_{justification}

for this choice is that

_r(i)

is a

_{strictly positive,}

_{strictly’decreasing}

function of i. If

X

l

,

X

2

, ...

X&dquo;,

are

independent,

then

Var(í

1m

)

=

q(0) /m.

To obtain

an indication of the effect of the correlation on

Var(í

1m

),

an ’effective number’ of

independent

observations can be assessed

as j

m

=

5

m

(0)/ Vâr(í

1m

)

’

When the

elements of the Gibbs chain are

_{independent, !m}

= m.

RESULTS

Estimates of the

_empirical

distribution function of various

_parameters

of the model for each of the 2 chains of the Gibbs

sampler

are shown in

figures

1-5. For

under 0.10.

_{Similarly, figure}

3 indicates that there is

90%

posterior

probability

that

heritability

in the

_underlying

scale

_(h

₂

=

4J£ / (J£ + Jfl ) )

lies between 0.24 and

_0.49,

and the median of the

_posterior

distribution is 0.35.

_Although

this distribution is

slightly skewed,

the estimate of the median agrees well with the ML

type

estimate of

_heritability

of

0.35, reported

in Andersen et al

_(1988).

_Figure

4

_depicts

estimates

of distribution functions for the mean

₍

_J1

₎

and for each of 3 sire effects

_(a

_l

_,

_a2,

_a

₃

_).

Figure

5

_gives

_{corresponding}

distributions for 2 threshold

parameters

(t

2

,

and

_t

₃

_).

The

_figures

fall in 3

_categories.

The distribution functions obtained from chains 1

and 2 coincide for each of the variables

_Q

_a,

_h2,

_al, a2 and a3, where the sire effects a

l

, a2 and a3

pertain

to 3 males with

_31,

5 and 158

_{offspring, respectively.}

A small deviation between chains 1 and 2 is observed for

_Jfl

and

J1, and a

_larger

deviation is observed for the threshold

_parameters

_{(fig 5).}

The Gibbs sequence for the threshold

parameters

showed very slow

mixing properties.

For

_example,

for threshold

_2,

the autocorrelations between

_sampled

values were

_0.785,

0.663 and

0.315,

for

_lags

between

_5,

10 and 50

_{samples, respectively.}

The reason is that the

_sampled

value for a

_given

threshold is bounded

_by

the values of the

_{neighbouring underlying}

variables U. If these are _very

close,

the value of the threshold in

_subsequent

samples

is

_likely

to

_change

_very

_slowly.

Under the

_{parameterization}

where 1 of the thresholds is substituted

_by

the residual

variance,

the autocorrelations associated with the

_{lags above,}

between

_samples

from the

_marginal

_posterior

distribution of

e 2,

were

_0.078,

0.064 and

_0.032,

_{respectively.}

Another scheme that _may accelerate

mixing

is to

sample

jointly

from the threshold and the

_liability.

For sire

_{effects, lag}

autocorrelations were close to zero.

A

_comparison

between

_marginal

posterior

means

(average

of the 30 000

samples)

estimated from the 2 chains is shown in table I. The difference between chains 1

and 2 is in all cases within Monte-Carlo

_sampling

_error,

_{which was}

estimated within

chains, using

[34].

The ’effective number’ of observations

_u

_Jm

for the means of

marginal

posterior

distributions is close to 30 000 for

_{or a 2,}

, la a2 and a3, but is

about 3 000 for

_Q

_e

_{and p}

and between 200 and 400 for

_t

₂

_{, ... , ts.}

The

_{marginal posterior}

distributions for _J1,

_t

₂

_{, ... , t}

₅

_,

_{, al a2,} a3are well

approxi-mated

_by

normal distributions. The

_posterior

means and standard deviations of these

_marginal

distributions can be

_compared

to estimates

_reported

in Andersen et

al

_(1988),

who used a

_2-stage

_procedure.

The authors first estimated variances

_using

an

_REML-type

estimator

_(Harville

and

Mee,

1984).

Secondly,

assuming

that the estimated variances were the true _ones, fixed effects and sire effects were estimated as

_{suggested by}

Gianola and

_Foulley

(1983).

For

_example,

for the 3 sires with

_158,

31 and 5

_{offspring respectively,}

the

_2-stage

_procedure

_yielded

estimates of sire effects

(approximate

posterior

standard

_deviations)

of 0.30

_(0.09),

0.17

_(0.17)

and -0.093

(0.26),

respectively.

The

_present

_Bayesian

_approach

with the Gibbs

_{sampler, yielded}

estimates of

_marginal

_posterior

means and standard deviations for these sires of 0.304

_(0.088),

0.177

_(0.166),

and -0.092

_(0.263),

_respectively

_(table

_I).

DISCUSSION

here likelihood inference with numerical

_integration

is a

_competing

alternative. For this model and

_{data, marginal}

_posterior

distributions of sire effects are well

ap-proximated by

normal distributions. On the other

hand,

with little information per

random

_effect,

_eg, animal

_models,

the

_normality

of

_marginal

_posterior

distributions when variances are not known is

_unlikely

to hold. A

_strength

of the

_Bayesian

ap-proach

via Gibbs

_sampling

is that inferences can be made from

marginal

_posterior

distributions in small

_{sample problems,}

without

_resorting

to

_asymptotic

approxi-mations.

_Further,

the Gibbs

_sampler

can accommodate

_{multivariately}

distributed random

_effects,

such as is the case with animal

_models,

and this cannot be

imple-mented with numerical

_integration

_techniques.

It seems

_important

to

_investigate

threshold models

_further,

_especially

with _sparse data structures

consisting

of many fixed effects with few observations per subclass.

This case was studied

_by

Moreno et al

_(manuscript

submitted for

_publication)

in the

_{binary model,}

where

_they

_{investigated frequency}

_properties

of

_Bayesian

point

estimators in a simulation

study. They

showed that

_improper

uniform

_prior

distributions for the fixed effects lead to an _average of the

_{marginal posterior}

mean of

_heritability

which was

_larger

than the simulated value.

They

obtained better

agreement

when fixed effects were

assigned

_proper normal

_prior

distributions. The use of ’non-informative’

_{improper prior}

distributions is

_discouraged

on several

grounds by,

among

others, Berger

and Bernardo

(1992),

as this can lead to

_improper

posterior

distributions. It seems that the

_disagreement

between simulated and estimated values in Moreno et al is due to lack of information and not due to

impropriety

of

_posterior

distributions.

_Thus,

the bias

_persists,

_though

_smaller,

when all

parameters

of the model are

assigned

_proper

_prior

distributions

(Moreno,

contain data

_falling

into

_only

one of the 2 dichotomies. There is no information in the data

_(in

the Fisherian

_sense)

to estimate these fixed effects and the likelihood

is ill-conditioned. In the case of these _sparse data

_structures,

the choice of the

_prior

distribution for the fixed effects _may well be the most critical

_part

of the

_problem.

This needs to be studied further.

Data

_augmentation

in the Gibbs

_sampler

led to conditional

_posterior

distribu-tions which are _easy to

_sample

from. This facilitates

_programming.

We have noted

though,

that threshold

_parameters

have very slow

mixing properties,

and this is

probably

related to the data

augmentation approach

used in this

study

(Liu

et

al,

1994).

With our

_data,

the

_{parameterization}

in terms of the residual variance

resulted in smaller autocorrelations between

_samples

of

_Q

_e

than between

_samples

of the thresholds. A scheme that is

likely

to accelerate

_mixing

is to

_sample

_jointly

from the threshold and

_liability.

This

_step

_may necessitate other Monte-Carlo

sam-pling techniques

such as a

_Metropolis

_algorithm

_{(Tanner, 1993),}

since

_sampling

is from a non-standard distribution. Alternative

_{computational strategies}

and param-eterizations of the model _may be more critical with animal models.

_Here,

there

is

_typically

little information on additive

_genetic

effects,

and these are correlated. These

_properties

slow down _convergence of the Gibbs chain.

The methods described in this paper can be

adapted easily

to draw inferences about

_genetic

_change

when selection is for

_categorical

data. Sorensen et al

₍₁₉₉₄₎

described how to make inferences about _response to selection in the context of the Gaussian model. In the threshold

_model,

the

_only

difference is that observed data are

_{replaced by}

the unobserved

_underlying

variable U

(liability).

In order to make inferences about response to

selection,

the

parameterization

must be in terms of an animal model.

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APPENDIX

Here it is shown that there is a one-to-one

_relationship

between 2 chosen param-eterizations of the threshold

model,

and that

_they

lead to the same

probability

distribution.

As a

_{starting point}

assume that the conditional distribution of

liability

is:

where e =

(b

l

, ... , bp,

a l

, ...

aq),

b

l

is an

_intercept

common to all

observations,

and associated with C

_categories

there are thresholds

_ti,

i =

_{0, 1, ... , C,}

with

to

= -oo and

_t

_c

= oo. We assume that the _p fixed effects are estimable. In order to make the

parameters

identifiable,

in what we call the standard

_{parameterization,}

we define:

Note that

_by

_setting

_f

_l

=

_0,

then

_t

_l

= 0. In _terms of this

_{parameterization,}

the

There are _{p + q +} C &mdash; 1 identifiable unknown

_parameters

which are:

In the alternative

_{parameterization,}

one can define

Ui

=

a

i

Ji,

such that:

where 8

=

a,6,a2

=

a2U2,g,

=

l

,

l

i

and

t

l

=

f

l

.

The number of identifiable

parameters

is of course the same as before. In this _paper we chose to set

_{tc_}

₁

=

f

2

,

where

_f

₂

>

_f

_l

is an

arbitrary

known constant.

The

_probability

distributions are

_{given by:}

P(Y

=

!0, t)

=

P(0-l

< U

i

s

t! !0, t)

=

P(T

j

-

l

< U

i

s

t! !0, t)

(since

the

relationship

between

(0, t)

and

(0, t)

is

_one-to-one)

=

P(t!_1

< Ui !

t! !0, t) _