Bayesian inference in the semiparametric log normal frailty model using Gibbs sampling

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Bayesian inference in the semiparametric log normal

frailty model using Gibbs sampling

Inge Riis Korsgaard, Per Madsen, Just Jensen

To cite this version:

Original

article

Bayesian

inference

in the

_{semiparametric}

_log

normal

frailty

model

_using

Gibbs

_sampling

Inge

Riis

_Korsgaard*

Per

Madsen Just Jensen

Department

of Animal

Breeding

and

Genetics,

Research Centre

_Foulum,

Danish Institute of

_{Agricultural Sciences,}

P.O. Box 50, DK-8830

Tjele,

Denmark

(Received

16 October

1997; accepted

23

_April

₁₉₉₈₎

Abstract - In this paper, a full

_{Bayesian analysis}

is carried out in a

_{semiparametric}

log

normal

_frailty

model for survival data

_using

Gibbs

sampling.

The full conditional

posterior

distributions

describing

the Gibbs

sampler

are either known distributions or

shown to be

_log

concave, so that

adaptive rejection sampling

can be used.

Using

data

augmentation, marginal posterior

distributions of

_breeding

values of animals with and without records are obtained. As an

_example,

disease data on future AI-bulls from the Danish

performance testing

programme were

_analysed.

The trait considered was ’time from

_entering

test until first time a

respiratory

disease occurred’. Bulls

without a

_respiratory

disease

_during

the test and those tested without disease at

date of

_analysing

data had

right

censored records. The results showed that the

hazard decreased with

_increasing

age at

entering

test and with

_increasing

degree

of

_{heterozygosity}

due to

crossbreeding.

Additive effects of _gene

_importation

had no

influence. There was

_genetic

variation in

_{log frailty}

as well as variation due to herd of

origin by

period

and year

by

season.

_©

Inra/Elsevier,

Paris

survival _analysis

_/

semiparametric log normal frailty

model / Gibbs

sampling

/

animal

_{model / disease}

data on _performance tested bulls

*

Correspondence

and

_reprints

E-mail:

_{snfirk@genetics.sh.dk}

or

_{IngeR.Korsgaard@agrsci.dk}

Résumé - Inférence Bayésienne dans un modèle de survie semiparamétrique

log-normal à partir de l’échantillonnage de Gibbs. Une

_{analyse complètement}

Bayésienne

utilisant

_{l’échantillonnage}

de Gibbs a été effectuée dans un modèle de survie

_{semiparamétrique log-normal.}

Les distributions conditionnelles a

_posteriori

mises à

profit

par

l’échantillonnage

de Gibbs ont

été,

soit des distributions connues, soit des distributions

log-concaves

de telle _{sorte que}

_{l’échantillonnage}

avec

_rejet

adaptatif

a _pu être utilisé. En utilisant la simulation des données _manquantes, on

avec ou sans données. Un

_{exemple analysé}

a concerné les données de santé des futurs

taureaux d’insémination dans les stations danoises de contrôle de

performance.

Les

taureaux sans maladie

respiratoire

ou n’en _{ayant pas} encore eu à la date de

l’analyse

ont été considérés comme _porteurs d’une information censurée à droite. Les résultats

ont montré que le

risque

instantané décroissait quant

l’âge

à l’entrée en station ou le

degré d’hétérozygotie

lié au croisement croissaient. Les effets additifs des différentes sources de

gènes importés

n’ont pas eu d’influence. Le

_risque

instantané de maladie a été trouvé soumis à des influences

_génétiques

et non

_génétiques

(troupeau

_d’origine

et

_{année-saison).}

_©

_{Inra/Elsevier,}

Paris

analyse de survie

_/

modèle semi-paramétrique

/

échantillonnage de Gibbs

_/

modèle animal

_/

résistance aux maladies

1. INTRODUCTION

When survival

data,

the time until a certain event

_happens,

is

_analysed,

_very often the hazard function is modelled. The hazard

function,

A

i

(t),

of an animal

i,

denotes the instantaneous

_probability

of

_failing

at time

_t,

if risk exists.

In Cox’s

_proportional

hazards model

_[5]

it is assumed that

_A

_i

_{(t) = A}

_o

_(t)

exp{x!,6},

where,

in

_{semiparametric models,}

_A

_O

_(t)

is _any

_arbitrary

baseline hazard function common to all animals. Covariates of animal

_i,

xi, are

_supposed

to act

_{multiplicatively}

on the hazard function

_by

_exp{x!,6},

where

_,Q

is a

vector of

_regression

parameters.

In

_fully

_parametric

models the baseline hazard function is also

_{parameterized.}

The

_proportional

hazard model assumes that

conditional on

_covariates,

the event times are

_independent

and attention is

focused on the effects of the

explanatory

variables. The baseline hazard function

is then

_regarded

as a nuisance factor.

Frailty

models are mixed models for survival data. In

frailty

models it is

assumed that there is an unobserved random

_variable,

a

_frailty

_variable,

which is assumed to act

_{multiplicatively}

on the hazard function. Sometimes a

frailty

variable is introduced to make correct inference on

_regression

_parameters.

In

other situations the

_parameters

of the

_frailty

distribution are of

_major

interest. In shared

_frailty

_models,

introduced

_by

_Vaupel

et al.

_(32),

_groups of individ-uals

_(or

several survival times on the same

individual)

share the same

_frailty

variable. Frailties of two individuals have a correlation

equal

to 1 if

_they

come

from the same _group and

_equal

to 0 if

_they

come from different _groups.

_Mainly

for reasons of mathematical

convenience,

the

frailty

variable is often assumed

to follow a _gamma distribution. In the animal

breeding

_literature,

this method

has been used to fit sire models for survival data

_using

_fully

_parametric

models

(e.g.

[8, 10]).

Several _papers deal with correlated _gamma

_frailty

models

_(e.g.

_[22,

_{26, 30,}

31!).

In these models individual frailties are linear combinations of

independent

gamma distributed random variables constructed to

_give

the desired variance covariance matrix _among frailties. From a mathematical

point

of view these

models are convenient because the EM

_algorithm

[7]

can be used to estimate the

parameters.

Because of the infinitesimal model often assumed in

_quantitative

genetics,

frailties may be

log

normally distributed;

thereby

conditional random

immediate to use the EM

algorithm

in

log normally

distributed

frailty

models as stated

_by

several authors and shown in

Korsgaard

!21!.

In this paper we show how a full

_{Bayesian analysis}

can be carried out in

a

_{semiparametric}

log

normal

frailty

model

using

Gibbs

sampling

and

adap-tive

_rejection

_sampling.

It is shown that

_{by using}

data

_{augmentation, marginal}

posterior

distributions of

_breeding

values of animals without records can be

ob-tained. The work is very much

_inspired

_by

the works of Kalbfleisch

_!19!,

_Clayton

!4!,

Gauderman and Thomas

_!11!

and

_Dellaportas

and Smith

_!6!.

Kalbfleisch

_[19]

presented

a

_{Bayesian analysis}

of the

semiparametric regression

model. Gibbs

sampling

was used

_by

_Clayton

[4]

for

_Bayesian

inference in the

semiparamet-ric gamma

frailty

model and

_by

Gauderman and Thomas

_[11]

for inference in

a related

_{semiparametric}

log

normal

frailty

model with

emphasis

on

applica-tions in

_genetic

_{epidemiology. Finally}

_Dellaportas

and Smith

_[6]

demonstrated that Gibbs

sampling

in

_conjunction

with

_{adaptive rejection sampling gives}

a

straightforward computational procedure

for

_Bayesian

inferences in the Weibull

proportional

hazards model.

The

_{semiparametric log}

normal

frailty

model is defined in section 2 of this

paper. In this

part

we show how a full

_{Bayesian analysis}

is carried out in the

special

case of the

log

normal

frailty

model,

where the model of

log

frailty

is a variance

_component

model. The full conditional

posterior

distributions

_required

for

_using

Gibbs

_sampling

are derived for a

_given

set of

_prior

distributions. In

section

_3,

we

_analyse

disease data on

performance

tested bulls as an

example

and section 4 contains a discussion.

2. BAYESIAN INFERENCE IN THE SEMIPARAMETRIC LOG NORMAL FRAILTY MODEL - USING GIBBS SAMPLING

Let

_T

_i

and

C

i

be the random variables

representing

the survival time and the

_censoring

time of animal

_i,

respectively.

Then data on animal i are

(y

2

,

6

i

),

where y

2

is the observed value

of Y

=

min{T

i

,

C

d

and

6

i

is an indicator random

variable,

equal

to 1 if

T

i

<

,

i

C

and 0 if

C

i

<

_T

_i

_.

In the

_{semiparametric frailty}

model,

it is assumed

_that,

conditional on

frailty

Z

i

= _zi, the hazard

function,

Ài(t),

of

Ti;

i =

_{1, ... ,}

_n, is

_{given by}

where

_A

_h

_(t)

is the common baseline hazard function of animals that

belong

to

the hth

_stratum,

h =

1, ... , H,

where H is the number of strata.

x

i

(t)

is a vector of

possible time-dependent

covariates of animal i

_and

is the

corresponding

vector of

_regression

_{parameters. Z}

_i

is the

frailty

variable of animal i. This is

an unobserved random variable assumed to act

_{multiplicatively}

on the hazard

function. A

large value,

zi, of

Z

i

increases the hazard of animal i

throughout

the whole time

_period.

Definition:

let w =

(wl, ... , w

n

)’;

if

w I E - N

n

(0,

E)

and the

frailty

variable

Zi

in

_equation

₍₁₎

be

_given

_by

_Z

_i

=

},

exp f w

2

i.e.

Z

Z

is

log normally

distributed;

i =

1, ... ,

n. Then the model

_given

_by

_equation

(1)

is called a

_{semiparametric}

This is the definition of a

_{semiparametric}

_log

normal

frailty

model in broad

generality.

However, special

attention is

_given

to a subclass of models where

the distribution of

_{log frailty}

is

_{given by}

a variance

_component

model:

or in scalar

_form,

_wi =

Uj+ ai+ ei

where j

is

the class of the random

effect,

u, that animal

i belongs

to; j

E

_{{1, ... , q}.}

ai is the random additive

genetic

value and ei the random value of environmental effect not

already

taken into account. It is assumed that

_ula

_’

_-

_Nq(O,

_Iq

_U

_{’), a[a§ -}

_N

_N

_(0,

_Aa!)

and

e!er! !!(0,In.cr!).

Q

u

and

_Q

_a

_.

_Q!

and

_Q

_a

are known

_design

matrices of

dimension n x _q and n x

_N,

_{respectively,}

where N is the total number of

animals

_defining

the additive

_genetic

_relationship

_matrix,

_A,

and n is the number of animals with records.

_Here,

_(u,

_a’),

_(a,

_or’)

and

_(e,

_U2

₎

are assumed

to be

_{mutually independent.}

Generalizations will be discussed later. From

equation

(2),

the hazard of

_T

_i

is:

assuming

that the covariates are time

_independent

and that there is no

stratification. The vector of

parameters

and

hyperparameters

of the model

is aJ =

(AoO,;3, u,a!,a,a!,e,a!),

where

_A

_o

_(t)

= It

A

o

(u)du

is the

_integrated

hazard function.

Note that

_{log frailty,}

_wi, of animal

_i,

is an unobserved

_quantity

which

is modelled. This is

_analogous

to the threshold model

_(e.g.

_[28]),

where an

unobserved

_quantity,

the

_liability,

is modelled. In the threshold

_model,

a

categorical

trait is

_considered,

but

_heritability

is defined for the

_liability

of the trait. In the

_{semiparametric}

_log

normal

_frailty

model the trait is a survival

time,

but

_heritability

is defined for

_{log frailty}

of the trait. The

_{semiparametric}

log

normal

_frailty

model is not a

log

linear model for the survival times

_T

_i

_,

i =

1, ... , n.

The

only log

linear models that are also

_proportional

hazards

models are the Weibull

_regression

models

_(including

_{exponential regression}

models),

where the error term is

_e/p,

with _p

_being

a

_parameter

of the Weibull

distribution and

_having

the extreme value distribution

_!20!.

Without restriction

on the baseline

hazard,

the

proportional

hazard model

postulates

no direct

relationship

between covariates

_(and

_frailty)

and time itself. This is unlike the threshold

_model,

where the observed value is determined

_by

a

_grouping

on the

underlying

scale.

2.1. Prior distributions

In order to _carry out a full

Bayesian analysis,

the

_prior

distributions of all

parameters

and

_{hyperparameters}

in the model must be

specified.

A

_priori,

it is assumed

_(by

definition of the

_log

normal

_frailty

_model)

that _u,

_given

the

hyper-parameter

(

7

u 2,

follows a multivariate normal distribution:

U

I

U2

u -

Nq(O,I9Qu).

priori

elements in

/3

are assumed to be

_independent

and each is assumed to fol-low an

_improper

uniform distribution over the real

numbers;

i.e.

p({3

b

)

oc

_1;

b =

1,...,.B,

where B is the dimension of

!3.

The

_{hyperparameters}

_{a£, a}

_§

_a and

_Q

_e

_are

assumed to follow

_independent

inverse gamma

distributions;

i.e.

a! ’&dquo; IG(¡.¿u, lIu), a! ’&dquo;

’

¿

a

,

IG(¡

v

a

)

and

_or2

_-

_e

_,

_¿

_’

_IG(¡

_v

_e

_),

where , u¿’¡ lIu, pa, va,

and,a,,

ve, are values

assigned according

to

prior

belief. The convention used for

inverse _gamma distributions is

_given

in the

_Appendix.

The baseline hazard

func-tion

_{>’0 (t)}

will be

_{approximated by}

a

_step

function on a set of intervals defined

by

the different ordered survival

times,

0 <

_t(

₁

₎

< ... <

t(

M

)

< oo:

>’o(t) =

Aom

for

_{t(,!_1)}

< t:=:;

_t(!,);

m =

_1,...,

_M,

with

t<

o

>

= 0 and M the number of dif

ferent uncensored survival times. The

_integrated

hazard function is then

con-tinuous and

_piecewise

linear. A

priori

it is assumed that

_{!oi, ... , A}

OM

are

in-dependent

and that the

_prior

distribution of

A

om

is

_given

_by

_p(A

_om

₎

oc

>’

0

’;’;

m = 1, ... , M.

The

prior

distribution of

Ao

m

=

Ao (t<

m

> ) -

Ao(t(.,))

-M

Aom(t(m) - t(m-,))

is then

_p(A

_om

₎

a

,)-’

o

(A

and

p(Aoi, ... ,

AoM)

oc

II

A

o

m,

m=1 1

by

having

assumed

_independence

of

_{!ol, ... , >’O}

M

a

_priori.

Based on these

as-sumptions and, assuming

furthermore that a

_priori

(A

ol

, ... ,

Ao,!,l),

!3,

(u,

u u 2),

(a, a’)

and

_(e,

_Q

_e)

are

_{mutually independent,}

the

_prior

distribution

of V)

can

be written

2.2. Likelihood and

_{joint posterior}

distribution

The usual convention that survival times tied to

_{censoring times,}

pre-cede the

censoring

times is

_adopted.

_Furthermore,

as in Breslow

_[3],

it is

as-sumed that

censoring occurring

in the interval

_[t(

_m-

₁

₎

_t(m))

occurs at

_t(,,,-

₁

_);

m =

1,...,

M ₊

1,

with

t(

M+i

)

= _oo.

Under the

assumption, where,

conditional on _u, a and _e,

censoring

is

independent

(e.g. [1, 2]),

the

_partial

conditional

_{(censoring omitted)}

likelihood

(e.g.

(15!).

Under the

_{assumptions given above, equation}

₍₅₎

becomes

_ _ r _ _ i

where

_D(t(m»)

is the set of animals that failed at time

_t!&dquo;,!,

_d(t(

_m»

₎

is the number of animals that failed at time

_t!&dquo;,!,

and

_{R(t!&dquo;,!)}

is the set of animals at risk of

_failing

at time

_t(

_m

₎

_’

Furthermore

_{assuming that,}

conditional on _u, a

and _e,

_censoring

is non-informative for

_!,

then the

_{joint posterior}

distribution

of o

is,

using

Bayes’

theorem,

obtained up to

_{proportionality by}

_multiplying

the conditional likelihood and the

_prior

distribution

_{of 0}

where

_{p((y, 8}

_{) 11/i)}

is the conditional likelihood

_given

_by

_equation

₍₆₎

and

_p(qp)

is

the

_prior

distribution of

parameters

and

_{hyperparameters given by equation}

_(4).

2.3.

_{Marginal posterior}

distributions and Gibbs

_sampling

If _cp is a

_parameter

or a subset of

_parameters

of interest from

_1/i,

the

marginal

posterior

distribution of cp is obtained

by integrating

out the

remaining

param-eters from the

_{joint posterior}

distribution. If this can not be

_performed

ana-lytically

for one or more

_parameters

of

_interest,

Gibbs

_sampling

[12, 14]

can

be used to obtain

samples

from the

joint posterior distribution,

and

_thereby

also from any

marginal

posterior

distribution of interest. Gibbs

sampling

is

an iterative method for

_generation

of

samples

from a multivariate distribution

which has its roots in the

_{Metropolis-Hastings}

_algorithm

_{[17, 24!.}

The Gibbs

sampler produces

realizations from a

_{joint posterior}

distribution

_{by sampling}

repeatedly

from the full conditional

posterior

distributions of the

_parameters

in the model. Geman and Geman

_[14]

showed

that,

under mild

_conditions,

and after a

_large

number of

_iterations,

_samples

obtained are from the

_{joint posterior}

distribution.

2.4. Full conditional

_posterior

distributions

In order to

_implement

the Gibbs

_sampler,

the full conditional

_posterior

distributions of all the

parameters

in 1/i

must be derived. The

_following

notation is used: that

_1/i

_B

_<p

_{denotes 1/i}

_except

_{cp; e.g.} if _cp =

{3,

then

1/i

V3

is

(A

01’

... ,

A

om

,

u,

o’!,

a,

<r!,

e,

o, e 2)

The full conditional

posterior

distribution of cp

given

data and all the

remaining

parameters,

1/iB<p’

is

proportional

to the

joint

posterior

distribution

_{of 1/i given by}

_equation

_(7).

where

_Of

=

!!

exp{ai+ei+x!,8}Aom

and

_d(u

_j

₎

is the number of animals !n,a!.&dquo;,,! < y;,

that failed from the

_jth

class of u and

S( Uj)

is the set of animals

_belonging

to the

_jth

class of u. For

_i,

an animal with

records,

the full conditional

posterior

distribution of ai is

given by

where

_{Of =}

_L

_{exp{uj+et+x!}Aomand{!4’’-’}aretheelementsofA !.}

m:t!m! Yi

For an

_{animal, i,}

without

_record,

the full conditional

_posterior

distribution of

a

i follows a normal distribution

_according

to

/ B

The full conditional

_posterior

distribution of

ei,

i =

_{1, ... , n,}

_is,

_up to

propor-tionality,

given

by

where

_Of

=

L

exp{ Uj

_{+ ai}

-f- xi/3!Ao!&dquo;,

and the full conditional

poste-ma!&dquo;,! < y;

rior distribution of each

_regression

_parameter

_{,!6, b =}

_{1, ... ,}

B is

_{given by}

The full conditional

_posterior

distribution of each of the

_{hyperparameters}

<7!, <r!

and

afl is

inverse gamma,

according

to:

and

Sampling

from _{gamma, inverse gamma} and

_normalely

distributed random variables is

_{straightforward.}

The full conditional

_posterior

distribution of _u!, of ai, for

i,

an animal with

_records,

of _ei and of

_regression

_parameters,

_given

by equations

(8), (9), (11)

and

_(12),

_{respectively,}

can all be shown

_[21]

to be

_log

_concave, and therefore

_{adaptive rejection}

_sampling

_[16]

can be used

to

_sample

from these distributions.

_Adaptive

_rejection

_sampling

is useful in

order to

_{sample efficiently}

from densities of

complicated

algebraic

form. It is

a method for

_{rejection sampling}

from _any univariate

_log-concave

_probability

density

function,

which need

_only

be

_specified

_up to

_{proportionality.}

3. AN EXAMPLE

3.1. Data

As an

_example,

disease data on future AI-bulls from the Danish

_performance

testing

programme for beef traits of

dairy

and dual purpose breeds were

analysed.

The trait considered was ’time from

_entering

test until first time

a

_respiratory

disease occurred’. The bulls of the Danish Red breed were all

performance

tested in the

_15-year

_period

1982-1996 and entered the

_Aalestrup

test station between 23 and 74

_days

of _age. Bulls which did not

_experience

a

respiratory

disease

_during

the test

_period

or which were still

undergoing testing,

on the date of data

analysis

have

right

censored records. For these

_animals,

it

is

_only

known that the time at first occurrence of a

_{respiratory disease, T}

_i

_,

will

be

greater

than the time at

_{censoring, C}

_i

_,

that

_is,

either the time at the end of the test

₍₃₃₆

_days

of

_age)

or the time at the date of data

_analysis

or the

time at

_being

culled before end of test

_(a

_very rare

_event).

Data on animal

_i;

i =

_{1, ... , n}

is

(y; ,

6

i

),

where y

i

is the observed value of

_Y

=

min{T

i

,

C

i

}

and

6

i

is a random indicator

variable, equal

to 1 if a

_respiratory

disease occurred

during

test,

and 0 otherwise. Data on all animals is

(y, 6).

3.2. Model

It is assumed that the hazard

_function,

_A

_i

_(t),

of

_T

_i

_,

is

_given

_by

where t is time

_{(in days)}

from

_entering

test. In

_{(17), A}

_o

_(t)

is the baseline hazard

function;

x’

=

(

X

i

l

,

X i 2

, Xi3,

!i4)

is a vector of covariates of animal

_i;

xilranges

between 23 and 74

_days

of age in the data and is the animal’s _age at

_entering

test;

xiz ranges between 0.0 and 1.0 and xi3 ranges between 0.0 and 0.78125

and are

_proportions

of _genes from

foreign

_populations

(American

Brown Swiss

and Red Holstein

_cattle)

and xi4

(which

ranges between 0.0 and

1.0)

is the

degree

of

_{heterozygosity}

due to

_{crossbreeding. x}

_ii

is included in order to take into account that bulls are

_entering

test at different _ages; Xi2 and x23 in order to

take additive effects of gene

importation

into account and xi4 in order to take

account of heterosis due to dominance.

_{

_3’

_

(0

1

, ... , Q4)

is the

corresponding

vector of

_regression

_{parameters. Z}

_i

=

exp{h!

_{+ ks+ ai+}

e

i

is

the

log normally

distributed

_frailty

variable of animal i.

_h

_j

is the effect of the

_jth

herd of

origin

number of herd of

_origin

_by

_{period combination,}

and sk is the effect of

entering

test in the kth yearseason

(one

season is 1

_month),

k =

1,...,

K,

where K is

the number of yearseasons. ai is an additive

_genetic

effect of animal i and ei is an effect of environment not

_already

taken into

_account;

i =

_{1, ... ,}

_n, where

n is the number of animals with records. In this

example

J is

_540,

K is 170 and n is 1 635. The

_relationship

among the test bulls was traced back as far as

possible, leading

to a total of N = 5 083 animals

defining

the additive

genetic

relationship

matrix.

3.3

_{Implementation}

of the Gibbs

_sampler

and results

The Gibbs

_sampler

was

_implemented

with

_prior

distributions

_according

to

the

previous

section. The

prior

distributions of the

_{hyperparameters}

_{a 2,}

_{as, or}

2 and

_or2

_were

_given

_by

inverse _gamma distributions with

_parameters

and

That

_is,

the

_prior

means were of

afl and

Q

a

_{were 0.1 and the}

_prior

means of

0

’;

s 2

and

_Q

_e

were

0.8. The

prior

variance of all the

hyperparameters

is 10 000. The

following

starting

values were

_assigned

to the

_parameters

h!°! _

_{(0, ... , 0)’,}

2

(

0

)

=

0.1,

s!°>

=

(0, ... , 0)’,

as !°!

=

0.8,

a(°) =

0

)’,

0

, ... ,

(

2

(

0

)-

0

.

1

,

e!°! _

_{(0, ... , 0)’,}

u

2

(0)

=

0.8,

!3!°>

=

(0,0,0,0)’.

Sampling

was carried out from

the

respective

full conditional

posterior

distributions in the

_following

order,

describing

one round of the Gibbs

_sampler:

1)

sample

1

1

°r&dquo;,;

m =

_1,...,

M from the

gamma distribution

given

by

equation

(16);

2)

sample

_{h!; j}

=

_{1, ... ,}

J from

_equation

(8)

with u

j

= h

j

and

using

adaptive rejection sampling;

3)

sample

_afl

from the _{inverse gamma} distribution

_given

_by

_equation

₍₁₃₎

with,

a2

=

Oh,

q =

J,

u = h and

(pu, 1

/u)

=

(p

h

,

Vh

);

4)

sample

ai from the normal distribution

given by

equation

(10)

if i is

an animal without

records;

if i is an animal with

records,

ai is

sampled

from

equation

(9)

with

_h

_j

+ s,! substituted for uj in

Of

and

using

adaptive

rejection

sampling;

5)

sample

Q

a

from

the _{inverse gamma} distribution

_{given by equation}

_(14);

6)

sample

ei; i =

_{1, ... ,}

n from

equation

(11) with h

j

+ sk substituted for

Uj in

Of

using

adaptive rejection sampling;

7)

sample

Q

e

from

the inverse gamma distribution

given

by

equation

(15);

8)

sample

(3

b

;

b =

1, 2, 3,

from

equation

(12)

with

h

j

_{+ Sk} substituted for

9)

sample

sk k =

_{1, ... ,}

K from

_equation

(8)

with _uj = _sk and

using adaptive rejection

sampling;

10)

sample

u2

from

the inverse gamma distribution

given

by

(13)

with

a£ = 0’;,

q =

K,

u = s and

u,

M

(

v

u

) =

(u

s

,

v

s

).

After 40 000 rounds of the Gibbs

sampler,

8 000

_samples

of model

_parameters

were saved with a

_sampling

interval of

_20;

i.e. a total chain

_length

of 200 000.

After each round of the Gibbs

_sampler,

the

_following

standardized

_parameters,

of

log frailty,

were

_computed

where

Q

z

=

cr!

₊

a/

₊

a§

₊

ae

is the variance of

log frailty

(not

of survival

time).

Summary

statistics of selected

_parameters

are shown in table 1.

The rate of

_mixing

of the Gibbs

_sampler

was

_{investigated by}

_estimating

lag-correlations in a standard time series

_analysis.

_Lag

1 and

_lag

10 correlations

(lag

1

_corresponds

to 20 rounds of the Gibbs

_sampler)

are

_given

in table I.

_N

_e

is the effective

_{sample size,}

derived from the method of

_batching

_{(e.g. !13!).}

The chain of

_samples

from the

_marginal

_posterior

distribution of

_Q

_a

has _very slow

_{mixing properties.}

This is reflected in the standardized

_parameters

as

well,

The

_marginal

_{posterior density}

of

_a

_and

the standardized

_parameters

_h

₂

_,

c2,

c2

and

e

2

_(of

log

frailty)

are shown in

figure

1. The densities were estimated

by

the

_methodology

of Scott

_!27!.

The mean of the

_{marginal posterior density}

of

h

2

is 0.14 where

h

2

is

heritability

of

_log

_frailty.

If herd of

_origin

_{by period}

and _year

_by

season had been considered as fixed effects rather than random

effects,

then

_heritability

of

_log

frailty

would have been

_higher.

The

_marginal

_posterior

densities of

{3

1

,

,

2

/?

Q3

and are shown in

figure

2.

Because the

_marginal

_posterior

mean of

₍₃

₁

_,

the effect of _age at

_entering

_test,

is

negative

(-0.0061),

the hazard is decreased

by increasing

age at

_entering

test. That

_is,

for a bull

_entering

test at 23

_days

of _age, the conditional hazard is

always

exp{-0.0061

x

23}/exp{-0.0061

x

74}

=

0.87/0.64

= 1.36

higher,

than

that of a bull

_entering

test at 74

_days

of age; conditional on

frailty

and other

covariates

_being

the same for the two bulls.

_Similarly,

because the

_marginal

posterior

mean of

(

3

4

,

the effect of

heterozygosity,

is

_negative

_(-0.22),

the hazard is decreased

_{by increasing}

the

_degree

of

_{heterozygosity.}

The

_marginal

posterior

means of additive effects of _gene

_immigration

from American Brown

Swiss and Red Holstein cattle are close to 0.0. The

_marginal

_posterior

mean of

h

2

,

the

_heritability

of

_{log frailty,}

is

_0.1406;

of

_c!,

the

_proportion

of variation in

log frailty

due to herd of

_origin

_by

period

combination is 0.0913 and of

_c!,

the

proportion

of variation in

_log

_frailty

caused

_by

a _year

_by

season effect is 0.2766.

4. DISCUSSION

This _paper illustrates how a

_Bayesian

analysis

can be carried out in the

semiparametric log

normal

_frailty

model

_using

Gibbs

_sampling.

In the version

of the Gibbs

_sampler

_implemented

_here,

_samples

were

_repeatedly

taken from univariate full conditional

_posterior

distributions. This is

_only

one

_possible

implementation.

With

_highly

correlated univariate

_components

it could be

preferable

to

_sample

from the

_joint

conditional

_posterior

distribution of these

components.

The

_advantage

of this method is its

_greater

_speed

of convergence to

the

_joint

_posterior

distribution

_[23).

The

_methodology

is

_quite

_general

and could

obviously

be used in full

_parametric

models and in models with stratification

and/or

time-dependent

covariates. It is time

_consuming

to

_sample

thousands of observations from thousands of

_{distributions; but,}

_analysing

the

_relatively

small dataset in section 3 left us

_optimistic

about

_{possibilities}

for

_analysing

larger

datasets.

Another

_possibility

is to

_perform

a

Bayesian

analysis

using Laplace

integra-tion. This was carried out

_{by Ducrocq}

and Casella

_[9]

with

_{special emphasis}

on

_obtaining

the

_marginal

_posterior

distribution of

_parameters,

_T, of the

dis-tribution of

_frailty

terms. Their

_prior

distributions of

_frailty

terms were either gamma or

log

normal.

_They

_point

out that the

_computations

_may

_quickly

be-come too

_heavy

and propose to summarize the

approximate

marginal

posterior

of T

through

its first 3 moments

_using

the Gram-Charlier series

_expansion

of a function. The moments are found

_using

numerical

_integration

based on

Gauss-Hermite

quadrature

followed

_by

an iterative

_strategy

to obtain

precise

estimates. When

_frailty

terms are _gamma

distributed,

these can be

integrated

out

_{algebraically}

to obtain the

_marginal

posterior

distribution of the

remaining

parameters.

From a likelihood

_point

of

_view,

_gamma distributed frailties can

This is so in correlated _gamma

_frailty

models as well

_(e.g.

!22!)

but not with

_log

normally

distributed

_frailty

terms. If T is a vector of variance

_components

of

log frailty

terms from a

_log

normal

_frailty

model such as

_equation

(2),

Laplace

integration

may be considered too

_costly

_(9!.

These authors

_{suggested letting}

some

_frailty

terms _{be gamma} distributed and others

_{log normally}

distributed in

order to

_integrate

out

_{algebraically}

some

_frailty

terms.

_{Implementation}

of the Gibbs

_sampler

in a

_log

normal

_frailty

model avoids

_making

the

_{mathematically}

tractable but somewhat artificial choice of a _gamma distribution of

_frailty

terms.

In this _paper some of the

_parameters

are modelled with

_{improper prior}

as

_pointed

out

by

Hobert and Casella

_!18!,

the mathematical

intractability

that necessitates use of the Gibbs

sampler

also makes

demonstrating

the

_propriety

of the

_posterior

distribution a difficult task. The fact that the full conditional

posterior

distributions

_defining

the Gibbs

_sampler

are all _proper distributions

does not

_guarantee

that the

_{joint posterior}

distribution will be _proper. The

question

of which

_{improper prior}

distributions will

yield

proper

joint posterior

distributions in hierarchical linear mixed models was addressed

_by

Hobert and

Casella

_(18!.

The

_{important question}

of

propriety

of the

posterior

distribution

using improper prior

distributions must be dealt with in a

_{Bayesian analysis}

using

a

_{Laplace integration}

as well. One _way to avoid

_{improper posteriors}

is to

use _proper

_prior

distributions.

The

_analysis

could have been carried out without

_augmenting

_[29]

_by

additive

_genetic

values of animals without records. The number of animals

defining

the additive

_genetic

_relationship

matrix A was

_{5 083,}

but

_only

1635

had records. Let a be the

_vector

of additive

_genetic

values of animals with

records,

then

_taking

the

part

A

of A

_relating

to animals with

_records,

the

analysis

could have been carried out

_using

the

_prior

_£[a

_§

_-

_{N.!(0, AQa).}

The number of distributions

_needing

_sampling

from would have been 5 083-1635 lower at the _expense of

_{obtaining marginal}

_posterior

distributions of

_breeding

values on animals without records. However

A-

1

,

is much denser than

_A-

₁

_,

and is more difficult to

_compute.

The model of

_{log frailty, specified}

_by

_equation

_(2),

can

easily

be

generalized

to include more

independent

variance

_components

and/or

the

_assumption

of

independence

can be

_relaxed;

for

_example

in

_equation

_(2),

u could

_represent

a

maternal effect.

_Assuming

that

_{(u’, a’)’1 G rv N}

_2N

_{(0, G}

Q9

_A),

where G is the

2 x 2 matrix of additive

_genetic

_covariances,

and that the

_prior

distribution of G is inverse Wishart

_distributed,

then the full conditional

_posterior

distribution of

_G,

_required

for Gibbs

_sampling,

will also be inverse Wishart distributed.

Furthermore,

it should be

_possible

to carry out a multivariate

_analysis

of a

survival

_trait,

a

_quantitative

trait and a

_{categorical trait,}

by

_assuming

that

the

_{log frailty}

of the survival

_trait,

the

_quantitative

trait and the

_liability

of the

_categorical

trait follow a multivariate normal distribution. It should also

be

_possible

to

_generalize

to an

_arbitrary

number of survival

_traits,

_quantitative

traits and

_categorical

traits.

By definition,

the trait considered in the

_example:

’time until first occurrence

of a

_respiratory

disease

during

test’ can occur at most once for each animal. These data _are,

however,

only

a subset of the data

_being

collected on bulls

during

the test

_period.

Each

repeated

occurrence of a

respiratory

disease is

recorded,

as well as other

_categories

of diseases and several

_quantitative

traits

and other traits of interest. Oakes

_[25]

_gives

a _survey of

frailty

models for

multiple

event time

_data,

and it would be

_interesting

to extend the

log

normal

frailty

models to allow for

_multiple

survival times for each animal as well as a

multivariate

_analysis

of censored survival time data and other traits of interest.

ACKNOWLEDGEMENTS

We wish to thank Anders Holst

_{Andersen, Department}

of Theoretical

_Statistics,

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New

York,

1992.

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D.A.,

Andersen

_{S., Gianola D., Korsgaard LR., Bayesian}

inference

in threshold models

using

Gibbs

sampling,

Genet. Sel. Evol. 27

₍₁₉₉₅₎

229-249.

[29]

Tanner

_{M.A., Wong W.H.,}

The calculation of

posterior

distributions

by

data

augmentation,

J. Am. Stat. Assoc. 82

₍₁₉₈₇₎

528-540.

[30]

Yashin

A.,

Iachine

_I.,

Survival of related individuals: an extension of some fundamental results of

_{heterogeneity analysis,}

Math.

_Popul.

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₍₁₉₉₅₎

321-340.

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Yashin

_{A.L, Vaupel J.W.,}

Iachine

_LA.,

Correlated individual

_frailty:

an

ad-vantageous

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to survival

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of bivariate

_data,

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Vaupel J.W.,

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16

₍₁₉₇₉₎

439-454.

APPENDIX: Convention used for gamma, inverse gamma and

log

normal distributions

The

_following

convention is used for gamma and inverse gamma

distri-butions: let

_{X - r(a, 0),}

with

_density

_p(x)

=

e-

3

(

x/

l

-

a

x

[f(

a

),8

a

r

1

.

Then

E(X)

=

a,8

and

V(X) = a,0

2

=

,8E(X).

Y =

X-’

has _an inverse _or inverted

gamma distribution Y -

IG(a,,8)

with

density

p(y)

=

-(

a)

y

y

e

l

&dquo;+

W

-1

[f(a),8

a

r

1

and

_E(Y)

_{_}

_{[(a -}

_1),8]-

₁

for a > 1 and

_V(Y)

₌

_{[(a -}

₁₎

₂

(

a

-

2

)

0

2

]-l

=

_(E(I’))

₂

_{(a -}

₂₎

_for

a > 2.

If X -

_N(p,,

_a

₂

_),

then Y =

exp{X}

is

said _to have _a

log

normal

distribution;