Bayesian analysis of calving ease scores and birth weights

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Bayesian analysis of calving ease scores and birth

weights

Cs Wang, Rl Quaas, Ej Pollak

To cite this version:

Bayesian analysis

of

_calving

ease scores

and birth

_weights

CS Wang*

RL

_Quaas

EJ

Pollak

Morrison

_{Hall, Department of}

Animal

_Science,

Cornell

_University,

Ithaca,

NY

_14853,

USA

(Received

3

_{January 1996; accepted}

13

_February

₁₉₉₇₎

Summary - In a

_typical

two _stage

_{procedure, breeding}

value

_prediction

for

_calving

ease

in a threshold model is conditioned on estimated

_genetic

and residual covariance matrices.

These covariance matrices are

traditionally

estimated

_{using analytical approximations.}

A

Gibbs

_sampler

for

_making

full

_Bayesian

inferences about fixed

_{effects, breeding values,}

thresholds and

_genetic

and residual covariance matrices to

_{analyze jointly}

a discrete

trait with

_multiple

ordered

categories

(calving

ease

scores)

and a

continuously

Gaussian

distributed trait

_(birth

_weights)

is described. The Gibbs

_sampler

is

_{implemented by}

drawing

from a set of densities -

_(truncated)

_normal,

uniform and inverted Wishart

-making implementation

of Gibbs

_{sampling straightforward.}

The method should be useful for

_{estimating genetic}

_parameters based on features of their

_{marginal posterior}

densities

taking

into full account uncertainties in

_estimating

other _parameters. For

routine,

large-scale estimation of location _parameters

_(breeding

_values),

Gibbs

_sampling

is

_impractical.

The

_{joint posterior}

mode

_given

the

_posterior

mean estimates of thresholds and

_dispersion

parameters is

_suggested.

An

_analysis

of simulated

_calving

ease scores and birth

_weights

is described.

dystocia

/

beef cattle

_/

threshold

_{model / Bayesian}

method

_/

Gibbs sampling

Résumé - _{Analyse bayésienne} des notes de difficultés de _vêlage et des poids de naissance. Dans une

_procédure

typique à deux

_étapes,

l’évaluation

_génétique

_pour la

diff’-cculté

de

_vêlage

dans un modèle à seuil est conditionnée par les matrices de covariance

génétiques

et résiduelles. Ces matrices de covariance sont habituellement estimées au

travers

_{d’approximations analytiques.}

On décrit

_{l’échantillonnage}

de Gibbs permettant

d’effectuer

des

_{inférences bayésiennes complètes}

à propos des

effets fixes,

des valeurs

génétiques,

des

_seuils,

et des matrices de covariance

_génétiques

et

_{résiduelles,}

_pour

_analyser

conjointement un caractère discret à

_{catégories multiples}

ordonnées

_(note

de

_difficulté

de

vêlage)

et un caractère continu _gaussien

(poids

de

naissance).

_{L’échantillonnage}

de Gibbs

est assez

_simple

à partir de densités de divers types : normale

_(tronquée),

_uniforme

et

Wishart inverse. La méthode est utile pour estimer les

_{paramètres génétiques}

à partir

de leurs distributions

_marginales

a _posteriori,

_après

_prise en _compte des incertitudes

*

Correspondence

and reprints: Pfizer Central

_{Research, T201,}

Eastern Point

Rd, Groton,

concernant les autres

_{paramètres. L’échantillonnage}

de Gibbs n’est pas

faisable

en routine pour estimer les valeurs

génétiques.

On

_suggère

le mode de la distribution

_conjointe

a

posteriori,

pour des valeurs des seuils et des

_paramètres

de

_{dispersion correspondant}

à leurs moyennes a

_posteriori.

On décrit une

_analyse

de notes de

_difficulté

de

vêlage

et de

poids

de naissance simulés.

dystocie

/

bovins à viande

_/

modèle à seuil

_/

méthode bayésienne

/

échantillonnage

de Gibbs

INTRODUCTION

Calving

ease is considered a calf trait and recorded

_subjectively

as one of several

exclusive ordered

_categories.

For

_example,

for American Simmental

cattle,

calving

ease is scored as 1

_(natural

_calving,

no

assistance),

2

_(easy

_pull),

3

_{(hard pull)}

or 4

_(mechanical

force or

_Cesarean).

Calf size

_{(birth weight)}

affects ease of birth: the

_bigger

the calf

_is,

the more

likely

the birth will be difficult

(Koger

et

_al,

_1967;

Pollak,

1975).

In this _paper, we consider

_{joint modeling}

of

_calving

ease scores and birth

_weights

using

the threshold model

concept

of

_Wright

_(1934).

In a threshold

model,

an

underlying

continuous variable is

postulated

for

_calving

ease. A set of thresholds

divides this continuous variable into the discrete

_calving

ease scores

_actually

recorded. Gianola

₍₁₉₈₂₎

and Gianola and

_Foulley

₍₁₉₈₃₎

considered

Bayesian

analysis

of

_single

trait threshold models

_assuming

known

_genetic

variance. Harville and Mee

₍₁₉₈₄₎

and

_Foulley

et al

_{(1987) gave}

_approximate

methods for variance

component

estimation.

_Foulley

et al

₍₁₉₈₃₎

_developed

a method to deal with a

binary

trait and two continuous traits without

_allowing

for

_missing

_data,

while Janss and

Foulley

(1993)

extended the method to handle data with

missing

patterns.

In 1990 at Cornell

_University,

a

_system

for routine sire evaluation of

_calving

ease scores and birth

_{weights jointly allowing}

for all

_{possible missing}

data framework was

implemented.

This

_system

assumed a

_sire-mgs

(maternal grandsire)

linear

model for the

_{underlying scale;}

it

_predicts

the

_frequency

of unassisted births for American Simmental cattle

_(Pollak

et

_al,

1995 _pers

_comm).

This evaluation

system

also assumed that

_genetic

and residual covariance matrices and thresholds were

known. Variance

_components

were estimated

(Dong

et

_al,

₁₉₉₁₎

_by

extension of

Foulley

et al

_(1987).

Hoeschele et al

₍₁₉₉₅₎

described further extensions of

Foulley

et al

₍₁₉₈₃₎

and Janss and

_Foulley

₍₁₉₉₃₎

to a situation of one

_multiple

ordered

categorical

trait and several continuous traits.

A

_difficulty

in

_estimating

_parameters

under threshold models is that the like-lihood or

_{marginal posterior}

distributions do not have closed forms and approx-imations are used. With the

_help

of Monte Carlo

_methods,

in

_particular

Gibbs

sampling

(Geman

and

_{Geman, 1984;}

Gelfand et

_al,

_1990),

these

_{approximations}

are no

longer

needed.

_Wang

et al

_{(1993, 1994a,b)}

described

_{making Bayesian}

in-ferences in a univariate linear model in an animal

_breeding

context

_using

Gibbs

sampling.

Sorensen et al

₍₁₉₉₄₎

demonstrated how inference about response to

se-lection in a linear model can be made.

_Berger

et al

₍₁₉₉₅₎

_applied

the methods of Sorensen et al

₍₁₉₉₄₎

and

_Wang

et al

_(1994b)

to

_analyze

a selection

_{experiment of}

the scope to multitrait linear models.

Bayesian analysis

of univariate thresholds via Gibbs

_sampling

in an animal

_breeding

context was

_recently

described

_by

Sorensen

et al

₍₁₉₉₅₎

_{by extending}

Albert and Chib

_(1993).

For a

_binary

_trait,

Hoeschele and

Tier

₍₁₉₉₅₎

_{compared frequency}

_properties

of three variance

_component

estimators: mode of

_approximate

_marginal

likelihood

_(Foulley

et

_al,

_1987),

_marginal

_posterior

mode and mean via Gibbs

_sampling.

Jensen

₍₁₉₉₄₎

_analyzed

simulated data of

one

binary

trait and one continuous trait via Gibbs

_sampling

under a

_Bayesian

framework.

_Wang

et al

₍₁₉₉₅₎

_gave a

Bayesian

method to

_analyze

one

_multiple

or-dered

categorical

trait and one continuous trait with Gibbs

_sampling.

Van Tassell

et al

₍₁₉₉₆₎

_presented

_Bayesian

_analysis

of

_twinning

and ovulation rate

_using

Gibbs

sampling.

The _purpose of this _{paper is} to extend the work of Sorensen et al

₍₁₉₉₅₎

and

Wang

et al

₍₁₉₉₅₎

to one

_multiple

ordered

categorical

trait

_{(calving ease)}

and one

continuous trait

_{(birth weight)}

with all

_{possible missing}

patterns

of data under a

Bayesian setting

via Gibbs

_sampling.

A set of full conditional

_posterior

densities will be derived in closed form

_{facilitating straightforward}

implementation

of Gibbs

sampling.

Simulated data are

_analyzed

to illustrate the

_methodology.

MODEL

Let

Y

lo

be a vector of birth

weights

(BW),

with o

denoting

observed

record,

and

Y

20

be

_calving

ease scores

(CE,

recorded as one of four scores

(1

= no

_assistance,

2 =

easy

pull,

3 = hard

pull

and 4 = mechanical force or

_Cesarean),

_U

_lo

=

_Y

_lo

and

_U

_2o

be the

_{corresponding}

_underlying

variable for observed

_calving

ease _scores,

which is also known as

_augmented

’data’

(Tanner, 1993).

A

_sir!mgs

model used

for the American Simmental

_calving

ease sire evaluation

(Quaas, 1994)

is assumed on the

_underlying

scale

(an

animal model can be assumed

_{by appropriately defining}

terms):

where ai is sire effect

(direct),

and mmgs is maternal

grandsire

effect

(1/4

direct

BV

_plus

_1/2

maternal

_BV)

and elo and e20 are residual effects.

_AgeDam

is _age of

dam effect and CG is

_contemporary

_group effect. Note that maternal effects for BW

are not modeled for the Simmental

_population

because the maternal contribution to the total

_genetic

variance was found to be

_negligible

(Garrick

et

_al,

_1989).

In matrix notation:

For reason of _easy identification of conditional

_posterior

distribution of the residual

covariance matrix

_{later, augmented}

data are further

_expanded

to include residuals associated with

_missing

data. Denote

_U’

=

[U!o e!m]’ U’

=

!U2o e!m]’ el

=

le

l

missing

BW and

_CE,

with m

_denoting

_{missing records, respectively.}

Then

[1]

can

be written as

where U contains

_U

_l

and

U

z

,

W is

_composed

of the

design

matrices -

_X

_i

_s,

_Z

_i

_s

and rows of zeros associated with

_{missing data,}

e contains location

_parameters

13

1

,

13

2

,

aI, a2and , zm and e, the residuals el and ez, with all the elements

properly

ordered and matched. For

_U,

we assume

where R is a block

diagonal

matrix

containing n

submatrices of residual covariances

(R

o

)

for the

_record(s)

of a

particular

animal with dimension

2, ie,

R =

R

o

_®

1,,

if the data are sorted

_by

animal and

_trait,

and n is the number of animals with at least one trait recorded.

A uniform

_prior

distribution is

assigned

to

_[3

_i

_s,

such that

_(eg,

Gianola et

_al,

_1990;

Wang

et al

_1994a)

which is similar to

_{treating 0}

as fixed in a traditional sense. We assume for the bull

effects

_(genetic):

where a _{contains al,} a2and m,

q is

the number of bulls

_(sires

and

_mgs),

G =

G

o

l8iA

with

_Go

=

Igij 1,

i, j =

1, 2, 3,

the _{covariance matrix among} three

_genetic

effects for

a

particular

animal and A is the numerator

_relationship

_{matrix among} sires and mgs.

To describe

_prior

_uncertainty

about

Go,

an inverted Wishart distribution

(John-son and

_{Kotz, 1972;}

Jensen et

_{al, 1994;}

Van Tassell and Van

_Vleck,

₁₉₉₆₎

is

_assigned

where

_Sg

is the location

parameter matrix;

vg is the scalar

shape

parameter

(degrees

of

_{belief); Sg}

=

E(G

o

iSg, Vg).

A

_large

value of _vg indicates relative

certainty

that

Go

is similar to

_Sg;

a small

_value,

_{uncertainty, ie,}

a

_relatively

flat distribution.

(The

subscript

3 of IW indicates the order of the covariance

_matrix.)

_Similarly

for

_R

_o

_,

The final

_parameters

are the thresholds: t =

(t

l

,

_,

₂

_t

t

3

),

with

_to

= -00 and

t

4

= oo. These are assumed to be distributed as order statistics from a uniform distribution in the interval

_{!tm;n, t}

_max]

_(Sorensen

et

_al,

_1995):

where

_I(.)

is an indicator function and

c = 4 in our _case, the number of

_categories.

Applying

Bayesian theorem,

the

_{joint posterior}

_density

of all the

parameters

including

the

_augmented

data

_(8,

_t,

_Go,

,

o

R

elm,

U

2

)

given

the observed data

(Y’ = !Yio!’io!)

and

_prior

_parameters,

_{assuming prior independence}

of

_t,

_Go

and

R

o

is:

if both BW and CE observed

if

_only

BW observed

if

_only

CE observed

with wli

and/or

W2i the incidence vectors associated with

8

1

and/or

0

2

for the ith animal’s

_record(s),

respectively.

The first term in

_[8],

_p(Y

_2oI

_U

_2o

_{, t),}

has a

degenerate density

(Albert

and

Chib,

1993;

Sorensen et

_al,

_1995):

where

_I(.)

is an indicator function. For

example,

for a

_particular

CE score

₍₌

k),

we have

One way to ensure

_{identifiability}

of

_parameters

is to set constraints.

_Assuming

full column rank for the incidence matrix

_W,

two constraints need be

specified.

Usually,

one threshold and the residual variance for CE on the

underlying

scale are set to 0 and

_1,

_respectively

_(Harville

and

Mee,

1984).

An

_equivalent

parame-terization

_(Sorensen

et

_al,

₁₉₉₅₎

is to fix two thresholds and to estimate the CE residual variance. We followed the latter because it allows easy

specification

of the

conditional

_density

of

R

o

.

The two

_{parameterizations, though equivalent,}

_may not

yield

the same

_{joint posterior density owing}

to different sets of

_{priors specified.}

Inference about location and

_dispersion

_parameters

will be based on the

_joint

posterior

density

[9],

or on their

_respective

marginal

_posterior

densities. For

Similarly,

inference about

_Go

is based on:

These densities cannot be derived

_{analytically.}

Monte Carlo

methods,

such as Gibbs

sampling,

draw

_samples

from

_!9!.

Such

_samples,

if considered

_jointly,

are from the

joint posterior

distribution or, viewed

marginally,

from an

_appropriate

marginal

posterior

distribution. Inferences can be based on these drawn

_samples.

Inferences

about functions of

_parameters,

such as heritabilities and

_{genetic correlations,}

can

be made based on transformed

_samples.

Fully

conditional

_posterior

densities

_(Gibbs

_sampler)

The Gibbs

_sampler

consists of a set of

_fully

conditional

_posterior

densities of unknown

_parameters

in the

_model,

ie,

the conditional

_density

of a

_parameter

_given

all other

_parameters

and the data. These can be derived from the

_{joint posterior}

density

[8]

or

!9! .

For location

_parameters

_(0),

we

_keep

terms in

_[8]

that are functions of 0 such

that:

where S2 =

L ! G_1 A

-1

’

with blocks of Os

_{corresponding}

to

_j3

_(Gianola

et

_al,

lo

Gol(&A-11,

l

1990).

This is a normal

_density,

so

where

6

satisfies Henderson’s mixed model

_equations

_(MME)

_(Henderson,

_1973,

1984):

where C =

{CZ! },

i, j = 1, 2, ... , N,

is the coefficient matrix of the

_MME,

with

_C

_jj

_s

as blocks of

C,

and b =

{b

i

},

i = 1, 2,..., N is

the

corresponding right-hand.

The

conditional

_posterior

distribution for the location

_parameters

is:

v

i

-

_Ciil,

e

i

and

_O

_j

are

_subvectors,

_possibly

_scalars,

of 0 and

_0-

_i

is e with

_O

_i

deleted. If

O

i

is a

scalar,

[15]

is the scalar version of the

sampler

for the location

parameters

(Wang

et

_al,

_1994a).

Note the

_similarity

of

_[15.1]

to an

update

in

(block)

Gauss-Seidel iteration. It _may be

_advantageous

to

_sample

a subvector

_jointly

to

speed

up convergence of Gibbs chain. For

example, sampling

all

genetic

effects for

an animal _may reduce serial correlations _among Gibbs

_samples

(Van

Tassell,

_1994;

Garcia-Cortes and

Sorensen,

1996).

From

_!9!,

the full conditional

_posterior

density

of the

_genetic

covariance

_matrix,

as

in Gaussian linear models

_(Jensen

et

_al,

_1994;

Van Tassell and Van

_Vleck,

_1996),

is

Similarly,

the

_fully

conditional

posterior

density

of the residual covariance matrix is

for

_SS

_e

in

_!17!.

From

_[8],

in

_general,

These distributions

_depend

on which combination of records is observed for a

calf: BW

_only,

CE

_only

or both BW and CE. For a

_particular

_calf,

if a BW is

observed

_(U

_lo

_,i

=

Y

l

.,i)

but CE is

_not,

we need

only

to

sample

e2m,i for

CE,

the

distribution is

_only

involved with

_{p(UI8, Ro),}

which follows a univariate normal distribution with

density:

where

_0(.)

is a normal

density function;

/

-t

=

,i

lo

be

=

b(u

lo

,

i

-

w!i8¡);

_w12 is the

incidence vector associated with

A

1

;

Q

2

= _{- 22r}

_rî2/rll;

_{b =}

r

12/

r

l

_{r2!}

=

Ro.

If both BW

_(U

_li

=

Y

ii

)

and CE

(Y

2i

=

k))

_are

observed,

then

only

U

2o

,

i

needs

to be

_sampled,

This is in a form of univariate truncated normal

_(TN)

distribution such that

with tk-1 <

_{U20,i !}

tk

,

where

p

=

w!i82

₊

-

b(U1o

i

,

w!0i), !

as in

_[18]

and w21 is the incidence vector associated with

₈

₂

_.

If

_only

a CE

(Y

2i

=

k)

is

observed,

then both

e lm ,

i and

U

20

,

1

need to be

sampled

Finally,

the conditional

posterior

distribution of a threshold is uniform

(Albert

and

Chib, 1993;

Sorensen et

_al,

_1995),

if CE is not

_missing:

if CE is

_missing.

As mentioned

_previously,

for t =

(t

l

,

,

2

t

t

3

),

there is

only

one estimable

_threshold,

which to estimate is

_arbitrary.

We took:

Note that

t

l

<

t

2

.

If

_only

three

_categories

of CE scores are

_available,

there is no need to estimate thresholds under this

parameterization.

If the fourth

_category

was _{rare, it} would be

_tempting

to combine scores into three

_categories

to avoid

estimating

thresholds.

GIBBS SAMPLING AND POST-GIBBS ANALYSIS

Densities

_[15]-[18]

_(or

_[19]

or

[20]),

and

[21]

(or

[21.1])

constitute the Gibbs

sampler

for our model. Gibbs

_{sampling repeatedly}

draws

samples

from this set of full conditional

_posterior

distributions. After

_burning-in,

such drawn numbers

are random

samples, though dependent,

from the

joint posterior

density

!9!.

Let

the Gibbs

_samples

of

_length

m for a

_particular

_parameter,

say for the direct

genetic

variance

_component

_gll for

_BW,

be x =

{xi},

i =

1, 2 ... , m.

An estimate of the

mean of the

_marginal

_posterior

density,

p(g

nI

Y),

is:

and the

_posterior

variance can be estimated

_by:

Modes and medians can also be used to estimate location

_parameter

of a

_posterior

density

(Wang

et

_al,

_1993),

_though

_usually

_requiring

more Gibbs

samples

because

Monte Carlo errors. Because the Gibbs

samples

are

_correlated,

one _way to estimate Monte Carlo errors is to

_adopt

standard time series

_{analysis techniques}

as

_suggested

by Geyer

(1992)

and used

_by

Sorensen et al

_(1995).

ROUTINE GENETIC EVALUATION

The

_preceding

sections describe a

Bayesian

_analysis

via Gibbs

_sampling

for

infer-ences about all the

parameters

in the model

_including

fixed

_effects,

_{(functions of)}

breeding values,

genetic

and residual covariance

_matrices,

and thresholds based

on

_{marginal posterior}

densities. This is sensible because all uncertainties in

esti-mating

other

_parameters

are taken into account when inference is made about a

particular

parameter

of

_interest,

_say for the

_breeding

value of a sire.

_However,

it is

computationally

expensive

to _carry out

_large

scale

_analyses

_routinely.

A

_practical

compromise

is to estimate covariance matrices and thresholds

_using

a full

_Bayesian

analysis

via Gibbs

_sampling

once and

subsequently

to estimate location

_parameters

based on conditional densities of location

_parameters

_given

the estimated

_dispersion

and threshold

_parameters.

As data

_accumulate,

covariance matrices and thresholds

are reestimated.

_Explicitly,

we

_suggest

a

_two-stage

_procedure

that

_might

be useful for a

large

scale routine

_genetic

evaluation _program:

1)

Estimate

Go,

R

o

and t

_using

mean

_(mode

or

median),

based on their

respective

marginal

posterior

densities,

p(Go!Y), p(R

oI

Y)

and

_p(t!Y),

via Gibbs

sampling, dropping

prior

parameters

Sg,

S

e

,

V9 and Ve in notation for

convenience;

and

2)

Estimate location

_parameters

based on the

following

conditional

_density:

which is an

_{approximation}

to the

_{corresponding}

_marginal

_density:

if the

_{marginal density,}

_p(t,

_Go,

_{Ro !Y),}

is

_symmetric

or

_peaked

(Box

and

_{Tiao, 1973;}

Gianola and

_Fernando,

_1986).

In other

_words,

if there is sufficient information in the data to estimate

_G

_o

_{, R}

_o

and t

_well,

then

_[25]

is a

good approximation

to

_!26!.

This would be the case if the data set is one of the national data

_bases,

for

_example,

the American Simmental data set. Note that the

_underlying

variable for CE

_(U

₂₀

₎

and the residual vector associated with the

_missing

data

_(e

_im

and

_e

_2m

₎

have been

integrated

out of the

_joint

_posterior

_density

_!9!;

that W matrix no

longer

contains

blocks of Os

_{corresponding}

to the

_{missing data, however,}

the same notation is

_kept

below. Note also that o and m

_denoting

observed and

_missing

data are

dropped

from the notation because

_missing

data no

longer

play

a role.

The

_{joint posterior}

mode of

_[25]

can be considered as

_point

estimates for

_e,

Foulley,

1983).

There are at least two _ways to

_compute

the MAP estimates:

expectation-maximization

(EM)

(Zhao,

1987;

Quaas, 1994,

1996)

and

Newton-Raphson

(Gianola

and

_{Foulley, 1983; Foulley et al,}

_1983).

The EM iteration

_equation

_{(Quaas, 1996)}

is:

where the coefficient matrix is

_exactly

the same as that in the usual MME with S2

_containing

_Go

₁

_&reg;

A-’

and blocks of Os

_{corresponding}

to the fixed

_effects,

the

superscript

in

_[27]

indicates iteration

_number,

and

U’ _

1 ut 1 fit 21 -

For a

particular

record,

with

Y

2i

=

_k,

if both BW and CE observed if

_only

CE observed

if both BW and CE observed

if

_only

CE observed with b as in

_[18],

The

_{Newton-Raphson}

iteration

_equation

_(Janss

and

_Foulley,

_1993;

_Quaas,

_1994,

1996;

Hoeschele et

al,

1995),

following closely

the notation of

_Quaas

_(1996),

is:

where,

for a

particular record,

with p

as in

[28.1],

Q

as in

_!28.2!,

_P

_k

as in

_[28.3!,

and

Structurally,

R

in

_[29]

is the same as R in

[2]

but

_consisting

of

R

o

matrices as

with b as in

[18]

and rn the residual variance of BW. It is clear that

R

o

depends

on

0, t, R

o

and

_Y;

thus it is record

_specific.

If an animal is

_missing

BW or

CE,

f

l

o

1

is a scalar: _-y or

rill,

respectively.

A

Bayesian analog

of the Beef

Improvement

Federation measure of

_{’accuracy’}

of

a bull’s

_genetic

_prediction

(BIF, 1990)

is:

If information contained in the data conflicts with

_prior

_belief,

then

_posterior

variance could be

_larger

than

prior

variance

_resulting

in a

_negative

_{accuracy. This}

is

_peculiar

to a

_{frequentist, particularly}

to a

_{producer, why}

after

collecting

data on

his

_animal,

the

_uncertainty

about his animal’s BV has increased! Posterior variances of

_breeding

values are

_usually

approximated,

based on

large

sample

theory, by

the

inverse of Fisher’s

_(expected)

information matrix or

_by

the inverse of

_negative

Hessian matrix. The latter

approximation

is:

where l =

_log{[25]}.

This is the inverse of the coefficient matrix of

_[29].

Note that

_[27],

the inverse of the coefficient matrix used for

_EM,

is not a

_large

_sample

approximation

to the

_posterior

variance matrix of

_(25!,

or, at

least,

not a _very

good

one. We shall return to this

_point

in the numerical

_example

section below.

NUMERICAL EXAMPLE

A data set

_representing

continuous BW and discrete CE scores with ordered

categories, 1-4,

of ’Simmental calves’ was simulated and

analyzed

to illustrate the

methodology.

Model

The records were simulated under the sire-maternal

_grandsire

model

[1].

Model

equations

for the kth calf with ith bull as sire and

_jth

bull as maternal

_grandsire

were

for BW with

_only

direct effects and CE

_underlying

variable with both direct and maternal

_{effects, respectively.}

The random vectors

_[a

_lk

_,

_{a2k ,}

_m

_2k]

were drawn from

a

N(0, Go)

distribution,

_ie,

bulls were unrelated.

Similarly

_!2lijk

e

2zjk]

-

N(0,

Ro).

Parameter values were similar to

_previous

estimates from the Simmental data

_(Dong

were scaled to obtain

R

o

and

_Go.

_Assuming

the

_phenotypic

SD for BW was nine

units and

_hb

_w

= 0.4 _!

afl!

=

1/4

x 0.4 x 81. The CE

_components,

_aa2 and _am2’

were assumed

equal

and

computed by

solving

h!e

=

x a e2 2

2a x 2 4

+

a2

for

a; =

!a2 2

=

a;’ . 2

2Q! -I- Uez

The residual SD which establishes the scale of

U

2

was set to ten to be

_comparable

with

_BW;

_hfl!

= 0.2. The

dispersion

matrices were

As found in

previous

analyses

of Simmental data

_(Zhao,

1986, 1987; Quaas

et

_al,

1988),

thresholds were

equally spaced

at 1 SD intervals: ti =

_0,

_t

₂

= 10 and _t3 = 20.

Corresponding

to

!65%

unassisted births of Simmental calves out of

_2-year-old

dams

_{(unpublished data):}

p

2 =

-!’’(0.65) =

-3.8532.

Thus,

P

k

=

«){tk-/-l2)/ae2}-«){tk-l -/-l2)/ae2} =

0.6500, 0.2670,

0.0744 and

0.0085,

for k =

_{1, 2, 3, 4,}

_{respectively.}

For

_BW,

_!,1 = 85.

Calving

ease scores were

assigned

such that

_Y

_2zj£

= k if

1

k

_

t

<

_U

₂

_{ijk <}

_t

_k

_.

Design

There were 75 bulls in three batches of 25. Bulls in the first batch were MGS of the calves sired

_by

bulls in the second

batch;

bulls 26-50 had two _progeny from

daughters

of each of the first 25 bulls.

Likewise,

bulls 51-75

_{(third batch)}

had two progeny from

daughters

of each of the second 25 bulls

(second batch).

Thus bulls 1-50 were each MGS of 50

calves;

bulls 26-75 were each sires of 50

_calves;

only

bulls 26-50

_(second

_batch)

were both sires and MGS. There were a total of 2500

calves,

each with a BW and a CE score.

Gibbs

_sampling

Priors for 1-1, t and

R

o

were uniform while that for

Go

was the inverted Wishart

in

_[5]

with

_Sg

a

_diagonal

matrix of the

_genetic

variances used for simulation and vg = 5. The latter

’slightly

informative’

prior

was

adopted

because

during

initial

testing

of the programs, a

_{nearly singular}

SSg

would cause the

_sampler

to crash. The last

Go

would look reasonable but would have an

effectively

zero

eigenvalue.

The informative

_prior

was

incorporated

to ensure

_(SSg

+

_Sg)

was

_positive

definite.

Subsequently,

the

_problem

was found in the simulation program where both

h

2

were almost _zero;

_Go

was

_nearly

singular!

The

safeguard,

however,

was left in the

program.

The initial

_dispersion

matrices were

diagonal

matrices of the variances used for

simulation,

bull effects were null and the _average BW was

assigned

to _!1. Thresholds

and _p2 were

computed

from the observed

frequencies:

_p2 = 10 x

!-!-1(Cl)!

and

t

From these

_starting

values the

parameters

were

repeatedly sampled

in the

following

order:

_U

_2ijk

from

_[19],

t

3

from

_{(21!, (}

₁

_p

_-L

_J

₂

₎

_jointly

from

_[15],

_(a

₁₂

a2i

m

2

)

jointly

from

_(15],

_Go

from

_IW

₃

_(SSg+Sg,

_{75+5), (16),}

and

_R

_o

from

_IW

₂

_(SS

_e

_,

2

500-3),

(17!.

Each bull’s three effects were

_{sampled jointly}

as were the two _ps. Most of

the results to be

_presented

came from a

_single

chain of 10 500

samples;

all _summary

statistics were

_computed

from the last 10 000. We feel a

_single

chain of this

length

is sufficient to demonstrate the

_procedures

but would

_probably

not suffice for a real

application.

A few references will be made to results from

_replicated

Gibbs chains and

_replicated

data. Our purpose

here, however,

was not to

_study

Monte Carlo

error,

single long

chain versus

multiple

short chains nor the small

sample properties

of

_posterior

means; no

_{systematic study}

was carried out on these

replications.

Comparisons

of

_{frequentist properties}

of estimators of certain

parameters

are found

in Jensen

₍₁₉₉₄₎

and Hoeschele and Tier

_(1995).

The most time

_consuming

_step

was

sampling

the CE

underlying

variables. This was due

_partly

to the number of these but also because of the way the truncated

normals in

_[19]

were

generated,

ie,

normals were

sampled

until one was obtained

within the _range determined

by

the CE score. The time for a

complete cycle

could

almost

triple;

this

_variability

was

_entirely

due to

_generating

the truncated normals. A more efficient truncated normal

_generator

might

be necessary for a

_large

data set in which the

_probability

of some

_categories

_{is very}

_small,

eg, a score of four for a

heifer calf out of a mature dam. In several million

records,

there will be a handful

of these and

_they

will cause

problems.

Joint conditional

_posterior

modes

Joint

_posterior

modes

_{of p}

and the bull effects

_given

_{t, R}

o

and

_Go

_[25]

were

computed by

both the EM

_[27]

and

_{Newton-Raphson}

_[29]

algorithms.

The values of

_t,

_R

_o

_,

and

_Go

were fixed at the estimates of their

_posterior

means from the

Gibbs

sampler.

Starting

values were the same as in the Gibbs

sampler.

Iteration was continued until

_log

_io

of the maximum absolute

change

of any bull effect was

< -10 at which

point

the modal estimates

of p

and bull effects

_{computed by}

the alternative

_algorithms

differed

_by

<

10-

l0

.

NUMERICAL RESULTS

Visual

_inspection

of

_plots

of

parameters

(or

functions of

_parameters)

against sample

number

_suggests

that a burn-in of 500

cycles

was

_probably

_unnecessary

(eg,

fig

1).

The

parameters

in

_figure

1 were chosen to illustrate the

markedly

different

_patterns

observed as a result of correlations _among

sequential

Gibbs

samples.

There were

marked differences _among

_parameters

in these serial correlations

_{(table II)}

which

were

particularly high

for the

threshold,

t

3

,

and also for the CE residual

variance,

r 22

. In

contrast,

mixing

for the other _{rij, all 9ij and the bull effects} was much more

rapid. However,

overall _convergence of a Gibbs chain is tied to

_parameters

with slow

_mixing

_properties.

In another

_words,

one cannot declare that a Gibbs chain is

converged

for some

_parameters

but not others.

Estimates of

_posterior

means and SDs are

_presented

in table I. The

_parameters

simulate the data. This was not the _case,

_however,

for the

parameters

affecting

CE. The

_dispersion

_parameters,

_especially,

were

_markedly

smaller than the ’true’ values.

Too much cannot be made of this from a

single

sample

and we cannot conclude that such a

_pattern

is a _consequence of

_using

the

_marginal

posterior

mean as an

neither t

k

nor _aú is identifiable but

(t

k

-

tk_1)/Q!e

is identifiable. The ’true’ thresholds were

_{equally spaced}

1 residual SD

apart;

analogous

functions of

t

l

, t

2

(fixed)

and the

_posterior

means of

_t

₃

and r22 are 0.996 and

_{0.954, quite}

close to the ’true’ values.

Likewise,

estimated heritabilities and

_{genetic correlations,}

table

_III,

differed

_considerably

less from the ’true’ values than did the

_{(co)variances.}

For this small

_example

_precise

estimates of the

_marginal

_posterior

densities were

not

_attempted.

Coarse

_histograms

were

_examined;

most were

slightly

skewed _away

from zero; in

figure

2 are

typical

examples.

None were

_’peaked’

in the _eyes of the

authors.

Though

the EM and

Newton-Raphson

algorithms

both _gave the same values for

the

_joint

conditional modes of

_[25],

there was a marked difference in the

_pattern

of convergence: linear versus

_quadratic

_{(fig 3).}

The EM

_algorithm

took six times as

many

rounds,

but

_elapsed

times were close: EM < two times

_longer.

For a

large,

field data

_application

it is

_questionable

as to which

_algorithm

will be

_faster;

we do

not

_expect

a

large

difference. While EM is easier to

_{code, approximations}

of SDs are not a

_by-product

as

_they

are if the Hessian is

_computed.

to the

_marginal

_posterior

means of

[26]

despite

the

_posterior

densities for

_t

₃

and

dispersion

parameters

being

neither

symmetric

nor

_particularly

_peaked.

The

similarity

of

_marginal

mean and

_joint

conditional mode is

graphically

illustrated for

the CE-D effect in

_figure

4. The

_regression

of mode on mean is

_slightly

_greater

than

unity, 1.08,

reflecting

the

greater

spread

of modes when there is no

_uncertainty

in thresholds nor

_dispersion

_parameters.

This was seen for all three bull effects.

Correlations,

within bull

_batch,

were also

_computed

between true and estimated

(mode

or

mean)

bull effects. These correlations caused debate among the authors.

The

_Bayesian

amongst

us was not sure how to

_interpret

them whereas the

_lapsed

frequentists

had no

_difficulty

whatsoever: a

big

number is

good;

a small number

is not so

_good.

_Except

for CE-D for batch 1 bulls and CE-MGS for batch 3 bulls

where the estimation came

_entirely

from correlated

_information,

these correlations seem

_satisfyingly

_large.

Of some interest was the observation that the correlations

between true and estimated direct

effects,

both BW and

_CE-D,

were a bit

_higher

for

the batch 3 bulls than for the batch 2 bulls. Both had 50 progeny but the latter also had 50

_{grandprogeny.}

What seems to be

_adding

information from

_{grandprogeny,}

apparently,

is noise. This could be due to

_{sampling, though}

the same

_pattern

was seen in three

_independent

_samples

of simulated data.

The

_posterior

SD were also examined. These were

approximated

in three _ways:

from the coefficient matrix of the MME used in EM

_[27!,

from the Hessian used in

Newton-Raphson

[29]

and

_by

Gibbs

_sampling.

In

_applications

these are sometimes

used

_{interchangeably but}

they

are

_not,

of _course, the same. The first is the

_posterior

SD of

_{p(elU 1,}

Û2,

G_o, R_o)

_or

the

_{frequentist’s}

standard error of

_prediction

(SEP)

for BLUP

_{p(elU 1,}

-

f J

2

,

Go,

R

o

)

note that this assumes the CE

underlying

variable

Carlo error. These are

_plotted

for the bulls’ BW effects in

_figure

5. It is

_immediately

obvious that the first behaves

_quite

_differently

to either of the latter two. The former

depends only

on

_{Go, R}

_o

and the information that is

_available,

hence a different

value for each of the three batches of bulls. The second also

depends

on

Go

and

R

o

(and

3

,)

t

but also on

, â

2

l

j!, â

and m2

_minimally,

hence the values vary a bit but

noticeably

so

only

for the bulls with _progeny.

(Considerably

more variation was seen

for the CE effects and the SDs were

_larger,

in

_general,

_reflecting

the fact that

_U

₂

was

not

_actually

_observed.)

In marked

_contrast,

the

_marginal

_posterior

SD are all over

the

_place.

This is not

_just

due to _{Monte Carlo error,} see

_figure

6 where results from

two

_independent

Gibbs

_samples

are

_plotted.

It is due to the

_dependence

of the SD on

the

_posterior

means. This is shown

_graphically

in

_figure

7.

_Presumably

this

pattern

is due

_primarily

to the

_uncertainty

of the

dispersion

parameters.

Heuristically,

a

bull _{with average progeny BW will have} zero

â

1k

regardless

of the value of

_(or

uncertainty

about)

hb

w

whereas the

_uncertainty

about

_(SD)

_a

_lk

for a bull whose

progeny BW

depart

markedly

from average will

depend

on

_uncertainty

about

hb

w

.

DISCUSSION AND CONCLUSIONS

Following

closely

the work of Sorensen et al

_(1995),

we have extended the Gibbs

sampling

scheme to make full

_Bayesian

inferences about

location, dispersion,

and thresholds from

_modeling

one

_multiple

ordered

_categorical

trait

_(CE)

and one

about

_genetic

and residual covariance matrices and thresholds are based on their

respective

marginal

posterior

densities in a unified fashion without

_analytical

ap-proximations,

in contrast to traditional methods based on

_{approximations}

(Foulley

et

_{al, 1987;}

Harville and

_Mee,

_1984).

Our model can be

expanded

to include

heterogeneous

variances in a similar _way as for linear models

(Gianola

et

_al,

_1992).

_Foulley

and Gianola

₍₁₉₉₆₎

_expanded

a

_log-linear

structural model to describe

_{heterogeneous}

variances

_(Foulley

et

_al,

1990;

San Cristobal et

_al,

₁₉₉₃₎

to a

single-trait

threshold model. This idea can be

extended to model

_{heterogeneous}

variances in our situation as well.

It is

_advantageous

to estimate location

parameters

including

fixed effects and

breeding

values

_jointly

with

_dispersion

_parameters

because uncertainties in esti-mation of

_dispersion

_parameters

can be taken into

_account,

_particularly

in small

populations. However,

in real

_genetic

evaluation

systems

with data sets of millions of

_records,

_{joint modeling}

_may be neither

_possible

nor _necessary. With

_good

esti-mates for

_dispersion

_parameters

and thresholds from the

_{joint posterior}

_density

_[9]

from

_large

data

_sets,

we _argue that the conditional

_posterior

_density

of location

parameters

[25]

with such estimated

_dispersion

_parameters

as fixed values is a

good

approximation

to the

_{marginal posterior}

_density

of location

parameters

!26!.

In our

numerical

_example

the

_joint

mode of

_[25]

_approximated

well the

_marginal

_posterior

The

_posterior

SD of

_[26],

however,

was not well

_{approximated by}

the inverse of the coefficient matrix of

_[27]

or

[29].

The latter are

_large

_{sample approximations;}

our

data were not numerous.

A _{main purpose} of the data

_augmentation

used in the _paper was to result in

fully

conditional

_posterior

distributions of

parameters

in standard

_recognizable

forms such that

_samples

were

_easily

drawn. We chose to

_augment

the

_underlying

variables of observed CE

_(U

_2o

₎

and the residuals associated with

_missing

BW and

CE

_(e

_lm

and

_e

_2m

₎

_l

the whole

_augmented

data vector was

(e!m U!oe!m)’

Another

way is to

augment

the

_underlying

variables of observed CE

_(U

_2o

₎

and that of

missing

CE

_(U

_2m

_),

and continuous data

_{corresponding}

to

_missing

BW

_(Y

_l

_,

_r

_,);

the whole

_augmented

data vector is

_(Y!m!2o!2m)-

Sorensen

₍₁₉₉₆₎

_gave a

_parallel

treatment of the

problem

under the latter data

_{augmentation.}

In

_general,

the whole

_design

matrix W is considered to be fixed. In other

_words,

the

_design

vector w

i associated with an observed or a

_missing

(either

BW or

CE)

calf record is

augmentation

was,

implicitly, equivalent

to

_treating

wis associated with

missing

BW and CE as random and to

_assigning

uniform

_priors

to

_them,

and

_integrating

them out of the

_{joint posterior density,}

as

_opposed

to Sorensen

₍₁₉₉₆₎

in which the

joint posterior

density

was conditioned on those wis of

missing

BW and CE. We

conjecture

that our

_approach

has

_{computational}

_advantages

while not

_sacrificing

theoretical

_simplicity,

_particularly

when the

_missing

data rate is

_high.

For

_example,

for American

_Simmental,

about

_1/3

of the records have either BW or CE

_missing.

It is clear also that other combinations of data

_augmentation

are

_possible,

such as

(

e

lm

U

2

o

U

2m)

(Y!m U!oe!m)

or

(ei

m

e2

o

ez

m

).

If no data

_augmentation

is

used,

fully

conditional distributions of certain

_parameters

may not be in

_recognizable

forms and alternative

_{sampling procedures}

such as

_rejection

_sampling

need to be used.

Zeger

and Karim

₍₁₉₉₁₎

presented algorithms

for a

_single

trait threshold model

without data

_{augmentation.}

There is a

_danger

in

_{using improper priors}

in a

_Bayesian

analysis.

In a linear

model

_setting,

some

_{improper priors}

induce

_{improper joint posterior}

densities even

though

the

_fully

conditional

_posterior

densities are well defined

(Hobert

and

Casella,

1996).

We do not know whether or not the uniform

_priors

we used in the numerical

example

for p, t and

R

o

will induce a _proper

_{joint posterior}

_density

_!9!.

The fact that no difficulties were encountered in

_analysis

does not

_necessarily

mean that

[9]

was suitable. The safe _{way may} be to use a noninformative but _proper

_prior

in an

ACKNOWLEDGMENTS

We thank the American Simmental

_Association,

Bozeman

_MT,

for

_{partially supporting}

this work. We thank D Sorensen for

_sending

us a _copy of his lecture notes

_(Sorensen,

1996)

in which an alternative data

_augmentation

is

_detailed,

and are indebted to him

along

with an _anonymous reviewer for numerous constructive criticisms which

_improved

the

_presentation

of the paper

considerably.

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