Inequality of maximum a posteriori estimators with equivalent sire and animal models for threshold traits

HAL Id: hal-00894106

https://hal.archives-ouvertes.fr/hal-00894106

Submitted on 1 Jan 1995

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Inequality of maximum a posteriori estimators with

equivalent sire and animal models for threshold traits

M Mayer

To cite this version:

Original

article

Inequality

of

maximum

a

posteriori

estimators

with

_equivalent

sire

and animal models

for

threshold traits

M

_Mayer

University of Nairobi, Department of

Animal

_Production,

PO Box 29053,

Nairobi, Kenya

(Received

25

_{May 1994; accepted}

17

_May

₁₉₉₅₎

Summary - It is demonstrated that the use of

equivalent

sire and animal models to

describe the

_underlying

variable of a threshold trait leads to different maximum a

_posteriori

estimators. In the

_{simple example}

data sets examined in this

_study,

the use of an animal

model shrank the maximum a _posteriori estimators towards zero

_compared

with the

estimators under a sire model. The differences between the 2 estimators were

_particularly

noticeable when the

_heritability

on the

_underlying

scale was

_high

and

_they

increased with

increasing heritability. Moreover,

it was shown that even the

ranking

of

breeding

animals may be different when

applying

the 2 different estimators as

_ranking

criteria.

threshold trait

_/

genetic evaluation

_/

maximum a _posteriori estimation

Résumé - _Inégalité entre des estimateurs du maximum a _posteriori avec des modèles équivalents père et animal pour des caractères à seuil. Il est démontré que l’utilisation

des modèles

_{équivalents père}

et

_animal,

_pour décrire la variable

_sous-jacente

d’un caractère à

_seuil,

conduit à

_différents

estimateurs du maximum a

_posteriori.

Dans les

_exemples

simples analysés

dans cette

_étude,

en _comparant l’utilisation de ces 2 modèles on constate

que les estimateurs du ma!imum a

_posteriori

tendent à se

_rapprocher

de 0

_lorsqu’on

utilise le modèle animal. Les

_différences

entre les 2 estimateurs sont

_{particuLièrement}

notables

_quand

l’héritabilité sur l’échelle

_sous-jacente

est

élevée,

et elles augmentent avec

l’héritabilité. De

plus,

il est aussi montré que même le classement des

reproducteurs

peut

être

_différent

selon l’estimateur utilisé comme critère de classement.

caractère à seuil

_/

évaluation génétique

/

estimation du maximum a _posteriori

INTRODUCTION

New

_procedures

based on the threshold

_concept

have

_recently

been

_developed

methods introduced

_by

Gianola and

_Foulley

_(1983),

Stiratelli et al

₍₁₉₈₄₎

and Harville and Mee

₍₁₉₈₄₎

are

_essentially

the same and were derived

_mainly

in a

Bayesian

framework. With these

methods,

location

_parameters

in the

_underlying

scale are estimated

_by

_maximizing

a

_{joint posterior density.}

The estimators are

therefore

_designated

as maximum a

_posteriori

estimators. Studies on the

_properties

of maximum a

_posteriori

estimators and

_comparisons

with the estimators and

predictors

of linear model methods have

previously

been based on sire models

to describe the variable on the

underlying

scale

(eg,

Meijering

and

Gianola,

1985;

H6schele,

1988;

Weller et

_{al, 1988;}

Renand et

_{al, 1990; Mayer,}

_1991).

For

continuously

distributed traits the use of an animal model is the state of the art for the

_genetic

evaluation of

_breeding

animals. A

_logical

_step

would be to use an animal model as well to describe the

_underlying

variable in

_genetic

evaluation with

threshold traits. In the

_present

paper some

_properties

of maximum a

_posteriori

estimators are studied in the context of

_applying

an animal model to describe the

underlying

variable of threshold traits. METHODS

Example

data set

Consider the

_following

_simple

data structure

₍₂

_sires,

each with 1

_progeny)

for a

dichotomous character with the 2 realizations

_M

_o

₍₀₎

and

_{M, (1):}

Progeny

1 exhibits the realization

_M

_o

and _progeny 2 the realization

_M

_l

_.

Assume

that we want to estimate the

_breeding

values of the sires for the trait M. Maximum a

_posteriori

estimation

To estimate the

_breeding

values of the

sires,

the threshold

concept

is used and we

follow the

_Bayesian

_approach

_along

the lines introduced

_by

Gianola and

_Foulley

(1983),

Stiratelli et al

_(1984),

and Harville and Mee

_(1984).

Let the data be

arranged

as a 2 x 2

_{contingency table,}

where the 2 rows

_represent,

as

_usual,

the

sire

_effects,

and the 2 columns

_represent

the realizations

_M

_o

and

_M

_i

_.

The elements of the

_contingency

table in the ith row and

_jth

column are

_designated

as _nij. The

The rth

experimental

unit in row i is characterized

by

an

underlying

continuously

distributed

variable,

which is rendered discrete

through

a fixed threshold.

Let us assume that the variance

_components

in the

description

of the

underlying

variable are

_known,

at least to

_{proportionality. Using}

a ’flat’

_prior

for the threshold

(t),

so as to mimic the traditional mixed model

_analysis,

and

_assuming

that the sire effects on the

_underlying

scale

_(u

_l

_{, u}

₂

₎

a

_priori

are

_{normally, identically}

and

independently

distributed with mean 0 and variance

Q

u,

the

_{joint posterior density}

has the form

₍

_C1

=

constant):

Using

a

probit-transformation

and a

purely

additive

_genetic

model for the

underlying

variable

_(4l:

standardized normal cumulative

_density

_function):

The residual standard deviation

₍

_Qe

₎

is taken as the unit of measurement on the

underlying

scale,

so that Qe = 1.

The maximum a

_posteriori

estimators are the values which maximize the

_joint

posterior

density

g(/3).

This is

_equivalent

to the maximation of the

_{log-posterior}

density,

which can be done more

_easily.

The first derivatives of the

_{log-posterior}

density

with

_respect

to t and ui are not linear functions of these

_parameters

and

must be solved

_iteratively

_using

numerical

_techniques

such as

Newton-Raphson

or

Fisher’s

_scoring

_procedure.

The

_{Newton-Raphson}

_algorithm

leads to the

_following

iterative scheme:

where

_(0:

standardized normal

_{probability density}

_function):

The

_{joint posterior density}

_g((3)

is

_symmetric

in the sense that

g(t

=

z i , U1 = _X2,U2 =

.r3!!,!,!)

=

9

(t

= -X1 , U1 = -X3,U2 =

-!2!!,!,!).

From this

_expression

it follows

_directly

that the

_posterior

_density

has its maximum where

t = 0 and _{U1 + U2} = 0.

Making

use of this

_result,

the iterative

equation

[2]

reduces to:

The estimated

_breeding

values of the sires

_equal

2u !k)

at _{convergence. For}

compatibility

reasons with the maximum a

_posteriori

estimators in the

following

section,

which are based on an animal model to describe the

_underlying

variable and where the environmental standard deviation

₍

_QE

₎

is taken as the unit of

measurement,

the estimator in

_[3]

_may also be

_expressed

in units of QE

by

multiplying

u!k)

by

[(1 -

4

c

)/(1 -

2

/

1

)]-

C

or

(1 -

3<!)’!,

_{respectively,}

where c

is the intraclass correlation

_Q

_u/(1+Qu).

Moreover,

it is convenient to _express the maximum a

_posteriori

estimators in terms of the

_phenotypic

standard deviation ap =

(

U2

₊

!e)1!2

by

multiplying

the estimator from

[3]

by

[1/(l -

c

)]-

l

/

2

or

(

a2

+

1)-1!2,

respectively.

Use of an

_equivalent

animal model

Up

to now we have dealt with a sire model to describe the

_underlying

variable

for the threshold trait. Let the rows of the

contingency

table now

_represent

the

progenies

in which the observations are made and we use an

equivalent

animal

model to describe the basis variable and to estimate the

_breeding

values of the sires. _{For that purpose let}

v’

=

[!1!2 !3 !4],

where _vl and _v2

_represent

the additive

genetic

values on the

_underlying

scale of the

_progenies

1 and

_2,

and v3 and v4

represent

the additive

_genetic

effects of their sires 1 and 2.

Under these conditions v is a

priori normally

distributed with mean 0 and variance-covariance matrix

_{Acr 2,}

where A is the numerator

_relationship

matrix and

_a2

_the

additive

_genetic

variance on the

underlying

scale. For the

example

data

set the

_relationship

matrix has the

_following

form:

The likelihood function

_invariably

follows a

product

binomial distribution. So

the

_{joint posterior density}

is now:

with

and the environmental variance

_(!E)

as the unit of measurement on the

_underlying

scale.

For the

_joint

_posterior

_density

_[5]

the

_{Newton-Raphson}

_algorithm

leads after a

where:

It can be

_easily

seen that for

vi

k

)

and v2!!

_{respectively,}

this

_equation

_system

is

formally

very similar to the iterative

_equation

_(2!.

_Further,

it shows the result that

V

i - Vi

/2

and

V4

2

/2,

ie the estimated

_breeding

values of the sires

equal

half of the estimated

_genetic

effects of their

_progenies.

To _express the maximum a

_posteriori

estimators in units of the

_phenotypic

standard deviation

_they

have to

be

_multiplied,

as in the case of the sire model

by

[1/(1 -

c)!-1!2

or

(ua

+

1

)-

1

/

2

,

respectively.

More

_complex

data structure

A more

_complex

data

structure,

viz the

_hypothetical

data set

_already

used

_by

Gianola and

_Foulley

₍₁₉₈₃₎

to illustrate the maximum a

_posteriori

estimation

procedure,

was

_employed

to demonstrate the differences between the estimators based on either a sire or an animal model. The data consist of

_calving

ease scores

from 28 male and female calves born in 2

_herd-years

from heifers and cows mated to

4 sires.

_Calving

ease was scored as a _{response in 1} of 3 ordered

_categories

(normal

birth, slight difficulty,

extreme

_difficulty).

The data are

_arranged

into a

_contingency

table in table I.

Two different

_heritability

values on the

_underlying

scale

(h

2

= 0.20 and

h

2

=

0.60)

were

_postulated.

For the maximum a

_posteriori

estimation of the location

parame-ters of the

_underlying

variable,

the

computer

program

package

TMMCAT

_(Mayer,

1991,

1994a)

was used.

Different sire

_rankings

even the

_ranking

of the sires _may be different

_depending

_upon the model

_type

chosen. To examine this

_question,

data for 3 sires were

_{systematically generated}

and

_analyzed.

The number of observations _nij of sire i

_(i

=

1,2,3)

in

response

category j

(j

=

1, 2)

_was varied from 0 to 8. A

simple

random

one-way model

(sire

model,

animal

_model)

for the

_underlying

variable and a

_heritability

value of 0.60

was

_{postulated. Again}

for the

_analyses,

the

_computer

_program TMMCAT was used.

RESULTS

Example

data set

Figure

1 shows the

_relationship

between the estimated

_breeding

value

_(EBV)

of sire

1 and

_heritability

for each of the models. The upper

graph

represents

the estimates

taking

the environmental standard deviation as the unit of

_measurement,

while in

the lower

_graph

the estimates are

_expressed

in units of the

_phenotypic

standard deviation.

It is obvious that the EBV is different whether a sire model or an

_equivalent

animal model is used to describe the

_underlying

variable. The difference between

the EBVs is noticeable at a

heritability

of about 0.60 and increases with

_increasing

the EBV based on a sire model shows an almost linear

_relationship

with the

heritability

even if with

_{increasing heritability}

the

_{slope slightly}

decreases. With a

higher

intraclass correlation on the

underlying

scale as with the animal

model,

this

exhibits a maximum at a

_heritability

value of about 0.81 and decreases

rapidly

thereafter.

Hypothetical

data set

_by

Gianola and

_Foulley

Table II shows the maximum a

_posteriori

estimates of the

_parameters

of the

hypothetical

data set in table I for the 2 different model

_types

_(sire

model and

animal

_model).

The estimates are

_expressed

in units of the

_phenotypic

standard deviation. The estimable linear function

h

+

H

i

+ A

h

+

S

7

&dquo;,

represents

the baseline.

Further,

in table II the estimates of the contrasts

_t

₂

_{- t}

_l

_(threshold

2 - threshold

_1),

H2 - H

i

(herd-year

2 - herd-year

1),

A!-Ah

(cow - heifer calving), 8f -8m (female

- male

calf)

are shown.

Moreover,

table II contains the EBVs of the

_sires,

which in the case of the sire

model,

for reasons of

comparison,

were calculated as twice the estimated sire effects.

If !j

+

!i

represents

the estimator of a

_particular

estimable combination i of the

explanatory

effects,

then the

_probability

of a _response in the

jth

_category

(pz!)

can

be

_{estimated as}

_p

_il

=

<1>[(t1

₊

!i)/Q),

i2

p

=

!!(t2

₊

.Ài)/u] -

<1>[(t

1

₊

!i)/!!

and

A

3

= 1 -

!P[(F2

₊

!t)/o’];

where _0’ is the _error standard deviation

(sire model)

_or the

environmental standard deviation

_{(animal model).}

For the

_heritability

value of 0.20 the estimators from the sire model and the animal model are

_quite

similar. The absolute estimates from the animal model are a little

smaller,

with the

_exception

of the estimated distance between the thresholds. The

_greatest

relative difference is shown

_by

the contrast between cow effect and

heifer effect. When a

heritability

of 0.60 is

applied,

the differences between the

model are

_{smaller, only}

the estimated distance between the thresholds is

_greater

and the

greatest

relative difference is found for the contrast between cow effect and

heifer effect.

Different sire

_rankings

Table III shows 3 data structures for which different sire

_rankings

under the different

model

_types

_(sire

model and animal

_model)

were identified.

The estimates based on the animal model are

_again

smaller than the estimates

from the sire model. The different

_rankings

under the 2 model

_types

are

_interesting.

Under the sire model the EBVs of the sires 3 are

_clearly

_greater

than the values

of the sires

_2,

whereas under an animal model it is the other _way around and the

EBVs of the sires 3 are

_clearly

smaller than the EBVs of the sires 2. Other data

structures with different sire

_rankings

were identified and table III shows

_only

a

few

_examples.

It

_might

be

_expected

that with

_{higher heritability}

values and more

complex

data

_designs

this

topsy-turvy

situation would be even more marked.

DISCUSSION

Studies on the

_properties

of maximum a

_posteriori

estimators for threshold traits have been based on sire models to describe the

_underlying

variable. Interest has

mainly

focused on

_comparisons

of maximum a

_posteriori

estimators with the

estimators and

predictors

of linear model methods. As

_regards

the

_EBVs,

or more

exactly

sire

_evaluation,

the

_comparisons

have been based on

_empirical

product-moment

_{correlations,}

sire

_rankings,

and the realized

_genetic

response obtained from

truncation selection.

_Meijering

and Gianola

₍₁₉₈₅₎

showed that with a balanced

random _one-way classification and

_binary

_responses, the

_application

of

quasi-best

linear unbiased

_prediction

_(QBLUP)

and maximum a

_posteriori

estimation

(MAPE)

in the mean true

_breeding

values of sires with either

_QBLUP

or MAPE as

selection rules were not

_significant.

In simulation studies where the data were

generated by

a _one-way sire model with variable progeny _group

_size,

there were

noticeable differences in

_efficiency

between

_QBLUP

and MAPE

_only

when the incidence was

_high

_(>

90

_%,

_binary

_response)

and with

_high

_{(> 0.20)}

_heritability

(Meijering

and

Gianola, 1985; H6schele,

1988).

When

_applying

mixed models for data

_simulation,

MAPE was

_generally

found to be

_superior

in

_comparisons

with

QBLUP.

This

_{superiority depends}

on several factors: differences in

incidence,

intraclass

_correlation,

unbalancedness of the

_layout,

and differences in the fixed

effects and the

proportion

of selected candidates

_(Meijering

and

_Gianola,

_1985;

H6schele, 1988;

Mayer,

1991).

The

_relationship

between the relative

_superiority

of

MAPE and the

_proportion

of selected candidates was not found to be of a

_simple

kind

_{(Mayer, 1991).}

Somewhat in contrast to these

_findings

in the simulation studies

were the results of

_{investigations}

where

_{product-moment}

correlations between

MAPE and

_QBLUP

estimators were calculated

(Jensen,

_1986;

_Djemali

et

_{al, 1987;}

H6schele,

1988;

Weller et

al, 1988;

Renand et

al,

1990).

The correlation coefficients

throughout

were _very

_high.

This is even more

_astonishing

because

_{theoretically}

there

is no linear

relationship

between MAPE and

QBLUP,

and so even with identical sire

_ranking

the correlation coefficient is smaller than one.

Generally,

a sire model can be considered as a

special

case of an animal

model,

making

very restrictive

supositions.

For

_continuously

distributed

_traits,

the use of an animal model is

_currently

the state of the art for

_genetic

evaluation of

_breeding

animals.

_Basically,

all the

_practical

and theoretical reasons in favour of the use of

an animal model instead of a sire model for traits

_showing

a continuous distribution

are also valid for threshold traits.

However,

in the

present

paper it was

clearly

shown that with threshold

_traits,

the

maximum a

_posteriori

estimators of the

_parameters

of the

_underlying

variable or the

estimators of the

breeding

value are different whether a sire model or an

equivalent

animal model is used to describe the

_underlying

variable. This

_is,

of course, an

unfavourable behaviour of the estimation

method,

but as

_long

as the

_ranking

of

breeding

animals is not

_affected,

it would not be so

_problematic

from a

_breeding

point

of view. As mentioned

_{above, QBLUP}

and MAPE

_yield

_exactly

the same

ranking

of sires for a _one-way random model with

_binary

_responses and constant progeny group size

although

there is no strict linear

relationship

between

QBLUP

and MAPE.

_However,

as was shown in this

_study,

with maximum a

_posteriori

estimation,

even the

ranking

of

breeding

animals _may

depend

on the model

_type

chosen.

The result that the use of

equivalent

sire and animal models for the

_underlying

variable and the use of the same estimation method lead to different _{estimators may} be counterintuitive. Under the animal model the maximum a

posteriori

estimators are obtained

_{by computing}

the mode of the

_{joint posterior}

distribution of the

’environmental’ effects and the

_breeding

values of all the animals. Under the sire

model the mode of the

_{marginal posterior}

_density

for the sires is

_computed.

In the

_analyses

_{of the}

_{simple example}

data set in this

_study,

the

_{’marginal’}

posterior

density

for

_the

sires

_equals

the

_{joint posterior}

_density,

after

_integrating

out the

JJ

g(t,

Vl, V2, V3,

!4 !!

8v

2’

h

1

)8v

2

It is well known that the

posterior

densities in

this

_study

are

_only

_{asymptotically}

normal and

_generally

do not have

_typical

forms

of distribution

_(see,

for

_example,

_Foulley

et

_al,

_1990).

Therefore the mode of the

marginal posterior density

function is different from the mode of the

_{joint posterior}

distribution. The situation is different in the linear model case or more

_precisely

when the likelihood follows a normal distribution. As shown

_by

several authors

(eg,

Gianola et

al,

1990)

then,

under the

_assumption

that the variance

components

are

_known,

the

_{joint posterior}

_density

is multivariate

_normal,

ie modes and means are identical and

_marginal

means can be derived

_directly

from the vector of

_joint

means.

If the maximum a

_posteriori

estimates

_depend

on the model

_type

used to describe the

_underlying

variable,

as shown in this paper, there may arise the

question

as to

which is the better estimator. In animal

_breeding,

the best selection

rule,

in the

sense of

_maximizing

the

_expected

value of the mean of the

_breeding

values of a

fixed number of individuals selected from a fixed number of

candidates,

is the

mean of the

_{marginal posterior}

distribution of the

_breeding

values

(Goffinet,

_1983;

Fernando and

_Gianola,

_1986).

_Only

because its calculation is

_{usually thought}

to be unfeasible for threshold traits

_(Foulley

et

_{al, 1990;}

Knuiman and

_Laird,

_1990),

are

joint posterior

modes or maximum a

_posteriori

estimators considered and

_regarded

as an

_{approximation}

to the

posterior

mean. For convenience and

_simplicity,

let us assume that in the

analysis

of the

_simple

example

data set in this

_study,

the value of the threshold is known to be t = 0. The

_marginal

_posterior

mean can then be

calculated,

not

_{only by}

numerical _means, but also

_{analytically.}

Under the conditions that the

_breeding

values and the environmental effects are

_{normally, identically}

and

_{independently}

distributed with mean 0 and variances

Q

a

and uf respectively,

and

_mutually

_independent,

it follows from well-known statistical

_properties

of the normal distribution that the

_marginal

_expected

value

_E[alM

_o]

is:

Using

the

_{additive-genetic}

transmission

_model,

where p is a _progeny of sire s

and dam

_d,

and _am is a random variable with mean 0 and variance

u’12

_describing

the Mendelian

_sampling:

the

_{marginal breeding}

value of sire 1

₍

_U1

_),

conditional to his progeny

exhibiting

the

realization

_Mo(n2!)

is:

In table IV this

_marginal

mean estimator is

_compared

with the maximum a

posteriori

estimators under the 2 different model

types

(sire

model and animal

model).

In the

_simple

_example

data set of this

_study

the maximum a

_posteriori

estimator under the sire model is much closer to the

_marginal

mean estimator. This is

very poor when

compared

with the

marginal

mean estimator. In a more

_thorough

study

of

_marginal

mean estimation

_{by Mayer}

(1994b),

similar results were found

in the

_analysis

of the

_calving

ease data set

_presented

in table I.

_Thus,

a normal

density approximation

may not be

_justified

for

posterior

distributions when _nij values are small as under an animal model. Further

Mayer

(1994b)

showed that

the standard deviation of the

_marginal

_posterior

_density

was

_clearly

_greater

than

the

_approximate

standard deviation of the maximum a

_posteriori

estimates derived

from the observed or estimated Fisher information matrix.

ACKNOWLEDGMENTS

The author is

_grateful

to the anonymous reviewers for their

helpful suggestions

and

comments on the

_manuscript.

REFERENCES

Djemali M, Berger PJ,

Freeman AE

₍₁₉₈₇₎

Ordered

_categorical

sire evaluation for

_dystocia

in Holsteins. J

_Dairy

Sci

_70,

2374-2384

Fernando

RL,

Gianola D

₍₁₉₈₆₎

_Optimal

_properties of the conditional mean as a selection

criterion. Theor

_Appl

Genet

_72,

822-825

Foulley JL,

Gianola

D,

Im S

₍₁₉₉₀₎

Genetic evaluation for discrete

polygenic

traits in animal

_breeding.

In: Advances in Statistical Methods

_for

Genetic

_{Improvement of}

Livestock

_(D

Gianola,

K

_Hammond,

_eds),

_{Springer-Verlag, Berlin,}

361-409

Gianola D,

Foulley

JL

₍₁₉₈₃₎

Sire evaluation for ordered

_categorical

data with a threshold

model. Genet Sel

Evol

15, 201-224

Gianola

_D,

Im

S,

Macedo FW

₍₁₉₉₀₎

A framework for

_prediction

of

_breeding

value. In: Advances in Statistical Methods

_for

Genetic

_{Improvement of}

Livestock

_(D

_Gianola,

K

Hammond,

eds),

Springer-Verlag, Berlin,

210-238

Goffinet B

₍₁₉₈₃₎

Selection on selected records. Genet Sel Evol

15,

91-97

Harville

_DA,

Mee RW

₍₁₉₈₄₎

A mixed model

procedure

for

analyzing

ordered

_categorical

data. Biometrics

_40,

393-408

Jensen J

₍₁₉₈₆₎

Sire evaluation for _type traits with linear and non-linear

_procedures.

Livest Prod Sci 15, 165-171

Knuiman

_MW,

Laird NM

₍₁₉₉₀₎

Parameter estimation in variance component models for

binary

response data. In: Advances in Statistical Methods

_for

Genetic

_{Improvement of}

Livestock

_(D

_Gianola,

K

_Hammond,

_eds),

_{Springer-Verlag, Berlin,}

177-189

Mayer

M

₍₁₉₉₁₎

_Schatzung

von Zuchtwerten and Parametern bei

kategorischen

Schwellen-merkmalen in der Tierzucht. Habilitation

_thesis,

Hannover School of

_Veterinary

Sciences,

Hannover

Mayer

M

_(1994a)

TMMCAT - a _{computer program}

_using

non-linear mixed model threshold _concepts to

_analyse

ordered

_categorical

data. In: Proc 5th World

_Congr

Genet

A

PP

L

Livest Prod

_22,

49-50

Mayer

M

_(1994b)

Numerical

integration techniques

for point and interval estimation of

_breeding

values for threshold traits. In:

l!5th

Annual

_Meeting

_of

the

European

Association

_for

Animal Production.

_Edinburgh,

5-8

_{September 1994}

Meijering A,

Gianola D

₍₁₉₈₅₎

Liner versus nonlinear methods of sire evaluation for

categorical

traits: a simulation

_study.

Genet Sel Evol

_17,

115-132

Renand

_G,

Janss

_LLG,

Gaillard J

₍₁₉₉₀₎

Sire evaluation for direct effects on

_{dystocia by}

linear and threshold models. In: Proc

_4rd

World

_Congr

Genet

_Appl

Livest Prod 465-467 Stiratelli

R,

Laird

N,

Ware JH

₍₁₉₈₄₎

Random effects models for serial observations with

binary

response. Biometrics 40, 967-971

Weller JI, Misztal

_I,

Gianola D

₍₁₉₈₈₎

Genetic

_analysis

of

_dystocia

and calf

_mortality

in