Statistical analysis of ordered categorical data via a structural heteroskedastic threshold model

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Statistical analysis of ordered categorical data via a

structural heteroskedastic threshold model

Jl Foulley, D Gianola

To cite this version:

Original

article

Statistical

_analysis

of

ordered

_categorical

data via

a

structural heteroskedastic

threshold model

JL

_Foulley

D

Gianola

2

1

Station de

_génétique

quantitative et

_appliquée,

Institut national de la recherche

agronomique,

centre de recherches de

_Jouy,

78352

Jouy-en-Josas cedex, France;

2

Department of

Meat and Animal

_{Science, University of Wisconsin-Madison,}

Madison, WI 53706,

USA

(Received

2 October

_{1995; accepted}

25 March

₁₉₉₆₎

Summary - In the standard threshold

_model,

differences _among statistical

subpopulations

in the distribution of ordered

_{polychotomous}

_responses are modeled via differences in

location _parameters of an

_underlying

normal scale. A new model is

_{proposed whereby}

subpopulations

can also differ in

_dispersion

(scaling)

_parameters.

_{Heterogeneity}

in such parameters is described

_using

a structural linear model and a

_loglink

function

_involving

continuous or discrete covariates. Inference

(estimation,

_{testing procedures, goodness}

of

fit)

about parameters in fixed-effects models is based on likelihood

_{procedures. Bayesian}

techniques

are also described to deal with mixed-effects model structures. An

application

to

calving

ease scores in the US Simmental breed is

_presented;

the heteroskedastic threshold model had a better

goodness

of fit than the standard one.

threshold character

_/

_{heteroskedasticity}

_/

maximum

_likelihood/

mixed linear model

_/

calving difficulty

Résumé - Analyse statistique de variables discrètes ordonnées par un modèle à seuils hétéroscédastique. Dans le modèle à seuils

_classique,

les

_différences

de

_réponses

entre

sous-populations

selon des

_catégories

discrètes ordonnées sont modélisées _par des

différences

entre

_paramètres

de _position mesurés sur une variable normale

sous-jacente. L’approche présentée

ici _{suppose que} ces

_{sous-populations diffèrent}

_{aussi par} leurs

paramètres

de

dispersion

(ou

paramètres

d’échelle).

L’hétérogénéité

de ces

paramètres

est

décrite _par un modèle linéaire structurel et une

fonction

de lien

logarithmique impliquant

des covariables discrètes ou continues.

_{L’inférence}

_(estimation,

_{qualité d’ajustement,}

test

d’hypothèse)

sur les

paramètres

dans les modèles à

effets fixes

est basée sur les méthodes du

maximum de vraisemblance. Des

_{techniques bayésiennes}

sont

_{également proposées}

_pour le

traitement des modèles linéaires mixtes. Une

_application

aux notes de

_difficultés

de

_vêlage

en race Simmental américaine est

_présentée.

Le modèle à seuils

hétéroscédastiqué

améliore dans ce cas la

_qualité

de

_{l’ajustement}

des données _{par rapport} au modèle standard.

caractères à seuils

_/

hétéroscédasticité

_/

maximum de vraisemblance

_/

modèle linéaire

INTRODUCTION

An

_appealing

model for the

_analysis

of ordered

_categorical

data is the so-called

threshold model.

_Although

introduced in

_population

and

_quantitative

_genetics

by Wright

(1934a,b)

and discussed later

_by

_Dempster

and Lerner

₍₁₉₅₀₎

and Robertson

_(1950),

it dates back to Pearson

_(1900),

Galton

₍₁₈₈₉₎

and Fechner

(1860).

This model has received attention in various areas such as human

_genetics

and

_{susceptibility}

to disease

_(Falconer,

1965;

Curnow and

_{Smith, 1975), population}

biology

(Bulmer

and

_{Bull, 1982; Roff,}

_{1994), neurophysiology}

_{(Brillinger, 1985),}

animal

_breeding

_{(Gianola, 1982),}

_survey

_analysis

_(Grosbas,

_1987),

_{psychological}

and social sciences

_(Edwards

and

Thurstone, 1952;

McKelvey

and

Zavoina,

1975),

and econometrics

_(Kaplan

and

_{Urwitz, 1979;}

_Levy,

_1980;

_Bryant

and

_{Gerner, 1982;}

Maddala,

1983).

The threshold model

_postulates

an

underlying

(liability)

normal

distribution

rendered discrete via threshold values. The

_probability

of response in a

_given

category

can be

_expressed

as the difference between normal cumulative distribution

functions

_having

as

_arguments

the _upper and lower thresholds minus the mean

liability

for

_{subpopulation}

divided

_by

the

_{corresponding}

standard deviation.

Usually

the location

_parameter

₍

_?7i

₎

for a

_{subpopulation}

is

_expressed

as a linear

function

77i

=

t’O

of _some

explanatory

variables

_(row

incidence vector

_ti)

_(see

theory

of

_generalized

linear

models,

McCullagh

and

Nelder, 1989;

Fahrmeir and

Tutz, 1994).

The vector of unknowns

_(e)

may include both fixed and random

effects and statistical

_procedures

have been

_developed

to make inferences about

such a mixed-model structure

_(Gianola

and

_Foulley,

1983;

Harville and

Mee, 1984;

Gilmour et

_al,

_1987).

In all these

_studies,

the standard deviations

_(also

called the

scaling

parameters)

are assumed to be known and

_equal,

or

_proportional

to known

quantities

(Foulley,

1987;

Misztal et

al,

1988).

The purpose of this paper is to extend the standard threshold model

(S-TM)

to a model

allowing

for

heteroskedasticity

(H-TM)

with

_modeling

of the unknown

scaling

parameters.

For

_simplicity,

the

_theory

will be

_{presented using}

a fixed-effects

model and likelihood

procedures

for inference. Mixed-model extensions based on

Bayesian

techniques

will also be outlined. The

_theory

will be illustrated with an

example

on

_{calving difficulty}

scores in Simmental cattle from the USA.

THEORY

Statistical model

The overall

_population

is assumed to be stratified into several

_{subpopulations (eg,}

subclasses of _sex,

_parity,

_age,

_{genotypes, etc)}

indexed

by

i =

_{1, 2, ... ,}

I

_representing

potential

sources of variation. Let J be the number of ordered _response

_categories

indexed

_by

_j,

_{and yi+ =}

_{

_Yij+

_}

be the

_(J

x

₁₎

vector _{whose element y2!+} is the

total number of responses in

category j

for subpopulation

i. The vector y2+ can be

and

₍₃₎

I - 1 contrasts _among

_log-scaling

_parameters

_(eg,

_ln(Q

_i

_{) -}

_ln(<7i)

for

i =

2, ... , I)

or,

equivalently,

I - 1 standard deviation ratios

_{(eg, O}

_&dquo;d

_O

_&dquo;d,

with

one of these

_arbitrarily

set to a fixed value

_(eg,

_<7i =

1).

This makes a total of

2I + J - 3 identifiable

parameters,

so that the full H-TM reduces to the saturated

model for J = 3

categories,

see

_examples

in Falconer

(1960),

_chapter

18.

More

_parsimonious

models can also be envisioned. For

instance,

in a two-way crossclassified

layout

with A rows and B columns

(I

=

AB),

there _are 16

additive models that can be used to describe the location

₍

_?7i

₎

and the

_{scaling (o}

_j

₎

parameters.

The

_simplest

one would have a common mean and standard deviation

for the I = AB

populations.

The _most

complete

one would include the main effects

of A and B factors for both the location and

_dispersion

parameters.

Here there are

2(A+B)+J-5

estimable

_parameters,

_ie,

J-1 thresholds

_plus

twice

_(A-1)+(B-1).

Under an additive model for the location

_parameters

_71i, it is

possible

to fit the H-TM to

_binary

data. For the crossclassified

_layout

with A rows and B

_columns,

there are AB -

2(A

+

_B)

+ 3

_degrees

of freedom available which means that we

need A

_{(or B) !}

4 to fit an additive model

_using

all the levels of A and B at both the location and

_dispersion

levels.

_Finally,

it must be noted that in this

_particular

case,

dispersion

parameters

act as substitutes of interaction effects for location

parameters.

Estimation

Let T =

{7!}

for j

=

1, 2, ... ,

J - 1 and _a =

(

T

’, (3’,

b’)’.

In fixed-effects models

with multinomial

data,

inferences about a can be based on likelihood

procedures.

Here,

the

_log

likelihood

_{L(a; y)}

can be

_expressed,

_apart

from an additive

_constant,

as: T T

with, given

!4!,

[5]

and

_!6!,

The maximum likelihood

_(ML)

estimator of a can be

computed using

a

second-order

_algorithm.

A convenient choice for multinomial data is the

_scoring

_algorithm,

because Fisher’s information measure is

_simple

here. The

_system

of

_equations

to

solve

_iteratively

can be written as:

-

-where

_{U(a; y)}

=

<9L((x;y)/<9<x

and

J(a)

_

-E

[å2L(a;y)/åaåa’]

are the score

function and Fisher’s information matrix

_{respectively;}

k is iterate number.

Analyt-ical

_expressions

for the elements of

_{U(a; y)}

and

_J(a)

are

_given

in

Appendix

1.

These are

generalizations

of formulae

given by

Gianola and

Foulley (1983)

and

Misztal et al

_(1988).

In some

_instances,

one _may consider a

backtracking procedure

(Denis

and

_Schnabel,

₁₉₈₃₎

to reach _convergence,

_ie,

at the

_beginning

of the iterative

process,

compute

a!k+1!

as

a[k+1]

=

ark]

₊

,cJ[k+1]!a[k+1]

with 0 _<

w[

k+1

] ::::;

1.

A constant value of w = 0.8 has been

_satisfactory

in all the

_examples

run so far

(over

the ni observations made in

subpopulation i)

of indicator vectors yir =

(Yilr i Yi2r i... i Yijri ... i YiJr)l

such that

_!r=l

1 or 0

_depending

on whether a

response for observation

(r)

in

population

(i)

is in

category

(j)

or not.

Given ni

_independent

repetitions

of

Yin

the sum _{yi+ is}

_{multinomially}

distributed

j

with

_{parameters n}

_i

_{= !}

_yij+ and

_probability

vector

_Ii

_i

=

{lIi

j

}.

j = l

In the threshold

_model,

the

_{probabilities}

_1I

_jj

are connected to the

_underlying

normal variables

_X

_ir

with threshold values _Tj via the statement

with To = -oo and _{Tj =}

₊

₀₀

_,

so that there are J - 1 finite thresholds.

With

_Xi

_r

_rv

_N(!2,Q2),

this becomes:

where

_!(.)

is the CDF of the standardized normal distribution.

The mean

_liability

₍

_?

_7i)

for the ith

_{subpopulation}

is modeled as in Gianola and

Foulley (1983)

and Harville and Mee

_(1984),

and as in

_generalized

linear models

(McCullagh

and

Nelder,

1989)

in terms of the linear

_predictor

Here,

the vector

_(p

x

₁₎

of unknowns

₍₀₎

involves fixed effects

_only

and _xiis the

corresponding

(p x 1)

vector of

qualitative

or

quantitative

covariates.

In the

H-TM,

a structure is

_imposed

on the

_scaling

_parameters.

As in

_Foulley

et al

_{(1990, 1992)}

and

_Foulley

and

_Quaas

_(1995),

the natural

_logarithm

of Qi is written as a linear combination of some unknown

_(r

x

₁₎

real-valued vector of

parameters

(6),

1

p’

being

the

_{corresponding}

row incidence vector of

_qualitative

or continuous

covariates.

Identifiability

of

_parameters

In the case of I

_{subpopulations}

and J

_categories,

there is a maximum of

_{I(J - 1)}

identifiable

_{(or estimable) parameters}

if the

_margins

ni are fixed

_{by sampling.}

These

are the

_parameters

of the so-called saturated model.

What is the most

_complete

H-TM

_(or

’full’

_model)

that can be fitted to the data

using

the

_approach

described here? One can estimate:

₍₁₎

J - 1 finite threshold

Goodness of fit

The two usual

_statistics,

Pearson’s

X

Z

and the

_(scaled)

deviance D* can be used

to check the overall

_adequacy

of a model. These are

where

_fig

=

77

tj

((x)

is the ML estimate of

1I

j

,

and

Above,

D* is based on the likelihood ratio statistic for

_fitting

the entertained

model

_against

a saturated model

_having

as _many

parameters

as there are

alge-braically

independent

variables in the data

_vector,

_ie,

_{1(J -}

₁₎

here. Data should

be

_grouped

as much as

_possible

for the

_asymptotic

_chi-square

distribution to hold in

_[9]

and

_[10]

_(McCullagh

and

_{Nelder, 1989;}

Fahrmeir and

_Tutz,

_1994).

The

_degrees

of freedom to consider here are

I(J - 1) (saturated model)

minus

((J - 1)

+

_rank(X)

+

_rank(P)]

_(model

under

_study),

where X and P are the

inci-dence matrices for

(3

and b

_{respectively.}

It should be noted

that

_[9]

and

_[10]

can be

computed

as

particular

cases of the _power

divergent

statistics introduced

_by

Read

and Cressie

_(1984).

Hypothesis testing

Tests of

_hypotheses

_{about y} =

6’)’

can be carried out via either Wald’s test or

the likelihood

ratio

_(or

_deviance)

test. The first

procedure

relies on the

_properties

of

_consistency

and

_asymptotic

_normality

of the ML estimator.

For linear

_hypotheses

of the form

_H

_o

_:

_K’y

= m

_against

its alternative

_H

_l

=

H

o

,

the

Wald

statistic

is:

which under

_H

_o

has an

_asymptotic

_chi-square

distribution with

rank(K) degrees

of

freedom.

_Above,

_r(y)

is an

_appropriate

block of the inverse of Fisher’s information

matrix evaluated at _y=

y

, where

y

is the ML estimator.

The likelihood ratio statistic

_(LRS)

allows

_testing

nested

_hypotheses

of the form

Ho :

y E

_no

_against

_H

₁

_:

_y - E n

no

where

no

and ,f2 are the restricted and

unrestricted

_parameter

_spaces

_{respectively,}

_pertaining

to

_H

_o

and

_H

_o

U

_H

_l

_.

The

LRS is:

where

y

and

_{y are}

the ML estimators of y under the restricted and unrestricted

models

_{respectively.}

The criterion

!#

also can be

_computed

as the difference in

This is

_equivalent

to what is

_usually

done in ANOVA

_except

that residual sums of squares are

replaced

_by

deviances.

Under

_H

_o

_,

A#

has an

_asymptotic

_chi-square

distribution with r =

dim(D)

-

dim(

Do

)

degrees

of freedom. Under the same null

hypothesis,

the Wald and LR

statistics

have the same

_asymptotic

distribution.

_However,

Wald’s

statistic

is based on a

_{quadratic approximation}

of the

_{loglikelihood}

around its maximum.

Including

random effects

In many

applications,

the

_y

_i

_{r ’s}

cannot be assumed to be

_{independent repetitions}

due to some cluster structure in the data. This is the case in

quantitative

_genetics

and

animal

_breeding

with

_genetically

related

animals,

common environmental effects

and

_repeated

measurements on the same individuals.

Correlations can be accounted for

_conveniently

via a mixed model structure on

the

_’T7!S,

written now as

where the fixed

component

x!13

is as

before,

and u is a

(q

x

₁₎

vector of Gaussian

random effects with

_{corresponding}

incidence row vector

_zi.

For

_simplicity,

we will consider a _one-way random

model, ie,

u !

N(O,

Ao,

u

2

)

(A

is a

positive

definite matrix of known elements such as

kinship

coefficients),

but the extension to several

_u-components

is

_{straightforward.}

The random

_part

of the location is rewritten as in

_Foulley

and

Quaas

(1995)

as

Z!O&dquo;Ui u*

where u* is a

vector

of standard normal

_deviation,

_{and au,} is the _square root of the

_u-component

of

_variance,

the value of which _may be

_specific

to

_{subpopulation}

i. For

_instance,

the sire variance _{may vary}

_according

to the environment in which the progeny of the

sires is raised.

_Furthermore,

it will be assumed that the ratio

_0’

_{.,,, /a}

_i

_,

_{where a}

_i

is now

the residual

variance,

is constant across

subpopulations.

In a sire

by

environment

layout,

this is tantamount to

_assuming

_homogeneous

intraclass correlations

_(or

heritability)

across

_{environments,}

which seems to be a reasonable

assumption

in

practice

(Visscher, 1992).

Thus,

the

_argument

_h2!

of the normal CDF in

_[4]

and

_[7]

becomes

where _p =

<

7

u,

/

<T:.

In the fixed

_model,

_parameters

T,

(3

and b were estimated

_by

maximum

likelihood. Given p, a natural extension would be to estimate these and u*

_by

the mode of their

_{joint posterior}

distribution

_(MAP).

To mimic a mixed-model

structure,

one can take flat

_priors

on _T,

₁₃

and b. The

only

informative

_prior

is then

on u

*

,

ie,

u* rv

_{N(O, A).}

Thus

MAP solutions can be

_computed

with minor modifications from

_[8].

The

_only

changes

to

_implement

are to

_replace:

_(i)

a =

’, (3’,

T

(

6’)’

_by

0 =

(

T

’,

0, 6’,

u#’)’

with

u#

=

_pu

_*

_;

(ii)

X

=J;xi,X2,...,x!,...,x//

by

S =

(S1, · .. , Si, ... , SI)’

with

S

’

=

to the random effects on the left hand side

and _p-

2

A -

lU[k]

to the

_u#-part

of the

right

hand side

_(see

_Appendix

_1).

A test

_example

is shown in

_Appendix

2.

A further

_step

would be to estimate _p

_using

an EM

_marginal

maximum likelihood

procedure

based on

This _may involve either an

_approximate

calculation of the conditional

expecta-tion of the

_quadratic

in

u#

as in Harville and Mee

(1984),

Hoeschele et al

₍₁₉₈₇₎

and

Foulley

et al

_(1990),

or a Monte-Carlo calculation of this conditional

_expectation

using,

for

_example,

a Gibbs

_sampling

scheme

_{(Natarajan, 1995).}

Alternative pro-cedures for

_estimating

_p

_might

be also

_envisioned,

such as

the

iterated

_re-weighted

REML of

_Engel

et al

_(1995).

NUMERICAL APPLICATION

Material

The data set

_analyzed

was a

_contingency

table of

_calving

_difficulty

scores

_(from

1 to

₄₎

recorded on

purebred

US-Simmental cows distributed

according

to

sex of calf

(males, females)

and _age of dam at

_calving

_{in years.} Scores 3 and 4 were

_pooled

on account of the low

frequency

of score 4. Nine levels were considered for _age

of dam: < 2 years,

2.0-2.5, 2.5-3.0, 3.0-3.5, 3.5-4.0, 4.0-4.5, 4.5-5.0, 5.0-8.0,

and

> _{8.0 years. In the}

_analysis

of the

_{scaling parameters,}

six levels were considered

for this factor: < 2 _years,

_{2.0-2.5, 2.5-3.0, 3.0-4.0, 4.0-8.0,}

and > _{8.0 years. The} distribution of the 363 859 records

_by

_{sex-age of} dam combinations is

_displayed

in table

I,

as well as the

frequencies

of the three

_categories

of

_calving

scores. The raw data reveal the usual

pattern

of

_{highest calving difficulty}

in male calves out of

younger dams.

However,

more can be said about the

_phenomenon.

Method

Data were

_analyzed

with standard

(S-TM)

and heteroskedastic

(H-TM)

threshold

models. Location and

_scaling

parameters

were described

_using

fixed models

involv-ing

sex

(S)

and age of dam

(A)

as factors of variation. In both _cases, inference was

based on maximum

likelihood

procedures.

A

_log-link

function was used for

_scaling

parameters.

With J = 3

categories,

the _most

highly parameterized

S-TM

that

can be

fitted

for the location structure includes J - 1 = 2 threshold values

(or, equivalently,

the

difference between thresholds

₍

₇₂

_-

₇₁

₎

and a baseline

_population

effect

/

-

l

),

_plus

sex

(one contrast),

age of dam

(eight contrasts)

and their interaction

(eight

contrasts)

as elements of

_(3;

this

gives r(X)

= 17 which

yields

19 _as the total number of

The H-TM to start with was as in the S-TM for location

_parameters

_(3.

With

respect

to

_dispersion

_{parameters 6,}

the model was an additive _one, with sex

_(S

_*

_:

one

_contrast)

and _age of dam

_(A

_*

_:

five

_contrasts)

so that

_r(P)

= 6

(

Q

= 1 in

male calves and < _{2.0 year} old

_dams).

Thus,

the total number of

parameters

was

19 + 6 = 25

and,

the

df

_were

equal

_to 36 - 25 = 11.

RESULTS

All factors considered in the S-TM were

significant

(P

<

_0.01),

_especially

the sex

by

age of dam interactions

(except

the first one, as shown in table

_II).

_Hence,

the

model cannot be

_simplified

further. This means that differences between sexes were

not constant across _age of dam

_subclasses,

_contrary

to results of a

_previous

_study

in Simmental also obtained with a fixed S-TM

(Quaas

et

_al,

_1988).

Differences

in

_liability

between male and female calves decreased with _age of dam. However the S-TM did not fit well to the

data,

as the Pearson statistic

(or

deviance)

was

X

2

= 419 on 17

_degrees

of

freedom,

_resulting

in a nil P-value. An examination of

the Pearson residuals indicated that the S-TM leads to an underestimation of the

As shown in table

_II,

_fitting

the H-TM decreased the

X

2

and deviance

_by

a

factor of 20 and led to a

satisfactory

fit. The

significance

of _many interactions

vanished,

and this was reflected in the LRS

(P

<

_0.088)

for the

_hypothesis

of no

S x A interaction in the most

_{parameterized}

model. Several models were tried and

tested as shown in table III. The

_scaling

_parameters

_depended

on

the

_age of the

dam,

the

effect

of sex

_being

not

_significant

_(P

<

_0.163).

Relative to the baseline

population,

the standard

deviation increased

_by

a factor of about

_{1.05, 1.15, 1.25,}

1.40 and 1.50 for cows of

_{2.0-2.5, 2.5-3.0, 3.0-4.0, 4.0-8.0,}

and > _{8.0 years} of _age

at

_{calving respectively (table IV).}

The H-TM made differences between sex liabilities across _ages of dam

practically

constant as the

interaction

effects were

negligible

relative to the main effects. The difference between male and female calves was about 0.5.

_Eventually,

a model

Logistic

heteroskedastic models have been considered

_{by McCullagh (1980)}

and

Derquenne

(1995).

Formulae are

_given

in

_Appendix

1 to deal with this distribution.

When the Simmental data are

_analyzed

with the

_logistic

(table VI),

the

data. Wald’s and deviance statistics were in _very

good

_agreement

in that

_respect,

with P values of 0.08 and 0.16

_{respectively,}

for the SA interaction. It should be observed that this heteroskedastic model has even fewer

parameters

₍₁₆₎

than the

two-way

S-TM considered

initially

(19 parameters).

In

_spite

of

this,

the Pearson’s

chi-square

(and

also the

_deviance)

was reduced from about 419

(table

II)

to 32

(table V)

with a P-value of 0.04. This fit is remarkable for this

_large

data set

(N

=

363859),

where _one would

_expect

_many models to be

rejected.

Although

the H-TM _may have

_captured

some extra hidden variation due to

ignoring

random

_effects,

it is

_unlikely

that the _poor fit of the S-TM can be attributed

solely

to the

_{overdispersion phenomenon resulting}

from

_{ignoring genetic}

and other

clustering

effects. The

_large

value of the ratio of the observed

X

2

to its

_expected

value

_(419/17

=

25)

suggests that

the

dependency

of the

_{probabilities}

_77,!

with

respect

to sex of calf and _age of dam is not described

_{properly by}

a model

with

constant variance. Whether the poor fit of the S-TM is the result of

ignoring

random

effects,

heterogeneous variance,

or

_{both, require}

further

_{study, perhaps}

simulation.

These

results

_suggest

that in beef cattle

_breeding

the

_goodness

of fit of a constant

variance threshold model for

_calving

ease can be

_improved

by

_{incorporating}

scale

effects _{for age} of dam either as discrete

_classes,

as in this

study,

or

alternatively

Qi

as a

_polynomial

_regression

of

_log

ai on _age.

DISCUSSION

Other distributional

_assumptions

The

_underlying

distribution was

_supposed

to be normal which is a standard

assumption

of threshold models in a

_genetic

context

_{(Gianola, 1982).}

_However,

other

distributions

_might

have been considered for

_modeling

_77!

in

_!3!.

A classical

choice,

especially

in

_{epidemiology,}

would be the

_logistic

_{distribution with mean 77i}

probit.

Interestingly,

there is not much difference between the

_complete

_(S+A+SA)

and the additive model

_(S

+

_A),

the interaction

_(SA)

_{being non-significant}

_(P

=

0.30). Taking

into account the variation in variance in addition to that

_explained

by

the additive model on location

_parameters

does not

improve

the fit

_greatly.

In that

_respect,

the main source of variation turned out to be sex rather than _age of

dam.

Other

_options

include the t-distribution

_(Albert

and

_Chib,

_1993),

the

_Edgeworth

series distribution

_{(Prentice, 1976)}

and other non-normal classes of distribution functions

_{(Singh, 1987).}

In

_fact,

the t distribution

_tv(!2,

_s

₂

₎

with

_spread

_parameter

8

2

and v

degrees

of freedom is the

marginal

distribution of a mixture of a normal

distribution

_!V(!,<7?),

with

_Q2

_randomly

_varying

_according

to a scaled inverted

X

2

(v,s

2

)

distribution (Zellner,

1976).

Therefore,

a threshold model based on such a t distribution is embedded in our

_{procedure by taking}

a _one-way random model for

_Incr?,

_ie,

_ln<r?

=

In

8

2

_{+ ai} with the

_density

function

p(a

i

lv)

of the random

variable ai as

_presented

in

_Foulley

and

_(auaas

_(1995,

formulae 21 and

_22);

see also

Gianola and Sorensen

₍₁₉₉₆₎

for a

_{specific study}

of the threshold model based on

the t-distribution in animal

_breeding.

Relationships

with variable thresholds

Conceptually, heterogeneity

of the

_a§s

is viewed here in the same way as in

Gaussian linear models since it

_applies

to an

_underlying

random variable that is

normally

distributed.

However,

the

_underlying

variables are not

observable,

and

the

_{corresponding}

real line includes cutoff

_points,

the

_thresholds,

that make the

outcomes discrete. It is of interest to address the

question

of how

changes

in

dispersion

can be

interpreted

with

_respect

to the threshold

concept.

Let us illustrate this

_by

a

_{simple example involving}

J = 3

categories,

and _a

one-way classification model

(i

=

1, 2, ... , I)

as, for

example,

sex of calf in the Simmental

breed. We will assume that the

_origin

is at the first

_threshold,

and that the unit of measurement is the standard deviation within males

₍

_QM

_{= 1).}

The difference

where

II

M1

,

lI

M2

are the

_{probabilities of response}

in the first and second

categories,

respectively,

for male calves. A similar

_expression

is

obtained

for female calves

_(F),

so

This is

_precisely

the ratio of the difference between thresholds 1 and 2 that would be obtained when evaluated

_separately

in each sex. Thus

Formulae

_[19]

and

_[20]

are

_analogous

to

_{expressions given by}

Wright

_{(1934b) (the}

reciprocal

of the distance between the thresholds on this scale

_gives

the standard

deviation on the

_postulated

scale on which the

thresholds

are

_{separated by}

a unit

distance,

p

545)

and Falconer

(1989,

formula

18.5,

p

307)

except

that these authors

set to

_unity

the difference between thresholds in the baseline

_population,

rather than the standard

deviation,

which we find more

_appealing

_{conceptually.}

In the case of the Simmental data shown in

table

I,

applying

formulae

_[19]

and

[20]

using

observed

frequencies

of _responses

_gives:

If more than three

_categories

are

_observed,

this formula also holds for the

differences T3 - 72, T4 - 73,..., - -17JTJ-2, so that the ratio between standard deviations in

_{subpopulation}

_(i)

and a reference

_population

(R)

can be

_expressed

as:

which involves

_{(J - 2)}

_{algebraically}

_independent

_equalities.

In the case of three

_categories

and a

_single

classification,

the saturated model

has 21

_parameters

₍

_T2

_-

_T

_{i ,}

_{, 1J.L U2-. - ;/!7)}

_{<!2/<7’i) - - -}

_ai!a1,

... ,

arla1).

Numerical

values

_{of a}

_i!

_a

₁

_computed

from

_[20]

are also ML estimates

_{(eg, â}

_2/

_¡

_h

=

0.973).

Formula

_[21]

indicates that there is a link between H-TM and models with

variable thresholds

_{(Terza, 1985).}

As

_compared

to

_these,

the main features of the H-TM are:

i)

a

multiplicative

model on ratios of standard deviations or differences between

thresholds,

rather than a linear model on such

differences;

ii)

a lower dimensional

_{parameterization}

due to the

_{proportionality}

_assumption

made in

_[18]

rather

than a

category-specific

_{parameterization,}

ie:

where

_{6 j}

is the vector of unknowns

_pertaining

to the difference

_{(T! -}

_Ti

_).

For J =

3,

the _two models

_generate

the _same number of

_parameters

but

they

_are

Further extensions

The H-TM opens new

_perspectives

for the

analysis

of ordinal _responses.

Interesting

extensions _may include:

i) implementing

other inference

_approaches

for mixed models such as Gilmour’s

procedure

based on

quasi-likelihood,

or a

fully Bayesian analysis

of

parameters

using

Monte-Carlo Markov-Chain methods

_along

the lines of Sorensen et al

_(1995);

ii)

assessing

the

_potential

increase in response to selection

by selecting

on

estimated

breeding

values calculated from an H-TM versus an

S-TM;

iii)

incorporating

a mixed linear model on

_{log-variances}

as described in San Cristobal et al

₍₁₉₉₃₎

and

_Foulley

and

_Quaas

₍₁₉₉₅₎

for Gaussian

_{observations;}

iv)

carrying

out a

_joint

_analysis

of continuous and ordered

polychotomous

traits

as

already proposed

for the S-TM

_{by Foulley}

et al

_(1983),

Janss and

_{Foulley (1993)}

and Hoeschele et al

_(1995).

Further research is also needed at the theoretical level to look at the

_sampling

properties

of estimators based on

_{mis-specified}

models. For

_instance,

one _may be

interested

in the

_asymptotic

_properties

of the ML estimator of

₍

_T

_{’, 0’,}

_6’l’

derived under the

_assumption

of

_independence

of the

_y

_i

_,’s

when this

_hypothesis

does not hold. This

_problem

has been discussed in

_{general by}

White

_(1982),

and it may be

conjectured

from the results of

_Liang

et al

₍₁₉₉₂₎

that the ML estimators of

these

parameters

remain consistent. It

_might

also be worthwhile to assess the effect of

departures

from

_independence

on

_testing

procedures.

The

generalized

chi-square

procedure

for

_goodness

of fit derived

_by

McLaren et al

₍₁₉₉₄₎

_might

be useful in

that

_respect

for

_analyses

based on

_{large samples.}

ACKNOWLEDGMENTS

Part of this research was conducted while JL

_Foulley

was a

_Visiting

Professor at the

University

of Wisconsin-Madison. He

_{greatly acknowledges}

the support of this institution and of the Direction des

_productions

animales and Direction des relations

_{internationales,}

INRA. The

_manuscript

is

_largely

drawn from invited lectures

_{given by}

the senior author

at the Midwestern Animal Science

_Meeting

_(Des

_{Moines, Iowa,}

_April

12

_1995),

the

University

of Wisconsin-Madison

₍₁₂

April

1995),

Cornell

_University

₍₁₅

_May

₁₉₉₅₎

and the

_University

of Illinois

₍₂₈

_August

_1995).

_Special

thanks are

_expressed

to all those who contributed

_{substantially}

to the discussions

_{generated thereby,}

and also to B Klei and B

Cunningham

from the US Simmental Association for

providing

the data set on

calving

ease scores

_analyzed

in this

study.

The H-TM has been

_applied

to

_{calving performance}

of other cattle breeds

_(US

_Holstein,

French Blonde

_{d’Aquitaine,}

_{Charolais, Limousin,}

Maine

Anjou).

It has also been

applied

to

_prolificacy

records in

sheep.

We are

grateful

to J

Berger

(Iowa

State

_University,

_Ames),

F

_Ménissier,

J

Sapa

(INRA

Genetics,

Jouy)

and JP

Poivey

(INRA

Genetics,

Toulouse)

for

_providing

the

_{corresponding}

data sets.

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M

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A non-normal class of distribution for dose

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DA,

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₍₁₉₈₅₎

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S

_(1934a)

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₍₁₉₇₆₎

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APPENDIX 1

Expressions

for the score function U and the information matrix J

Concerning

derivatives with

_respect

to the vector

(3

of location

parameters,

one

has:

Finally,

U can be

expressed

as:

with

_expressions

for

_£&dquo;

_vp and v8

given

in

(A.l!, [A.2]

and

(A.3!.

Elements of the information matrix

_J(a)

include the

_expectations

of minus the second derivatives. The

_following

derivatives will be considered:

threshold-threshold; (3-threshold-threshold;

6-threshold; 13 - 13; (3-6;

and S - b.

(3-Threshold

derivatives

The

_expectation

of the first term vanishes because

_E(yz!)

=

nj

1

Ijj.

where T =

{t!!} is

_a

(J - 1)

x

_{(J -}

_{1) symmetric}

band matrix

_having

as elements:

t

ij

=

E(-å

2

L/årJ),

and

_t

_j

_,

_j+1

=

’

)

j+1

j

år

L/år

2

E(-å

_given

_in

[A.4]

and

(A.5!.

These

_expressions

can be extended to

obtain the

MAP of

parameters

in a mixed-model structure

_{by replacing}

(i)

13 by

e =

((3!,

u#’)’

with

u# r-

N(0,

P

Z

A)

(p

2

=

U

2

i

/

0

,?

=

constant);

(ii)

X by

S =

₍

_Sl

S2 .... , Si,...,

)’

S

I

with

s!

=

(x!, (Jiz

D

;

and

_making

the

_appropriate

_adjustments

for

_prior

information as shown below:

APPENDIX 2