Probability statements about the transmitting ability of progeny-tested sires for an all-or-none trait with an application to twinning in cattle

HAL Id: hal-00893808

https://hal.archives-ouvertes.fr/hal-00893808

Submitted on 1 Jan 1989

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.

Probability statements about the transmitting ability of

progeny-tested sires for an all-or-none trait with an

application to twinning in cattle

J.L. Foulley, S. Im

To cite this version:

Original

article

Probability

statements

about the

_transmitting

ability

of

_{progeny-tested}

sires

for

an

all-or-none

trait with

an

application

to

twinning

in

cattle

J.L.

_Foulley

S.

Im2

Institut National de la Recherche

_Agronomique,

station de _{génétique quantitative} et

appliquée, centre de recherches de _Jouy, 78350 _{Jouy-en-Josas;}

Institut

National de la Recherche _Agronomique, laboratoire de _biométrie, centre de recherches de Toudouse BP _27, 31326 Castanet-Tolosan _Cedex, France

(received 20 _{September 1988, accepted} 17 _April

₁₉₈₉₎

Summary - This paper compares three statistical

procedures

for

_{making probability}

statements about the true _{transmitting ability} of

_{progeny-tested}

sires for an all-or-none

polygenic trait. Method I is based on the beta binomial model whereas methods II and III result from Bayesian approaches to the _{threshold-liability} model of Sewall _Wright.

An _application to lower bounds of the _{transmitting ability} of _superior sires with a _high

twinning

rate in their

_daughter

_{progeny is presented.} Results of different methods are in

good

agreement. The

_flexibility

of these different methods with _respect to more _complex

structures of data is discussed.

genetic evaluation - all-or-none traits - beta binomial model - threshold model

-Bayesian methods

Résumé - Enoncés _{probabilistes} relatifs à la valeur _génétique transmise de _pères testés sur _{descendance pour} un caractère tout-ou-rien avec une _application à la

gémellité chez les bovins. Cet article compare 3 procédures statistiques en vue de la

formulation d’énoncés _{probabilistes}

_relatifs

à la valeur _génétique transmise de _pères testés

sur descendance _pour un caractère polygénique tout-ou-rien. La méthode I repose sur le modèle bêta binômial alors que les méthodes II et III découlent _{d’approches bayésiennes} du modèle à seuils de S. _Wright. Une _application concernant la borne

_inférieure

de la valeur _génétique transmise de _pères d’élite _présentant un taux de

_gémellité

élevé chez leurs

filles est _présentée. Une bonne concordance des résultats entre méthodes est observée. La

flexibilité

de ces

_différentes

méthodes vis-à-vis de structures de données _{plus complexes} est

abordée en discussion.

INTRODUCTION

Genetic evaluation for all-or-none traits is

usually

carried out via Henderson’s mixed model

_procedures

_{(Henderson, 1973)}

_having

_{optimum properties}

for the Gaussian linear mixed model. Even

_though

a linear

_{approach taking}

into account some

specific

features of binomial or multinomial

_{sampling procedures}

can be worked out in

_{multi-population analysis}

_(Schaeffer

&

Wilton, 1976;

Berger

&

Freeman, 1976;

Beitler &

Landis, 1985;

Im et

ad.,

1987),

these methods suffer from severe statistical

drawbacks

_(Gianola,

_1982;

_Meijering

&

_Gianola,

_1985;

_Foulley,

_1987).

_Especially

as

distribution

_properties

of

_predictors

and of

_prediction

errors are unknown for

_regular

or

improved

Blup

procedures applied

to all-or-none

_traits,

it would therefore be

dangerous

to base

_probability

statements on the _property of a normal

_spread

of

genetic

evaluations or of true

breeding

values

given

the estimated

breeding

value.

The aim of this _{paper is} to

_investigate

alternative statistical methods for that purpose.

Emphasis

will be

placed

on

making

probability

statements about

true

_transmitting

_ability

_(TA)

of

_superior

sires

_{progeny-tested}

for some

binary

characteristic

_having

a multifactorial mode of inheritance. Numerical

_applications

will be devoted to sires with a

high twinning

rate in their

daughter

progeny.

METHODS

The methods

_presented

here are derived from statistical sire evaluation

_procedures

which are based on

_specific

features of the distribution involved in the

_sampling

processes of such

binary

data. Three methods

_(referred

to as

_I,

II and

III)

will be

described in relation to recent works in this area. The first method is based on the

beta binomial model

_{(Im, 1982)}

and the two other ones on

_{Bayesian approaches}

(Foulley

et

_al.,

₁₉₈₈₎

to the

_{threshold-liability}

model due to

_Wright

₍₁₉₃₄

a and

b).

All three methods assume a conditional binomial distribution B

(n,

T

r)

of

_binary

outcomes

_(i.e.

n _progeny

performance

of a

_sire)

_given

the true value 7r of a

probability

parameter

(here

the sire’s true

_breeding

value or

_transmitting

ability).

The three methods differ in

_regard

to the

_modelling

of 7r

_itself,

either

_directly

(beta

binomial)

or

_indirectly

_{(threshold-liability model),}

and

_consequently

in

_describing

the

_prior

distributions of _parameters

involved,

v.i.z. !r itself or location _parameters on an

_underlying

scale.

Method I ’

Let _{yij =} 0 or 1 be the

_performance

of the

_jth

_progeny

_{( j}

=

_{1, 2, ...,}

n

i

)

_out of the ith sire

_(i

=

1, 2,

..., q)

and

n

i unrelated dams.

Let

J

r

i

designate

the true

transmitting

ability

(

7

r

i

)

of sire i.

A

_priori,

the

_7r

_i

_’s

are assumed to be

_{independently}

and

_identically

distributed

Conditional on true value -7ri, the distribution of

binary

responses among progeny

of a

given

sire is taken as binomial B

,

i

(n

7

r

i

).

These distributions are

conditionally

independent

among sires so that the likelihood is the

_product

binomial

where the circle stands for a summation over the

_{corresponding}

_subscript

(here

y

2o =

Ej

Y

ij)

and

capital

letters indicate random variables.

As the

prior

and the likelihood are

_conjugate

(Cox

&

Hinkley,

1974,

_p.

308),

the

posterior

distribution remains in the beta

_family

and can be written as:

! , ¡’ , ... -- I

with

_normalizing

constant

The

_density

in

₍₃₎

is a

_product

of _q

independent

beta densities.

Ignoring

subscripts,

the

_posterior

_density

for a

_given

sire is:

This Beta distribution B

_{(a, b)}

can be

_{conveniently expressed}

with a

reparame-terization in terms of a

_prior

mean 7r,, =

a/(a

₊

!3),

_an intra class correlation

p

b =

(a +,3 +

1)-

1

and the observed

frequency

_p =

y

o/

n.

The conditional distribu-tion of 7r

_given

_{n, p,} 1r and pb is

_then

with

expectation

which will be noted 7r so as to reflect both its

_{interpretation}

as a

_Bayesian

estimator

of 1

f as well as its

equivalence

with the best linear

predictor

or selection index

(Henderson, 1973).

Using

7r and

_letting

a

b

=

pb

1

-

1 with

À

b

interpretable

as a

ratio of within to between sire _components of

_variances,

the distribution in

_(4b)

can be viewed as a function of _n, 7r and

_A

_b

_,

that

_is,

conditional on n,

!6

and

1

?

,

the

distribution of 7r is:

with

expectation

and variance

Probability

statements about true values of TA

_given

the data

_(n

and p or

_7i’)

expressions

(4b)

or

(5)

of the

_posterior

_density

of -7r. Notice that formula

_(6a)

also represents the

_probability

of response

Pr(Y

ik

=

1

ni, p

i

)

for _a future _progeny

(k)

out of sire

_(i)

with an observed

_frequency

of _{response pi} in ni

offspring.

These

_probability

statements can be made for

_specific

sires

_given

_{their progeny}

test data

_(n,

_p or

_11’)

and the characteristics of the

_{corresponding population,}

such

as the mean incidence 1fand the intraclass coefficient pb as a _parameter of

_genetic

diversity.

To allow for

_comparisons

_among

_methods,

this _pb, or

_equivalently

the

ratio

_A

_b

_,

will be

_expressed

_according

to Im’s

₍₁₉₈₇₎

results which relate intraclass

coefficients

on the

_binary

_(p

_b

₎

and

_underlying

_(p)

scales in a

_population

in which

the incidence of the trait is 7r,,

(see

next

_paragraph).

In the case of

_twinning,

interest is

_usually

in

superior

sires

_having

estimated

transmitting ability

(ETA)

values above the mean 7ro. Attention will then be

devoted to the lower TA bound 1fm which is exceeded with a

probability

a

_i.e,

to 1fm, such that:

This involves

_computing

x E

_{[0, 1]}

values of the so-called

_incomplete

beta

function defined as, in classical notations

Details about numerical

_procedures

used to that _respect are

_given

in

appendix

A.

In

_addition,

more

general

results can be

produced

for instance in terms of

(n, 1

?

)

values such that formula

₍₇₎

holds for

_given

values of a&dquo;t

(TA

lower

bound)

and a

(probability level):

see

appendix

A.

Method 11

This method is derived from

_genetic

evaluation

_procedures

for discrete traits

introduced

_{recently by}

several authors. All these

_{procedures postulate}

the

_Wright

threshold

_liability

_concept. We restrict our attention here to

_Bayesian

inference

approaches proposed independently by

Gianola &

_Foulley

_(1983),

Harville & Mee

(1984),

Stiratelli et al.

₍₁₉₈₄₎

and Zellner & Rossi

_(1984).

Although

the

_methodology

is _very

_general

vis-a-vis data _structures, for the sake

of

_simplicity

_only

its

_{unipopulation}

version

_(p

_model)

will be considered in this

paper.

Let

_l

_ij

be a

_conceptual

_underlying

variable associated with the

_binary

_response

y2! of the

jth

progeny of the ith sire. The variable

12!

is modelled as: .

where 1/i is the location parameter associated with the

_population

of _progeny out

of sire i and the

_e

_ij

_’s

are NID

_(0,

o,’)

within sire deviations.

Conditional on _qj, the

_probability

that a _progeny

_responds

in one of the two

exclusive

_categories

coded

_[0]

and

_[1]

_respectively

is written as:

where T is the value of the

threshold,

a the within sire standard deviation and

It is convenient to _put the

_origin

at the threshold and set ue to

_unity,

i.e.

&dquo;standardize&dquo; the threshold model

_(Harville

&

Mee,

1984)

the

_expression

for _7ri[o] can be written as

and that for _7ri[l] as:

In what

_follows,

and to

_simplify

_notation,

_jpjrJ will be referred to as 7ri .

Letting

tt=

fail

be _an

(q

x

₁₎

_vector, a natural choice for the

_prior

distribution

of

IL

under

_{polygenic inheritance,}

is:

where A is

equal

to twice Malecot’s

_{genetic relationship}

matrix for the _q sires

(A

= I in method

_I),

U2

_is

_{the sire component of variance and}

p

o the

general

phenotypic

mean in the

_underlying

scale.

These _{parameters po} and

_u2

_{are linked} to the overall incidence 7r via:

- . - .. ,

or,

equivalently,

defining

Q2 =

0

-; +

U2

with

Q

e

=

1,

Similarly,

the sire variance

_Q!6

in the

_binary

scale can be related to the

_underlying

distribution via

where

_{!2(x, y; r)}

is the standardized bivariate cumulative

_density

function with

mean 0 and correlation

_{r, jl}

_p

_o

_j

+

Q

u)i/2

and p in

(13b)

is the intraclass

correlation coefficient _p =

a;/a2.

The variance in

(13b)

_can be obtained

directly by

a

_probability

_argument or as the limit of a formula

_{given by Foulley}

et al.

₍₁₉₈₈₎

for

the variance of the observed

_frequency

pi when the progeny group size n tends to

infinity.

Notice also that it differs from the classical

_expression

_<p2(jí,)a;

_proposed

by

Dempster

& Lerner

_{(1950), 0(.)}

_designating

the standardized normal

_density

function,

which is a first order

_{Taylor expansion}

of

_(13b)

about p = 0.

where

_z!

is a

₍₁

x

m)

row vector

_having

1 in the ith column and 0 elsewhere.

The

_logposterior

_density

L

_(p.;

_{y, !Co,}

_a2)

can be minimized with _respect to _p.

_by

a

_scoring

_algorithm

of the

_general

form

The value

_{of t}

_L

in the t-th iteration can be

_{computed by solving}

the non-linear

system

where W and v are an

_(n

x

_n)

_diagonal

matrix and an

(n x 1)

vector

_respectively

having

elements

Define A =

&OElig;; / &OElig;;

=

1/ &OElig;;

and u= (L -

p

o

l

as an

(m x 1)

vector of sire deviations. Then the _system to be solved

_A

_IL

= A u becomes

An

_interesting

feature of the

_posterior

distribution in

₍₁₄₎

is its

_asymptotic

normality

where

_(.1.

_*

is the mode of the

_{posterior density}

_of

_IL

in

₍₁₄₎

and I

_(IL)

is defined as

lim

_{[I (}

_E

_{t) /}

_no!-1

can be

replaced

for test statistics

by

a consistent estimator such as

-no[I(!*)!-1

where I

(t

L

)

is evaluated

_{at t}

_L

_

t

L

*

.

The variance of the

_limiting

normal distribution is

_usually

taken to be

_(Berger,

1985,

p.

224)

where

_wi

_stands

for wi evaluated at the mode

_/-Li.

*

Letting

pm _

<I>-l (

1I

&dquo;

m

)

be the parameter value in the

_underlying

scale

cor-responding

to 1I&dquo;m in

(7),

the

probability

that the true sire

_TA,

_pz of sire i

(ETA

=

tt*)

exceeds t

t

(or

equivalently

!r >

_1I

_&dquo;

_m

₎

can be

_expressed

as

For

_given

values of n, p

(or 7

?

)

and

Àb( 7f 0,

011

),

this

probability

can be

_computed,

given

7fm, and

compared

to the

_{corresponding probability}

level obtained with the

beta binomial model.

_{Alternatively,}

one can determine the lower bound 7f m such that Pr

₍

₇

_r

_i

>

_7fm)

= _a

fixed,

by

taking

pm =

f

-Li -

Î

;/

2

ép-l(0:).

Notice that

_computing

the

_probability

in

_(20),

_based

on the

_posterior

distribution

of the true

_TA,

_f

_(p

_n,

_{p, po,}

_A)

is

_equivalent

to

_computing

Pr

₍

₇

_r

>

_7fm

₎

over

the distribution of the

_probability

of response 7r =

4)(,U)

for a future _progeny of sires

_having

an ETA

_equal

to

_tt

_*

and a true TA distributed

_according

to

_(19).

This distribution in the observed scale would

_probably

be more

_appealing

for

practitioners.

This is

_especially

clear as far as ETA’s are concerned and one _may

alternatively

to

_tt

_*

_,

consider as a sire

_evaluation,

the

_expectation

of !r =

_!(!)

with

respect to the

_{density of}

_A

in

_(19),

_say

_!!2.

This

_expectation

is:

However,

the whole distribution of 7r =

4b(/,t)

remains less tractable

numerically

than that

_{of p}

in

₍₁₉₎

due to its

_following

form:

Method III

This method is also derived from the threshold

_liability

model but

_employs

asymptotic properties

at an earlier

_stage.

Let us

consider,

as

previously,

the observed

frequency

of _{response pi} in ni

progeny of sire i.

_{Conditionally}

to the true

TA,

),

i

r

7

(

pi has an

_asymptotic

normal

distribution,

i.e.:

The normit transformed

_{of p}

_i

_,

mi = 14D-

(p

i

)

has a conditional distribution which

is also

_{asymptotically}

normal.

_Following

a classical theorem in

_asymptotic

theory

(see

for

_instance,

formula

_6a.23,

_{page 386} in

_Rao,

₁₉₇₃₎

and

_knowing

that:

one

_{has, given}

_ai

Assuming

as in section II B

_that,

a

_priori,

the

_f-Li’S

are i.i.d !N

(p.!, o

D

leads

The

_expectation

_(jiz)

and variance

_(ci)

of the distribution in

₍₂₅₎

can be

easily

expressed analytically

as

(see

for instance Cox &

Hinkley, 1974,

formula

22,

_page

373).

with

Alternative formulae can be derived so as to mimic usual selection index

expressions,

which are,

ignoring subscripts

... ,- - !

where CD is

_{given by}

the usual formula for the coefficient of

determination,

i.e. CD

=

n/n

₊

k)

where the scalar k is defined as:

In method II and with a definition of CD restricted to var

(f-Lly,a-;) = (1- C D)a;,

this coefficient is:

Notice that k can be

_interpreted

as in selection index

theory

as the ratio of a

within sire to a sire

variance,

since

_{p(l -}

_p)/(p2(.)

is the

asymptotic

variance of the normit transformation of the

_frequency

of response in

progenies

of a

_{given sire,}

conditionally

on the sire’s true

_underlying

TA.

Substituting

jiz

and ci for

(26a

and

b)

or

₍₂₇

a and

b)

for

_p*

and _Ii

respectively

in

₍₂₀₎

enables us either to determine 1rm for a

_given

a

probability

level

knowing

p

o

,

ufl

(or

equivalently

7r,, and

!),

p and n, or to _compute the

_probability

level a

such that p.i exceeds a

_given

threshold

_Again,

the distribution of 7r =

iI?(f-L)

in

the observed scale

_{corresponding}

to the

_{density of p,}

in

₍₂₅₎

can be obtained

_using

the formula

_(22).

Its

expectation

has the same

_expression

as in

(21) with Îii

and ci

replacing f

-

L

i

and

7z

respectively.

NUMERICAL APPLICATION

Procedures described in this _paper are

_applied

to the

_problem

of

_screening

_superior

sires with a

_high

_twinning

rate in their female _progeny.

There are

_{relatively large}

differences among cattle

populations

with respect to the

_prevalence

-7r,, of

twinning

(see

for instance

Maijala

&

Syvajarvi,

1977).

Two

values of 7r,,, say 2.5 and 4% are considered in this

application.

These values of

_twinning

rate are those observed in 2nd and mature

(3rd

to

7th)

calvings respectively

in such breeds as Charolais and

Maine-Anjou

(Frebling

et

al.,

1987).

The first value of or1

(0.025)

can be viewed as a _progeny test of young

of its

_extremely

low rate

_(0.007).

The second value

_(0.04)

is an illustration of an

evaluation _system of service sires based on mature

_calvings.

Genetic variation for occurrence of

_multiple

births in cattle was assumed to

have an

_heritability

coefficient in the

_underlying

scale

_equal

to

_0.25,

_according

to

estimates

_published

_{by Syrstad}

_(1974).

In this

_study,

h2 values of the

_underlying

continuous variable are

_higher

than those in the observed scale

_{and, especially}

more

stable over

_parities

as

theoretically expected

for such

_binary

traits.

_Using

this value

in

_(13a

and

_b)

leads to _!o

_equal

to -2.0242 and _-1.8081, and h2 in the

_binary

scale

equal

to 0.0394 and 0.053 for 7!-o

equal

to 0.025 and

0.04,

respectively.

Notice,

as

already

pointed

out

_by

Im

₍₁₉₈₇₎

that h2 values

reported

here are

slightly higher

than those which would be obtained

_(0.0350

and

_0.0483)

from the classical formula of

_Dempster

& Lerner

_(1950).

First

_application

The first

_application

deals with 5

_specific

sires known for their

_{high twinning}

rate

in

_daughters.

Table I shows lower bounds

₍

_TT

_m)

of the TA in

_twinning

of these 5 sires

_knowing

their _progeny test

_performance

_{(!,, p)}

in mature

_calvings

at 2 different

probability levels,

a = 0.90 and 0.95. In both _cases, results _are in

_good

_agreement

across methods

_with,

as

_expected

due to

asymptotic approximations,

higher

values of 7r m

being

obtained with method II

and,

to a

larger

extent with III. Differences between I and

II,

however,

are of little

_importance

_given

the

_large

values of n.

Differences _among methods are also reflected in ETA values on the observed scale

with values for _{methods II and III very}

_slightly

less

_regressed

towards the mean

incidence 7r = 0.04 than in method I. On the other

hand,

this is a

_good

_example

of a

change

of

ranking

according

to criteria used. B and C have close ETA and !r,&dquo;,,

values

_{although they}

differ

_largely

in

_frequency

_(p).

E is lower than D in _p, close to

D in ETA but

_larger

in Tfm due to a _greater number of progeny.

Second

_application

For the two 7r

frequencies,

tables II and III show the values of progeny group size

(n)

which

_provides

an a = 0.9

probability

level for different combinations of _!r&dquo;t,

the TA taken as a minimum and

7i’,

the ETA value

according

to formulae

_(5b)

and

_(6).

Minimum TA and ETA values were

_expressed

in _percent as well as, for

practical

purposes, deviations from 7r in Qub units

(uu

b

= 1 standard deviation in true TA in the 0 &mdash;

_{1 scale).}

Results are shown in tables II and III for !&dquo;,,

_varying

from +0.25 to +2.75 Qub and ETA from +1.00 to 3.00 oub with an

elementary

increment of 0.25.

The

_higher

the

_ETA’s,

the lower are _progeny numbers for a

_given

value of

Tfm-For

_instance,

for _7ro = 0.025 and _!r&dquo;,, = 0.048

(or

equivalently

+1.50o-it

t

,)

_progeny

numbers

_providing

a 0.9

_probability

level are

_{5199, 1291, 546, 279,}

152 and 81

when the ETA _goes from + 1.75 to 3.00 uub.

Corresponding

figures

are

_{3631, 895,}

375, 188,

100 and 51

_respectively

when 7r

o

=

0.04,

i. e. about 60-70% of

previous

quantities.

This

_special

case is illustrated for 7r = 0.025 in

figure

1 with _a

graph

of the beta

_prior

_density

and

_posterior

distributions

_{corresponding}

to 7rm = 0.048

For a

_given

ETA

_value,

the closer this value _{to 7r,n,} the

_higher

_{the progeny group} size needed to reach a

_probability

level of 0.9. For such a difference

(ETA

minus

7

r

m

)

taken as fixed in uub

units,

variations in the progeny number n are rather less

pronounced.

These variations

_(An)

in n are

_proportional

to variations

_(A£)

in ETA within _{the range} of values considered. For

_instance,

for 7? - 7r m = 0.75 uu_b, An = 504 - ₃₃₀ = 174 when 7? varies from ₊ 1.00 to ₊ 2.00

uu b

at 7r, = 0.025 and

An = 671 - 504 = 167 with 7?

_going

from _{+ 2.00 to + 3.00}

( TUb

.

Clearly,

these

variations result from the

_dependency

of the

posterior

variance on ETA as reflected

in formula

_(6b)

_contrarily

to what

_happened

in the Gaussian linear model. For

?!-! = 0.025 and h2= 0.25 and a =

_0.9,

_{73 to}

_167,

_{157 to}

_{325, 330}

_to

_671,

904 _to 1663 and 4041 to _{7062 progeny} are

_required

to exceed a minimum TA value

_equal

to

ETA minus

1.25, 1.00, 0.75, 0.50

and 0.25 (TUb,

respectively. Corresponding figures

can be found in table III for 7r = 0.04.

_Coming

back _to the _case of

_twinning,

practical

interest will be for 7rmvalues around +1.50 Qub. This

corresponds

to sires

having

an ETA

_equal

to 2.25 to 2.50 (TUb

(A

difference ETA -7rm of 1.00 to 1.25 Q

u

b seems reasonable in

_practice).

A _progeny number of 279 to 546 and 188 to 375

is needed

_depending

on 7? - 7r! for 7r... = 0.025 and

_0.04,

_{respectively.}

For each _{n, 7rnt,}

_ETA,

₎

_o

_r

₇

₍

_b

_À

_combination,

the

_{corresponding}

_probability

that

of

_fact,

methods

_I,

II III can be

_{compared only}

when

_using

the same amount of

information,

v.i.z. the same n and p values on the one

_hand,

and the same

_prior

expectations

and variances

_(formulae

13a and

_b)

on the other hand.

_Probability

values for methods II and III are shown in Tables II and III.

The _agreement between the 3 methods is

_generally

_{good, especially}

between the beta, binomial model and the

_{Bayesian approach}

of the threshold model. For 71&dquo;

m < +2.00 and

ETA <

+2.25 uub , the difference between the 2

probabilities

never

exceeds 0.01. As

_pointed

out

_previously

_{(first application),}

distributions

_employed

to make

_probability

statements in II and III are both

_{asymptotically}

normal on

the

_{underlying scale,}

and

_consequently

underestimate the real

_posterior

variance

and overestimate the

_probability

that the true TA is

_higher

than 71&dquo;m.

Clearly,

this

drawback is more severe for method III than method

II,

especially

for

_high

values of

the ETA as shown in Tables II and III and in

_figure

2 with the

_graph

of distributions

corresponding

to Jro=

0.025,

hz=

0.25, n

= 38 and

p =

5/38.

DISCUSSION-CONCLUSION

Adaptation

for other

_practical

situations

The situation adressed in this paper was to make

_probability

statements about the

true TA of

_{&dquo;superior&dquo;}

sires vis-a-vis a minimum

₍

₇₁

_&dquo;

_m

₎

value for a rare

_interesting

cha,racteristic. The

_procedure

described can also be viewed as

making probability

statements about TA’s of &dquo;inferior&dquo; sires vis-h-vis an _{upper TA limit for} a

frequent

unfavourable trait. This is

_likely

to occur in

practical

animal

breeding, especially

for the so-called

_secondary

traits

_(e.g.

_fertility

or

_dystocia

in

cattle)

where selection

usually

operates

against

the _poorest sires and dams. In some instances for such

detrimental traits at a low

_prevalence,

interest

_might

be towards the occurrence

of TA’s in the lower tail of the distribution. The

_{procedure reported}

here can be

easily

accomodated for this case in

_changing

the

_{inequality sign}

in formula

₍₇₎

and

in

_taking

an

_opposite

_argument for

«1>(.)

in

(20).

Threshold

_liability

_concept

and beta binomial

_approaches

In the

_example

studied and _{within the range} of _parameters

_{considered, probability}

statements made

_by

methods I and II were _very similar as

_clearly

illustrated

_by

figure

2. This may

help

reconcile the beta binomial

approach praised by

statisticians

(see,

among others

Im,

1982)

with the threshold

liability

concept put ahead

by

quantitative geneticists

especially

those

_working

in human

_genetics

_(see

for instance Curnow &

_Smith,

_1975;

_Falconer,

_1965;

_Fraser,

_1980),

even under

complex

segregation

patterns

(Lalouel

et

_al.,

_1983).

_However,

the threshold

_liability

model offers more

_flexibility

than the beta binomial. In

_particular,

it can be

easily adapted

to mixed model structures

_involving

a

_{multipopulation analysis}

as well as several

random sources of variation.

_Foulley

et al.

_{(1988) highlighted}

how inferential issues in such structures can be handled from a unified

_{perspective using}

the

Bayesian

paradigm.

Thus,

taking

into account several records _per animal

_(e.g.

_multiple

vs

single

births at different

_parities)

can be achieved with this

methodology using

with

_possible

_missing

data _patterns

_(Foulley

&

_Gianola,

_198G)

as well as correlated

information on normal continuous traits

_(Foulley

et

al.,

1983).

Unfortunately,

the beta binomial model in its _present state of

_development

remains confined to a

_single

population

analysis

with one random factor

(Williams, 1988).

Gilmour’s

_approach

Although

the

_approach

of Gilmour et

_al.,

₍₁₉₈₅₎

to the threshold model has its

own

_rationality

via its connection to the

_methodology

of

_generalized

linear models

and

_{quasi-likelihood,}

its

_{justification}

for

_{predicting breeding}

values as

_compared

to

_Bayesian

methods is still

_questionable

_(Foulley

et

_{al., 1988;}

Knuiman &

_Laird,

1988).

This is

_especially

true with _respect to distribution

_properties

of TA

_predictors

or those of

_prediction

errors for which no formal statistical

_properties

can be claimed

for Gilmour’s method.

Distribution

_properties

Distribution

_properties

of the true TA

_given

the data were derived in this

_study

under the

_{implicit assumption}

that the beta binomial model is true in method I

and the threshold _concept is also true in methods II and III. Another issue on

comparing

these methods is to

_investigate

which is the best.

_Ways

of

_challenging

models is a difficult

_topic

which is

_beyond

the _scope of this _paper

_(see

for instance

Smith,

1986).

For

_categorical

data and in an animal

_breeding

_context,

_suggestions

on how to _compare non-nested models were made

_{by Foulley}

_(1987),

which are

based on the

_predictive

distribution of a future data set

_given

the data observed at

present.

Genetic

_charges

The

_reasoning

followed

_throughout

this paper is a conditional view

_knowing

the

progeny test data information on sires. This leads to

_probability

statements about the true TA of

_specific

sires.

_Discussing

_{progeny group numbers} for a

_planned

progeny test _programme can involve a different

_approach

_(e.g.

_Curnow,

₁₉₈₄₎

especially

when

_looking

a

_priori

at this issue with no

_specific

_progeny-test results.

One could make some statements about a distributional form for ETA’s and values

of the selection differential and

_point

of truncation. As

_pointed

out

_by

Hill

_(1977),

genetic

change

due to

_selecting

_superior

sires on their ETA’s under extra constraints

with _respect to such factors as

_{inbreeding levels, testing}

facilities and other costs appears to be a natural

_approach

_(e.g.

_Curnow,

_1984).

_Response

_(R)

to one

generation

of

_upward

truncation selection on

1

?

(1

?

_>

1

?s

)

is a random variable 1r

having

in the beta binomial model the

_following

conditional

_{density given}

7T > _1rs.

account, not

_only

the

_posterior

_density

of 7r

_given

_progeny test results but also the

marginal density

of the ETA’s and

_integrate

7r out in the latter and their

product

over the selection _space.

These are not _easy

_{manipulations}

from an

_analytical

_point

of

_view,

_especially

with the threshold model. More research is therefore needed to derive

_original

results in that area.

ACKNOWLEDGEMENTS

The authors are

_grateful

to Mr Dan Waldron

_(University

of Illinois at

_Urbana)

and Mrs Annick Bouroche

_(Unite

de

documentation,

Jouy-en-Josas)

for their

_English

revision of the

_manuscript.

Thanks are also extented to _{the three anonymous}

referees for their

_helpful

comments.

REFERENCES

Abramowitz M. &

_Stegun

I.A.

₍₁₉₇₂₎

Handbook

_of

mathematical

_functions.

9th ed. Dover

_{publications,}

New York

Beitler P. & Landis J.R.

₍₁₉₈₅₎

A mixed-effects model for

_categorical

data. Biometrics

_41,

991-1000

Berger

J.O.

₍₁₉₈₅₎

Statistical decision

_theory

and

_{bayesian analysis.}

2nd edition.

Springer-Verlag,

New York

Berger

J.P.,

Freeman A.E.

₍₁₉₇₈₎

Prediction of sire merit for

_calving

_difficulty.

J.

_Dairy

Sci.

_61,

1146-1150

Cox D.R. &

_Hinkley

D.V.

₍₁₉₇₄₎

Theoretical statistics

_Chapman

&

_Hall,

London Curnow R.N.

₍₁₉₈₄₎

_{Progeny-testing}

for all-or-none traits when a multifactorial model

_applies.

Biometrics

40,

375-382

Curnow R.N. & Smith C.

₍₁₉₇₅₎

Multifactorial models for familial diseases in man.

J.R. Statist. Soc.

_A,

_138,

131-169

Dempster

E.R. & Lerner I.M.

₍₁₉₅₀₎

_Heritability

of threshold characters. Genetics

35, 212-236

Falconer D.S.

₍₁₉₆₅₎

The inheritance of

_liability

to certain diseases estimated from

the incidence _among relatives. Ann. Hum. Genet.

_29,

51-76

Foulley

J.L.

₍₁₉₈₇₎

Methodes d’evaluation des

_{reproducteurs}

_pour des caractères discrets a d6terminisme

_polyg6nique

en s6lection animale. These d’Etat. Universite

de

_Paris-Orsay

Foulley

J.L. & Gianola D.

₍₁₉₈₆₎

Sire evaluation for

_{multiple binary}

_responses when information is

_missing

on some traits. J.

_Dairy

Sci.

69,

2681-2695

Foulley

J.L.,

Gianola D. &

_Thompson

R.

₍₁₉₈₃₎

Prediction of

_genetic

merit from

data on

_binary

and

_quantitative

variates with an

application

to

_calving

_difficulty,

birth

_weight

and

_{pelvic opening.}

Genet. Sel. Evod.

_15,

401-424

Foulley

J.L.,

Gianola D. & Im S.

₍₁₉₈₈₎

Genetic evaluation for discrete

_polygenic

traits in

animal

_breeding.

In: Advances in Statistical Methods

_for

Genetic

Im-prove!rcent

of

Livestock.

_February

_{16-20, 1987,}

Armidale. Australia.

_Springer

_Verlag

Fraser F.C.

₍₁₉₈₀₎

The William Allan Memorial Award Adress: Evolution of a

palatable

multifactorial threshold model. Am. J. Hum. Genet.

32,

796-813

Frebling

J.,

Gillard P. & Menissier F.

₍₁₉₈₇₎

Resultats

_{pr6liminaires}

sur

l’aptitude

naturelle a la

_gemellite

dans un _troupeau de vaches Charodaises et

_Maine-Anjou

et

de leur

_{filles. INRA,}

mimeo

Gianola D.

₍₁₉₈₂₎

_Theory

and

_analysis

of threshold characters. J. Anim. Sci.

_54,

1079-1096

Gianola D. &

_Foulley

J.L.

₍₁₉₈₃₎

Sire evaluation for ordered

_categorical

data with

a threshold model. Genet. Sel. Evol.

_15,

201-224

Gilmour

_A.R.,

Anderson R.D. & Rae A.L.

₍₁₉₈₅₎

The

_analysis

of binomial data

_by

a

_generalized

linear mixed model. Biometrika _72, 593-599

Harville D.A. & Mee R.W.

₍₁₉₈₄₎

A mixed model

_procedure

for

_analyzing

ordered

categorical

data. Biometrics

_40,

393-408

Henderson C.R.

₍₁₉₇₃₎

Sire evaluation and

_genetic

trends.

_{Proceedings of the}

animal

breeding

and

_{genetics symposium}

in honor

_of

Dr. J.L. Lush. American

_society

of animal science and american

_dairy

science

_association,

_Champaign,

_Illinois,

10-41 Hill W.G.

₍₁₉₇₇₎

Comments on statistical

_efficiency

in _{bull progeny}

_testing

for

calving difficulty.

Livest. Prod. Sci.

4,

203-207

H6schele

_I.,

_Foulley

J.L. & Gianola D.

₍₁₉₈₆₎

Genetic evaluation for

_{multiple binary}

responses. Genet. Sel. Evol.

18,

299-321

Im S.

₍₁₉₈₂₎

Contribution d 1’etude des tables de

_contingence

d

_param!tres

aleatoires: utilisation en biom6trie. These 3e

_cycle,

Universite

_{Paul-Sabatier,}

Toulouse

Im S.

₍₁₉₈₆₎

Mutual

_independence

in a mixed effects model letter to the editor. Biometrics

_42,

997

Im S.

₍₁₉₈₇₎

A note on

heritability

of a dichotomous trait. Biom. J.

_4,

407-411

Im

_{S., Foulley J.L.,}

& Gianola D.

₍₁₉₈₇₎

A linear model for

_genetic

evaluation on

categorical

traits. 82nd Annual

_Meeting

of the American

_Dairy

Science Association Columbia

_Missouri,

June 21-24. J.

_Dairy

Sci. _70,

_Suppl.

_1,

124 abstract

Johnson N.L. & Kotz S.

₍₁₉₇₂₎

Distributions in Statistics Continuous Univariance

Distributions,

vol. 3. John

_Wiley

&

_Sons,

New York

Knuiman M.W. & Laird N.M.

₍₁₉₈₈₎

Parameter estimation in variance component

models for

binary

response data. In: Advances in Statistical Methods

for

Genetic

Improvement

of

Livestock.

_February

_{16-20, 1987,}

Armidale. Australia.

_Springer

Verlag

(forthcoming)

Lalouel

_M.,

Rao

_D.C.,

Morton N.E. & Elston R.C.

₍₁₉₈₃₎

A unified model. J. Hum. Genet.

_35,

816-826

Maijala

K. &

_Syvajarvi

₍₁₉₇₇₎

On the

_possibility

of

_developing

_multiparous

cattle

by

selection. Z. Tierz.

_{Zuechtungsbiol.}

_94,

136-150

Meijering

A. & Gianola D.

₍₁₉₈₅₎

Observations on sire evaluation with

categorical

data

_using

heteroscedastic mixed linear models. J.

_Dairy

_Sci,

_68,

1226-1232 Pearson K.

₍₁₉₆₈₎

Tables

_of

the

incomplete

beta

_function.

2nd ed.

_Cambridge

University Press, Cambridge

Rao C.R.

₍₁₉₇₃₎

Linear statistical

_inferences

and its

_{applications.}

2nd ed. John

Wiley

&

_Sons,

New York

Schaeffer L.R. & Wilton J.W.

₍₁₉₇₆₎

Methods of sire evaluation for

_calving

ease.

Smith A.F.M.

₍₁₉₈₆₎

Some

_Bayesian

_thoughts

on

_modelling

and model choice. The Statistician

_35,

97-102

Stiratelli

_R.,

Laird N. & Ware J.H.

₍₁₉₈₄₎

Random effects models for serial observations with

_binary

response. Biometrics

40,

961-971

Syrstad

O.

₍₁₉₇₄₎

Inheritance of

_multiple

births in cattle. Livest. Prod. Sci.

11,

373-380

Williams D.A.

₍₁₉₈₈₎

Extra binomial variation in

_toxicology.

XIVth International biometric

_{conference, Namur, July}

_{18-23, 1988,}

invited _papers, 301-316

Wright

S.

_(1934a)

An

_analysis

of

_variability

in number of

_digits

in an inbred strain

of

_{guinea pig.}

Genetics

_19,

506-536

Wright

S.

_(1934b)

The results of crosses between inbred strains of

_{guinea pigs}

differing

in number of

_digits.

Genetics

_19,

537-551

Zellner A. & Rossi P.E.

₍₁₉₈₄₎

_{Bayesian analysis}

of dichotomous

_quantal

response

models. J. Econ.

_25,

365-393

APPENDIX A: ALGORITHM FOR SOLVING

_EQUATION

₍₇₎

.

Analytical

approximations

to the inverse function of the

_incomplete

data function

are

_proposed

in calculus books: see, for

instance,

formula

26.5.22,

page 945 in

Abramowitz &

_{Stegun (1972).}

To

_improve

the accuracy of these

approximations,

we

employed

an

_algorithm

based on finite difference

_techniques.

It consists in

_iterating

from

_(t)

to

_(t

+

1)

with

Starting

values are xo =

0;

(a, b)

o

I

= 0 and x_l = value of the inverse

probability

function

_{given by}

e.g. Abramowitz

& Stegun.

In this

_equation,

the

_incomplete

beta function is calculated from Peizer and

Pratt’s

_{approximation}

as recommended

_by

Pearson

(1968,

formulae

_{47, 48,}

49 page

XXX)

and Johnson & Kotz

_(1972,

vol.

_III,

_pp.

_48,

_49).

For the values of a and

b encountered in this

_study,

this

_{approximation}

is _{very accurate;} the difference

Itrue-approximated values)

is less than

10-

3

_according

to the

_previous

authors.

Computationally, solving

equation

(7)

in terms of

_(n,1?)

_requires

_obtaining

the

parameter a =

(n

₊

’x

b

)1?

of the

incomplete

beta function

I

z

(a, b)

= 1 - _a,

given

x values of _{a, x} and the ratio

a/b

=

!r/(1 -

7r).

This _can be worked _out with _an

algorithm

similar to

_[Al]

in which a is substituted for x.

For the sake of

_comparison,

n was

_predicted

in the fashion

_described,

and

also, using

a second

_{approximation}

to the

_incomplete

beta function

_(formulae

26.5.21,

page

945)

in Abramowitz &

Stegun

(1972).

In the

examples considered,

the