Evaluation des pères dans le cas de paternité incertaine

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Evaluation des pères dans le cas de paternité incertaine

Jean Louis Foulley, Daniel Gianola, D. Planchenault

To cite this version:

Jean Louis Foulley, Daniel Gianola, D. Planchenault. Evaluation des pères dans le cas de paternité

incertaine. Genetics Selection Evolution, BioMed Central, 1987, 19 (1), pp.83-102. �hal-02717943�

Sire evaluation with

uncertain

_paternity

J.L. FOULLEY D. GIANOLA D. PLANCHENAULT’

*

1.N.R.A., Station de

_Génétique

_{quantitative et}

_appliqu

_L

₆

_e

Centre de Recherches

Zootechniques,

F 78350

_{Jouy-en-Josas}

* ’’

Department of

Animal Sciences, University

of

Illinois, Urbana, Illinois 61801, U.S.A.

***

Insiimi

_d’Elevage

et de Médecine Vétérinaire des

_Pays

_Tropicaux,

10, rue Pierre-Curie, F 94704

_{Maisons-Alfort}

Cedex

Summary

A sire evaluation

procedure

is

proposed

for situations in which there is

uncertainty

with respect to the

assignment

of progeny to sires. The method

requires

the

specification

of the

prior

probabilities

P;j that progeny i is out of sire

_j.

Inferences about location _{parameters («} fixed >

environmental and _{group effects and}

_transmitting

abilities of

_sires)

are based on

_Bayesian

statistical

procedures.

Modal values of the

_posterior

distribution of these parameters are taken as

_point

estimators.

_Finding

this mode entails

solving

a nonlinear system of

_equations

and several

algo-rithms are

_suggested.

The

_methodology

is described for univariate evaluations obtained from

normal or

_binary

traits. Estimation of unknown variances is also addressed. A small numerical

example

is

_presented

to illustrate the

procedure.

Potential

_applications

to livestock

_breeding

are

discussed.

Key

words : Sire evaluation, uncertain

paternity, Bayesian

methods.

Résumé

Evaluation des

_pères

dans le cas de

_paternité

incertaine

Une méthode d’évaluation des

pères

est

proposée

en situation d’incertitude vis-à-vis de

l’assignation

des descendants à leurs

_pères.

La méthode

_requiert

la

spécification

des

_{probabilités}

a

priori

pij que le descendant i

provienne

du

_{père j.}

L’inférence des

_paramètres

de

_position

(effets

« _groupe » et de milieu, considérés comme fixes et valeurs

_génétiques

transmises des

_pères)

est

basée sur des

procédures

statistiques

bayésiennes.

Les valeurs modales de la distribution a

posteriori

de

_{ces paramètres}

ont été

prises

comme estimateurs

ponctuels.

La recherche du mode nécessite la résolution d’un

_{système d’équations}

non linéaire pour

_{lequel plusieurs algorithmes}

sont

proposés.

La

_{méthodologie}

est

_développée

dans le cadre _{univariate pour} des caractères normaux et

binaires. Le cas de variances inconnues est

_également

abordé. Un

_petit

_exemple

_numérique

est

présenté

à titre d’illustration. Enfin, les

applications possibles

aux

_espèces

_domestiques

sont

discutées.

I. Introduction

There are situations such as m

_{multiple-sire}

_matings

under

_pastoral

conditions where sire evaluation is

_complicated

because of

_uncertainty

with

_respect

to the

assign-ment _{of progeny} to sires.

_Using

information from red blood cell

types,

major

histocom-patibility

markers or

_precise

records on

_breeding

_period

and

_{gestation length,}

it is

possible

to

_specify

the

_{probabilities}

_(p

_;j

₎

that a

_{given offspring (i}

=

_1,

...,

n)

has been

sired

_by

different males

_(j

=

_1,

...,

m).

In the absence of such

information,

it is reasonable to state that individual males in a

_given

set, e.g., bulls

breeding

in the same

paddock,

are sires with

_{equal probability.}

This

_problem

was studied

_by

PmVEY & ELSEN

(1984)

within the framework of selection index and its restrictive

_assumptions.

The purpose of this paper is to

_present

a more

_general

and flexible

_methodology

able to

cope with several sources of variation

_including

unknown fixed effects and variance

components.

The

_procedure

is

_along

the lines of linear and nonlinear mixed model

methodology (H

ENDERSON

,

1973 ;

GIANOLA &

F

OULLEY

,

1983a,

b).

Continuous and discontinuous variation are _{examined in this paper} to _{illustrate the power and}

_generality

of the

_approach.

II.

_Normally

distributed data

A.

_Methodology

Consider the usual univariate linear model :

where y is a vector of

_{records, [3}

is an I x 1 vector of « fixed » effects

(e.g., genetic

groups, « nuisance » environmental

factors),

u is an m x 1 vector of random

_transmitting

abilities of

sires,

X and Z are instance

matrices,

and e is a vector of residuals. The matrices X and Z are known

_{(non-random),}

if the sires of the progeny with records in

y are identified. In other

words,

the above model holds

_{conditionally}

on X and Z. Let

_T

_i

_j

define the situation in which

_{male j}

is the true sire of _{progeny i. The} conditional _{distribution of the record y,}

_given

_Yi

_j

_,

the location

_parameters

_p

and u and

the residual variance

_U2

can be written as

where NIID stands for

normal,

independent

and

_identically

distributed ;

_z

_ij

is an m x 1

vector

_having

a 1 in

_{position j}

and 0’s elsewhere. Put

_{wi, =}

_[x,,

_zi

_j]

_,

0’ =

_{[(3’, u’]}

and define _laij =

w’,,O.

Inferences about 0 can be obtained

_conveniently

via

_Bayes

theorem,

and this has also been done in other

_genetic

evaluation

_{problems (RB}

_NNINGEN

_,

1971 ;

D

EMPFLE

,

1977 ;

L

EFORT

,

1980 ;

GIANOLA &

F

ERNANDO

,

1986).

The

_prior

distribution of 0 is «

naturally

» taken as the

_conjugate

of

_[1]

_(Cox

&

HII

NKLEY

,

1974)

so

where a’ =

_[8’,

_0]

and

It will be assumed from now on that

_{prior knowledge}

about

(3

is vague so as to

mimic the traditional mixed model

_analysis.

Hence,

the

_prior

distribution of 0 is

_strictly

proportional

to the

_marginal

_prior

distribution of u.

_However,

the notation of

_[2]

above is retained to

_present

a more

_{general expression}

for the

_posterior

distribution of the

vector 0. The matrix

_X

_u

= _A

U2

,

where A is the matrix of additive

_{relationships}

between

sires,

and

_u’

_is

the variance between

sires,

equal

to one

_quarter

of the additive

_genetic

variance.

Because the observations are

_{conditionally}

_independent,

the likelihood function can

be written as :

because I p

ij

= _1. The _mean of _the distribution in

[3B]

is

i

where

_P

_{; =}

_[

_Pil’

.. ,

PiP

&dquo;&dquo;

p

i.]

is a 1 x m row vector

containing

the

probabilities

p,,

of

5£i;

(progeny

i out of sire

_j).

As shown in

_Appendix

A,

the variance of the distribution

[3B]

is

The

_posterior

distribution of 0

_(assuming

that the

_dispersion

parameters

are

known,

can be written

from,

[1],

[2], [3A]

and

_[3B]

as

which is not in the form of a normal distribution.

_Hence,

the mean of this distribution

cannot be a linear function of the data.

The selection rule which maximizes the

_{expected transmitting ability}

of a fixed number of selected sires is the mean of the

_posterior

distribution

_{[4] (G}

_OFFINET

&

E

LSEN

,

1984 ;

FERNANDO &

G

IANOLA

,

1986).

Because the

_expected

value of this distribution is difficult to obtain in closed

_form,

we calculate the modal value of 8 and

rule in the sense described

above ;

this is a reasonable

_{approximation}

as

_sample

size

increases

(Z

ELLNER

,

1971).

B.

_Computations

Finding

the maximum of

_[4]

with

_respect

to 0

_requires

_setting

to 0 the first derivatives of

_[4]

with

_respect

to this vector.

_Letting

_L(O)

be the

_{log-posterior density,}

we obtain :

and (!

(.)

is the standard normal

_density

function. Observe that

_q

_;j

is the

_posterior

probability

that progeny i is out of sire

_j,

and that this

_probability

is maximum when the residual y; -

w;!6

is null. This is so because in this instance the model under

_:

_£

_;j

would fit

_perfectly

to the data.

_Equating

_[5]

to 0

_gives

a nonlinear

system

of

_equations

on 0 so an iterative

_procedure

is

_required

to solve it.

Although

several

_algorithms

can be used for this purpose, the

_simple

form of

[5]

suggests

to

_implement

a functional iteration.

_Setting

[5]

to 0 and

_rearranging

_{yields :}

because

_prior

information about

_{(3 is}

_{vague and}

2q

;j

=

_{1 ; !}

₌

u’I(T’!

=

(4/h 2) _1,

_where

h

z

2 is

_{heritability.}

Note that the

coefficient

matrix and the

_right-hand

sides

_depend

on 0 as

q

i

,

is a function of

(3

and u ; this is clear from

[6].

Defining :

Q

=

{q

¡j

} :

an n x m matrix of

_{posterior probabilities,}

..

and :

D

c

=

_Diag

{Iq

¡J

:

an m x m

_diagonal

_matrix,

whose elements can be

_thought

of as the i

posterior expected

value of the number of progeny of sire

j,

the above

_system

can be written in terms of the iterative scheme :

where

_[k]

indicates the iterate number. In

_[8],

the matrices

Q

and

_D

_c

are evaluated at

One

_possible

way of

starting

iteration is to take

_q,j!

=

p

ij

for all values of i and

j.

Thus

₍

_y

= P =

{p;!},

and

₁₎

₁

_:’1

=

_A!

=

_Diag

flp

ijl

,

and these values can be viewed as the

« natural » ones to

_adopt

_prior

to the

data.

In

_{practice, uncertainty}

is

_only

with

respect

to a small subset of the sires that need

to _{be evaluated. The progeny} can _{be classified into 2 groups :}

_I&dquo;

_pertaining

to

individuals

_having

sires

_{unambiguously}

identified,

and

₁

₂

_{corresponding}

to _progeny with

parentage

under «

dispute

».

_Similarly,

sires can be allocated to _{2 groups :}

_J&dquo;

with all their progeny in set

_I&dquo;

and

J,,

with some _{progeny in}

_I,

and some _{progeny in}

₁

₂

_.

The data vector can be

_partitioned

into three

_mutually

exclusive and exhaustive

compo-nents :

because

the set

_{i

E

_i,

_{f1 j}

E

_J,}

is

_empty.

The vector of

_transmitting

abilities can be

partitioned

as

_[u&dquo;

u

2]

,

_{corresponding}

to sires in

_J,

and

_J,,

_{respectively,}

so.

Likewise

correspond

to the three

_partitions

in

_[9]

above. Further

with

_Z&dquo;

_{_}

_lp

_ij

= _{0 or}

11, Z

12

=

fp

ij

= _{0 or}

1}, Q22 = 10

<

_q,

<

_11,

_P

₂₂

=

₁₀

_<

p

ij

<

_11,

as _per

the

_partitions

in

_[9].

_Using

this

notation,

equations [8]

become :

where

_D!zz

is a

_diagonal

matrix with elements calculated as before but for the _progeny

and sires in the third

_partition

of

_[9].

_Again,

iteration can be started

_by

_replacing

the

«

_posterior

_{» Q}

and D matrices in

[ll],

_by

their «

_prior

»

_{counterparts,}

P and

A,

of

appropriate

order. The above

_equations

illustrate

_clearly

the modifications needed in the mixed model

_equations

to take into account uncertain

_paternity.

The

_portions

in the coefficient matrix and

_right-hand

sides

_pertaining

to records where

_paternity

is

unambi-guous

(y&dquo;

and

y

jz

)

are the usual ones. The incidence matrix

_Z22

that would arise if

paternity

of animals with records in Y22 were

_certain,

is

_{replaced by}

a matrix

Q

of

posterior probabilities.

These are

_updated

_during

the course of iteration to take into

account the contribution of the data.

Likewise,

Z!2Z22

is

_replaced

_by

the D

_matrix,

which is a function of the

_{posterior probabilities}

_q

_ij

_,

as

_already

indicated. Because

_Q

₂₂

is

usually

a small

matrix,

_[8]

or

_[11]

_{will converge}

_rapidly.

If functional iteration is slow to

converge,

algorithms

such as

_{Newton-Raphson}

can be

_{employed (Appendix B).}

III.

_Binary

data

A.

_Methodology

The data are now

_binary

_responses so _yi = _{0 or 1. The model used here is based} on

the

_concept

of «

_liability

»

originally developed by

WRIGHT

(1934),

where it is assumed

that there is an

_underlying

normal variable rendered

binary

via an

_abrupt

threshold. Genetic evaluation

_procedures

based on threshold models have been discussed

_by

several authors

_(G

_IANOLA

&

F

OULLEY

,

1983a,b ;

FOULLEY et

_{aI. , 1983 ;}

FOULLEY &

GI

A

NOL

A

,

1984 ;

HARVILLE &

_M

_EE

_,

1984 ;

ILMOURG et

_C

_{lI. ,}

1985 ;

HBSCHELE et

1

1I. ,

1986).

The notation of the

_preceding

section is

retained,

with the

_{understanding}

that the

parameters

are now those of the

_underlying

distribution. The conditional distribution of

a

_binary

_{response is taken} as :

where

_<1>(.)

is the standardized normal cumulative distribution function. The

_parameter

IJ

-ij is the difference between the threshold and the mean of the statistical «

sub-population

» defined

by

indexes

i, j

J (GIANOLA

&

_F

_OULLEY

_,

1983a)

expressed

in units of

standard deviation.

_Assuming

the

_prior

distribution is as in

_[2]

and

_replacing

the normal

density

in

_[3B]

_by

_[12],

the

_{posterior density}

can be written as :

because the residual standard deviation is

_equal

to 1.

Finding

the 9 - mode of

_[13]

involves

_solving

a

_system

with a

_higher

order of

nonlinearity

than the one

_stemming

from

_[5]

so

_{Newton-Raphson}

is used here instead

Letting

the

_{Newton-Raphson equations}

can be written after

_algebra

as :

where the variance ratio À =0

_{11 <r.}

because the residual variance is

_{unity, .:1pl}

_k

_{l =}

_pl

_k

_l

-

P! ’!,

.:1u lk ! = ulkl -

U

[k

-1

1 ,

and

_l

_m

_,

_In

are vectors of ones of

_appropriate

order. One

_possible

_way

to start iteration would be to use

_equations

_[8]

with

_Q

_{replaced by}

P,

D,

replaced by

.:1c,

and y

replaced by

a vector of 0 and 1’s

_indicating

the absence or _{presence of the}

attribute in the progeny in

question.

The values of

₁₃

and u so obtained would be used

to calculate

_1T¡j

and

_r

_ij

in

_[16]

and

_[17]

to then

_{proceed iterating}

with

_[18]

above.

B.

_Analogy

with the normal case

Write _7,, in

_[16]

as

The

_expression

_q!

is

_{directly comparable}

to

_q

_ij

of

_[6]

for the normal case. Both can be

interpreted

as the

_{posterior probabilities}

_{that progeny i is} out of sire

_j,

and are similar

to formulae

_arising

in multivariate classification

_problems

_(L

_INDEMAN

et

_{al., 1980,}

_p.

196).

In the discrete case and

_given

_Y

_ij

_,

if _uij is

_large

_{progeny i would be}

_expected

to

respond

with

_{high probability}

in the first

category

_{and q*,}

will be

_larger

when the response is

actually

in the first rather than in the second

category.

The

_expression

for

v

jj

(with

a minus

sign)

is the « normal score » discussed

_by

GIANOLA & FOULLEY

(1983a,

p.

216 ;

1983b,

p.

143).

IV. Estimation of unknown variances

The

_point

estimators of location described above are the modes of

_posterior

in the situation of

_binary

_{responses. When these variances} are

unknown,

Box & TIAO

(1973)

and O’Hncnrr

₍₁₉₇₆₎

have

_given

_arguments

_indicating

that inferences could be made from the distribution

_f(Olul

=

8

j,

_u!

=

8[

),

where the variances are

_replaced

_by

the

modal values of the

_{marginal posterior}

distribution of the variances. In the absence of

prior

information about the

variances,

these modal values are those obtained from the method of restricted maximum likelihood

_(H

_ARVILLE

_,

1974,

1977).

This

_approach

was

employed by

GrnrroLn et al.

₍₁₉₈₆₎

in the context of

_{optimum prediction}

of

_breeding

values and these authors view the

_resulting

_predictors

as

_belonging

to the class of

empirical Bayes

estimators. The

_{general principles}

involved in

_finding

the modal values of the

_posterior

distribution of the variances are

_given

below.

F

OULLEY et al.

₍₁₉₈₆₎

and GIANOLA et al.

₍₁₉₈₆₎

showed that maximization of

_f(

_G.

₁

_,

u:.ly)

with

_respect

to the variances in the absence of

_prior

information about these

parameters

leads to the

_{equations :}

where

_E!

indicates

_expectation

with

_respect

to the distribution

_f(ul<T},

_Q

_{;, y).}

Further,

and now

_taking

_expectation

with

_respect

to

f(6!aj’,

Q

u, y),

we need to

_{satisfy :}

The derivation is based on the

_{decomposition}

of the

_posterior

distribution of all

unknowns,

f([3,

u,

_u,

₂

_,,

cr.21y),

into

It should be noted that the likelihood function does not

_depend

on

_{u 2,}

which is true

both in the normal and

_binary

cases.

_Also,

when flat

_priors

are taken for the

_variances,

f(I.T!)

and

_f(<7!)

do not _appear in the above

_{decomposition.}

Solving

[19}

and

_{[20] simultaneously}

for the unknown variances leads to an iterative

scheme

_involving

the

_{expressions :}

where

o k is iterate

_number,

9 C is the inverse of the coefficient matrix in

_{Newton-Raphson (Appendix B),}

or

of

_[18]

when observations are

_binary,

o

C!,,

is the submatrix of C

_{corresponding}

to the

u-effects,

o M is the coefficient matrix in

_[8]

or

[18]

without

_A-’X,

. W =

[X, Q]

It should be noted that in the

_binary

case the residual variance is not estimated because it is taken as

_equal

to one. The derivation of

_[22]

is

_given

in

_Appendix

C.

_Equation

[21],

however,

holds in both cases. The conditional

_expectations

are taken as if the

« true » values of the variance

_components

were those found in the

_previous

iteration. As

_pointed

out

_by

GIANOLA et al.

_(1986),

_[20]

and

_[21]

arise in the EM

_algorithm

(D

EMPSTER

et

_al.,

₁₉₇₇₎

when

_applied

to estimation

_by

restricted maximum

likelihood,

and the

_resulting

estimates are never

_negative.

V. Numerical

application

A small data set from a _progeny test of Blonde

_{d’Aquitaine}

sires carried out in France was used to illustrate the methods

_presented

_{in this paper. The data} set is the

same as the one utilized

_by

FOULLEY et al.

_(1983),

with some’

_{modifications,}

as

illustrated in table 1. There were 47

_calving

records

_including

information on

_region

of

origin

of the

_heifer,

_calving

season, sex and sire of

_calf,

and birth

_weight

(BW)

and

calving

ease

_(CE)

as _{response variables.} CE was recorded as an all-or-none trait with

«

easy » and « difficult »

_calvings

coded as 0 or

1,

respectively.

As shown in table

1,

paternity

was uncertain in the case of records

1,

2,

3 and 39. For the first three

records,

information on

_{breeding periods}

and

_{gestation lengths}

led to an

_assignment

to

natural service sires 7 and 8 of

_{probabilities}

_equal

to

1

and

3

,

_{respectively.}

In the

4 4

case of record

39,

artificial insemination sires 1 and 2 were

_assigned

_{probabilities}

of

1 1

and 2 ,

respectively.

2 2

A. Model

Birth

_weight

was

_regarded

as

_following

a normal

distribution,

and CE was treated

as a binomial trait. Both traits were

_{analyzed using}

the model

where

_H

_;

is the effect of

_region

i of

_origin

of heifer

_(i

=

_1,

_2),

A

j

is the effect of the

_jth

season of

_calving

_(j

=

_1,

2), S,

is the effect of _sex of calf k

(k

= _{1 for males or 2 for}

females), f,

is the

_{transmitting ability}

of the lth sire of heifer

₍₁

=

1,

.. ,

8),

and

e

ijkl

is a

residual with variance

_uj.

The vectors

_p

and u were

Prior

_knowledge

about

!3

was assumed to _{be vague.}

_Heritability

was .25 for both

traits,

and

₍

_T

_e2

was 5

_kg

for BW and 1 for

_CE,

the discrete trait. In

_forming

the

relationship

matrix

A,

it was assumed that the artificial insemination sires

_{(1 through 6)}

were

_unrelated,

and that the natural service sires 7 and 8 were non-inbred sons of 5

In this

_example,

the sets needed to define

_[9]

were :

_1,

=

11,

2, 3,

39} (with

I,

being

the

_complement),

_{J, =}

{2,

3, 4,

5}

and

_{J, =}

_{l,

6,

7,

8}.

Thus,

the matrix

_P,,

in

_[10]

was

For

BW,

the nonlinear

_system

_[11]

was solved

_using

3

algorithms :

functional

iteration,

and

_{Newton-Raphson}

and

_scoring

as described in

Appendix

B. For

CE,

computations

were carried out with

_{[18] ;}

_starting

values were calculated as

discussed

earlier.

Iteration

_stopped

when the square root of the average

squared

correction was less than

10-5

. Variance

_components

were estimated for both traits

using

the

_procedures

outlined in section III.

Sire evaluations

_ignoring

_uncertainty

on

_paternity

were also calculated so as to

further illustrate the

_procedures.

This was done

_{by assigning progenies}

1, 2,

3 to sire 1 and record 39 to sire 6.

B. Results

Results of the

_analysis

conducted for BW are

_presented

in table 2.

_Irrespective

of

the

_algorithm

_used,

the

_stopping

rule of 10-1 was satisfied in 4 iterations. The fact that

the

_algorithms

were

_equally

fast to _{converge is}

_undoubtedly

related to the limited

extent of

_{nonlinearity,}

as

_only

4 out of 39 records had

_ambiguous

_{parentage. Further,}

a

«

sharp

»

_assignment

of

probabilities

(&mdash;vs. &mdash;)

was

made in 3 out of the 4 records. In B4 4 4 /

this data set, from a

_practical

_point

of view iteration could have

_stopped

at the second round. Differences between the

_analyses

conducted

_{ignoring uncertainty}

and

_taking

it into account were minimal. Sire 1 was the most affected because 3 records

_assigned

to

him in the case of certain

_paternity

were

_assigned

to sires 7 or 8 when

_paternity

was

uncertain.

The

_analysis

of

_calving<

ease is shown in table 3. When

_paternity

was

certain,

5

iterates were

_required

to _{converge. On the other}

_hand,

13 iterations were

_required

when

_uncertainty

was taken into account. This is so because with a

_binary

trait there are 2 sources of

_nonlinearity

when

_paternity

is uncertain : one due to the fact that the model is

_nonlinear,

and the second due to the

_uncertainty

itself. The second source of

nonlinearity

was

_responsible

for the 8 additional iterations.

Estimates of variance

_components

in this

_example

were

<1!

= _{0 and}

<1;

= 22.81 for

BW,

and afl =

.096 for CE. The latter value

_gives

an estimate of

_heritability

of .35 in the

_underlying

scale. For

BW,

more than 400 iterates were needed for estimates of variance

_components

to _{converge, and 193 iterations} were

_required

for CE. It is well

known that the EM

_algorithm

is

_extremely

slow to _converge

_(T

_HOMPSON

_,

_1979),

especially

in small

_samples.

However,

alternative

_{parameterizations}

of the model or

numerical shortcuts

_(e.g.,

_S

_CHAEFFER

_,

_{1979 ;}

MISZTAL &

S

CHAEFFER

,

1986)

can be used

VI. Discussion

The

_impact

of the extent of misidentification on sire evaluation and on estimates of

genetic

parameters

was studied

_by

VAN VLECK

_(1970a,b)

and BoNmTt

_(1975).

These authors found that misidentification of sires biased downwards estimates of

_heritability

and of

_expected

_genetic

_{progress. Biases in evaluation of sires increased} as the fraction

of misidentified animals increased.

The

_approach

followed in the

_present

_study,

as in PomEY & ELSEN

(1984),

is to

directly

take into account in the

_{analysis uncertainty}

on the

_assignment

_{of progeny} to

sires so as to

_improve

_prediction

of

_breeding

values.

However,

PowEV & ELSEN

₍₁₉₈₄₎

studied the

_problem

in a selection index framework which

_{requires knowledge}

of means

and variances. The issue was adressed here in a more

_general

manner so as to

accommodate different

_types

of distribution

_(normal

or

_binomial),

and less restrictive

states of

_knowledge

vis-a-vis fixed effects and variance

_components.

Using

the

_{Bayesian paradigm}

as in GIANOLA et al.

_(1986),

leads to inferences based

on a

_posterior

distribution with the

_uncertainty

«

integrated

» or

_averaged

out. With the

p

and u

_components

of the mode of the

_posterior

distribution taken as

_point

estimators and

_predictors

of fixed and random

effects,

respectively,

a nonlinear

_system

_of equa-tions is obtained.

_Algorithms

for

_solving

these

_equations

are _{discussed in the paper.}

Variance

_components

were estimated from the

_{joint posterior}

distribution of the variances after

_taking

into account

_uncertainty

in the

_assignment

_{of progeny} to sires. The

_point

estimators chosen were the modal values of this

distribution ;

expressions

for

computing

the estimates

_iteratively

were

_presented.

Is it

_possible

to use

_directly

the mixed model

_equations

to obtain standard best linear unbiased

_predictors

when

_paternity

is uncertain ? The best linear unbiased

predictor

of u is

_{given by}

where V is the variance covariance matrix of the records

_(H

_ENDERSON

_,

₁

_973).

From results in

_Appendix

_A,

the

_diagonal

elements of V in the continuous case are

and the

_{off-diagonals}

are

where

_a

_jr

is the additive

_relationship

between

_{sires j}

and

_j’.

It follows that V is not in theformZGZ’ +

_{R needed to put V-’}

= R-1 -

_R-

₁

_Z(Z’R-

_I

_Z

₊

_{G-’)-’Z’R-’ so as to establish}

the

_equivalence

between the best linear unbiased

_predictor

above and the results

_given

by

the mixed model

_equations

_(H

_ENDERSON

_,

_1984).

It is not obvious how to treat the

problem

of uncertain

_paternity

_using

standard

_techniques.

The

_Bayesian

_solution pre-sented

_here,

on the other

_hand,

offers a clear answer. POIVEY & ELSEN

₍₁₉₈₄₎

discussed situations in which the methods

_presented

here could be

_applied.

These include :

_i)

females

_{exposed simultaneously}

or

_successively

in time to _{groups of}

_{males ;}

_{ii) joint}

use

of artificial and natural

_breeding

in

_sheep

flocks and cattle herds in

_conjunction

with

estrous

_{synchronization}

_{techniques ;}

and

_{iii) heterospermic}

_progeny

_testing

with ambi-guous

parentage.

A

_requirement

of the

_procedure

is the

_{specification}

of

_prior

probabili-ties

_p

_ij

which can be based on external information such as biochemical

_{po)ymorphysms}

or, more

_likely

under

extensive

_conditions,

records on

_breeding

dates and

_gestation

lengths.

In

_particular,

_{the methods described here may be}

_potentially

useful in situa-tions where natural service

sires

are used

_extensively,

_e.g.,

_{pastoral production}

_systems.

The

_computations

are

_feasible,

at least for univariate sire evaluations carried out under the

_assumption

of

_normality

and with

_genetic

_parameters

assumed known. Extensions to

the multivariate situation can be done without

_great

_conceptual

_difficulty.

Received

_February

27,

1986.

Accepted

June

_24,

1986.

Acknowledgements

This research was conducted while J.L. FOULLEY was a

_George

A. Miller

_Visiting

Scholar at

the

_University

of Illinois, on sabbatical leave from INRA. J.L. Fouc.r.EV wishes to thank the Direction des Productions animales and Direction des relations internationales, INRA, for their

support. D. GIANOLA

_acknowledges

_support from the Illinois

_{Agriculture Experiment}

Station and of Grant No. US-805-84 from BARD-The United States-Israel Binational

_Agricultural

Research

and

_Development

Fund.

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&

Appendix

A

Variance-covariance structure

_of

the data

_{(frequentist viewpoint)}

The

_starting

_point

is

_[3B].

As shown in the text,

omitting

the

_conditioning

on

₍

_T

_.

₂

for the sake of

_{simplicity :}

The variance of the distribution can be obtained

_by

_{writing :}

which follows from

_[1]

and

_[All.

Likewise

where the covariance is taken with

respect

to the

_joint

distribution of

_-Tii

and

_Y,.,.

The first term in the above

equations

is

_clearly

null because the observations are

conditio-nally independent. Assuming

P(:£

¡j

fl

_:£

_i’k

₎

₌

_Pi(Pik’

we

_get

We consider now the variance-covariance structure

unconditionally

on 0.

_Applying

the same

strategy,

one can write :

’

The second term is the variance of _jn; taken with

_respect

to the distribution of 0.

Arguing

from a classical

_viewpoint

_(P

fixed,

u

_random)

we have :

From !A2]

because

_Z’

_j

A

_Z,

= ₁

_(if

_{sires are not}

_inbred),

_and

Yp

ij

z

ij

=

p,

Collecting [A5]

and

[A6]

into

_[A4]

_gives

Finally,

we consider the unconditional covariances between records y; and y;..

Writing :

we

observe

as before that the first term is null.

_Also,

from

_{[All :}

It should be observed that

_Var(y)

cannot be written as

_R

_Q

_e

+

_PAP’u!

because the

diagonal

elements of this last matrix

_expression

are not

_equal

to

_[A7]

except

in the trivial case P =

_Z,

_i.e.,

_when

_paternity

is certain.

Appendix

B I

Newton-Raphson

and

_scoring

_{algorithms for}

normal data The

_{Newton-Raphson algorithm}

consists in

_iterating

with :

where 0!&dquo;! =

O

lk

) -

8!’&dquo; and O

lk

)

is the solution _at iteration k. The first derivatives are

given

in

_[5]

and the second derivatives are :

From the definition of

_q;!

in

_[6],

we have :

Using

this in

_[B2]

above and

_{rearranging gives :}

Some

_{simplification}

in the calculations can be achieved

_{by replacing}

_r;!

_by

its

expectation

taken

_{conditionally}

on

0,

Q

?

and

_;

_£i

_j’

From

_[B4]

we obtain

_{directly :}

This used in

_[B5]

above

_yields

a «

scoring

»

_algorithm

for

_solving

_{the nonlinear}

system

of

_equations.

The

_system

_[B5]

can be written in matrix notation

_using

the matrices

R, E!,

and

_E,

of page 89 with

r,

j

as in

[B4]

instead of

[17].

Also,

define matrices

The _system

_[B5]

becomes then

The

_algorithm

is also described for the case where uncertain

_paternity

is

_only

with respect to a small

_proportion

of the sires evaluated.

Here,

we

_partition

R in the same

With the above

notation,

the

_system

in B6 can be written as :

The above

_equations

indicate the

_parts

of the

system

that need to be amended to

take uncertain

_paternity

into account. As

before,

the

_nonlinearity

stems from the contribution of the vector _Yn to information about the unknown

parameters.

In order

to start

iteration,

one may take RIOj =

_P,

E!

OI

=

A,,

E’&dquo;’

=

_A!,

and values of the _vectors

_(3,

u, and u, obtained

by

applying

linear mixed model

_methodology

upon the vectors Y and

Yi,-Appendix

C

Derivation

_of

the

_algorithm

used

_{for estimating}

_Qr

with normal data The estimator of

_a,

₂

needs to

_satisfy

_[20].

From

_[3A]

and

_[3B]

Using

this result in

_[Cl]

and then in

_[201

_{gives :}

where

_q

_ij

is as in

[6],

and where

_E,

indicates

_expectation

taken with

_respect

to the conditional distribution

_f(Olu,,

_{a!, y).}

The

_expectation

in

_[C2]

is difficult to obtain because

_q

_ij

is a function of 0. If _qij is

_regarded

as a constant a

_{rearrangement}

of

_[C2]

gives :

As done

_by

HARVILLE & MEE

₍₁₉₈₄₎

in the context of threshold

models,

we

_replace

E(¡L;,

=

w’, 6!j’,

a’,

y) by

the mode

6

calculated

_{using equations [8] (or}

the

_expressions

described in

Appendix

_B).

Thus, the formula above becomes :

Now,

Var(0(y) =

C

_u

_j

_,

where C is the inverse of the coefficient matrix in

_[B6]

or

[B7]

evaluated at

6.

_Further,

at the

maximum,

[5]

must be null which is satisfied when

[7]

is satisfied.

_Multiplying

both sides of

_[7]

_{by 6}

_(and

_remembering

that a flat

_prior

is

used for

_!3)

_yields

With this in

mind,

we obtain the

_following

for the

_components

of

_[C3]

where M is the coefficient matrix in

_[8J

without A-’. ’.