On sire evaluation with uncertain paternity

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On sire evaluation with uncertain paternity

Jl Foulley, R Thompson, D Gianola

To cite this version:

Short

note

On

sire evaluation

with uncertain

_paternity

JL

_Foulley

R

_Thompson

D

Gianola

1

INRA,

Station de _{Génétique Quantitative,} 78350 _{Jouy-en-Josas, France;}

2

Institute

_of

Animal _Physiology and Genetics Research, Edinburgh EH9 3JN, UK;

3

Department of Animal Science, Urbana, IL _61801, USA

(Received

28 November 1989; accepted 21 _May

₁₉₉₀₎

Foulley

et al

_{(1987) -}

from now on referred to as FGP - have described a first order

algorithm

(functional iteration)

for

_computing

the maximum a

_posteriori

(MAP)

estimator of the location

_parameter

0 of a normal distribution in situations where there is

_uncertainty

with _respect to the

_assignment

of data to some effects

_{(eg sire)}

of 0. This

_algorithm

has a

simple form,

related to the mixed model

_equations,

and is easy to _program and to

_apply.

Second order

_algorithms

can also be used for

computing

MAP estimates of 0. These

_algorithms

are needed for

_getting

estimates

of the

_asymptotic

accuracy of these modal

estimators,

or for variance _component estimation.

The

_objective

of this note is to correct formulae needed for such

_algorithms

_given

by FGP,

and to describe an alternative

computing

procedure

based on the method of

_scoring.

Let

_L(O)

be the

_logposterior

density;

the first derivatives can be written as:

where:

W

tj

being

an incidence column vector

_(p,l)

_pertaining

to the ith observation

(i

=

_1,2,...,

_n),

_{given j is}

the _true

_sire,

and

*

Differentiating

(1)

again,

one obtains the

_expression

for the

_negative

Hessian of

L(O), ie:

where:

R!k

is an

_(n

x

_n)

_diagonal

matrix

_pertaining

to

_{sires j}

and k with ith element

or, explicity:

where qi! is the same as in FGP and

_jk

₆

is the Kronecker

delta, equal

to 1

_{if j}

₌

k or 0 otherwise.

Note that these formulae are

_slightly

different from those

_{given by}

FGP

(Appen-dix

_B,

p

99).

Actually,

formula

(5)

reduces to their

_expression

_[B4]

_{when j}

₌

k and to:

The

_{Newton-Raphson algorithm}

can be written as:

Letting

W! _

(X, z

j

)

where X and _zj are

(n, p)

and

(n, m)

incidence matrices

(given j

being

the true

_sire)

_pertaining

to the

_,0

and u elements of 9 =

_{(!3’, u’)’,}

this _system can be

_expressed

more

explicitly

as:

and,

as before:

Another

_possibility

would be to

_develop

a

_{scoring procedure}

_{by taking}

in

₍₄₎

the unconditional

_expectation

_{of ri,!! based} on the

_following

_expression:

or, more

_explicitly:

where _ajk is the

_jk

element of the numerator

_relationship

matrix

_{A; if j}

=

k,

formulae

_(6a

and

_b)

_apply

with _zij = _zik and

a

hj = _ahk

respectively.

Finally,

it must be

_kept

in mind that

_{&dquo;regular&dquo;}

mixed model

_equations

can also be used as an alternative to

₍₄₎

as shown

_{recently by}

Im

(1989).

The same corrections

_apply

to

_Foulley

and Elsen

₍₁₉₈₈₎

on _p 233. Their

expression

[23b]

should read:

with,

in

_[24b]:

and,

in

_[26c]:

or:

The

_expression

in

₍₁₃₎

_replaces

that in

_[25].

Formula

_[26b]

for _Vml

_is

unaltered

and reduces to v&dquo;,1 =

_(y.&dquo;

₁

_-

IL

,,,,)/u2

_in

the normal _case.

REFERENCES

Foulley JL,

Gianola

D,

Planchenault

₍₁₉₈₇₎

Sire evaluation with uncertain

pater-nity.

Genet Sel Evol

_19,

83-102

Foulley

JL,

Elsen JM

₍₁₉₈₈₎

Posterior

_probability

at a

_major

locus based on

progeny-test results for discrete characters. Genet Sel

Evol 20,

227-238