Assessment of a Poisson animal model for embryo yield in a simulated multiple ovulation-embryo transfer scheme

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Assessment of a Poisson animal model for embryo yield

in a simulated multiple ovulation-embryo transfer

scheme

Rj Tempelman, D Gianola

To cite this version:

Original

article

Assessment of

a

Poisson

animal model

for

_{embryo yield}

in

a

simulated

multiple

ovulation-embryo

transfer

scheme

RJ

_Tempelman

₁

D

Gianola

2

1

University of Wisconsin-Madison, Department of Dairy Science;

2

University of Wisconsin-Madison, Department of

Meat and Animal

_Science,

1675,

Observatory Drive, Madison,

WI 53706, USA

(Received

16 March 1993;

accepted

10

_January

₁₉₉₄₎

Summary - Estimation and

prediction techniques

for Poisson and linear animal models were

_compared

in a simulation

study

where observations were modelled as

_embryo

_yields

having

a Poisson residual distribution. In a _one-way model

_(fixed

mean

_plus

random animal

effect)

with

_genetic

variance

_(0’;)

_equal

to 0.056 or 0.125 on a

_log

linear

scale,

Poisson

marginal

maximum likelihood

_(MML)

_{gave estimates} of

₀

_{’; with}

smaller

_empirical

bias and mean

_squared

error

_(MSE)

than restricted maximum likelihood

_(REML)

_analyses

of raw and

_{log-transformed}

data.

_Likewise,

estimates of residual variance

_(the

average

Poisson

_parameter)

were _poorer when the estimation was

_by

REML. These results were

anticipated

as there is no

_appropriate

variance

_{decomposition independent}

of location

parameters in Che linear model. Predictions of random effects obtained from the mode of the

_{joint posterior}

distribution of fixed and random effects under the Poisson mixed-model tended to have smaller

empirical

bias and MSE than best linear unbiased

prediction

(BLUP).

Although

the latter method does not take into account

nonlinearity

and does

not make use of the

_assumption

that the residual distribution was

Poisson, predictions

were

_essentially

unbiased. After

log

transformation of the

records, however,

BLUP led

to

unsatisfactory predictions.

When

_{embryo yields}

of zero were

ignored

in the

analysis,

Poisson animal models

accounting

for truncation

_outperformed

REML and BLUP. A mixed-model simulation

involving

one fixed factor

_{(15 levels)}

and 2 random factors for 4 sets of variance _components was also carried out; in this

_study,

REML was not included

in view of

highly heterogeneous

nature of variances

_generated

on the observed scale. Poisson MML estimates of variance _components were

_{seemingly unbiased, suggesting}

that statistical information in the

_sample

about the variances was

_adequate.

Best linear unbiased estimation

_(BLUE)

of fixed effects had _greater

_empirical

bias and MSE than the Poisson estimates from the

joint posterior distribution,

with differences between the

*

2

_{analyses increasing}

with the

_genetic

variance and with the true values of the fixed effects.

_Although

differences in

_prediction

of random effects between BLUP and Poisson

joint

modes were

_{small, they}

were often

_significant

and in favor of those obtained with the Poisson mixed model. In

_conclusion,

if the residual distribution is

_Poisson,

and if the

relationship

between the Poisson _parameter and the fixed and random effects is

_{log linear,}

REML and BLUE may lead to _poor

_inferences,

whereas the BLUP of

_breeding

values is

remarkably

robust to the

_departure

from

_linearity

in terms of average bias and MSE.

Poisson distribution

_/

_{embryo yield}

_/

_generalized linear mixed

model / variance

component estimation

_/

counts

Résumé -

Évaluation

d’un modèle individuel poissonnien pour le nombre d’embryons dans un schéma d’ovulation _multiple et de transfert _{d’embryons.} Des

techniques

d’estimation et de

prédiction

pour des modèles poissonniens et linéaires ont été

comparées

par simulation de nombres

d’embryons supposés

suivre une distribution résiduelle de Pois-son. Dans un modèle à un

_facteur

_(moyenne

_fixée

et

_effet

individuel

_aléatoire)

avec des

variances

_génétiques

_{(Q! )}

_égales

à 0, 056 ou 0,125 sur une échelle

loglinéaire,

la méthode de

maximisation de la vraisemblance

marginale

(MML)

de Poisson donne des estimées de

_ou

2

ayant un biais _empirique et une erreur

_quadratique

_moyenne

(MSE)

inférieurs

à

l’analyse

des données

brutes,

ou

transformées

en

logarithmes,

_par le maximum de vraisemblance

restreinte

_(REML).

De

même,

la variance résiduelle

_(le

paramètre

de Poisson

_moyen)

était moins bien estimée _par le REML. Ce résultat était

_prévisible,

car il n’existe pas de

décomposition appropriée

de la variance

_{indépendante}

des

_paramètres

de

_position

dans le modèle linéaire. Les

_prédictions

des

_effets

aléatoires obtenues à

partir

du mode de la distribution

_conjointe

a

_posteriori

des

_{effets fixés}

et aléatoires sous un modèle mixte

pois-sonien tendent à avoir un biais

_empirique

et une MSE

_inférieurs

à la meilleure

_prédiction

linéaire sans biais

(BLUP).

Bien _que cette dernière méthode ne _prenne en _compte ni la non-linéarité ni

_{l’hypothèse}

d’une distribution résiduelle de

_Poisson,

les

_prédictions

sont

sans biais notable. Le BL UP

_{appliqué après transformation logarithmique}

des données con-duit

_cependant

à des

_prédictions

non

_{satisfaisantes. Quand}

les valeurs nulles du nombre

d’embryons

sont

_ignorées

dans

_l’analyse,

les modèles individuels poissonniens prenant en

compte la troncature donnent de meilleurs résultats que le REML et le BL UP. Une

simu-lation de modèle mixte à un

facteur fixé

(15

niveaux)

et

_{2 facteurs}

aléatoires

_pour

₄

en-sembles de composantes de variance a

_également

été

_réalisée;

dans cette

_étude,

le REML n’était pas inclus à cause de la nature hautement

_{hétérogène}

des variances

_générées

sur l’échelle d’observation. Les estimées MML

_{poissonniennes}

sont apparemment non

biaisées,

ce

_{qui suggère}

_que

_{l’information statistique}

sur les variances contenue dans l’échantillon

est

adéquate.

La meilleure estimation linéaire sans biais

(BLUE)

des

effets fixés

a un biais empirique et une MSE

supérieurs

aux estimées de Poisson dérivées de la distribution

conjointe

a

_posteriori,

avec des

différences

entre les

2 analyses

qui augmentent avec la

va-riance

_génétique

et les vraies valeurs des

_{effets fixés.}

Bien que les

différences

soient

faibles

entre les

effets

aléatoires

prédits

par le BL UP et par les modes

conjoints

poissonniens,

elles sont souvent

significatives

et en

_faveur

de ces dernières. En

conclusion,

si la

distri-bution résiduelle est _{poissonnienne,} et si la relation entre le

_paramètre

de Poisson et les

effets fixés

et aléatoires est

_{loglinéaire,}

REML et BLUE peuvent conduire à des

_inférences

de mauvaise

_qualité,

alors que le BL UP des valeurs

génétiques

se _comporte d’une manière

remarquablement

robuste

_face

aux écarts à la

_linéarité,

en termes de biais _{moyen et} de MSE.

distribution de Poisson

_{/ nombre}

_d’embryons

_/

modèle linéaire mixte _{généralisé}

_/

INTRODUCTION

Reproductive

technology

is

_important

in the

_{genetic improvement}

of

_dairy

cattle. For

_{example, multiple}

ovulation and

_embryo

transfer

_(MOET)

schemes may aid in

accelerating

the rate of

_genetic

_progress attained with artificial insemination and

progeny

testing

of bulls in the

past

30 _years

_(Nicholas

and

Smith,

1983).

An

_important

bottleneck of MOET

_{technology, however,}

is the

_{high variability}

in

quantity

and

_quality

of

_embryos

collected from

_{superovulated}

donor dams

_(Lohuis

et

al, 1990 ;

Liboriussen and

_Christensen,

_{1990; Hahn, 1992; Hasler,}

_1992).

Keller and

Teepker

(1990)

simulated the effect of

_variability

in number of

_{embryos following}

superovulation

on the effectiveness of nucleus

_breeding

schemes and concluded that increases of _up to

40%

in

_embryo

_recovery rate

_(percentage

of cows

_producing

no

transferable

_embryos)

could more than halve female-realized selection

differentials,

the effect

_being

_greatest

for small nucleus units. Similar results were found

_by

Ruane

(1991).

In these

_studies,

it was assumed that residual variation in

_embryo

_yields

was

_normal,

and that

_yield

in

_{subsequent superovulatory}

flushes was

_independent

of

that in a

_previous

_flush,

ie absence of

genetic

or

_permanent

environmental variation for

_{embryo yield.}

Optimizing

embryo

yields

could be

_important

for other reasons as well. For

instance,

with

_greater

_yields,

_{the gap} in

_{genetic gain}

between closed and open

nucleus

_breeding

schemes could be narrowed

_{(Meuwissen, 1991).}

_Furthermore,

because of

_{possible antagonisms}

between

_production

and

_{reproduction,}

_{it may} be

necessary to use some selection

_intensity

to maintain

_reproductive

_performance

(Freeman, 1986).

Also,

if

_yield

promotants,

such as bovine

somatotropin,

are

adopted,

the relative economic

_importance

of

_production

and

_{reproduction,}

with

respect

to

_{genetic selection,}

will

_probably

shift towards

_{reproduction. Finally,}

if

cytoplasmic

or nonadditive

_genetic

effects turn out to be

_important,

it would be desirable to increase

_{embryo yields by selection,}

so as to

_produce

the

_appropriate

family

structures

_(Van

Raden et

al,

1992)

needed to

_{fully exploit}

these effects. Lohuis et al

₍₁₉₉₀₎

found a zero

_heritability

of

_{embryo yield}

in

_dairy

cattle.

Using

restricted maximum likelihood

_(REML),

Hahn

₍₁₉₉₂₎

estimated heritabilities of 6 and

4%

for number of

_ova/embryos

recovered and number of transferable

embryos

recovered,

respectively,

in

_Holsteins;

_{corresponding repeatabilities}

were

23 and

15%.

Natural

_twinning

_ability

_may be

_closely

related to

_{superovulatory}

response in

dairy cattle,

as cow families with

_high

_twinning

rates tend to have a

high

ovarian

_sensitivity

to

_{gonadotropins,}

such as PMSG and FSH

(Morris

and

Day,

1986).

Heritabilities of

twinning

rate in Israeli Holsteins were found to be

_2%,

using REML,

and

10%

employing

a threshold model

(Ron

et

_al,

_1990).

Best linear unbiased

_prediction

_(BLUP)

of

_breeding

_values,

best linear unbiased estimation

_(BLUE)

of fixed

effects,

and REML estimation of

_genetic

parameters

are

_widely

used in animal

_breeding

research.

_However,

these methods are most

appropriate

when the data are

normally

distributed. The distribution of

embryo

yields

is not

_normal,

and it is

_unlikely

that it can be rendered normal

_by

a

transformation,

particularly

when mean

yields

are low and

embryo

recovery failure

rates are

_high.

_Analysis

of discrete data with linear

_models,

such as those

_employed

in BLUE or

REML,

often results in

_spurious

interactions which

biologically

do not

It seems sensible to consider nonlinear forms of

_analysis

for

_{embryo yield.}

These

may be

computationally

more intensive than BLUP and

_REML,

but can offer more

flexibility.

The

study

of Ron et al

₍₁₉₉₀₎

_suggests

that nonlinear models for

_twinning

ability

may have the

potential

of

capturing genetic

variance for

reproduction

that

would not be usable

_by

selection otherwise. For

_example,

threshold models have

been

_suggested

for

_genetic

_{analysis of categorial}

_traits,

such as

_calving

ease

(Gianola

and

_Foulley,

_1983;

Harville and

Mee,

1984).

In these

models,

gene substitutions are

viewed as

_occurring

in a

_underlying

normal scale.

_However,

the

_relationship

between

the outward variate

_(which

is scored

_{categorically,}

eg,

’easy’

versus ’difficult’

_calving)

and the

_underlying

variable is nonlinear and mediated

_by

fixed thresholds. Selection

for

_categorical

traits

_using

_predictions

of

_breeding

values obtained with nonlinear threshold models was shown

_by

simulation to

_give

_{up to}

_{12% greater}

_genetic

_gain

in

a

_single

_cycle

of selection than that obtained with linear

predictors (Meijering

and

Gianola,

1985).

Because

_{genetic gain}

is

_cumulative,

this increase may be substantial.

The use of better models could also

_improve

(eg,

smaller mean

_squared

error

(MSE))

estimates of differences in

_embryo

_yield

between treatments.

In the context of

_{embryo yield,}

an alternative to the threshold model is an

analysis

based on the Poisson distribution. This is considered to be more suitable

for the

_analysis

of variates where the outcome is a count that may take values

between zero and

infinity.

A Poisson mixed-effects model has been

developed by

Foulley

et al

_(1987).

From this

model,

it is

_possible

to obtain estimates of

_genetic

parameters

and

_predictors

of

_breeding

values.

The

_objective

of this

_study

was to _compare the standard mixed linear model with the Poisson

_technique

of

_Foulley

et al

_(1987),

via

_simulation,

for the

_analysis

of

embryo yield

in

_dairy

cattle.

_Emphasis

was on

_{sampling performance}

of estimators

of variance

_components

_(REML

versus

_marginal

maximum

_{likelihood, MML,}

for the Poisson

_model),

of estimators of fixed effects and of

predictors

of

_breeding

values

_(BLUE

and BLUP evaluated at average REML estimates of

_variance,

versus

Poisson

_posterior

modes evaluated at the true values of

_variance).

AN OVERVIEW OF THE POISSON MIXED MODEL

Under Poisson

_sampling,

the

_probability

of

_observing

a certain

_{embryo yield}

response

(y

i

)

on female i as a function of the vector of

_parameters

9 can be written

as:

with

The Poisson mixed model introduced

_by

Foulley

et al

₍₁₉₈₇₎

makes use of the link function of

_generalized

linear models

_(McCullagh

and

_Nelder,

_1989).

This function allows the

_modelling

the Poisson

_parameter

A

i

for female i in terms of 0. This

parameterization

differs from that

_presented

in

_Foulley

et al

₍₁₉₈₇₎

who modelled Poisson

_parameters

for individual

_offspring

of each

_{female, allowing}

for extension

in

_Foulley

and Im

₍₁₉₉₃₎

and Perez-Enciso et al

_(1993).

In the Poisson

_model,

the link is the

_logarithmic

function.

such that

Above,

0’ =

_![3’,

_{u’], 13}

is a _p x 1 vector of fixed

_effects,

u is a _q x 1 vector of

breeding

values,

and

_w’

=

[x!

z!]

is an incidence row vector

_relating

0 _{to r¡i.} Let X =

fx’l

and Z =

fz’l

be incidence matrices of order n x p and _n x _q,

_{respectively,}

such that:

In a

_Bayesian

_context,

_Foulley

et al

₍₁₉₈₇₎

_employ

the

_prior

densities

and

where A is the matrix of additive

relationships

between animals and

_Q

_u

_is

the additive

_genetic

variance. Given

_o,’,

_Foulley

et al

₍₁₉₈₇₎

calculate the mode of the

joint posterior

distribution of

₍₃

and u with the

_algorithm

where t denotes iterate

_number,

and where y is the vector of observations. Note that the last term in

_[8]

can

be

_regarded

as a vector of standardized

_(with

_respect

to the conditional Poisson

variance)

residuals.

Marginal

maximum likelihood

_(MML),

a

_{generalization}

of

_REML,

has been

suggested

for

_estimating

variance

_components

in nonlinear models

_(Foulley

et

_al,

1987;

H6schele et

al,

1987).

An

_{expectation-maximization}

_(EM)

_type

iterative

where T =

trace(A-

1

C

u

&dquo;),

such that

and u is the

u-component

solution to

_[7]

_{upon convergence} for a

_given

o,’

_value.

In

!9!,

k

_pertains

to the iteration

_number,

and iterations continue until the difference between successive iterates of

_[9],

_separated

_by

nested iterates of

_[7],

becomes

arbitrarily

small. The above

_{implementation}

of MML is not

_exact,

and arises from

the

_{approximation}

_(Foulley

et

al,

1990)

SIMULATION EXPERIMENTS

A _one-way random effects model

Embryo yields

in two MOET closed nucleus herd

_breeding

schemes were simulated.

Breeding

values

_(u)

for

_{embryo yields}

for n, and nd base

population

sires and

dams, respectively,

were drawn from the distribution u N

N(0, I!u),

where

_0’

_;

_had

the values

_specified

later. The dams were

superovulated,

and the number of

_embryos

collected from each dam were

_{independent drawings}

from Poisson distributions with

parameters:

where 1 is a nd x 1 vector of _{ones, p} is a location

_parameter

and ud is the vector

of

_breeding

values of the nd dams. In nucleus

1,

f

..l

=

ln(2),

whereas in nucleus ₂

p =

ln(8).

Note from

[12]

that for _a

_given

_{donor dam}

_d

_i

_,

so, in view of the

assumptions,

which

_implies

that the location

_parameter

can be

interpreted

as the mean of the natural

_log

of the Poisson

_parameters

in the

_population

of donor dams. It should be

_noted,

as in

_Foulley

and Im

₍₁₉₉₃₎

that

Thus

The sex of the

_embryos

collected from the donor dams was

_assigned

at random

embryo

to age at first

_breeding

was !r = 0.70 in nucleus

1,

and 7r = 0.60 in nucleus 2. This is because research has

_suggested

that

_{embryo quality}

and

_yield

from a

_single

flush tend to be

_negatively

associated

_{(Hahn, 1992).}

Thus the

_expected

number of female

_embryos

_surviving

to _age at first

_{breeding produced by}

a

_given

donor dam

d

i

is,

for i =

_{1, 2, ... n}

_d

_,

and,

on _average,

The

_genetic

merit for

_embryo

_yield

for the ith female

_offspring,

_{uo! ,} was

_generated

by randomly selecting

and

_mating

sires and dams from the base

_population,

and

using

the

_{relationship:}

where _USi and _udi are the

_breeding

values of the sire and

_dam,

_{respectively,}

of

offspring

i,

and the third term is a Mendelian

_{segregation residual;}

z -

N(0,1).

As with the

_dams,

the vector of true Poisson

_parameters

for female

_offspring

was

71

0

=

exp[1p

₊

u

o]

where _uo

represents

the vector of

_daughters’

_genetic

values. The unit vector 1 in this case would have dimension

_equal

to the number of

_surviving

female

_{offspring. Embryo yields}

for

_daughters

were

sampled

from a

Poisson distribution with

_parameter

_equal

to the ith element of

_X

_o

_.

Four

populations

were

_simulated,

and each was

_replicated

30 times:

₁₎

nucleus 1

(g = In 2),

U2

= 0.056;

2)

nucleus 1

_(!

=

In 2),

Q

u

=

_0.125;

3)

nucleus 2

=

In 8),

Qu

= 0.056 ;

and

₄₎

nucleus 2

_(u

=

ln8), !

= 0.125. Features of the 2 nucleus herds are in table I. The

expected

nucleus size is

slightly

_greater

than

n

s +

_n

_d

₍₁

+

₍

₇

_{!,/2)exp(J.l)),}

ie about 218 cows in each of the 2

_schemes,

_plus

the

corresponding

number of sires. The values of

_{or were}

arrived at as follows:

_Foulley

et al

_{(1987), using}

a first order

_{approximation,}

introduced the

parameter

which can be viewed as a

_{’pseudo-heritability’. Using this,}

the values of

u

£

in

the

4

_populations

_correspond

to:

₁₎

‘h

2

’

=

_0.10;

2)

’

‘h

2

=

_0.20;

3)

‘h

2

’

=

_0.31;

and

4)

’h

2

,

= 0.50.

In each of the 30

_replicates

of each

_population,

variance

_components

for

_embryo

yield

were estimated

_employing

the

_following

methods:

1)

Poisson MML as in

Foulley

et al

_{(1987); 2)}

REML as if the data were

_normal;

₃₎

truncated Poisson MML

_excluding

counts of zero, and

using

the formulae of

_Foulley

et

al,

(1987);

4)

REML-0,

ie REML

_applied

to the data

_excluding

counts of _zero; and

₍₅₎

non-null responses while

discarding

the null _responses.

_Empirical

bias and mean

squared

error

(MSE)

of the

_estimates,

calculated from the 30

_replicates,

were

used for

_assessing

_performance

of the variance

_component

estimation

_procedure.

Because the

_probability

of

_observing

a zero count in a Poisson distribution with a mean of 8 is very

low,

the truncated Poisson and REML-0

analyses

were not

carried out in nucleus 2.

_Likewise,

_breeding

values were

predicted

_using

the

following

methods:

₁₎

the Poisson model as in

_[7]

with the true

_o!,

and

_taking

as

_predictors

A =

exp[1Q

₊

û],

where _u is the _vector of

breeding

values of

_sires,

dams,

and

daughters;

2)

BLUP

_(1!*

+

_u

_*

₎

in a linear model

_analysis

where the variance

components

were the _average of the 30 REML estimates obtained in the

_replications

and the asterisk denotes direct estimation of location

_parameters

on the observed

scale;

3)

a truncated Poisson

_analysis

with the true

a

and

_predictors

as in

_1);

4)

BLUP-0,

as in

₂₎

but

_excluding

zero

_counts,

and

_using

the _average of the

30 REML-0 estimates as true

_variances;

and

₅₎

_BLUP-LOG,

as in

₂₎

after

_excluding

zero counts and

_transforming

the

_remaining

records into

_logs.

The average of the

30 REML-LOG estimates of variance

_components

was used in this case.

BLUP-LOG

_predictors

of

_breeding

values were

expressed

as

exp[lti

+

u]

where p

and

u

are

solutions to the

_{corresponding}

mixed linear model

_{equations. Hence,}

all 5

_types

of

predictions

were

_comparable

because

_breeding

values are

_expressed

on the observed scale. As

_given

in

_!12!,

the vector of true Poisson

_parameters

or

_breeding

values

for all individuals was deemed to be A =

exp[1p

₊

u].

Average

bias and MSE of

prediction

of

_breeding

values of dams and

_daughters

were

_computed

within each

data set and these statistics were

_averaged

_again

over 30 further

_replicates.

Rank

correlations between different estimates of

_breeding

values were not considered as

they

are often _very

_large

in

_spite

of the fact that one model _may fit the data

substantially

better than the other

_{(Perez-Enciso}

et

_al,

_1993).

A mixed model with two random effects

The base

_population

consisted of 64 unrelated sires and 512 unrelated

_dams,

and the

_genetic

model was as before. The

probability

of a

daughter surviving

to age at

Embryo yields

on dams and

daughters

were

_generated

_by

_drawing

random

numbers from Poisson distributions with

_parameters:

where p

is a fixed effect common to all

observations,

H =

{H

i

}

is a 15 x 1 vector

of fixed

effects,

s =

{

Sj

} ’&dquo; N(0,Iu£)

is a 100 x 1 vector of unrelated ’service sire’

effects,

u =

j

Uk

} -

N(0,

)

U2

A

is a vector of

_breeding

values

independent

of service sire

_effects,

and

₀

_’;

and

₀

_’;

_are

_appropriate

variance

_components.

The values

_of

_{+ Hi}

were

_assigned

such that:

Thus,

in the absence of random

_effects,

the

_expected

_{embryo yield ranged}

from 1 to 15. Each of the 15 values of fl

+ H

i

had an

_equal

chance of

being assigned

to any

particular

record.

Service sire has been deemed to be an

_important

source of variation for

embryo

yield

in

_{superovulated dairy}

cows

(Lohuis

et

al, 1990; Hasler,

1992).

However,

no sizable

genetic

variance has been detected when

embryo yield

is viewed as a trait of the donor cow

(Lohuis

et

al, 1990; Hahn,

1992).

This influenced the choice of the 4 different combinations of true values for the variance

_components

considered. In all cases, the service sire

component

was twice as

_large

as the

genetic

component.

The sets of variance

components

chosen were:

(A)

u

£

=

0.0125,

=

0

.

0250;

(

B

)

Qu

=

0

.

0250

,

g

2

=

0500;

.

0

(

C

)

o

r2

=

0

.

0375

,

U2

=

0

.

0750;

and

_(D)

_U2

₌

_0.0500,

_a;

= 0.1000.

_Along

the lines of

_[14],

the

_genetic

variances

correspond

to

_{’pseudoheritabilities’}

of

_7.5-22%,

and to relative contributions of service sires to variance

_of

_{15-44% ;}

these calculations are based on the

_approximate

average true fixed

effect A

on the observed scale in the absence of

overdispersion:

For each of the 4 sets of variance

_parameters,

30

_replicates

were

generated

to assess the

_{sampling performance}

of Poisson MML in terms of

_empirical

bias

and _{square root} MSE. Relative bias was

empirical

bias as a

_percentage

of the

true variance

_component.

Coefficients of variation for REML and MML estimates

of variance

components

were used to

_provide

a direct

_comparison

as

_they

are

expressed

on different scales. REML estimates were also

_required

in order to compare estimates of fixed effects and

predictions

of random effects obtained

under a linear

_mpdel

_analysis

with those found under the Poisson model. MML and REML estimates were

_{computed by}

_{Laplacian integration}

(Tempelman

and

Gianola, 1993)

using

a Fortran program that

incorporated

a _sparse matrix

_solver,

SMPAK

_(Eisenstat

et

_al,

₁₉₈₂₎

and ITPACK subroutines

_(Kincaid

et

al,

1982)

to

set up the

_system

of

_{equations !7!.}

For

REML,

this

_corresponds

to the derivative-free

_algorithm

described

_by

Graser et al

₍₁₉₈₇₎

with a

_computing

_strategy

similar

to that in Boldman and Van Vleck

_(1991).

well

_defined)

to

_compute

estimates of fixed effects and

_predictions

of random effects in the linear model

_analysis;

for the Poisson

_model,

the true values of the variance

components

were used.

_Empirical

biases and MSEs of the estimates of fixed effects obtained with the linear and with the Poisson models were assessed from another

30

_replicates

within each set of variance

_components.

One more

_replicate

was then

generated

for each variance

_{component set,}

from which the

empirical

average bias and MSE of

_prediction

of service sire and animal random effects were evaluated.

In order to make

_comparisons

on the same

_scale,

the Poisson model

_predictands

of the random effects were defined to be

_b.exp(s)

for service sires and

_b!exp(u)

for

additive

_{genetic effects,}

_{respectively;}

b is the ’baseline’

_parameter:

In view of

_!15!,

so that

Hence,

The ’baseline’ value can then be

_interpreted

as the

_expected

value of the Poisson

parameter

of an observation made under the conditions of an

_{’average’}

level of

the fixed effects and in the absence of random effects. The Poisson mixed-model

predictions

were constructed

_{by replacing}

the unknown

_quantities

in

_{b, exp(s),}

and

exp(u)

by

the

_appropriate

solutions in

_[7].

In the linear mixed

_model,

the

_predictors

were defined to be:

and

for service sire and

_{genetic effects,}

_{respectively.}

Here the unit vectors 1 are of the

same dimension as the

_respective

vectors of random effects and the asterisk is used

to denote direct estimation of location

_parameters

on the observed scale.

Estimators for fixed effects were also

_expressed

on the observed scale. The true

values of the fixed effects were deemed to be i =

_exp(p

+

_H

_i

₎

for i =

_{1, 2, ...}

15 as

in

_!16!.

Estimators

for fixed effects under the Poisson model were therefore taken to

be

exp(ti+!)

for i =

1, 2, ... 15.

As the linear mixed model estimates

parameters

RESULTS AND DISCUSSION

One-way

model

Means and standard errors of estimates of the

_genetic

variance

₍

₀

_&dquo;)

for the five

procedures

are

_given

in table II and MSEs of the estimates are

_given

in table III.

Clearly,

estimates obtained with REML and REML-0 were

_{extremely biased;}

this

is so because the

_genetic

components

obtained are not on the

appropriate

scale of

measurement

_(ie

the canonical

_log

_scale).

The

_problem

was somewhat corrected

by

a

logarithmic

transformation of the records. For

E(A

i

) *

2 and

Q

u

=

0.056,

the

_REML-LOG,

Poisson and Poisson-truncated estimators were

_nearly

unbiased

(within

the limits of Monte-Carlo

variance),

but the Monte-Carlo standard errors were much

_larger

for REML-LOG. For

Q

2

u

=

_0.125,

the Poisson estimates _were biased downwards

_(P

<

_0.05)

for both values of

_E(A

_i

_),

while those of

REML-LOG were biased

upwards

and

significantly

so with

E(A; ) x5

8. In a _one-way sire threshold

_model,

H6schele et al

₍₁₉₈₇₎

also found downward biases for the MML

procedure.

In

_spite

of these small

biases, however,

the MSEs of the Poisson estimates

(table

III)

were much lower than those of REML-LOG. The _very

large

(relative

to

Poisson and

_REML-LOG)

MSEs of the REML and REML-0

_procedures

illustrate the

pitfalls

incurred in

_carrying

out a linear model

analysis

when the situation

dictates a nonlinear

_analysis,

or a transformation of the data.

A linear one-way random effects

model, however,

can be contrived in which case

it can be shown that REML may

actually

estimate somewhat

meaningful

variance

components

on the observed scale.

_Presuming

that

multiple

records on an individual

is

_possible,

the variance of

_{Yi! (with}

_{subscripts denoting}

the

_jth

record on the ith

individual)

can be

classically represented

as:

such that from

_[13c]

and results

presented

by

Foulley

and Im

_(1993):

The covariance between different records on the same individual

(ie

cov(Y!,

Y!!!)

can be used to

_represent

the variance of the random effects.

Given

_independent

Poisson

_sampling

conditional on _ui, the first term of the above

equation

is

_null,

and

Thus a _one-way random linear model that has the same first and second moments

as Y

ij

is

where

_Y2!

is the

_jth

record observed on the ith

_animal,

_is

the overall _mean,

_u*

is the random effect of the ith animal and

_{e !}

is the residual associated with the

jth

record on the ith animal. Here

_(i

_*

= exp()i+o-!/2)

_ui

has null mean and variance

_{a 2}

_*

=

exp(2p)exp(u

£

)

[exp (

g

2

)

1]

and

_eij

has null mean and variance

a e 2*

=

exp

(p

-f-

_Q!/2).

The

_empirical

mean REML estimates

_reported

in table II

closely

relate to the functionals for

_{or u 2}

_*

in

_!22b!,

in

_spite

of the violated

_independence

underestimated

_E(!i),

as

_expected

_{theoretically.}

In the Poisson

_model,

the residual

variance is the Poisson

_parameter

of the observation in

_question.

Hence the residual variance in a linear model

_analysis

would be

comparable

to

_E(A

_i

_).

The

log-transformed REML estimates

_(REML-LOG)

have no

_meaning

here because the

Poisson residual variance is

_generated

on the observed

scale,

_contrary

to the

_genetic

variance which arises on a

_logarithmic

scale.

_Generally,

the Poisson and REML

methods gave

seemingly

unbiased estimates of the true _average Poisson

_parameter.

However,

REML estimates of residual

_variance,

_rather,

of

_E(A

_i

_),

_appeared

to be

biased

_upwards

_(P

<

_0.01)

for the

_higher

_genetic

variance and

_higher

Poisson mean

population

(table IV).

The MSEs of Poisson estimates of _average residual variances

were much smaller than those obtained with

REML,

especially

in the

_populations

with a

higher

mean. REML-0 was even worse than

_REML,

both in terms of bias

(table IV)

and MSE

_{(table V).}

This is due to truncation of the distribution

_(eg,

Carriquiry

et

al,

1987)

which is not taken into account in REML-0. The truncated Poisson

_analysis

_gave

_upwards

biased estimates and had

_higher

MSE than the

standard Poisson method.

However,

truncated Poisson

outperformed

REML in an MSE sense in

_estimating

the _average residual

_variance,

in

_spite

of

_using

less data

(zero

counts not

_included).

Empirical

mean biases of

_predictions

of

_breeding

values for dams and

_daughters

are shown in table VI for

Q

u

= 0.056 and table VII for

Q

u

= 0.125. Poisson-based

methods and _{BLUP gave} unbiased estimates of

_breeding

values while BLUP-LOG

and BLUP-0

performed

poorly;

BLUP-LOG had a downward bias and BLUP-0 had an

_upward

bias. Predictions of

_breeding

values for the truncated Poisson

_analysis

were

_generally

unbiased.

Empirical

MSEs of

_predictions

of

_breeding

values are shown in tables VIII

(

U2

=

0.056)

and IX

(

Q

u

=

0.125).

Paired t-tests were used in

_assessing

the

performance

of the

_comparisons

Poisson versus BLUP

(and

BLUP-LOG)

and

Poisson truncated versus BLUP-0. BLUP-LOG and BLUP-0

_procedures

had the

largest

MSEs,

probably

due to their substantial

_empirical

bias. For

_{ufl =}

0.056

when

_{E(Ai ) x5}

_2,

and a smaller

_(P

<

_0.05)

MSE for

_{predicting daughters’ breeding}

values when

_{E(Ai) !!}

8. For

_ufl

= 0.125

_(table

_IX),

Poisson and BLUP had similar

MSEs when

_{E(Ai )}

_*

_2,

but Poisson had smaller MSEs than BLUP

_(P

<

_0.05)

for both dams and

_daughters

when

_{E(Aj )}

_x5

8. These small differences between BLUP and Poisson are somewhat

_surprising

in view of the different scales of

prediction,

and their

_practical

_significance

is an _open

_question.

It was also

surprising

that the

differences between BLUP and Poisson tended to be more

significant

with

E(Ai ) *

8

than with

_{E(Ai ) x}

₅

_2,

since it is known that the Poisson distribution

_approaches

a

normal distributions as

_A

_i

increases

_(Haight,

_1967).

However,

this may be due to

a

_higher

_power of the test when

_detecting

a

_larger

difference. Another

_explanation

may be that a lower

E(A

i

)

leads to a lower

’pseudoheritability’

and, hence,

a lower

degree

of association between

_phenotypes

and

_breeding

values. In this _case, the linear and Poisson models _may differ less when

_predicting

_breeding

values because of a

_{higher degree}

of

_shrinkage

towards zero. When counts of zero were

_excluded,

the Poisson-truncated method had

_always

smaller MSEs of

prediction

of

_breeding

values than the BLUP-0 method.

Mixed model

Because variance

_components

estimated

_by

MML and REML are on different

scales, empirical

coefficients of variation

_(CV)

were used to

_provide

a basis of

comparison

(fig

1).

Clearly,

REML estimates were more variable than their Poisson

counterparts.

No clear

_pattern

with

_respect

to

_increasing

values of the variance

components

emerged,

except

that CVs for both MML service sire and

_genetic

estimates seemed

_relatively

more stable while CVs for REML service sire estimates

steadily

increased for values of

_ufl

_larger

than 0.0250

_(ie

_0’

_{; =}

_0.0500).

For

count data with low _{means, variance}

_components

that are estimated under linear

mixed-effects

_models,

more

_general

than the one-way random effects

model,

are

heteroskedastic from one observation to the

next,

depending

on both

_experimental

design

and location

_parameters

_(Foulley

and

_Im,

_1993).

Relative biases of the MML estimates are

_given

in

_figure

2. Relative biases were

less than

4%

for all 4 sets of variance

components,

with no clear trend with

_respect

to the true values of variance

_components.

_Using

_t-tests,

these biases did not differ

from zero. Unlike results obtained with threshold models

_(eg,

H6schele et

_{al, 1987;}

Simianer and

Schaeffer,

1989),

a small subclass

(equal

to 1 in the Poisson animal

model)

did not lead to detectable bias of variance

_component

estimates in the Poisson mixed model.

Relative errors

(square

roots of the

_empirical

_MSEs,

_expressed

as a

_percentage

the size of the true variances were somewhat

_opposite

for service sire and

_genetic

variance

_component

estimates. Relative error for the

_genetic

_component

decreased as the true variance

_increased,

whereas the error of the service sire estimates

increased somewhat with the true value of the

_parameter.

The relative errors of

MML estimates were almost identical to their

empirical

CVs

_{(fig 1)}

because of the small

_bias,

as shown in

figure

2.

Empirical

biases of fixed-effect estimates obtained with linear and Poisson

components.

The 2 methods _gave estimates that were biased

upwards,

but the bias was

larger

for BLUE in all 4 cases. This

_apparent

_paradox

can be

_explained

and not

_exp(p

+

_H

_i

_).

Empirical

biases for the 2

_methods,

and their

_difference,

increased with

_increasing

values of fixed effects and with

_higher

values of variance

components.

The

_upward

biases of the Poisson estimates were

_generally

stable across sets of variance

_components,

_being

_always

less than 0.5.

_However,

the bias of the BLUEs of the fixed effects increased

_{substantially}

as variance increased.

BLUE estimates.

_Empirical

MSEs of the fixed effects estimates are

_depicted

in

figures

8-11 for each of the 4 sets of variance

_components.

As with

_empirical

biases,

MSEs of the linear model and Poisson

_estimates,

and the differences between the MSEs of the 2

_procedures

tended to increase with

_increasing

values of the variance

components

and with the true values of the fixed effects. The MSEs of the Poisson estimates were

_again

more stable across sets of variance

components

and,

although

tending

to increase with the value of the fixed

_effects,

were much smaller than those

of BLUE estimates. For

_example

_{(fig 11),}

when

_{embryo yield}

was around

_15,

the

MSE of BLUE was about 6 times

_larger

than that of Poisson estimates.

Because the values of fixed effects were constructed such that:

exp(

1

l + Hi

+d

- exp(

1

l + H

i

) = 1;

i = 1, 2, ... ,14

it was of interest to examine the extent to which the 2 estimators would

_capture

such

difference. If the estimates are

regressed against

true

values,

the ’best’ estimator should

_give

a

_slope

close to one. Such

_regressions

were assessed

_by

_ordinary

least-squares.

Empirical

biases and MSEs of the estimates of the

regression

effects

are

_given

in

_figures

12 and

_13,

_{respectively. Regression}

estimates obtained under

both models were

_slightly

biased

_downwards,

_particularly

_BLUE;

the absolute bias

increased somewhat as true variance

_components

increased in value. The differences

between the biases of the 2 estimators were found

_significant

in all cases

_using

a

paired

t-test

_(P

< 0.05 for variance

_component

set

_A,

and P < 0.01 for variance

component

sets

_{B, C,}

_D).

The differences in

_empirical

MSEs were in the same

direction,

ie BLUE had

_larger

MSEs.

Empirical

average biases and

empirical

average MSEs of the

predictions

of service sire effects are

_given

in

_figures

14 and

_15,

_{respectively.}

The statistics were

_slightly

(P

<

_0.05)

from 0

_only

at

_or2

₌ 0.0500. Based on

_paired

_t-tests,

_empirical

_average

MSEs were not different between the 2 methods.

Statistics associated with

_prediction

of

_genetic

effects are

_depicted

in

_figures

16

and

17,

with results

presented

separately

for base

generation

sires and

dams,

and for their female progeny

surviving

to _age at first

_{breeding. Poisson-predicted breeding}

values tended to be biased

_downwards,

whereas BLUP

_predictions

were biased

solutions were

_{significantly}

biased

(P

<

_0.01)

at the 2

_highest

variance

_components.

Dam and

_daughter

BLUP solutions were

_{significantly}

biased

(P

< 0.01 in all cases,

except

P < 0.05 for dams at

_{ufl =}

_0.0125).

Poisson dam solutions were biased

(P

<

_0.01)

_only

at

_U2

_-

_0.0250,

while Poisson

_daughter

solutions were biased at

o

l

2= u

0.0250

_(P

<

_0.01)

and a

=

0.0375

_(P

<

_0.05).

_Certainly,

all

predictions

reflect

_uncertainty

in the baseline estimates of fixed effects

_given

in

_[17]

and

_[20]

for

Poisson and BLUP

_{models, respectively,}

and

_upward

biases in BLUP

predictions

Differences in

_empirical

_average MSEs of

_prediction

of

_breeding

values between the 2

_procedures

are

depicted

in

figure

17.

Although

the differences between the 2 methods were

_small,

a

_paired

t-test _gave

_significant

differences for

_daughters

(P

< 0.01 for

_U2

=

0.0125,0.0375,

and

0.0500;

P _< 0.05 for

U

2

-

0.0250)

and for

base

_generation

dams,

except

at

_{or =}

0.0125

_(P

< 0.01 for

’7!

=

_{0.0250,0,0500;}

P < 0.05 for

₀

_’;

=

0.0375).

Differences in

empirical

average MSE of

predictions

of

CONCLUSIONS

This

_study

evaluated the

_{sampling performance}

of estimators of location and

dispersion

parameters

in Poisson mixed

_models,

and of

_predictors

of

_breeding

values

in the context of simulated MOET schemes. Records on

embryo yield

were drawn

from Poisson distributions for 4

_populations

characterized

_by

_appropriate

_parameter

values.

_Analyses

were carried out with the Poisson model and with a linear model

In

_general,

the estimates of

_parameters

and the

_predictions

of random effects obtained with the Poisson model were better than their linear model

_{counterparts.}

This is. not

_{surprising because,}

to

_{begin with,}

in the Poisson

_analysis,

the data

were

_analyzed

with the models

_employed

to

_generate

the records in the simulation.

Further,

if the distribution is indeed

_Poisson,

there is no

_appropriate

variance

decomposition independent

of location

_parameters

in the linear

_model,

which makes the variance

_component

estimators

_employed

with normal data somewhat

inadequate

for

_{estimating dispersion}

_parameters,

_particularly

in the mixed effects model. A

_{log-transformation}

_improved

_(relative

to untransformed

_REML)

the mean

squared

error

_performance

of REML estimates of

_{genetic variance,}

but worsened

the estimates of residual variance.

_Similarly,

fixed effects were estimated

by

BLUE

with a

_larger

bias and mean

_squared

error than when estimated

_by

the Poisson

model.

BLUP was found to be robust in

_predicting

random

_effects,

_although

Poisson

joint

modes were

_{significantly}

better with

_respect

to bias and MSE _{in many}

cases. This is

_remarkable,

_considering

that the BLUP

predictors

were based on

location-parameter-dependent

estimates of variance.

BLUE, however,

showed bias with

_increasing

values of fixed effects and variance

_components.

Truncated-Poisson estimators and

_{predictors always}

_outperformed

BLUP and REML when zero counts

were excluded from the

_analysis;

truncation of null counts _may be common in field records on traits such as litter size

(Perez-Enciso

et

_al,

_1993).

Estimates of variance

_components

obtained

by

MML in Poisson mixed models did

not exhibit the

_typical

bias due to small subclass sizes encountered often in threshold

models.

_Rather,

the downward biases of MML found in the one-way models may

et al

₍₁₉₈₇₎

attributed the ’small subclass’ bias in threshold models to

_inadequacy

of a normal

_{approximation}

invoked in MML. The same

_{approximation}

is made

when

_estimating

variance with the Poisson

_model,

but its _{consequences may} be less critical here.

In

_conclusion,

REML and BLUE do not

_perform

well when the

_assumption

of a

Poisson distribution

holds,

even if all

_pertinent

factors in the model are included

in the

_analysis.

BLUP, however,

was found to be

_{robust, although}

it had a

_slightly

inferior

_sampling

_performance.

This

_study

did not address the situation when the data are

_counts,

but the distribution is not Poisson. For this

_instance,

estimators

based on the

_assumption

of

_normality

_may

_perform

better than those that

_rely

on

the Poisson

_{distribution,}

due to central limit

_theory.

It should also be noted that if the

_operational

Poisson model fails to consider all

pertinent

explanatory

factors,

the conditional variance _may be

_{substantially}

_greater

than the conditional mean

of an

observation;

these 2

_parameters

are defined to be

_equal

as in

_!2!.

Possible

diagnostics

for

_{investigating}

this source of

_{overdispersion}

were discussed

_by

Dean

(1989).

ACKNOWLEDGMENTS

The authors thank M Perez-Enciso and I Misztal for

_making

available to them sparse

matrix subroutines for computations

involving relationship

matrices. The authors also thank R Fernando and T

_Wang

for

_{helpful suggestions}

and comments. The work was

partially

funded

by

grants from 21st

_Century

Genetics

_(Shawano,

WI,

USA),

Sire Power Inc

_{(Tunkhannock,}

PA,

USA)

and

by

the National Pork Producers Council

_(Des

Moines,

10,

USA).

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