Attenuating effects of preferential treatment with Student-t mixed linear models: a simulation study

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Attenuating effects of preferential treatment with

Student-t mixed linear models: a simulation study

Ismo Strandén, Daniel Gianola

To cite this version:

Original

article

Attenuating

effects

of

_preferential

treatment

with

Student-t

mixed

linear

models:

a

simulation

study

Ismo

Strandén

a

Daniel Gianola

a

Department

of Animal

_{Sciences, University}

of

_Wisconsin,

Madison,

WI 53706, USA

b

Animal Production

_{Research, Agricultural}

Research Centre -

_MTT,

31600

_Jokioinen,

Finland

(Received

14

_May

_1998;

_accepted

16 October

₁₉₉₈₎

Abstract - Preferential treatment of cows in four herds of a

_multiple

ovulation and

embryo

transfer scheme under selection was simulated. Prevalence and amount of

preferential

treatment

depended

on a function correlated with true

_breeding

value. Three mixed effect linear models were

_compared

in terms of their

_ability

to handle

preferential

treatment: the classical Gaussian

_model,

a model with multivariate

t-distributed errors clustered

_{by herd,}

and a model with

_independent

t-distributed

errors. In the models with t-distributed _errors, both the scale _parameters and the

degrees

of freedom were considered unknown. A

_{Bayesian analysis}

was carried out

for all three models via the Gibbs

_sampler,

and posterior means were used to infer

about _genetic

_{variance, herd-year effects, breeding}

values and realised response to

selection. Performance over _repeated

_sampling

was assessed via Monte Carlo mean

squared

error. In the absence of

preferential

treatment, the three models had a

similar

_performance.

When

_preferential

treatment was

_prevalent

and _strong, the

univariate t-model was the

best; hence,

the Gaussian

_assumption

for the errors was

_{clearly inappropriate.}

It _appears that some robust linear models can handle

preferential

treatment of animals better than the standard mixed effect linear model with Gaussian

_assumptions.

_©

_{Inra/Elsevier,}

Paris

dairy cattle

_{/ preferential}

treatment

_{/ simulation / Bayesian}

statistics

_{/ Student-t}

distribution

_{/ Gibbs}

_sampling

*

Correspondence

and reprints

E-mail: [email protected]

Résumé - Atténuation des effets de traitement _{préférentiel} dans un modèle

linéaire mixte à distribution de Student

_(t).

Étude

de simulation. On a simulé le

traitement

_{préférentiel}

de certaines vaches dans _{quatre troupeaux} de sélection utilisant

ont

_dépendu

d’une fonction corrélée à la valeur

_génétique

vraie. On a

comparé

trois modèles linéaires mixtes _pour leur

aptitude

à

_prendre

en _compte le traitement

préférentiel :

le modèle

classique Gaussien,

un modèle avec des erreurs t-multivariates

groupées

par troupeau et un modèle avec des erreurs t-distribuées

indépendantes.

Dans le modèle où les erreurs suivaient une distribution _t, les

paramètres

d’échelle et les

_degrés

de liberté ont été considérés inconnus. Une

analyse

bayésienne

a

été effectuée pour les trois modèles à partir de

l’échantillonnage

de Gibbs et les moyennes a _posteriori ont été utilisées _pour en inférer au

_sujet

de la variance

génétique,

des effets

troupeau-année,

des valeurs

_génétiques

et des

_réponses

réalisées à la sélection. La

_performance

des modèles a été évaluée au travers des erreurs

quadratiques

moyennes. En l’absence de traitement

préférentiel,

les trois modèles

ont eu une

_performance

similaire.

_Quand

le traitement

préférentiel

a été

_fréquent

et

d’effet important, le modèle t-univariate a été le meilleur et le modèle Gaussien a été

clairement

_inadapté.

Il

_apparaît

_que des modèles linéaires robustes _peuvent

_prendre

en

compte les traitements

_{préférentiels}

mieux _que les modèles linéaires mixtes Gaussiens

classiques. @

Inra/Elsevier,

Paris

bovins laitiers

_/

traitement _{préférentiel}

_/

simulation / statistique bayésienne /

distribution de Student

1. INTRODUCTION

Preferential treatment is _any

_management

_practice

that is

_applied

non-randomly

to animals within a

_contemporary

_group

[9].

For

_example,

better

housing

and

_feeding,

hormonal

_treatment,

_{longer milking}

intervals on test

_day

and

_{feeding according}

to

_production

are known to be

_{applied selectively}

in

dairy production.

Preferential treatment occurs in

_{dairy cattle, presumably}

to

increase the economic value of a cow or the

_probability

that it will be chosen as

a bull-dam. Several studies

(e.g. [17,

20!)

have found that

_genetic

evaluations

for milk

_yield

are inconsistent with

expectations

based on

_theory.

This _may

be due to

_inadequate

statistical

assumptions

or failure to account

_properly

for selection or

_preferential

treatment of cows.

Preferential treatment is often

_suspected

when no

_apparent

reasons exist for

such

_{discrepancies.}

Kuhn et al.

_[9]

simulated effects of

preferential

treatment

on ’animal model’

_genetic

evaluations. Mean

squared

error of

prediction

of

breeding

values increased as the extent of

_preferential

treatment increased. Kuhn and Freeman

_[10]

found that when the dam of a sire was treated

preferentially,

more than 30

_daughters

with untreated records were needed to

offset the bias in

_prediction

of

_breeding

value caused

_by

the dam’s information. Bias increased as the

_proportion

and number of

_{daughters receiving}

preferential

treatment increased. Bias decreased when all

_{daughters given preferential}

treatment were in the same

_herd;

this is so because the

_{’herd-year’}

effect in the model

_{captures part}

of the

preferential

treatment administered in a

_particular

herd-year.

In order to account for

preferential

treatment,

Harbers et al.

_[7]

included an

environmental correlation between related females in a

_genetic

evaluation model

for a MOET

(multiple

ovulation and

embryo

transfer)

scheme. This

improved

accuracy of cow evaluations when

preferential

treatment was mild.

_Weigel

et al.

[29]

simulated different

_strategies

of

_preferential

treatment and found that it

within a herd is treated

preferentially.

Burnside and

_Meyer

[3]

simulated effects

of bovine

_somatotropin

_(bST).

Sire evaluations were least accurate when bST administration was

targeted

to the best

producing

cows.

In the context of

prediction

(e.g. !8]),

a bias takes

place

when the

_expected

values of the

_predictand

and of the

_predi!tor

differ. Evaluation of bias

_requires

knowledge

of the true model

_but,

in

_practice,

this is not

_available,

so ad hoc

assessments of bias have been

_suggested.

Several studies

_[15,

_{16, 27,}

_28]

found

upward

’biases’ of cow’s

_pedigree

indexes for

_protein

or milk

yield

in Finnish

Ayrshire.

It is unclear if this

_discrepancy

is due to

_chance,

but

_preferential

treatment of dams of cows _may be a

_culprit.

On the other

_hand,

Powell

and Norman

_[19]

found that

_pedigree

indexes understated the first estimated

breeding

values of

_daughters

of _{proven sires} mated to lower

_producing

dams. Little work has been undertaken on how _{to cope} with

_preferential

treatment in

_practice,

at least from a statistical

_point

of view. Kuhn and Freeman

[11]

studied power transformations of records but this was, at

best, slightly

effective

in

_reducing

bias due to

_preferential

treatment. An alternative

_approach

is to

consider an error distribution with thicker tails than the

normal,

to allow for

more variation. A

_commonly

used one is the

_{t-distribution,}

which is

_symmetric

and

_leptokurtic.

It has been advocated because of its

_simplicity

_[12],

and because

_only

one

_parameter

(the

degrees

of

freedom)

is needed to describe robustness. A suitable robust distribution _may be

_capable

of

_attenuating

the

impact

of outliers on data

_{analysis. Many}

authors have

_employed

statistical

models with t-distributed residuals

_[4,

12,

13, 25,

31]

in linear and non-linear

regression models,

with

_varying

_degrees

of success. Use of the t-distribution in

the context of mixed effects or hierarchical models is

_relatively

recent

_!1,

_2,

_5,

6, 22-24, 26,

30].

Our

objective

was to assess

frequentist properties

of

Bayesian point

estima-tors obtained from mixed linear models where residuals were assumed to be either Gaussian or t-distributed. Milk

_production

records obtained in herds in

which some

preferential

treatment was

practised

were simulated. The

analy-sis focused on mean

_squared

error of estimation of

_{genetic variance,}

_herd-year

effects,

breeding

values and

_genetic

_response to selection.

2. STRUCTURE OF THE SIMULATION

2.1.

_{Conceptual population}

Milk

_production

records in a

_hypothetical

’adult’ MOET nucleus scheme

[18]

were simulated. The scheme extended the

_simple

hierarchical

_mating

structure

of Stranden et al.

_!21].

Our modification allowed bulls of the

_{previous generation}

to mate current

_generation

females. The nucleus consisted of 32 cows and

eight

bulls _{in every}

_generation.

In each

_generation,

_every nucleus cow

_produced

(by

multiple

ovulation and

_embryo

transfer to

_recipients)

_eight

_offspring,

four females and four males. An animal could be selected

only

once into the nucleus

as a

_parent

and unselected animals were culled. The females were selected

among those

offspring

to the nucleus that had

completed

a first lactation. Males

were selected within those that had been born in the

_{preceding generation.}

In

Thus,

males within a full-sib

_family

had the same estimated

_breeding

value and

three such males were

randomly

discarded. Each selected male was mated to

four cows, chosen

randomly

from those that had been selected as

replacements.

. 8 1 32 1 .

Selection pressure in males and females

_was

₃₂

₄

and

2g

= -,

respectively,

32 4 128 4

per

generation.

With this scheme carried out for four

_generations,

the data included 544 cows with records

(32

in the base

_plus

32 x 4 x 4 = 512 female

progeny)

and 32 sires with

_daughters

in

_production,

i.e. a total of 576 animals.

A

_diagram

of the simulated

population

is shown in

_figure

1.

Base

_generation

cows were

_assigned

to four herds in

_{equal numbers,}

i.e.

eight

cows _per herd. Female

_offspring

of a cow remained in the same herd as her

dam,

whereas sires were used across herds.

_Breeding

values of base

animals were drawn at random from

_N(0,0.25)

distribution. Records of the base animals were

_generated

_{by adding}

a

_herd-year

effect

_{(independently,}

normally

distributed)

to a

_breeding

value and to an

_{independently}

drawn

residual from

_N(0,0.75)

distribution. Records in

_{subsequent generations}

were

simulated

_similarly,

_except

that the

_breeding

value of an individual was formed

segregation

residual,

where

_a

₂

_is

the additive

_genetic

variance and

F

is

the _average

_inbreeding

coefficient of the

parents.

The selection criterion in

the

_breeding

scheme was BLUP of

_breeding

value with the true variance

components.

The statistical model included

_{the herd-year}

as a fixed effect and animal as a random effect

(but

ignored

preferential

treatment,

as discussed

later)

using

all

_{genetic relationships}

available up to the time of selection.

2.2. Preferential treatment

In

_practice,

_preferential

treatment takes

_place

in the course of a selection

programme so this is the _way that the

present

simulation

_proceeded.

None of the base

population

cows were treated

preferentially,

so there were 512 cows

eligible

to receive

_preferential

treatment. A scheme in which the

_preferential

treatment

_{assigned depended}

on the

’perceived’

breeding

value of an animal

(e.g.

based on a

_genetic

evaluation available before the animal

_produces

the

record)

was

_adopted.

The records were

_generated

as

where yij is the record of

animal j made

in

herd-year

i, h

i

is a

herd-year

effect,

Uj is the

breeding

value of animal

j,

and eij is an

independent

residual. The

preferential

treatment

_02!

was a stochastic effect

_taking

the values:

where

_!(.)

is the standard normal cumulative distribution

function,

pmin is a

constant smaller than the

_herd-year

effect

_h

_i

_,

_{and u/j =}

_!+(u!+v!)

/

_afl

+

_w

is a ’value’ function such that _{Ui rv}

A!(0,er!),

_{- jv}

N(0,

Q

v),

_{Cov(u_,,f}

_j

₎

=

_0,

so _{Wj rv}

N(À, 1).

In the

_preceding,

0’

;

is the variance of

_breeding

values and

w

j2

is an

’uncertainty’

variance. The ratio

O’! 2 describes

the

_uncertainty

the herd

!u

manager has about the true

_breeding

value of animal

_j.

For

_example,

if the breeder _{is very} uncertain about the

_breeding

value of the

_animal,

this ratio of variances should be

_high.

Three values of the

_uncertainty

were

considered,

0’2

1

1

=

——,

1,

100. The correlation between Wj and the

breeding

value Uj is

/ 100

j2

!l/2

C1

1 + ;! 2)-1/2

giving 0.995, 0.71

and 0.10 at values of the

_{uncertainty equal}

B

!/

to 100’ 1 and

₁₀₀

100,

respectively.

The

_preferential

treatment scheme in

_equation

₍₂₎

induces a correlation

be-tween related animals

_{Corr(u/j , Wjl)}

= a JJ ’10’2

u +

2 Cov( 2 v

J’ VI) J

_where

_ajj, is the

tween related animals

_{Corr(iu,,w,’) =}

—&dquo;—!—!——&dquo; v

, where a,,’ is the

a! + a!

additive

_relationship

between

_{animals j and}

_j’.

If the _Vj deviates are

inde-pendent,

then

_{Corr(u/j ,}

_u

_{/j, )}

=

a!!!!!!(!u

₊

o!)’

For

example, if j and

j’are

full-sibs and

_Q

_z

=

0’

;,

the

_breeding

value the

_higher

the amount

_preferential

treatment and the chance of

_receiving

it.

The constant pmin in

equation

(2)

controls the range of

production

associated

with

_preferential

treatment. It was set

_equal

to

-5

Qh

where ahis the standard

deviation of

_herd-year

effects. These were drawn from a normal distribution

with mean zero and variance

a

2

and two different values of the

herd-year

variance were considered:

afl

=

au

and

2

U

2

The _constant A controls the

proportion

of cows to be

_{preferentially}

treated. Normal distribution

_theory

can

be

used

to find a value of A such that a desired

_proportion

of cows receives

preferential

treatment. The

_proportion

of

preferentially

treated cows increases

with

.!,

because

_Pr(u/j

>

₀₎

increases

_{concomitantly.}

Three different

prevalences

of

_preferential

treatment were considered: 1 out of

10,

1 out of

32,

and 1 out

of 64 cows. These

_correspond

to A values of

_-1.2816,

-1.8627 and

-2.1539,

respectively.

It was intended to

_keep

the

_proportion

of

preferentially

treated animals

roughly

constant from

_generation

to

_generation.

To do _so, it must be noted that selection is

_expected

to increase mean

_breeding

value and to reduce

genetic

variance over time. In order to account for these

_effects,

the formula for w was

changed

to:

where u is the mean

_breeding

value of animals available for

preferential

treatment in the

_generation

to which

_{animal j belongs,}

and

_Su

_is

the additive

genetic

variance for individuals born in that

_generation.

The

_probability

distribution of the amount of

_preferential

treatment

_(Di!)

depends

on the values of _Qh and A as shown in the

Appendix.

The _average

amount of

preferential

treatment

_actually

_applied

was assessed via a

simula-tion of 1000

_replicates

of the MOET scheme. Mean increase

_(mean

of

₀₎

in

production

due to

_preferential

treatment under

_{varying prevalence}

of

prefer-ential treatment and amount of

_herd-year

variance is in table I. As

_intended,

production

increased with

_prevalence

of

_preferential

_treatment,

and with

_{or h* 2}

Average

value of

_preferential

treatment was not affected

_by

level of

_uncertainty

j

2

2 .

This is not shown in table

_1,

but it was

_expected

because the distribution

Q!

of

_A

_zj

does not

_depend

on this ratio.

Table I.

_Average

increase in simulated lactation

production

due to

preferential

treatment as a function of

herd-year

variance

(a

h

)

and of

prevalence

of

preferential

treatment

_(values

in

_parenthesis

are Monte Carlo standard errors from 1 000

_replicates

2.3. Statistical models and

_computations

Three linear statistical models were

_compared,

both with and without

pref-erential treatment

_incorporated

in the simulation. The

_objective

was to assess

the relative

_ability

of these models to handle

_{perturbations}

caused

_by

unknown

preferential

treatment. In all three

models,

the linear structure for the records included an unknown

_herd-year

effect

(treated

as fixed

_{computationally),}

the unknown

_breeding

value of the cow and a

_residual,

distributed

_according

to an

appropriate

error

_{distribution,}

as noted below. In the three

_models,

a

multivari-ate normal distribution

_N(0,

_Aa!),

where A is a 576 x 576

_{relationship matrix,}

was used for the

_{genetic effects,}

so there was no difference in this

_respect.

The

three

_{models, differing only}

in the error distribution were the

_following.

1)

G: a

_purely

Gaussian model with errors

Niid(0,

Q

e ).

2)

t-l: errors were

independently

and

_identically

distributed as

_{univariate-t,}

t

i

(0,

0’ e 2,

v,).

Here,

the variance of the distribution is

_U2V

_,I(V,

_e

_-

_2),

where

_Q

_e

is

a scale

_parameter

and v, are the unknown

_degrees

of freedom.

3)

t-H: within herd

_{i (i}

=

1,

2, 3,

4),

the error _{vector ei}had the

multivariate-t distribution

_t

_ni

_(0,

_I

_ni

_U2

_,

₎

_e

_v

where ni is the number of records in herd i.

Here,

Var(e

i

)

=

I!o!fe/(fe - 2).

Although

the _{errors are}

_{uncorrelated,}

they

are not

independent,

this

_being

a

_property

of the multivariate t-distribution. Error

vectors in different herds were

_{mutually independent, however,}

but with the

same

or and

_ve

_parameters.

We refer to this model as a ’herd-clustered’ one.

The G model is the usual _one; model t-1 _{discounts outliers yzj} on a ’case’

by

’case’

basis,

and model t-H discounts

_outlying

vectors _yz for the entire herd i. Because the ’value function’ _w! used to

_generate

_preferential

treatment

does not

_depend

on the

herd,

there is no

_apparent

reason

_why

model t-H should

outperform

model t-1. It should be noted that as Ve -j _oo, the two

t-distributions tend towards the Gaussian one.

A

_Bayesian

structure was

_adopted

for inference. Prior distributions were the same for all three models. Herd effects were

assigned

a uniform

_{prior and,}

as

noted,

a multivariate normal _process was used as a

_prior

distribution for the

breeding

values. The

_dispersion

_components

_Q

_u

_{and Q}

_e

_were

_assigned

inde-pendent

scaled inverted

_chi-square

distributions with four

_degrees

of freedom and mean

_equal

to the true variance

_component,

i.e. 0.75 for the residual

vari-ance and 0.25 for the

_genetic

variance. In the

_t-models,

the

_prior

for

_a

_is

for the scale of the distribution and not for the residual

_variance,

which is

a e

2v

,l(v, -

2)

as noted before. In the two models

_involving

the

_{t-distribution,}

the residual

_degrees

of freedom

_parameter

v, was considered unknown.

_Degrees

of freedom values allowed in the herd-clustered t-model were

_{4, 10,}

100 or 1

000,

all

_{equally likely,}

a

_priori.

In the univariate t-distribution

_model,

the _space

of v

e

was

_{4, 6, 8, 10,}

12 or

_14,

all

_{receiving equal prior}

_probability.

These values

were chosen

_arbitrarily.

It is

_possible

to use a continuous

_prior

for _v,

[23]

but

the discrete distribution

_employed

here facilitated

_{implementation.}

A Gibbs

sampler

was used to _{carry out} the

_{Bayesian computations employing}

the full conditional distributions described in Strandén

_[22].

Tests made in several sim-ulations with

_{varying starting}

values indicated that a burn-in

period

of 7 000

iterates with 70 000 Gibbs iterates thereafter

_(all

_samples

_kept)

was

_enough

About 60 min of CPU time were

required

to

_perform

70 000

_iterations,

for _any of the

models,

in an HP

_{9 000(3)}

_computer.

2.4.

_Frequentist

_comparison

Each

_replicate

of the simulation consisted of a data set

_generated

as _per the

scheme in

_figure

1 under the

appropriate assumptions

of

_preferential

treatment.

A

_{Bayesian analysis}

of the data set

according

to each of the three models was

carried out in each

_replicate.

Mean

_squared

errors of

_posterior

mean estimates

were

_computed,

over

replicates,

for:

a)

genetic

variance,

b)

herd-year effects,

and

_c)

breeding

values. Mean

squared

errors were also

_computed

for three

classes of

_breeding

values:

_sires,

cows who had been

preferentially

treated and cows without

preferential

treatment.

d)

An additional

end-point

of interest was mean

_squared

error of estimated _response to

_selection,

assessed

_by

_predicting

breeding

values

_{using posterior}

means from the three models contrasted. ’True’

response was the mean difference in true

breeding

value

(due

to selection

using

BLUP)

between animals born in the last

_generation

and those born in the first

generation.

Differences in mean

_squared

errors between models should reflect

the relative accuracy of estimation of

genetic

trend.

A

_’pilot

run’

_[14]

was conducted to assess the number of

replicates

needed to

attain

_enough

_precision

for a

_parameter

of interest. The

approximate

number

of

_{replications required}

to achieve an absolute

_precision

r for the confidence

interval

_given

a

_pilot

run of n

replicates

was found

_using:

where

_t

_i

_-

₁

_,

₁

_-

_a/2

is the value of a t-distribution with i -1

degrees

of freedom at

the

_{100(1 -}

_a)

_percentile

_{(’confidence’).}

Our

pilot study

consisted of

_carrying

n = 20

_replicates

for each of the three models. The number of

replications

re-quired

to achieve 0.05

_precision

with 95

%

confidence for the

genetic

variance

was less than 60 for most cases.

_Hence,

it was decided that all cases would

be

_replicated

60 times. Absolute

_precision

was recalculated after

60 replicates,

and a further 40

_replicates

were made for the schemes

involving

1/10

preva-lence of

preferential

treatment. One scheme

92

2 =

3, -—

2 = 100 )

required

an

(

a2 a2

u u

1)

100

/

additional 40

_replicates

to achieve the

_required

_precision.

Table II indicates the schemes and number of

replicates

performed..

Because of its

_{heavy computing requirements,}

the

_analysis

was

performed

using

a network of machines administered

by

Professor Miron

Livny

of the

Department

of

_Computer

_{Science, University}

of Wisconsin at Madison. This cluster was accessed

using

the Condor

_system,

which allows

running jobs

simultaneously

at _many

_computers

while the data and _program reside in one

computer.

Each

_replicate

of each model was a _process to be executed in this network of

_computers.

There were between 10 and 15

computers

available at

3. RESULTS AND DISCUSSION

3.1. Absence of

_preferential

treatment

The

_objective

here was to examine

_possible

losses in

_efficiency

due to

_using

the two t-distribution models when there is no

preferential

treatment and the

Gaussian

_assumption

holds

_{throughout. Averages}

and mean

squared

errors of

estimates of additive

_genetic

variance are

_given

in table III. The

_posterior

means

of

_afl

_for

each of the three models were

practically unbiased,

in

light

of the

Monte Carlo variation.

_However,

the mean

_squared

error was

_larger

for the two

t-models than for the Gaussian one.

_Hence,

if the Gaussian

_assumption

holds,

posterior

means of additive

_genetic

variance for the t-models are less accurate

than those from the G-model. The increase in mean

_squared

error over the

Gaussian model was about 5-6

%

for the t-H

model,

and 7-18

%

for the t-1

model.

Tables IV and

_Vgive

the

posterior

distributions of the

_degrees

of freedom for the two t-models in the absence of

_preferential

treatment. The

analysis

carried

out with the herd-clustered t-model

clearly

favoured a model with Gaussian

errors, as indicated

_by

a

posterior probability

of about 90

%

for the

degrees

of freedom

being

larger

than 10.

_Also,

the univariate t-model

_assigned

the

highest posterior probability,

about 40

_%,

to the

_largest

value of the

_degrees

of freedom

_(v

_e

=

14)

considered. The

posterior

distributions were not

_sharp,

this

_being

a function of the low informational content the data have about

v e

.

However,

both

analyses

favoured the

larger

values

of v

e

or,

equivalently,

the

Gaussian

assumption

for the errors. For

_example,

in the herd-clustered

t-model,

the

_posterior

odds ratio of v, = 1000 relative to v, = 4 was 17.7 and 29:7 for

2 =

1 and

ah

2 =

3,

respectively.

In the univariate

_t-model,

the odds ratio

(

of v, = 14 relative _{to ve} = 4 was 384 and 404 for the two values of the ratio

between herd and additive

_genetic

variances.

Mean

_squared

errors of estimates of location

_parameters

were similar in

all models

_{(table VI! ,}

_although

_slightly

smaller for the G-model. As

_expected,

mean

_squared

errors were

larger

for

breeding

values of cows

(smallest

amount of

In summary, in the absence of

_preferential

treatment and with the Gaussian

assumption

holding throughout,

the t-models were less accurate for estimation

of

_Q!,

but were as

_competitive

as the Gaussian model for estimation of

breeding

values and of

_genetic

trend.

3.2.

_{Preferentially}

treated data

3.2.1. Additive

_genetic

variance

Mean

_squared

error of estimates of additive

_genetic

variance are in

_figure

2.

Differences between models were clearest when

_preferential

treatment was more

_prevalent

(1/10)

and when the

herd-year

variance was

high

(this

affects

the distribution of

_0).

_Also,

differences between models were

_largest

when

uncertainty

about true

_breeding

values was

_low,

so the value function is a

high

correlate of

_breeding

value. There was little difference between the G and

the t-H

models,

but the univariate t-model had the best

_performance

when

prevalence

of

preferential

treatment was medium

(1 out

of 32

cows)

or

_high

(1 out

of 10

_cows).

The univariate t-model was robust to variation in the

uncertainty

parameter;

this was not the case for the G and the t-H

_models,

whose

performance

was

hampered

under severe forms of

preferential

treatment.

3.2.2. Posterior distribution of the

degrees

of freedom

Posterior

_{probabilities}

of the

_degrees

of freedom under the herd-clustered t-model were often

_higher

for the

_larger

values of _ve, thus

_supporting

the

Gaussian

_model,

_especially

when

_preferential

treatment was uncommon, or

uncertainty

was

high. Only

under medium

(1/32)

or

_high

(1/10)

_prevalence

of

_preferential

treatment and a

_{high herd-year}

variance the

largest

values of

this

_depended

on the level of

_uncertainty

and on the amount of

_herd-year

variance. For

_example,

when

_prevalence

of

_preferential

treatment was

_1/10

and with

₍

_T2

=

3a’,

low values of the

degrees

of freedom had

_higher

_posterior

probabilities

when

_uncertainty

was

low; however,

as

_{uncertainty increased,}

the

posterior

distribution tended to favour

_larger

values of the

_degrees

of freedom. Posterior

_{probabilities}

of v, for the univariate t-model are

_given

in

_figure

3.

Here,

posterior

distributions tended to be flat.

_{Higher probabilities}

were

as-signed

to the

_largest

values of the

_degrees

of freedom

_only

when

_preferential

treatment was rare and the

_herd-year

variance low. As in the herd-clustered

t-model,

high

uncertainty

often resulted in

_{higher probabilities}

_assigned

to the

of freedom values also received

_relatively

high

probabilities.

When

preferen-tial treatment was

prevalent

(1/10)

and the

herd-year

variance was

large,

the

posterior

distribution was

_sharp,

with a modal value of ve = 4 at all levels of

uncertainty.

This

_points

_away from a Gaussian distribution of the residuals.

With a small data set such as the one in this simulated MOET

scheme,

one

should not

_expect

the

_posterior

distribution of the

degrees

of freedom

param-eter to be

_{highly peaked. Nevertheless,}

the univariate t-model

recognised

the

non-Gaussian situation even when

prevalence

was rare

(1/64),

provided

that

the variance between herds was

_{relatively large.}

This is because the

expected

value of the

_preferential

treatment,

E(Di!),

increased with

_{or2 h,}

as illustrated in

3.2.3. Estimates of

_{herd-year effects, breeding}

values and

_genetic

response to selection

Average

of mean

_squared

error of estimates of

_herd-year

effects was similar for the three models

_except

when

_preferential

treatment was

prevalent

or

herd-year variance was

_high,

but it was

always

smallest for the univariate t-model

(figure l!).

When

preferential

treatment was common

(1/10),

the univariate

The _average of mean

_squared

error of estimates of all

_breeding

values is

shown in

_figure

5. This criterion was about the same with all models

_except

when

_preferential

treatment was common and the

_herd-year

variance

_high.

Here,

when

_uncertainty

was

_high,

there were no differences between the

_models,

but at low levels of

_uncertainty,

the univariate t-model was

markedly

_superior.

The

_picture

for mean

_squared

errors of estimates of sire and cow

_breeding

values and for

_genetic

_response was similar to that for of all

_{breeding values,}

so the

_figures

are not

_presented.

In all _cases, differences between models were

clear, favouring

the univariate t-model when

_preferential

treatment was more

prevalent

(1/10).

The same was true for

_{preferentially}

treated cows

_{(figure 6),}

but mean

_squared

errors were

_larger

than for

_breeding

values of cows that were

performance

than the Gaussian or herd-clustered models when

preferential

treatment was rare or

_{mildly prevalent,}

but it was

_superior

when such treatment

was common. In

_particular,

at the lowest level of

_uncertainty

and at the

_highest

herd-year variance,

the univariate t-model gave

predictions

of

_breeding

value of

_{preferentially}

treated cows that had a mean

_squared

error of about a third of that observed with the Gaussian model. In this

_situation,

the herd-clustered

.

4. CONCLUSIONS

In the absence of

_preferential

_treatment,

the t-models were as

good

as the

Gaussian model for

_estimating

_breeding

values and _response to selection. When

preferential

treatment was

_{mildly prevalent}

(1/32)

the models

performed

simi-larly.

However,

when

_preferential

treatment was common

(1/10)

and

especially

when the

_herd-year

variance was

large

relative to the additive

genetic

variance,

the univariate t-model was

_clearly

the

_best,

at least in terms of mean

_squared

error. Under

preferential

_treatment,

the

posterior

distribution of the

degrees

of freedom in the univariate t-model

_pointed

_away from the correctness of the Gaussian

assumption.

The univariate t-model was

_quite

robust to variation in

the simulation

parameters,

but it is unknown whether this robustness holds

across different forms of

_preferential

treatment.

This simulation could not differentiate

clearly

between the Gaussian and the herd-clustered

_t-models,

_although

the latter was

always slightly

better under

preferential

treatment. A reason for the lack of difference between these two

models may be the low number of herds in the simulation. With a few clusters

(herds)

the statistical information about the

degrees

of freedom is

_low,

so the

posterior

distribution of this

_parameter

cannot be estimated

_accurately.

In

_conclusion,

it appears that the univariate t-model can attenuate adverse effects of

_preferential

treatment as

applied

here. It leads to better inferences

about

breeding

values and

_genetic

trends than those obtained with the Gaussian

model,

especially

when

_preferential

treatment is

_prevalent,

at least under the conditions of the

_{study. If,}

on the other

hand, preferential

treatment is

non-existent,

or the

assumption

of a Gaussian distribution of the residuals seems

to be

_true,

there is little loss in

_efficiency

from

using

a robust

_model,

such as the univariate t. It is

encouraging

that a

_symmetric

error

distribution,

such as Student

_t,

improved

_upon the Gaussian one under a

single-tailed

form of

_preferential

treatment as in

_equation

(2).

This

_suggests

that a robust

asymmetric

distribution _may do even

_better,

but

_perhaps

at the _expense of

conceptual

and

_{computational simplicity.}

ACKNOWLEDGEMENT

We wish to thank W.G.

_{Hill, University}

of

Edinburgh,

for some useful comments.

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APPENDIX: Distribution of the

preferential

treatment variable When _w! is

_positive

and very

large,

A

ij

tends

to hi - pm,n,

so in this case

equation

(1)

becomes:

When _w! is

_negative,

_6.

_ij

=

_0,

_as indicated in

(1)

so _yij

_{= h}

₂

+ _uj + _e!.

Hence,

given h

i

,

the _{range in}

_production

records due to

_preferential

treatment

is

_expected

to be

_{hi -}

_pmin

_{= hi}

+

Qh

5

=

(

Z

i

₊

5)Œ

h

where _zi N

N(0, 1).

Unconditionally,

the

_expected

_{range is} then

_5o

_h

_.

For

_6.

_ij

defined in

_equation

(2),

the average

preferential

treatment

applied, conditionally

on

h

i

would be

where

₀

_,

_B

_(.)

is normal

_density

with mean A and variance 1. For A =

_0,

OA

(-)

is the standard normal

_density

_0(.).

Because

I

f

z

00

4)(w)o(w)dw

=

24)’(z) I

and

!(0)

=

1,

it follows that 2 so

E(A

jj

)

=

E(E(02!!hi)) _ -!Jmin·

Likewise,

_E(!7jlhi)

=

24

(h -Pmin)

2

for A = 0.

Thus,

!

With pm;&dquo; _

-5(]’

h

,

we have

E(Di!)

=

1.875(]’

h

and

Var(02!) !

4.068(]’!.