Models to estimate maternal effects for juvenile body weight in broiler chickens

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Models to estimate maternal effects for juvenile body

weight in broiler chickens

Anm Koerhuis, R Thompson

To cite this version:

Original

article

Models

to

estimate

maternal

effects for

juvenile body weight

in

broiler chickens

ANM

Koerhuis

R

_Thompson

!

Ross Breeders

_{Ltd, Newbridge,}

Midlothian EH28

8SZ, UK;

2

Institute

_{of Cell,}

Animal and

_{Population Biology,}

University of Edinburgh,

EH9

_{3JT, UK;}

3

Roslin Institute

_(Edinburgh),

Roslin,

Midlothian EH25

_9PS,

UK

(Received

21

_{May 1996; accepted}

7 March

₁₉₉₇₎

Summary - The estimation of

_genetic

and environmental maternal effects

by

restricted

maximum likelihood was considered for

_{juvenile body weight}

_(JBWT)

data on 139 534

and 174 668 broiler chickens from two

_populations.

Of the biometrical models

_usually

assumed in the estimation of maternal effects

_(’reduced

Willham’

_models),

a

_genetic

model

allowing

for direct and maternal

_genetic

effects with a covariance between them and a permanent environmental maternal effect

_provided

the best fit. The maternal heritabilities

(0.04

and

_0.02)

were low

_compared

to the direct heritabilities

_(0.32

and

_0.27),

the direct-maternal

_genetic

correlations

_(r

_AM

₎

were

_negative

and identical for both strains

_(-

_0.54)

and environmental maternal effects of full sibs

_(0.06

and

_0.05)

were

_{approximately}

a

factor of two greater than maternal half sibs

_(0.03

and

_0.02).

A

_possible

environmental

dam-offspring

covariance was accounted for in the mixed model

_by

₍₁₎

estimation of the covariance between the environmental maternal and the environmental residual effects

(c

EC

)

and

₍₂₎

a maternal

_phenotypic

effect

_{through regression}

on the mother’s

_phenotype

(F

m

,

’Falconer’

_model).

Whilst

increasing

the likelihoods

_{considerably,}

these extended models resulted in somewhat more

_negative

_rAM values

owing

to

_positive

estimates of CEC

(0.04-0.08

and

_0.03-0.09)

and Fm

(0.01-0.14

and

_0.01-0.11).

A more detailed fixed effects

model, accounting

for environmental effects due to individual

_{parental flocks,}

reduced

estimates of rAM

(-

0.18 to -

_0.33).

Results

_suggested

a limited

_importance

of maternal

genetic

effects

_exerting

a non-Mendelian influence on JBWT. The _present

_integrated

’Falconer-Willham’ models

_allowing

for both maternal

_genetic

_{(co)variances}

and maternal

action

_{through regression}

on the mother’s

_phenotype

in a mixed model

_setting

_might

offer attractive alternatives to the

_commonly

used ’Willham’ models for mammalian

_species

(eg,

beef

_cattle)

as was illustrated

by

their

superior goodness-of fit

to simulated data.

broiler chickens

_/

_{juvenile body weight}

_/

maternal effects

_/

restricted maximum likelihood

_/

animal model

*

Correspondence

and

_reprints.

**

Résumé - Modèles d’estimation des effets maternels sur le _{poids corporel jeune} des poulets de chair. L’estimation des

_effets

maternels

_génétiques

et non

_génétiques

sur le

poids

jeune

(JBWT)

a été

effectuée

_{par maximum} de vraisemblance restreinte sur

1

3

9 534

et

₁₇₄

668 données provenant de deux

_populations

de

_poulets

de chair. Parmi les modèles habituellement utilisés dans l’estimation des

_effets

maternels

_(modèles

«réduits» »

de

_Willham),

le meilleur

ajustement

a été obtenu avec un modèle

génétique

_permettant

des

effets génétiques

directs et maternels corrélés ainsi

_{qu’un effet}

maternel permanent non

_génétique.

Les héritabilités maternelles

(0, 04

et

_{0, 02)}

ont été

faibles

en comparaison

des héritabilités directes

_(0,32

et

_0,27),

les corrélations

génétiques

entre

_effets

directs et

maternels

_(r

_AM

₎

ont été

_négatives

et

_identiques

pour les deux souches

_{(- 0,54),}

les

_effets

maternels non

_génétiques

_pour les

pleins frères

(0,06

et

_0,05)

ont été environ deux

_fois

plus grands

que pour les

_demi-frères

_(0,03

et

_0,02).

On a tenu _compte d’une covariance

non

_{génétique possible}

entre mère et

_produit

dans le modèle mixte

_i)

en estimant la

covariance entre les

_effets

maternels non

génétiques

et les

effets

résiduels non

_génétiques

(u

EC

)

et

_ii)

en introduisant un

effet

maternel

_{phénotypique}

au travers de la

_régression

sur la

_phénotype

de la mère

(F

m

dans le modèle de

Falconer).

Bien

qu’ils

augmentent

considérablement les

vraisemblances,

ces modèles étendus ont abouti à des valeurs encore

plus négative

de rAM à cause d’estimées

positives

de QEC

(0, 04

à

0, OS

et

0, 03

à

0, 09)

et FIn

(O,Ol

à

0,14

et

O,Ol

à

_0,11).

Un modèle

_{plus dëtaillë pov,r}

les

_{effets fixés}

tenant compte

des

effets

de milieu propres aux _{troupeaux parentaux} a réduit les estimées de

rpM (- 0,18

à -

_0,33).

Les résultats ont

_suggéré

une _importance limitée des

effets

maternels

_génétiques

non mendéliens sur JBWT. Les modèles

_intégrés

«Falconer- Willham» » permettant à la

fois

des

_{co(variancés)}

maternelles

_génétiques

et une action maternelle via le

_phénotype

de la mère dans un modèle mixte

_{pourraient offrir}

des alternatives intéressantes aux modèles de « Willham» couramment utilisés pour les

mammifères

(par

exemple,

bovins

allaitants)

comme il

apparaît d’après

leur meilleur

ajustement

à des données simulées.

poulet de chair

_/

poids juvénile

/

effets maternels

_/

maximum de vraisemblance restreinte

_/

modèle animal

INTRODUCTION

At

_present,

estimation of maternal

_genetic

variances in animal

_breeding

is

_mainly

based on the biometrical model

suggested by

Willham

(1963).

This model of

maternal inheritance assumes a

_single

(unobserved)

maternal

_trait,

inherited in a

_purely

Mendelian

fashion, producing

a non-Mendelian effect on a

_separate

trait

in the

_offspring.

For

_instance,

the dam’s milk

_production

and

mothering ability

might

exert a combined non-Mendelian influence on

_{early growth}

rate of beef cattle

(Meyer, 1992a).

The

_practical

_application

of such models has been

_greatly

facilitated and hence

_{encouraged by}

derivative-free IAM-REML programs of

Meyer

(1989),

in which estimation of

_genetic

maternal effects

_according

to Willham

₍₁₉₆₃₎

forms

a standard feature.

Meyer

(1989),

however,

uses a ’reduced’ model

by

_assuming

absence of an environmental

dam-offspring

_covariance,

which is

likely

to

improve

the

_precision

of the often

_highly

confounded

_components

to be estimated but which

might

at the same time lead to biased estimates of the correlation between the direct and the maternal

_genetic

effects

_(r

_AM

₎

in

_particular

_(Koch,

_1972;

_Thompson,

1976;

Meyer,

1992a,

b).

Often the

_types

of covariances between relatives available

in the data do not have

_sufficiently

different

_expectations

to allow all

_components

of Willham’s

₍₁₉₆₃₎

model to be estimated

_(Thompson,

_{1976; Meyer,}

_1992b).

For

of a

growth

trait in beef

cattle, Meyer (1992b)

found that the environmental

dam-offspring

covariance should amount to at least

30%

of the

_permanent

environmental variance due to the dam before a likelihood ratio test would be

_expected

to

distinguish

it from zero. Greater data

_sets,

_however,

_{including multiple generations}

of observations and a

_variety

of

_types

of covariances between relatives

_{might provide}

sufficient contrast for the

_higher

number of

components

in an extended model to

be estimated more

_precisely.

Falconer

₍₁₉₆₅₎

considered the case where the

_phenotypic

value of the mother for

the character in

_question

influenced the value of the

_offspring

for the same

_character,

which results in an

_{environmentally}

caused

_{dam-offspring}

resemblance. To account for this resemblance

_{statistically,}

he included a

partial regression

coefficient in the

model,

which related

_daughters’

to mothers’

_phenotypic

values in the absence of

genetic

variation among the mothers. The

genetic

basis of the maternal effect is

ignored

in such a model.

_Thompson

(1976)

_investigated

Falconer’s

(1965)

_approach,

using

maximum likelihood

_methods,

as an alternative to Willham’s

₍₁₉₆₃₎

model with low

_precision

and

_{high sampling}

covariances between some estimates.

Lande and

_Kirkpatrick

₍₁₉₉₀₎

showed that Willham’s

₍₁₉₆₃₎

model fails to

account for

_cycles

of maternal effects as in Falconer’s

₍₁₉₆₅₎

model. Robinson

(1994)

demonstrated

_by

simulation that a

_negative

dam-offspring

_regression,

as

in Falconer’s model with a

_regression

coefficient of -

_0.2,

was fitted

_by

Willham’s

model

_partially

as a

_negative

rAM and as a

_permanent

environmental effect

using Meyer’s

IAM-REML _programs.

_{Consequently,}

she

_argued

that such

_negative

covariance

_{might explain}

the often

_disputed

_negative

rAM estimates.

Because of these mutual limitations it

_might

be

_interesting

to

_integrate

Falconer’s and Willham’s models in a mixed model

_setting

to enable consideration of both the

genetic

basis of the maternal effect and the maternal action

_through

_regression

on

the

_phenotype

of the mother

_(corrected

for BLUE solutions of fixed

_effects).

A

_great

amount of work has been carried out on the estimation of maternal

effects among domestic

livestock,

in

particular

for mammals

(see

Willham,

1980;

Mohiuddin, 1993; Robinson,

1996).

In

_poultry,

_however,

where maternal

_(egg)

effects on

_juvenile

broiler

_{body weight}

_(JBWT)

are

_apparent

_(Chambers,

1990),

no

_major

_attempts

have been made to

_partition

this maternal variance into

_genetic

and environmental

_components.

Also the

_sign

and

_magnitude

of rAM has not been

estimated

_according

to Willham’s

₍₁₉₆₃₎

model.

_Although

many studies have shown a

_positive

(phenotypic)

effect of _egg

weight

on JBWT

(Chambers, 1990).

Such

poultry

data may be suitable for the estimation of maternal

genetic

variances

_owing

to their size and structure with many

offspring

per dam and often many recorded

generations

available.

The

_objectives

of the

present

study

were to

_investigate

₍₁₎

the effect of estimation of the environmental

_{dam-offspring}

covariance on the other

(co)variance

compo-nents and

_resulting

parameters

(particularly r

AM

)

and on the likelihood of the

size-able data sets for JBWT in two

_meat-type

chicken

_populations

_by

IAM-REML

methods and

₍₂₎

the

_{goodness-of-fit}

of

_{Falconer-type}

and

_integrated

Falconer-Willham models to simulated data and these JBWT data and the

_resulting

MATERIAL AND METHODS

Data

Field data

The data on JBWT

_originated

from two commercial broiler

_{populations. Summary}

statistics are illustrated in table I. The data on strains A and B

represented

approximately

six and three

_{overlapping generations, respectively.}

Male and female JBWT SDs were somewhat

_{heterogeneous, presumably,}

because of a scale effect. Some

_{heterogeneity}

of raw CVs was

_apparent,

but

_disappeared

after

_{precorrection}

for effects of hatch week and age of the dam. Some data structure

aspects

are shown in table II.

Simulated data

Data were simulated to

study

the

_{goodness-of-fit}

of the various models to estimate maternal effects

_(see

the

_following)

and the differences between simulated and estimated

_(co)variance

_components.

The

_genetic

model was similar to the one

and a residual

_{effect, sampled}

from

N(0,100), N(0,20)

and

N(0,280), respectively.

Furthermore,

a

_regression

of - 0.1 on the

_phenotype

of the dam was assumed. The

base

_population

consisted of 110 animals. Ten sires were mated to 100 dams in a

nested

_design

with ten full sib

_{offspring produced by}

each sire-dam combination.

Parental candidates were

randomly assigned

from these thousand

offspring

to

generate

the next

_generation.

This hierarchical

mating

scheme was

repeated

for

eight

generations.

Models of

_analyses

Effects of location

Fixed effects fitted were hatch week

(198

and 90 levels for strains A and

B,

respectively),

sex

(two levels)

and _age of the dam when the egg was laid in 3-week

intervals

_(seven

_{levels) representing}

effects on _eggs

(eg,

size).

Considering

male and female JBWT as

_separate

traits

Table I _gave some evidence that the differential SDs of both sexes are due to the

dependence

of variance and mean, since

adjusted

CVs were

_homogeneous.

To

_fully

justify

evaluation of male and female JBWT as one trait in the

analysis

of maternal

effects,

however,

the two sexes were considered as

_separate

traits in a bivariate

analysis

in order to

_investigate

the

_genetic

_relationship

between these traits and hence the

_importance

of

_segregation

of sex-linked _genes

_affecting

JBWT in the

present

broiler

_populations.

In matrix notation the bivariate model can be

presented

as:

r.. 1

where,

for trait

_{i (i}

=

_1,2;

representing

JBWT on males and

females),

_yi is a

vector of

_{observations; b}

_i

is a vector of fixed

effects;

ai is a vector with random

additive

_genetic

animal

_effects;

ci is a vector with random maternal

_permanent

environmental

_effects;

ei is a vector with random residual

effects;

and

_Xi,

_Z

_a

_i

and

Z

ct

are incidence matrices

relating

the observations to the

respective

fixed and

random effects. The assumed variance-covariance structure is:

where

_o,

₂

_.a2.

and

_{o, 2.}

are the additive

_genetic,

the maternal

_permanent

corresponding

covariances between the male and female

JBWT;

A is the

relation-ship matrix;

I

i

is an

_identity

_matrix;

and B is a

_rectangular

matrix

_linking

male

and female progeny records to the dam. The

_algorithm

of

Thompson

et al

₍₁₉₉₅₎

was used. Their method reduces the model to univariate forms

_{by scaling}

and

trans-formation,

which diminishes

_{dimensionality}

and

_speeds

_{up convergence.}

A ’reduced’ Willham model

Initially

six different

_{genetic models, applied}

_{by Meyer}

_(1989),

were considered for

both strains.

Table III exhibits the random effects fitted and the

_(co)variance

components

estimated in each model. Model 1 was a

_purely

direct additive

model,

while model

2

_(with

sub-models

a,b

and

_c)

allowed for dams’

_permanent

environmental effects in addition. This environmental maternal

_component

was

_{slightly expanded by}

distinguishing

between a covariance of maternal half sibs

(c H

s

, 2

model

2a)

and full

sibs

_(cF

_S

_,

model

_2b).

_Fitting

both

_{simultaneously}

was considered also

(model 2c).

When

_{only fitting}

_{c 2s}

then

_{c 2s}

=

C2

(see

table

III),

since covariance

_amongst

maternal HSs also

_applies

to FSs. Model 3 included a maternal

_genetic

effect

in addition to the animals’ direct

_{genetic effects, assuming}

zero direct-maternal

covariance

_(<

_7AM

_)-

Model 4 was as model 3 but allowed for a non-zero _<7AM. Models

maternal

_permanent

environmental effects in addition

_(on

maternal HSs

_and/or

FSs).

The sub-models

_(1-5)

follow from the full mixed linear model

_{(model 6),}

which in matrix notation is:

where

_{y, b, uA, uM,}

c and e are vectors of

observations,

fixed

effects,

direct

breed-ing

values,

maternal

_breeding

_values,

random common maternal

permanent

en-vironmental

_effects,

and random environmental residual

_{effects, respectively;}

and

X,

Z

A

,

Z

M

and

Zc

are incidence matrices

relating

the observations to the

_respective

fixed and random effects. The variance-covariance structure is

where

_afl

_represents

either the covariance between FSs or maternal HSs.

An ’extended’ Willham model

Throughout

the

_previous

models a zero direct-maternal environmental covariance

(a

E

c )

was

_assumed,

which is

_{commonly practiced.}

_However,

the

_possibility

of a non-zero _QEC is real. The existence of a

_negative

_QEC, for

_example,

has been

_suggested

(eg,

Koch,

1972).

Ignoring

a

(non-zero)

QEC is

likely

to bias the

parameters

involved

in the estimation of maternal effects. In

particular

<7AM

might

be biased in a

downward direction when

_ignoring

a (TEC that is

negative. Therefore,

aEc was

included in all models in a second series of runs

(models

7-12)

to

_{study changes}

in estimated

_components

and

_parameters

and

_{goodness-of-fit.}

The

_(co)variance

structure now is

Consequently,

three maternal environmental covariances were

_conceivable,

a

covariance

_amongst

maternal half

_sibs,

a covariance

_amongst

full sibs and a

covariance between dam and

_offspring.

When

_only

_fitting

CECthen

4

s

=

C2 H

= CEC, since _CEC also

_applies

to the covariance

amongst

maternal HSs and FSs.

The

_(direct)

Falconer model

Falconer

₍₁₉₆₅₎

_suggested

a model

_including

a maternal effect

(F

m

)

as linear

func-tion of the mother’s

_phenotype

_(see

outline in

_Appendix).

_Thompson

₍₁₉₇₆₎

derived

the

_expectations

for

_QP

_and

_a

_£

in terms of

_F

_m

for the sources of

(co)variation

(y - F

m

y’).

In a mixed model

_setting

this model

_(ignoring

the dominance

compo-nent)

can be formulated in matrix notation as

where _{yp is} a vector with the dams’ observations and

Xp

is the incidence matrix

relating

these observations to the

_respective

fixed effects. An

_integrated

Falconer-Willham model

To account for

_possible

maternal

_pathways

_through

the dam’s

_phenotype

as well

as the

_{genetic origin}

of maternal effects an

_{integrated approach}

was

_investigated

in a third series of runs

(models

13-18).

The matrix

_{representation}

of the full linear

integrated

Falconer-Willham model that was considered is

which is Willham model

₍₂₎

and Falconer model

₍₃₎

_amalgamated.

The

_Appendix

provides

a derivation of the variance of _y.

For models with a maternal effect the fraction of the selection differential that

would be realised if selection were on

_phenotypic

values

(hA

+M

),

_ie,

the

_regression

of the sum of direct and maternal

_genotypes

on the

_phenotype

was calculated as

(Willham,

1963):

where

_QA

is the

direct

additive

_{genetic variance,}

_a

_M

is the maternal additive

genetic

variance and

_up

is the

_phenotypic

variance.

Methods of

_analyses

Henderson-III and

_{offspring-parent}

_regression

Henderson’s method III was

_applied

to the data to

_produce

estimates of variance due to sires

_(patHS)

and sire-dam combinations

_(FS).

A

_weighted

_average of

the individual

_generation

estimates was obtained

_{by weighing}

them

_inversely

proportional

to their

_sampling

variances. Covariances between

_offspring

and sire and

_{dam, respectively,}

were obtained

_{by weighted}

_regression

_analyses

_(with

the

degrees

of freedom as

_weights)

of average

offspring

on

_{parental performances,}

which were both deviated from OLS

_expectations

based on the effects of location. The

sources of

(co)variation

were

_equated

to their

_expectations

and the

_resulting

_system

of linear

_equations

was solved

_by

_{multiple regression}

for a series of values for

_F

_m

_,

thereby locating

the

_F

_m

that resulted in minimisation of the mean _square error or rather maximisation of the likelihood and the ’best’ estimates for

_o,2

_and

_{a A 2}

IAM-REML

IAM estimates of the

_(co)variance

_components

for both data sets were obtained

by

a derivative-free REML

_algorithm

based on _{programs written}

_{by Meyer}

(1989).

The programs were

_adapted

to include an environmental

_{dam-offspring}

covariance

com-ponent

and to enable the estimation of Falconer’s maternal

_phenotypic

_regression,

either on its own or

_integrated

in Willham’s model.

_Equations

in the mixed model matrix

_(MMM),

the coefficient matrix and the RHS’s

_augmented,

were reordered

using

a

multiple

minimum

_{degree reordering}

_(George

and

_Liu,

₁₉₈₀₎

to minimise

fill-in,

before Gaussian elimination was

performed

on MMM. The Downhill

Simplex

method

_(Nelder

and

Mead,

1965)

was used to locate the maximum

_log

likelihood

(log L).

Convergence

was assumed when the variance of the function values

_(-

_2log

L)

in the

_Simplex

was less than

.

g

10-

For a series of values for

_F

_m

_,

the likelihood of

the

_remaining

_parameters

in the Willham model was maximised

_given

these values of

_F

_m

_.

For the first

F

m

maximisation run the

scaling

factor for the residual variances

of animals with

_missing

maternal observations

_(s

_F

_,

see

_Appendix)

was set to

_unity

since sFis a function of

_F

_m

and the

_{(co)variances}

to be estimated. A second run was

performed,

for every value

of F,T&dquo; incorporating

a

scaling

factor as deduced from the

estimated

_(co)variance

_components

and

_F

_m

_(see

_equations

A2 and A3 in

_Appendix).

In this second run the likelihood was remaximised and

_adjusted

for the

changes

in the

_projected

data and the variance

_component

estimates. A second

_update

of sF

and

_subsequent

maximisation run led to

_{only negligible changes}

in likelihood and

was,

hence,

not

_performed

for these

_analyses.

For every other

F,T,

maximisation run

the initial SFvalue was chosen as a

_proportion

of the

_previous

maximised _sF value.

The Falconer

_{parameter F}

_m

_maximising

the likelihood was estimated

by quadratic

approximation

of the

_profile

likelihood surface of

_F

_m

_.

The

_accompanying

param-eters in the Willham model had maximum likelihood conditional to this value of

F

m

.

Likelihood ratio

tests,

with error

probability

of

5%,

were carried out to determine

whether maternal

_genetic

or

_permanent

environmental effects contributed

signifi-cantly

to the

_phenotypic

variance in JBWT for both strains.

Furthermore,

the

_asymptotic

_sampling

variances of 0&dquo; AM

(models

6c and

12c)

and QEC

(model 12c)

were obtained

_{by fitting}

_quadratic

_{Taylor polynomials}

to

their

_{profile log-likelihood}

curvatures

_(Smith

and

Graser,

1986).

The

_profile

likeli-hoods were

L

,,(

O

’

2

,

a

2

C

72

O&dquo;!IO&dquo; AMr¡,

y

),

(or2

,

7

L

la2 !Cr/!

_OEC?7,

0

’

2

1

0

’A

M

,,

Y)

and

₂

_a

₂

_{a2 ,}

_{o-g J}

_UECN

_,

_Y

₎

for QAM in models 6 and 12 and for UECin model

12,

respectively,

where h

represents

the fixed

_point

for which the

_{log-likelihood}

was

maximised.

RESULTS

Sex-linked variation in JBWT

Results of the bivariate

_{analyses considering}

male and

female

JBWT as different

traits are shown in table IV. Differences in male and female

_phenotypic

variances

were substantial as

_might

be

_expected

because of the

_large

differences in mean

were somewhat

_greater

than the male heritabilities. In birds the females are the

het-erogametic

sex. Female

offspring

_get

their sex-linked _genes

only

from their fathers.

Therefore,

if

_{significant sex-linkage}

is

_present,

_higher

male heritabilities

_might

be

anticipated,

which was not the case.

Also,

_{genetic relationships might}

be

expected

to deviate

_markedly

from

_{unity. However,}

the correlations were _very

_{high, although}

statistically

just

different from

_unity.

We can now with more confidence _say that

sex-linked genes did not

notably

contribute to the differential variation of male and female JBWT in the

_present

_{populations. Logarithmic}

transformation was

_applied

to alleviate the variance-mean

_dependency.

The

_comparison

of

_genetic

_parameters

of several models

_involving

maternal effects did not reveal any

important

discrepan-cies between the data on the arithmetic and the

_geometric

scales.

Hence,

analyses

of the data on the arithmetic scale will be

presented.

Conventional estimation

of

_{(co)variances,}

heritabilities and

the

Falconer

parameter

Heritability

estimates based on between sire variances

_{(paternal HS)}

were

equal

for

both

_populations

_(0.21)

and very similar to the

offspring-sire

regression

estimates

(0.20

and 0.19 for

_populations

A and

_B,

_{respectively) (see}

table

_V).

The

heritability

estimates based on FSs and

_{offspring-dam}

_regression

were

considerably higher.

For

_population

A the FS estimate was somewhat

higher

than the

_{offspring-dam}

_estimate,

whereas

_population

B showed the reverse. The

components

were

equated

to their

_expectations

for several

_F

_m

values

_{(table VI).}

The

’optimum’

F

m

estimates were

_positive

with 0.03 and 0.07 for

populations

A and

B,

respectively.

The derived

_heritability

estimates were 0.21 and 0.19 for

populations

IAM-REML estimation of maternal

_genetic

_parameters

Simulated data

The

_{goodness-of-fit}

of

Willham,

Falconer and

_integrated

models were tested to simulated data based on an

_integrated

Falconer-Willham

_model,

with a direct and

maternal

_genetic

effect with zero covariance and a maternal

_phenotypic

_effect,

assumed before

_by

Robinson

_(1994).

The results are shown in table VII. The likelihoods were deviated from

model 1,

which

represented

the

_{appropriate genetic}

model. The estimated

_components

were close to simulated

_components

for model 1. Model

_{2, representing}

a Willham model with direct and maternal

_genetic

effect with

non-zero covariance and a maternal environmental

_component,

estimated a

c

2

-effect

of 0.03 and a

_{significantly}

_negative

estimate for QAM

resulting

in a

_negative

_rAM

of -

0.56,

which was observed also

by

Robinson

(1994).

The likelihood ratio test

adjudged

the fit to be

_{significantly}

worse than model 1 at a confidence level of

99%.

effect,

was

_greater

than model 2 but

_{significantly}

less than model 1 with P < 0.05. The ’full’ Falconer-Willham model

_(model

_4),

_assuming

a non-zero _QAM,

_appeared

to fit better than the true

_model,

_although

the difference was not

_significant

at P =

0.05. The ’extended’ Willham model

_(model

₅₎

_{’picked up’}

most of the

_negative

environmental covariance between dam and

_offspring

as such.

_However,

the effect was

_partially

fitted as a

_negative

_(TAM

_leading

to an _rAM value of - 0.22. The

goodness-of-fit

of model 5 was similar to the true model.

Field data

Estimated

_phenotypic

variances and

_genetic

_parameters

for JBWT of both strains under a series of different

genetic

models

_together

with their likelihoods are

sum-marised in tables VIII and IX.

_Clearly,

_very

_significant

increases in

_{log-likelihood}

(over

model

₁₎

demonstrate that both environmental and

_genetic

maternal effects exist for both strains.

_Generally,

_genetic

_parameters

were

_quite

similar over

_strains,

despite

distinct differences in selection

_history.

Fitting

a maternal

_permanent

environmental effect

(with

the

_pertaining

variance

component

as

_proportion

of

QP

_being

referred to as

C2 H

s

for maternal half sibs

_(HSs)

and

48

for full sibs

_(FSs))

in model 2 resulted in

_{highly significant}

increases in the likelihood for both strains.

_Estimating

a

c

2

for

HSs and FSs

_{simultaneously}

resulted

in a

_{significantly}

better fit with the effect of FSs

_being

about a factor of two

_greater.

The presence of a maternal

heritability

(m

2

)

in addition to

h

2

_(model

₃₎

was much

more

_likely

than

_{model 1,}

but did not fit the data as well as model 2.

_Allowing

for a non-zero direct-maternal

_genetic

covariance

_(presented

as a

_proportion

of

o, 2

m

2

estimates

in model 5a decreased

_{substantially}

for strain A

_(from

0.07 to

_0.03)

and for strain B

_(from

0.05 to

_0.01);

and thus the maternal variance seemed to be more of a

(permanent)

environmental than

_{genetic origin. Estimating}

(YAM in

addition to model 5

_(model

₆₎

showed a similar

_pattern

for both strains in terms

this smaller

m

2

_parameter

was

accompanied by

a much more

_negative

CAM and

consequently

rAM relative to model 4.

Likelihoods increased

_{considerably by adopting}

CEC. All the cEc estimates were

positive

and

_consequently

the estimates of rAM tended to be more

_negative

and

heritability

estimates

_dropped

somewhat. For the models 12a and 12c the

m

2

estimate increased

_by

a factor of 1.5 to 2

_(from

0.04

_(0.04)

to 0.07

_(0.06)

in

population

A and from 0.03

_(0.02)

to 0.05

_(0.04)

in

_population

_B).

_Assuming

a zero CAM and a non-zero _cac

_{(model 11)}

fitted the data of

population

B better

than the reverse

assumption,

a non-zero _{CAM and} a zero _CEC

(model 6).

This was not the case for

_population

A.

_However,

the

highest

likelihood for both

populations

was attained

_{by assuming}

both these covariances to be non-zero

(in

model

12).

All the

F

m

values were

_positive

(in

models

13-18)

as were the C_{EC estimates}

in models 7-12. The models without a

c

2

_effect

(models

13,

15 and

16)

did not

fit as well as their

_counterparts

fitting

CEC

_(models

7,

9 and

_10,

_{respectively).}

The

_F

_m

estimates were

generally

similar for both

populations.

The

improvements

in likelihood relative to the models 7-12 were

_greater

for

_population

A. The

_F

_m

estimates for model 14b were similar to the estimates based on

multiple regression

of the

_analysis

of variance

components

(table

VII).

Generally,

m

2

and

CAM estimates increased somewhat and led to more

_negative

rAM values

compared

to the models

including

CEC.

Estimation

_{of sampling}

variation

of CAM

and cEc

Approximate profile

likelihood and

_(derived)

sampling

variances for CAM

_(in

models 6c and

_12c)

and cac

(in

model

_12c)

were

investigated

to obtain a

bet-ter

_insight

into the accuracy of CAM in model 6c

(assuming

zero

a

E

c)

_compared

to the accuracy that could be attained when the

potentially highly

confounded

com-ponents

CAM and CEC

(Meyer,

1992b)

were estimated

_together

(model 12c),

using

the

_present

sizeable data sets.

Figure

1

_depicts

the

_quartic

_{Taylor polynomial}

fitted to seven

_points

of the

_profile

likelihood for CAM

(with

R

2

=

100%).

The

resulting

approximate

profile

likelihood

shows that CAM is

highly unlikely

to be

positive

for both strains.

The

_approximate

_profile

likelihood curvatures for CAM and

C

EC

(both

quartic

as well with

R

2

=

100%)

in model 12c are shown in

_figure

2a and

b,

respectively.

Once

_again,

_profiles

show a similar

_pattern

for both strains and

also

the

profiles

for CAM

and

CEC act

fairly similarly

to the

_images

_(with

opposite sign

for the

values)

of CEC and c,!M,

respectively,

which

pointed

towards the presence of a

high

negative sampling

covariation between these

components.

The

figures

illustrate the low likelihood of a

_positive

_CAM on the one hand and the _very low likelihood of a

negative

CEC on the other hand.

The

sampling

errors

_approximated

from the above

profile

likelihood curves are

exhibited in table X.

_Generally,

the direct-maternal covariance

components

were

accurately

estimated for both

_strains,

with the

_sampling

error of CEC

being

roughly

half the size of the

approximation

for CAM. The accuracy of the CAM estimates for models 6c and 12c were

similar,

hence the

sampling

correlation of CAM with

C

EC

(in

model

_12c)

did not hinder much the

_precise

estimation of these

components

which was illustrated

_by

the similar curvatures of the

_profile

likelihoods for strains

A and B.

DISCUSSION

Sex-linkage

The

_segregation

of sex-linked genes

affecting

JBWT was found to be

_small,

which agrees with results summarised

by

Chambers

(1990).

Owing

to their

hemizygous

form these genes are

_likely

to be driven towards

_{fixation, especially}

in

_meat-type

poultry

with a

_long

and extensive selection

_history

for

growth

traits. The

_genetic

correlation between male and female JBWT

performance

was

_just

significantly

different from

_unity,

but this could

_easily

be attributable to endocrine differences between both sexes.

Analysis

of variance

The estimates of

F

m

,

found while

equating

the

_(co)variance

_components

to their

expectations

and

_{minimising MSE,}

were small

_(0.03

and

_0.07)

and similar to the values

_(0.04

and

_0.06)

found for its

_equivalent

in a mixed model

_setting,

model 14b.

The conventional

h

2

estimates

were,

however,

substantially

lower

(0.21

versus 0.30 and 0.19 versus 0.24 for the

_populations

A and

B, respectively).

The difference in

estimates was

larger

for

population

A. The data on

_population

A

_represented

six

generations

(three

more than

population

B)

and hence the numerator

_relationship

matrix accounted for more selection in this

_longer

time

_period.

Maternal effects estimation in a mixed model

setting

It was shown that inclusion of a maternal

_permanent

environmental effect

provided

a much better fit to the data

_(over

model

₁₎

and that inclusion of _any more

effects,

although statistically significant,

gave

relatively

a much smaller additional increase in

_log

L

_(over

model

_2).

This was reflected

_by

the direct

_heritability

_estimates,

which

fluctuated within a rather narrow _range for models 2-18

(except

for model

13)

of the maternal variance over environmental and

_genetic

maternal

_{(co)variances,}

although

some cross-substitution of the direct additive

_genetic

variance and hence

the direct

_heritability

with the direct-maternal

_{genetic covariance,}

in

_particular,

was

likely

to occur

(Thompson,

1976; Meyer, 1992b).

Dominance

might

have some

effect on estimates of maternal

_effects,

_although

dominance was found to be of little

importance

in broiler

_{body weight}

_(Koerhuis

et

_al,

_1997).

REML combines information on various collateral relatives and various

offspring-parent

regressions

in order to obtain one

_efficiently

_pooled

estimate for

h

2

with

minimum variance

_(Thompson,

_{1977; Hill,}

_1988).

The

_large

reduction in the

h

2

2 estimate in model 2

_compared

to

_{model 1, accompanied by}

_relatively

small

c

2

2

estimates,

suggested

a

_{high weighting}

of the between dam

_family

h

2

estimate,

relative to the between sire

_family

h

2

estimate.

This

_might

have been

expected

with such a

_large

_number,

on _average, of

_large

dam families in the data

_{(table II),}

_leading

to very accurate estimates on between dam

_family

variance. A lower

_weighting

of dam

_family

information is

_expected

for domesticated

_species

in

_general

and for

beef cattle in

_particular,

where dam families are much smaller

(eg,

_Meyer,

1992a).

The

h

2

estimates

in model 2 should be

_expected

to be closer to the Henderson-III sire

_{component h}

2

estimates. Chambers

₍₁₉₉₀₎

_pooled

53 sire

_{component h}

2

2

estimates from 23 studies

_resulting

in an _average value of 0.41. The

present

smaller

h

2

estimates

_might

be

_{explained by}

the much

_longer

and more extensive

selection

_period

the

_present

broiler

_populations

have

_undergone

in

_comparison

to

the

_populations

used in many

experiments,

bearing

in mind that the vast

_majority

of these studies was conducted two to three decades _ago. The smaller variance for strain B

_might,

beside

_genetic

strain

_differences,

be due to the lesser extent of correction for reduction in variance caused

_by

selection as

_only

three

_generations

were available for this strain

_compared

to six

_generations

for strain A.

_Furthermore,

Chambers’

₍₁₉₉₀₎

summarised estimates were often based on

_weights

at older _ages

(8,

9 or 10

_weeks).

It is not uncommon for heritabilities to increase with _age of

weight

owing

to

_diminishing

maternal influences.

Allowing

for _(JAM, resulted in a value of _rAM that was

_{considerably negative}

in model 6. This was somewhat

_surprising

since we

expected

a

_positive

_genetic

correlation between JBWT and _egg

_weight

_(Kinney,

_1969;

Koerhuis and

_McKay,

1996),

which is believed to increase the

_{offspring’s}

JBWT.

_Fitting

both CAM and a

EC

(model 12),

to account for

possible

downward bias of (JAM

(Koch,

1972;

Meyer,

1992b),

resulted in

_slightly

more

_negative

rAM estimates

owing

to

_positive

estimates of UEC - Cantet et al

(1988)

also obtained

large

negative

estimates of (JAM

accompanied by positive

estimates of UEC for

growth

traits in beef cattle.

However,

Cantet et al

₍₁₉₈₈₎

found

_negative

estimates for

F

m

(in

the range - 0.15 to -

_0.25),

whereas our estimates of

_F

_m

were

_{positive just}

like _OEc estimates and led to even more

_negative

rAM estimates. Cantet et al

₍₁₉₈₈₎

had a small data set and used conventional

_{methods, equating}

_separately

estimated covariances between relatives

to their

_expectations

and

_solving

the

_resulting

_system

of linear

equations.

This

ignores

the fact that the same animal

_might

have contributed to different

_types

of covariances and that different observational

_components

_might

have different

sampling

variances, ie,

combining

information in a

_non-optimal

_way

(Cantet

et

_al,

For our JBWT

_data,

the

_genetic

variance of maternal

_origin

could,

for the

_greater

part,

relate to _egg

_(shell)

_quality

rather than egg

size,

which could

explain

the

negative sign

of <TAM- Koerhuis et al

_(1997),

_following

suggestions

by

Lande and

Kirkpatrick

(1990),

fitted individual maternal

_pathways

related to the egg as

co-variates in an

offspring-parental

_{regression model,}

to

investigate

their

_importance

in

_causing

maternal variation in JBWT. Those results

implied

a

_negative

_partial

maternal effect of _egg

_weight

loss between the start and the 18th

_day

of

_incubation,

and are in

_agreement

with Robinson et al

₍₁₉₉₃₎

who

_reported

a

_negative

relation-ship

between

_{body weight}

and _egg

_(shell)

_quality,

an inferior

quality

_giving

rise to a

greater

loss of

_weight.

However,

this

_negative

_partial

effect was offset

_by

a

_positive

partial

maternal effect of egg

weight

at the 18th

_day

of

incubation,

and hence the

aggregate

maternal effect on JBWT was found to be small

_(Koerhuis

et

_al,

_1997).

A

_negative

<7AM would decrease the

efficiency

of

phenotypic

selection for JBWT

as

expressed by

the low

hA

+M

estimates for the models 12 and 18 with overall

superior

log

L. Selection on maternal

breeding

values for JBWT _may,

however,

not

be _very effective

_owing

to the low maternal

_{heritability.}

_Moreover,

it

_might

not be the

_{preferable approach}

since _egg

_(shell)

_quality

characteristics can

_readily

be

selected for

_directly

with

_higher

accuracy and

predictability

(Koerhuis

et

_al,

₁₉₉₇₎

and less

_delay,

because the

_expression

of the maternal

_{effect, although}

_occurring

later in

_life,

would not

_lag

a

_generation

behind the direct effect as is

normally

the case

(Willham, 1980). Nevertheless,

the

_{presented amalgamation}

of Falconer and

Willham models in a mixed model

setting

might

offer attractive alternatives to

Meyer’s

(1989)

models

_for,

eg, beef cattle as was illustrated

_by

results based on

simulated data

_{(table VII).}

Meyer

(1992b)

studied the

_sampling

behaviour of REML estimates of

(co)varian-ce

_components

due to additive

_genetic

and environmental maternal effects. She

showed that

_sampling

correlations between estimates were

high

and that sizeable

data sets are

_required

to allow

_reasonably

accurate estimates to be obtained. Results in the

present

study,

using

large

data

_sets,

illustrated the

_possibility

of

good sampling

properties

for both the

_genetic

and environmental direct-maternal

covariance

_components.

_Hence,

these

_poultry

data sets

_might

also increase the scope

for the

_application

of more detailed

models,

_eg,

_estimating

dominance variance and

variance due to new mutation in addition to

_genetic

and environmental maternal

effects,

yet

providing

sufficient contrast for the often

_highly

correlated

_genetic

parameters

involved,

to be estimated

_precisely.

More research is

needed, however,

with

_regard

to the

_{practicalities}

of such detailed maternal effects

_(and

_other)

models

for use in

_genetic

evaluation

methods

performed by

breeders.

The effect of more detailed fixed effect structures

Robinson

_{(1994, 1996)}

showed that additional variation

_(eg,

sire x

_year)

unac-counted for in the model affected estimates of maternal effects. Differences in

re-sults from Mackinnon et al

₍₁₉₉₁₎

and

Meyer

(1992a)

for the same data

_suggested

sensitivity

of maternal effects to different fixed effects models. In our data different

parental

flocks of different ages and farms contributed

offspring

to one hatch week.

The age difference was accounted for in the

_model,

but more

_specific

maternal

were identified. The effect of flock nested within hatch on the

_genetic

_parameters

in models 1 and 2 was small

_{(not presented).}

The effect on the

genetic

parameters

for the more

comprehensive

models

(5c, 6c, llc, 12c,

17c and

18c)

was

investigated

for both

_populations.

The

_phenotypic

and direct and maternal

_genetic

variances

were reduced

considerably

and were

accompanied by

rAM estimates much closer to zero

(see

table

XI).

The

h

2

estimates

were now _very similar to the estimates of

hA

+M

.

This limited

importance

of maternal effects

_exerting

a non-Mendelian influ-ence on JBWT is in closer

_agreement

with the results obtained

_by

Koerhuis et al

(1997).

The choice of the fixed effects model _appears to be

_paramount

for detailed maternal effects

_models,

but the increase in

_computing

time

_(four-fold

increase _per likelihood evaluation for the

_present

_data)

_might

often

_prevent

more refined fixed

effect structures to be

_occupied.

ACKNOWLEDGEMENTS

We

_acknowledge

the

_help

of referees in

_turning

the nebulous

_description

of the method into

_something

more

_{intelligible.}

RT

_acknowledges

the _support of MAFF in this work.

IACR-Rothamsted receives

_grant-aided

_support from the

_{Biotechnology}

and

_Biological

Sciences Research Council.

REFERENCES

Cantet

_RJC,

Kress

_DD,

Anderson

_DC,

Doornbos

_DE,

_{Burfening PJ,}

Blackwell RL

₍₁₉₈₈₎

Direct and maternal variances and covariances and maternal

_phenotypic

effects on

preweaning growth

of beef cattle. J Anim Sci

_66,

648-660

Chambers JR

₍₁₉₉₀₎

Genetics of

_growth

and meat

_production

in chickens. In:

_Poultry

Falconer DS

₍₁₉₆₅₎

Maternal effects and selection response. In: Genetics

today

(SJ

Geerts,

ed),

Pergamon Press,

New York

George A,

Liu JWH

₍₁₉₈₀₎

_Computer

Solutions

_{of Large Sparse}

Positive

_{Definite Systems.}

Prentice-Hall, Englewood Cliffs,

New York

Henderson CR

₍₁₉₇₆₎

A

_simple

method for

_computing

the inverse of a numerator

relationship

matrix used for

prediction

of

breeding

values. Biometrics

32,

69

Hill WG

₍₁₉₈₈₎

Considerations in the

_design

of animal

breeding experiments.

In: Proc.

Symp

Adv Stat Meth Genet

Improv

Livest

_(K

Hammond,

D

_Gianola,

_eds),

_Springer

Verlag, Heidelberg

Kinney

TB Jr

₍₁₉₆₉₎

A summary of

reported

estimates of heritabilities and of

genetic

and

phenotypic

correlations for traits of chickens.

_Agric

Handbook No 363,

Agric

Res

Service,

US

_{Dept Agric}

Koch RM

₍₁₉₇₂₎

The role of maternal effects in animal

_breeding.

VI. Maternal effects in

beef cattle. J Anim Sci

_35,

1316-1323

Koerhuis

_{ANM, McKay}

JC

₍₁₉₉₆₎

Restricted maximum likelihood estimation of

_genetic

parameters for egg

production

traits in relation to

juvenile body weight

in broiler

chickens. Livest Prod Sci

46,

117-127

Koerhuis

_ANM,

_McKay

JC,

Hill

WG, Thompson

R

₍₁₉₉₇₎

A

_{genetic analysis}

of egg

quality

traits and their maternal influence on

_{offspring-parental regressions}

of

juvenile body

weight performance

in broiler chickens. Livest Prod Sci

_(In

_press)

Lande

R, Kirkpatrick

M

₍₁₉₉₀₎

Selection response in traits with maternal inheritance.

Genet Res

_55,

189-197

Mackinnon

_{MJ, Meyer K,}

Hetzel DJS

₍₁₉₉₁₎

Genetic variation and covariation for

growth,

parasite

resistance and heat tolerance in

_tropical

cattle. Livest Prod Sci

_27,

105-122

Meyer

K

₍₁₉₈₉₎

Restricted maximum likelihood to estimate variance components for

animal models with several random effects

using

a derivative-free

_algorithm.

Genet

Sel Evol

_21,

317-340

Meyer

K

_(1992a)

Variance components due to direct and maternal effects for

_growth

traits of Australian beef cattle. Livest Prod Sci

31,

179-204

Meyer

K

_(1992b)

Bias and

_sampling

covariances of estimates of variance components due

to maternal effects. Genet Sel Evol

_24,

487-509

Mohiuddin G

₍₁₉₉₃₎

Estimates of

_genetic

and

phenotypic

parameters of some

_performance

traits in beef cattle. Anim Breed Abstr

_61,

495-522

Nelder

_JA,

Mead R

₍₁₉₆₅₎

A

simplex

method for function minimization.

Computer

J

_7,

147-151

Robinson DL

₍₁₉₉₄₎

Models which

_{might explain negative}

correlations between direct and maternal

_genetic

effects. In: Proc 5th World

_Congr

Genet

_Appl

Livest Prod

_Guelph

Canada.

Robinson DL

₍₁₉₉₆₎

Estimation and

_{interpretation}

of direct and maternal

genetic

param-eters for

_weights

of Australian

_Angus

cattle. Livest Prod Sci

45,

1-11 1

Robinson

_FE,

Wilson

_JL,

Yu

_MW,

Fasenko

GM,

Hardin RT

₍₁₉₉₃₎

The

_relationship

between

_{body weight}

and

reproductive

efficiency

in _meat-type chickens.

_Poultry

Sci

_72,

912-922

Smith

SP,

Graser H-U

₍₁₉₈₆₎

Estimating

variance components in a class of mixed models

by

restricted maximum likelihood. J

Dairy

Sci

_69,

1156-1165

Thompson

R

₍₁₉₇₆₎

The estimation of maternal

_genetic

variances. Biometrics

_32,

903-917

Thompson

R

₍₁₉₇₇₎

The estimation

_{of heritability}

with unbalanced data. II. Data available

on more than two

_generations.

Biometrics

_33,

497-504

Thompson R, Crump RE, Juga J,

Visscher PM

₍₁₉₉₅₎

_Estimating

variances and

covari-ances for bivariate animal models

_{using scaling}

and transformation. Genet Sel

Evol 27,