Bias and sampling covariances of estimates of variance components due to maternal effects

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Bias and sampling covariances of estimates of variance

components due to maternal effects

K Meyer

To cite this version:

Original

article

Bias

and

_sampling

covariances

of estimates of variance

_components

due

to

maternal

effects

K

_Meyer

Edinburgh University,

Institute

_{for Cell,}

Animal and

_{Population Biology,}

West Mains

Road, Edinburgh

EH9

3JT, Scotland, UK;

Unibersity

of

New

England,

Animal Genetics and

_{Breeding Unit,}

Armidale,

NSW

_2351,

Australia

(Received

13

_September

1991;

accepted

26

_{August 1992)}

Summary - The

_sampling

behaviour of Restricted Maximum Likelihood estimates of

(co)variance

components due to additive

_genetic

and environmental maternal effects is examined for balanced data with different

_family

structures. It is shown that

_sampling

correlations between estimates are

_high

and that sizeable data sets are

_required

to allow

reasonably

accurate estimates to be

obtained,

even for

_{designs specifically}

formulated for the estimation of maternal effects. Bias and

_resulting

mean _square error when

_fitting

the wrong model of

analysis

are

_{investigated, showing}

that an environmental

_{dam-offspring}

covariance,

which is often

_ignored

in the

_analysis

of

_growth

data for beef

cattle,

has to

be

quite

large

before its effect is

statistically

significant.

The

_efficacy

of

_embryo

transfer in

_{reducing sampling}

correlations direct and maternal

_genetic

_(co)variance

_components is illustrated.

maternal effect

_/

variance component

/

sampling covariance

Résumé - Biais et covariances _{d’échantillonnage} des estimées de composantes de variance dues à des effets maternels. Les

propriétés d’échantillonnage

des estimées du maximum de vraisemblance restreint des variances-covariances dues à des

_effets

maternels

génétiques additifs

et de milieu sont examinées sur des données d’un

_{dMpo!!<!/ e!MtK&re}

et avec

_différentes

structures

_familiales.

On montre que les corrélations

_{d’échantillonnage}

en-tre les estimées sont élevées et

qu’un

volume de données

_important

est

requis

pour obtenir

des estimées raisonnablement

précises,

même avec des

dispositifs

établis

spécifiquement

pour estimer des

effets

maternels. L’étude du biais et de l’erreur

quadratique

moyenne

milieu,

souvent

_ignorée

dans

_l’analyse

des données de croissance des bovins à

_viande,

doit être très

grande

pour que son

_effet

soit

_{statistiquement significatif. L’efficacité}

du

_transfert

d’embryon

pour réduire les corrélations

d’échantillonnage

entre les variances-covariances

génétiques

directes et maternelles est illustrée.

effet maternel

_/

_composante de variance

_/

covariance _{d’échantillonnage}

INTRODUCTION

The

_importance

of maternal

_effects,

both

_genetic

and

_{environmental,}

for the

_early

growth

and

_development

of mammals has

_long

been

_recognised.

For

_post-natal

growth,

these

_represent

_mainly

the dam’s milk

_production

and

_{mothering ability,}

though

effects of the uterine environment and extra-chromosomal inheritance _may contribute. Detailed biometrical models have been

_suggested.

Willham

₍₁₉₆₃₎

distinguished

between the animal’s and its

_mother’s,

ie direct and

_maternal,

additive

genetic,

dominance and environmental effects

_affecting

the individual’s

_phenotype.

Allowing

for direct-maternal covariances between each of the 3

_effects,

_{this gave} a

total of 9 causal

_(co)variance

_components

_contributing

to the resemblance between

relatives. Willham

₍₁₉₇₂₎

described an extension to include

_{grand-maternal}

effects

and recombination loss.

Estimation of maternal effects and the

_{pertaining genetic}

_parameters

is

inher-ently problematic.

Unless

_embryo

transfer or

crossfostering

has taken

_place,

direct and maternal effects are

_generally

confounded.

Moreover,

the

_expression

of

mater-nal effects is

_sex-limited,

occurs late in life of the female and

_{lags by}

one

_generation

(Willham, 1980).

Methods to estimate

_{(co)variances}

due to maternal effects have

been reviewed

_{by Foulley}

and Lefort

_(1978).

_Early

work relied on

_estimating

co-variances between relatives

_separately,

_equating

these to their

_expectations

and

solving

the

_resulting

_system

of linear

_equations.

_However,

this

_ignored

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 combined information in a

_non-optimal

_way. In

_addition,

_sampling

variances of estimates could not be derived

_(Foulley

and

_Lefort,

_1978).

Thompson

(1976)

presented

a maximum likelihood

(ML)

_procedure

which over-comes these

problems

and showed how it could be

_applied

to

_designs

found in the literature. He considered the ML method most useful when data were balanced due

to

_{computational requirements}

in the unbalanced case. Over the last

_decade,

ML

es-timation,

in

_particular

Restricted Maximum Likelihood

_(REML)

as first described

by

Patterson and

_Thompson

_(1971),

has found

_increasing

use in the estimation of

_(co)variance

_components

and

_genetic

_parameters.

_Especially

for animal

breed-ing applications

this almost

_invariably

involves unbalanced data.

_{Recently, analyses}

under the so-called animal

_model,

_fitting

a random effect for the additive

_genetic

value of each

_animal,

have become a standard

procedure.

To a

_large

_extent,

this

was facilitated

_by

the

_availability

of a derivative-free REML

_algorithm

_(Graser

et

Maternal

_effects,

both

_genetic

and

_{environmental,}

can be accommodated in

animal model

_analyses

_by

_fitting

_appropriate

random effects for each animal or

each dam with progeny in the data.

Conceptually,

this

_simplifies

the estimation of

genetic

parameters

for maternal effects. Rather than

_having

to determine the

types

of covariances between relatives

_arising

from the data and their

_{expectations,}

to estimate each of them and to

_equate

them to their

_{expectations,}

we can estimate maternal

_(co)variance

components

in the same _way as additive

_genetic

_{(co)variances}

with the animal

_{model, namely}

as variances due to random effects in the model

of

_analysis

_(or

covariances between

_them).

The derivative-free REML

_algorithm

extends

_readily

to this

type

of

_analyses

_{(Meyer, 1989).}

As

_{emphasised by Foulley}

and Lefort

_(1978),

estimates of

_genetic

_parameters

are

likely

to be

_imprecise.

_Thompson

₍₁₉₇₆₎

_suggested

_{that in the presence} of maternal

effects,

sampling

variances of estimates of the direct

_heritability

would be increased 3-5-fold over those which would be obtained if

_only

direct additive

_genetic

effects

existed.

_Special

_experimental

_designs

to estimate

_{(co)variances}

due to maternal

effects have been

described,

for

_instance,

_by

Eisen

₍₁₉₆₇₎

and Bondari et al

_(1978).

Thompson

(1976)

applied

his ML

_procedure

to these

_designs

and showed that for Bondari et al’s

₍₁₉₇₈₎

_data,

estimates of maternal

_components

had not

_only

_large

standard errors but also

_{high sampling}

correlations.

In the estimation of maternal effects for data from livestock

improvement

schemes,

non-additive

_genetic

effects and a direct-maternal environmental covari-ance have

_largely

been

_ignored.

In

_part,

this has been due to the fact that 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. Even for Bondari et al’s

₍₁₉₇₈₎

_experiment,

_providing

11

_types

of

rela-tionships

between

_{animals, Thompson}

₍₁₉₇₆₎

_emphasised

that

_only

7

_parameters,

6

(co)variances

and a linear function of the direct and maternal dominance variance

and the maternal environmental

_variance,

could be estimated. In field

data,

the

contrasts between relatives available are

_likely

to be

_fewer,

thus

_limiting

the scope

to

_separate

the various maternal

_components.

In the

_analysis

of

_pre-weaning

_growth

traits in beef

cattle,

components

estimated

have

_generally

been restricted to the direct additive

_genetic

variance

_{(o, A 2 ),}

the maternal additive

_genetic

variance

_{(0-2 m ),}

the direct-maternal additive

_genetic

covariance

_(0-

_AM

_),

the maternal environmental variance

_(o-b)

and the residual error

variance

_(a

₅

₎

or a subset

_thereof;

see

_Meyer

₍₁₉₉₂₎

for a

recent

_summary.

_Using

data from an

_experimental

herd which

supplied

various &dquo;unusual&dquo;

relationships,

Cantet et al

₍₁₉₈₈₎

_attempted

to estimate all

_components.

There has been concern

about a

_negative

direct-maternal environmental covariance

(0-

EC

)

in this case

(Koch,

1972)

which,

if

_ignored,

is

_likely

to bias estimates of the other

components

and

_{corresponding}

genetic

parameters,

in

_particular

the direct-maternal

_genetic

correlation

_(rA,!r).

_Summarising

literature results in- and

_excluding

information from the

_{dam-offspring}

_covariance,

the

only

observational

component

affected

_by

LTEC

, Baker

(1980)

reported

mean values of _rAM of -0.42 and 0.0 for birth

weight,

- 0.45 and -0.05 for

_{daily gain}

from birth to

_weaning

and -0.72 and -0.07 for

weaning weight, respectively.

While the modern methods of

_{analysis together}

with the

_availability

of

_high

parameters

due to maternal

_effects,

_{they might}

make it all too _easy to

_ignore

the inherent

_problems

of this kind of

_analyses

and to ensure that all

_parameters

fitted can be estimated

_accurately.

Unexpected

or inconsistent estimates have been attributed to

_{high sampling}

correlations between

parameters

or bias due

to some

_component

not taken into account _{without any}

_{quantification}

of their

magnitude

(eg

Meyer,

1992).

The

_objective

of this paper was to examine REML estimates of

_genetic

_parameters

due to maternal

_{effects, investigating}

both

_sampling

(co)variances

and

potential

bias due to

_fitting

_{the wrong} model of

_analysis.

MATERIAL AND METHODS

Theory

Consider a mixed liner

_model,

where _y,

_b,

u and e denote the vector of

_{observations,}

fixed

_effects,

random effects and residual errors,

respectively,

and X and Z are the incidence matrices

_pertaining

to b and u. Let V denote the variance matrix of _y. The REML

_log

likelihood

(.C)

is then

For the

_majority

of REML

_algorithms

_employed

in the

_analysis

of animal

breeding

data,

[2]

and its derivatives have been

_re-expressed

in terms

_arising

in the

mixed model

equations pertaining

to

_!1!.

An

_alternative,

based on the

_principle

of

constructing

independent

sums of squares

(SS)

and

crossproducts

(CP)

of the data

as for

_analyses

of

_{(co)variances,}

has been described

_{by Thompson}

_(1976,

_1977).

As

a

_{simple example,}

he considered data with a balanced hierarchical full-sib structure

and records available on both _parents and

offspring, showing

that the SS within

dams,

between dams within sires and between

_sires,

as utilised in an

_analysis

of

variance

_(for

data on

_offspring

_only),

could be extended to include information on

parents.

This was

_{accomplished by augmenting}

the later 2

_by

rows and columns

for dams and

sires,

yielding

a 2 x 2 and a 3 x 3

matrix,

respectively,

with the additional elements

_representing

_offspring

_parent

_CP,

and

_SS/CP

_among

_parents;

see

Thompson

(1977)

for a detailed

_description.

More

_generally,

let the data be

_{represented by}

_p

_independent

matrices of

_SS/CP

S!.,

each with associated

_degrees

of freedom

_{d! (k}

=

1, ... , P).

The

corresponding

matrices of mean squares and

products

are then

_M!;

=

S!/d!

with

expected

values

In the estimation of

_(co)variance

components,

V and the matrices

_V!

are

_usually

linear functions of the

_parameters

to be

_estimated,

A =

f Oi with

_i =

1, ... , t,

ie

REML estimates of 0 can then be determined as iterative solutions to

(Thompson, 1976)

with B =

lb

ij

and

_q =

fq

i

for

_{i, j = 1, ... , t,}

and

This is an

_{algorithm utilising}

second derivatives

_of

_log

G. At _convergence,

an estimate of the

_{large sample}

covariance matrix of

6

is

_{given by}

-2B-

1

.

As

emphasised

by Thompson

(1976),

B is

_singular

if a linear combination of the

matrices

_F!i

is zero for all

_k,

which

_implies

that not all

_parameters

can be estimated.

This

_methodology

can be

_{employed readily}

to examine the

_properties

for REML estimates for various models. Consider data

_consisting

of records for

_{f independent}

families. Hence

_{V!;, M!;}

and the

F!i

can be evaluated for one

_family

at a time. If

the data are

&dquo;balanced&dquo;,

ie all families are of size n and have the same

_structure,

these

_{calculations, involving}

matrices of size n x _n, are

_{required only}

_once, ie _p = 1.

Fitting

an overall mean as the

only

fixed

_effect,

the associated

_degrees

of freedom

of

S¡

are then

f &mdash;

1.

’

Let a

_{record y}

_j

for

_{animal j}

_with

dam

_j’

be determined

_by

the animal’s

_(direct)

additive

_genetic

value _a!, its dam’s maternal

_genetic

_{effect mj,,} its dam’s maternal environmental effect _Cj’ and a residual error _e!, ie:

with all

_remaining

covariances

_equal

to zero.

_Letting,

in

_turn,

maternal _{effects m!’}

and _Cj’ be

present

or absent and covariances O’Ayl and UEC be zero or

_not,

yields

a

total of 9 models of

_analysis

as summarised in table I.

Clearly,

M

k

in

_[4]

above represents the contribution of the data to

_{log G,}

ie relates to the &dquo;true&dquo; model

_describing

the data.

_Conversely,

_V!

is determined

_by

the &dquo;assumed&dquo; model of

_analysis,

ie the effect of

_fitting

an

_{inappropriate}

model can

be examined

_deriving

_V!.

under the _wrong model.

_Furthermore,

the information

contributed

_by

individual records can be assessed

_{by &dquo;omitting&dquo;}

these records from

the

_analysis

which

operationally

is

_simply

achieved

_{by setting}

the

_{corresponding}

rows and columns in

_V!

and

M

k

to zero.

Analyses

In

_total,

6

_family

structures were considered. The

_first,

denoted

_by

FS1,

was a

simple

hierarchical full-sib

_design

with records for both

_parents

and

_offspring

for

f

sires mated to d dams each with m

_offspring

_per

_dam,

ie

_f

families of size

n = 1 +

d(l

₊

m).

As shown in table

_II,

this

yielded only

5

_types

of covariances between

relatives,

ie not all 9 models of

_analysis

could be fitted.

_Linking

_pairs

of

such families

_{by assuming}

the sire of

_family

1 to be a full sib

_(FS

₂

_F)

or

_paternal

half sib

_(FS

₂

_H)

to one of the dams mated to sire 2 then added up to 3 further

relationships

(see

table

_II).

With s = 2 sires

per

family,

this gave a

_family

size of

n = 2(1 + d(i

+

m)).

The fourth

_design

examined was

_design

I of Bondari et al

_{(1978) .}

As

_depicted

in

_figure

_1,

this was created

_by

_mating

2 unrelated

_grand-dams

to the same

2 mated to a

random,

unrelated animal. From each of these

_matings,

2

offspring

were recorded. For Bondari’s

_design

I

_(B1),

records on

_{grand-parents}

and random mates were assumed

unknown,

yielding

a

family

size of n = 8 and 10

types

of

relationships

between

_{animals. Assuming,}

for this

_study,

the former to be known then increased the

_family

size for

_design

B1P to n = 13 and added

grand-parent

offspring

covariances to the observational

components

available.

The last

_design

chosen was Eisen’s

_design

1

_(E1).

For

_this,

each

_family

consisted

of s sires which were full-sibs and each sire was mated to

_d,

dams from an unrelated full-sib

_family

and to

_d

₂

dams from an unrelated half-sib

family.

Each dam had m

offspring

which

_yielded

a

_family

size of n =

s(l

₊

d

l

₊

d

2

)(1

₊

n)).

As shown in

table

_II,

this

_produced

a total of 13 different

_types

of

relationships

between animals.

Figure

2 illustrates the

_mating

structure for this

_design.

For each

_design

and set of

_genetic

_parameters

_considered,

the matrix of mean

squares and

products,

M!

was constructed

assuming

the

population

(co)variance

components

to be known and &dquo;estimates&dquo; under various models of

_analysis

were

Results obtained in this way are

_equivalent

to those obtained as means over

many

replicates. Large sample

values of

sampling

errors and

_sampling

correlations

between

parameter

estimates were then obtained from the inverse of the information

matrix,

F =

&mdash;2B!.

This is

commonly

referred to as the formation matrix

(Edwards, 1966).

Simulation was carried out

_by

_sampling

matrices

_M*

_from

an

appropriate

Wishart distribution with covariance matrix

M!;

and

_{f &mdash;}

1

_degrees

of freedom and

obtaining

estimates of

_(co)variance

_components

and their

_sampling

variances and

correlations

_using

the MSC

_algorithm.

_However,

this did not

_guarantee

estimates to

be within parameter space.

Hence,

if estimates out of bounds

_occurred,

estimation

was

_{repeated using}

a derivative-free

_(DF)

_{algorithm, calculating log}

G as

_given

in

_[4]

and

_locating

its maximum

_using

the

_{Simplex procedure}

due to Nelder and

Mead

_(1965).

This allowed estimates to be restrained to the

parameter

space

simply

by

assigning

a _very

_{large, negative}

value to

_log

G for

_{non-permissible}

vectors of

parameters

(Meyer, 1989).

Large sample

95%

confidence intervals were calculated as estimate !1.96x the

lower bound

_sampling

error obtained from the information matrix.

_{Corresponding}

likelihood based confidence limits

_(Cox

and

_Hinkley,

₁₉₇₄₎

were

determined,

as

described

_{by Meyer}

and Hill

_(1992),

as the

_points

on the

_profile

likelihood curve

for each

_parameter

for which the

_{log profile}

likelihood differed from the maximum

by &mdash;1.92,

ie the

_points

for which the likelihood ratio test criterion would be

_equal

to the

_X

_Z

value

_pertaining

to one

degree

of freedom and an error

probability

of

5%

RESULTS AND DISCUSSION

Sampling

covariances

Sampling

errors

(SE)

of

(co)variance

_component

estimates based on 2 000 records

from

_analyses

under Model

_6,

ie when both

_genetic

and environmental maternal effects are

_present

and there is a direct-maternal

_{genetic covariance,}

are summarised

in table III for data sets of 3

_designs,

and 2 sets of

_population

_{(co)variances.}

For

comparison,

values which would be obtained for

_{equal heritability}

and

_phenotypic

variance in the absence of maternal effects

_{(Model 1)}

are

_given.

The most

_striking

feature of table III is the

_magnitude

of

_sampling

errors even for a

_quite

_large

data set and for

_designs

like B1 and El which have been

especially

formulated for the estimation of maternal effects

_components.

In all

cases,

SE((j!)

under Model 6 is about twice that under Model 1.

FS2F

and El

yield considerably

more accurate estimates than B1 under

_{Model 1,}

with

_virtually

ie a

_high

direct

_heritability

and low

_negative

direct-maternal

_correlation,

and are

comparatively

even less variable for

_parameter

set

_II,

ie a low direct and medium

maternal

_heritability

and a moderate to

_high

_{positive genetic}

correlation.

Table IV

_gives

means and

_empirical

deviations of estimates of

_(co)variance

components

and their

_sampling

errors under Model 6 for 1000

replicates

for a data

set of size 2 000 for

_parameter

set 1. While MSC estimates _agree

_closely

with the

population values,

corresponding

mean DF estimates _are,

_by

_definition,

biased due to the restriction on the

_parameter

_space

_imposed.

This is

particularly

noticeable

for

_designs

FS2F and B1 with 355 and 258

_replicates

for which estimates needed

to be constrained.

_{Overall, however, corresponding}

estimates of the

_asymptotic

lower bound errors _appear little

affected:

means over all

replicate

and

_considering

replicates

within the

_parameter

_space

_only

_(MSC

_*

₎

show

_only

small

_differences,

except for

_FS2F,

and _agree with the

_population

values

_given

in table III.

_Moreover,

standard deviations over

_replicates

for these

_(not

_shown)

are small and

virtually

the same for MSC and

_{MSC!‘, ranging}

from 0.22

_(SE(â

₁

₎

for

_B1)

to 1.19

_(SE(âÄ

₁

₎

for

_FS2F).

In

_turn,

_empirical

standard deviations of MSC estimates _agree well with their

_{expected values,}

_being

on _average

_{slightly higher.}

Those of the DF

_estimates,

however,

are in

_parts

_{substantially}

_lower,

_{demonstrating}

_clearly

that

_constraining

estimates alters their

distribution,

ie that

_large

sample

theory

does not hold at the bounds of the

_parameter

space.

Table V

_presents

both

_{large sample}

_(LS)

and

_profile

likelihood

_(PL)

derived

for the

_population

_{(co)variances.}

As noted for other

_{examples by Meyer}

and Hill

(1992),

unless bounds of the

_parameter

space are

_{exceeded, predicted}

_lengths

of the

interval from the 2 methods _agree

_consistently

better than values for the

_position

of the confidence bounds. Lower PL limits for

_a2

_m

and

_a

₂

_c

for

_designs

FS2F and

B1 could not be determined

_(as

the

_{log profile}

likelihood curve to the left of the estimates was so flat that it did not deviate from the maximum

_by

_-1.92),

and

were thus set to _zero, the bound of the parameter space. While differences between

PL and

LS

intervals are small for all

_designs

for

_larger

data sets

_{(not shown),}

considerable deviations occur for the 2 000 record _case,

_particularly

for

_Q

_AM

and

Corresponding

empirical

and

_expected

_sampling

correlations between DF

esti-mates of

_(co)variance

_components

are contrasted in table VI. Mean

_expected

values

over

_replicates

were in all cases

_equal

_(to

the second or third decimal

_place)

to those

derived from the information matrix of the

_population

parameters.

While

empiri-cal values for

_larger

data sets

_{(not shown)}

_again

_{agree well with their theoretical}

counterparts,

those based on 2 000 records

deviate,

again

reflecting

the effect of

constraining

estimates to the

_parameter

space on their

_sampling

distribution.

De-viations are in

places

considerable for

parameter

set

II,

for which estimates from

773,

726 and 553

_replicates

for

_FS2F,

B1 and

_{E1, respectively,}

needed to be

cons-trained to the parameter space.

Overall,

however,

some of the

_sampling

correlations

_{(expected values)}

show

remarkably

little variation between

_designs

or differences between

_parameter

sets

considered.

_6;1

and

_û1

are

_{consistently highly negatively}

correlated,

with values of

! -0.8 to

_{-0.9; while}

_5!

and the maternal effects

_components,

_6’

and

_{iT- 2 ,}

show

comparatively

little

_(though

more

_variable)

_association,

correlations

_ranging

from

0 to m 0.4 and 0 to m

_-0.3,

_{respectively. Similarly,}

correlations between

_û1

and

û

1

are low and

_negative

and between

<7!.

and

3%

are low and

_positive

or

_negative

and close to zero.

Differences between

_designs

and the amount of information available to

_separate

not

_only

direct and maternal but also maternal

_genetic

and environmental

com-ponents

as

_apparent

in table

III, however,

are

clearly

exhibited in the correlations

among

û1, a’ m

and

9!..

While the correlation between

û

1

and

6

AM

is as

high

as -0.9 for

_FS2F,

it is reduced

in

_magnitude

to -0.8 for B1 and -0.7 to -0.6 for El.

_{Correspondingly,}

a

_high

_positive

correlation between

%

AM

and

_QC

for FS2F

_(!

0.9)

and Bl

_{(! 0.7)}

is reduced

_{substantially}

for El

_{(to m}

_0.4).

FAMILY STRUCTURE

The

_relationship

between

_family

structure and

_sampling

_{(co)variances}

for a

_given

number of records is further

_investigated

in table

_VII,

for

_analyses

under Models

1 and 4. The total

_genetic

variance is defined as

a5

=

92

A

₊

1/2

QM

₊

3/2

QAM

(Willham,

1963),

ie is the same as

o- A 2

for Model 1. As in table

_III,

differences

between

_designs

are small for

_{Model 1,}

but increase with the number of

_parameters

estimated. In

particular,

including

the direct-maternal

_genetic

covariance has a

pronounced

effect.

For FS1 with

_only

one

_offspring

_per dam

_{(column 6),}

there are no full-sibs in the

data.

_Though

the

_remaining

4 observational covariances between relatives still allow

all 4

_components

under Model 4 to be

_estimated,

this causes an almost

_complete

sampling

correlation between

_û

₁

and both

_{6’ E}

and

_û!M’

and

_{correspondingly}

_high

sampling

variances.

_Conversely,

with the same number of

_offspring

but

_only

one dam

per sire

(column 3)

there

are

no

paternal

half-sibs.

However,

as the

expectation

of

the

_pertaining

covariance involves

_only

_{a A 2}

estimates of the maternal

_components

are thus much less

_{affected, though}

SE(al A

and the

_sampling

correlation between

a2 A

and

_û

₁

are

_largest

_amongst

those for the FS1

_designs.

As noted above for

Model 1,

Bondari’s

_design

1

_gives

less accurate estimates

than most full-sib

_family

structures

_(unless

d = 1 or m =

1)

for the

simpler

models

involve data on

_only

2

_generations,

B1P includes records on

_{grand-parents,}

ie 3

generations

in total.

_Though

the coefficients of

_{a A 2}

and aAM in the

expectation

of the

_{grand-parent offspring}

covariances are

_{comparatively}

small

(see

table

II),

this

clearly

reduces the

_sampling

errors of all

_components

estimated and the

_magnitude

of

_sampling

correlations between

₆₂

_A

and both

_lTiI

and

_%5

_.

As for

_{FS1, sampling}

errors for E1 are

_markedly

increased when one or several of the covariances between

relatives are

_missing

_(s

= 1 or

_d

_i

= 1 _or

d

2

=

1),

the _more the _more

_parameters

are estimated.

_Sampling

correlations follow a similar

_pattern

as for FS1. Based on

8 000

_{records, design}

12

_provides

the most accurate _{estimates among the 12 data} structures examined. Some discussion on the

_optimal

choice of

_{s, d}

_l

and

d

2

for

Eisen’s

₍₁₉₆₇₎

_designs

is

_given

_by

_Thompson

_(1976).

BIAS AND MEAN

_SQUARE

ERROR

So

_{far, only analyses}

under the &dquo;true&dquo; model

_describing

the data have been

considered. In some

_instances,

_analyses

are carried _out,

_however,

_fitting

the _wrong model. A

_particular

_example

as discussed above is the

analysis

of

growth

traits in

beef cattle where an environmental correlation between a dam and her

_daughter,

though

assumed to

_exist,

is

_generally

_{ignored. Figure}

3

_shows

the effect of such environmental covariance

_(b

_EC

=

lTEG/lT!)

on the estimates of

_(co)variance

components

and the direct-maternal

_genetic

correlation

_(r

_AM

₎

under Model 6 when the true model

_describing

the data is Model

_9,

for

_parameter

set I and 3

_designs.

While

_(7!

and

_%5

are

generally

little

affected,

even for

large

(absolute)

values of

b

EC

,

all the maternal

_components

are

_{substantially}

biased. The

_pattern

of biases

differs between

_designs,

_reflecting

_clearly

the differences in covariances between

relatives available and the information contributed

_by

each of them. For

_design

FS1

_{(not shown),}

estimates of

_{o, A 2 ,Or}

_Am

and

_a5

were unbiased while

ô

’

1-

and

ô’!

were biased

_{by -2a}

_E

_c

and

_+2o-

_EC

_,

_respectively

(unless

estimates exceeded the

bounds of the

_parameter

space and were

constrained).

Figure

4 shows the

_{corresponding}

differences in

_{log £}

from

analyses

under models 6 and 9. For the

_parameter

set

_examined,

the

_magnitude

of

_b

_E

_c

needs to exceed 0.3 for

_design

El before a likelihood ratio test would be

_expected

to

_identify

a

significantly

better fit of Model 9 than of Model 6

_(at

an error

_probability

of

5%;

the dashed line in

_figure

2 marks the

_{significance level).}

While estimates from EFS2H

and EFS2F

_(not

_shown)

differ

_little,

the

_higher

coefficients in the observational

covariances due to the across

_family

_{relationships}

for FS2F

_clearly

increase the

scope to

identifiy

a non-zero QEC.

The effect of an over- or

underparameterized

model of

analysis

on estimates of

(co)variances,

their lower bound

_sampling

errors and the

_resulting

mean _square error

_(MSE),

defined as bias

_{squared plus prediction}

error

_variance,

is further

illustrated in table VIII.

_Clearly,

_estimating

a

(co)variance

when it is not

_present

increases the

_sampling

errors of all

components

unnecessarily. Similarly,

when the

bias introduced

_{by ignoring}

a

_component

is

_small,

MSEs under the wrong model

may be

considerably

smaller than under the correct model. As the deviations in

_log

G from the value under Model 9

show,

none of the

_analyses

would be

_expected

to

EMBRYO TRANSFER

With a dam

affecting

the

phenotype

of her

offspring

both

_through

half her

direct additive

_genetic

value and her maternal

_genotype

as well as her maternal

environmental

_{effect, high sampling}

_{correlations among} the

_genetic

and maternal

(co)variance

components

are

_invariable,

even with the best

_{experimental design.}

Fortunately,

modern

_reproductive

_technology

allows some of these correlations to

be reduced. As a

simple illustration,

consider the hierarchical full-sib

_design

(FS1)

with one _{sire per}

_family.

Assume now that the sire has been mated to

_only

one

out of the d dams with md full-sib

_{offspring resulting}

from this

_{mating. Further,}

assume that each dam raises m of these

_offspring

_{(design FS1ET).}

This

gives

rise

to 3 different

_{dam-offspring}

covariances,

namely:

- the &dquo;usual&dquo; covariance between

a dam and her

_offspring

raised

_{by her,}

with

expectation

_{afl /2}

+

₅

_UAM/

₄

+

_{or2 /2}

+ aEc I

- the covariance between

a dam and her

_offspring

raised

_{by another, recipient}

dam,

with

_expectation

_{a fl /2 + a}

_AA

_{f /4,}

ie the same as the

sire-offspring

_covariance;

and

- the covariance between

a

_recipient

dam and the

offspring

(of

another

dam)

which she

raised,

with

_expectation

aAM +

_(jÄ

_I

_/2

+ o-EC.

Similarly,

we now need to

_distinguish

between 4

_types

of covariances between full-sibs:

- the &dquo;usual&dquo; covariance between full-sibs raised

by

their

_genetic

_dam,

with

expectation

(j!/2

+ (jAM

+

U

2 m +

0

,2

c

- the covariance between full-sibs raised

by

the same

_recipient

dam

_(not

their

genetic

dam),

with

expectation

(j!/2

+

_U2

_m

_{+o, C}

₂

’ - the covariance between full-sibs raised

by

different

_dams,

with one of them

being

their

_genetic

_dam,

with

_expectation

_QA/

₂

+

_O

_-

_A

_{M /2;}

and

- the covariance between full-sibs raised

by

different

_{recipient dams,}

none of

which is their

_genetic

dam,

with

expectation

!A/2.

Table IX _compares the

_{expected sampling}

errors for FS1 and FS1ET for 3

family

structures and Table X contains the

_{corresponding sampling}

correlations. Results from

_analyses

under Models

3, 4,

5

_{(not shown)}

and 6 were contrasted. For Model

3,

with low correlations between

_&2

_m

and the other

_components,

FS1ET

_{yields slightly}

less accurate estimates than FS1.

_However,

as soon as a direct-maternal

_genetic

covariance is

_fitted,

FS1ET

_{gives considerably}

smaller

_sampling

errors than FS1 as

it reduces the

_{high sampling}

correlations between

_{&A 2}

and

_QM

_(Model

_4,

5 and

_6),

(j2

and

₆₂

_(Model

4 and

_5),

_a2

and

_%5

_{(Model 5),}

₆₂

_A

and

_%

_AM

_(Model

4 and

_6),

Q

N1

and

â

AM

_(Model

4 and

slightly

for Model

_6),

or

&2 A

and

_ûb

_(Model

5 and

_6).

Clearly,

however,

FS1ET does not allow

_genetic

and environmental maternal effects

to be

_separated

_any better than

FS1,

and

_sampling

correlations between

_{& m 2}

and

a

2

c

(Model

5 and

₆₎

are still

_large

and

negative.

Other

_{designs involving genetically}

more diverse &dquo;litter mates&dquo; and related

parents

or

_recipients

will

_provide

more _{types of} covariances between relatives

and thus allow even better

_separation

of

_genetic

and

environmental,

and direct

and maternal effects. While the

_expectation

of all observational

components

in

table II which involve

_{o, m 2}

also include

_{U2 A}

and 0-!!, the covariance between 2

them)

which are full-sibs or maternal

_half-sibs,

is

_solely

due to maternal

_genetic

effects

_(expectation

_{(TiE /4).}

CONCLUSIONS

It has been shown that estimates of

_(co)variance

_components

are

_subject

to

large

sampling

variances and

_{high sampling}

_{correlations,}

even for a &dquo;reduced&dquo;

model

_ignoring

dominance effects and

_family

structures

_providing

numerous

_types

of covariances between relatives which have been

_{specifically designed}

for the estimation of maternal effects. For small data sets and models of

_analysis

_fitting

both

_genetic

and maternal environmental effects or a direct-maternal

_covariance,

this

_frequently

induces the need to constrain estimates to the parameter space.

Consequently, large

sample

theory

predictions

of

_sampling

errors and correlations

estimates do not _{agree with}

_empirical

results. Further research is

_required

to evaluate the

_implications

of such

_{large sampling}

_{(co)variances}

on the _accuracy of selection indexes

_including

both direct and maternal

_effects,

ie the

_expected

loss in selection response because

inaccurately

estimated

_parameters

have been used

The

_efficiency

of search

_procedures

used in derivative-free REML

_algorithms

is

highly dependent

on the correlation structure of the

parameters

to be

estimated,

being

most effective if these are uncorrelated. The fact that

expected

sampling

correlations between some

_components

for a

_given

model of

_analysis

varied little

between

_designs

_(see

table

_VI)

_suggested

that a

_{reparameterisation}

to linear

functions of the

_(co)variance

_components

_might

_improve

_{the convergence} rate

of such

_algorithms.

_Inspection

of

_eigenvalues

and

_eigenvectors

of the formation

matrices, however,

failed to

_identify

_any

_{general guidelines.}

Examination of

bias,

sampling

variances and

_resulting

mean _square errors when

fitting

the wrong model of

analysis

showed

_that,

in some

_instances,

_ignoring

some

_component(s)

can lead to

_considerably

smaller MSE without

_biasing

the

(co)variances

estimated

_{substantially}

or

_reducing

the likelihood

_{significantly}

over

that under the true model. In

_{particular, investigating}

the effect of

_ignoring

an

appropriate

for a

_growth

trait in beef

cattle, suggested

that for a data set of size

8 000,

the 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

(at

5%

error

probability).

Results

_presented

here reinforce earlier

_warnings

about the

_inaccuracy

of

es-timates of maternal effects and the

_pertaining

variance

_components

_(Thompson,

1976;

Foulley

and

_Lefort,

_1978).

_Clearly,

use of an estimation

_procedure

with &dquo;built-in&dquo;

_optimality

characteristics like REML will not alleviate the need for

_large

data

sets

_supplying

numerous

_types

of covariances between relatives when

_attempting

to

estimate these

_components.

Use of modern

reproductive

techniques

such as

_embryo

transfer _may

_provide

data where direct and maternal effects are less confounded.

includ-ing

several

_generations

would

_provide

further contrast which

_{might help}

to reduce

the

_biologically

induced

_{high sampling}

correlations.

_Implications

for the scope of

fitting

more detailed

_{models, accounting,}

for

_instance,

for dominance

effects,

recom-bination loss or variance due to new

_mutation,

and of

_estimating

the

_appropriate

(co)variance

components

are somewhat

_{discouraging.}

ACKNOWLEDGMENTS

Financial support for this

study

was

_provided

under the MRC

_(Australia)

_grant UNE15 and

_by

the

_Agricultural

and Food Research Council

_(UK).

I am

_grateful

to WG Hill for

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_K,

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