Variance component analysis of skin and weight data for sheep subjected to rapid inbreeding

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Variance component analysis of skin and weight data for

sheep subjected to rapid inbreeding

Frank H. Shaw, J.A. Woolliams

To cite this version:

Original

article

Variance

_component

_analysis

of skin

and

_weight

data

for

_{sheep subjected}

to

_rapid

_inbreeding

Frank

H. Shaw

J.A.

Woolliams

a

Department

of

Ecology,

Evolution and

Behavior, University

of

Minnesota,

1987

Upper

Buford

_Circle,

St

Paul,

MN _55108, USA b

Roslin Institute

_(Edinburgh),

Roslin,

Midlothian EH25

9PS,

UK

(Received

5 November

1997; accepted

27 November

₁₉₉₈₎

Abstract - A variance _component

_analysis

was carried out on data from a 20-year experiment in the

_{rapid inbreeding}

of

_purebred

and crossbred lines of three hill breeds of

_sheep.

Parent

offspring

matings were made over several

_generations

to

_{produce inbreeding}

coefficients in lambs of up to 0.59. The traits chosen for

analysis

were the live

_weights

at 24 and 78 weeks of age and the ratio of the densities of

_secondary

and

_primary

skin follicles. A

_complete

model of intralocus allelic effects was carried out with both additive

_genetic

variance and dominance variance. The latter was

_partitioned

into _components

_arising

from loci which were

homozygous by

descent and those that were not.

_{Inbreeding depression}

was fitted as a covariate. This model has not been

_{attempted previously}

in livestock

_populations.

Crossbred animals were found to exhibit more dominance variance than

purebred

animals.

Though partitioning

of the dominance variance was

possible

in some of the

data sets

_considered,

estimation of the novel

quadratic

components was difficult and

provided

little evidence of

homozygous

dominance variance as

_{distinguished}

from the familiar random dominance variance

_(that

_arising

in

_randomly

mated

_{populations).}

A

_pooled

dominance model is

_proposed

in which inbred dominance effects have the

same variance as random dominance effects. For live

_weight

the results

_suggested

that the

_genetic

architecture involved many loci with deleterious recessive

alleles,

but for the ratio of follicle

_density

there was no clear

_explanation

for the results observed.

©

Inra/Elsevier,

Paris

inbreeding depression

/

dominance variance _/ restricted maximum likelihood _/ variance components

/

sheep

*

Résumé - Analyse des composantes de variance pour des données de poids et de peau concernant des moutons soumis à une _{consanguinité rapide.} Une

ana-lyse

des composantes de variance a été effectuée sur des données _provenant d’une

expérimentation

de 20 ans sur la

_{consanguinité rapide}

de

_lignées

_{pures et} croisées

issues de trois races ovines de _montagne. Des

_{accouplements}

entre _parents et descen-dants ont été effectués sur

_{plusieurs générations}

en vue de

produire

des coefficients

de

_{consanguinité}

élevés

_{(jusqu’à 0,59)}

chez les agneaux. Les caractères choisis pour

l’analyse

ont été les

_poids

vifs à 24 et 78 semaines

_d’âge

et le _rapport entre les densités de follicules cutanés secondaires et

_primaires.

Un modèle

complet

des effets

alléliques

intralocus a été établi avec à la fois une variance

_génétique

additive et une variance de dominance. Cette dernière a été

_{partitionnée}

en _{composantes provenant} de loci

homozygotes

par descendance mendélienne ou non. La

_dépression

de

_{consanguinité}

a

été considérée en covariable. Ce modèle n’a _pas été tenté

_{précédemment}

sur les

popu-lations d’animaux

_domestiques.

Les animaux croisés ont manifesté

_plus

de variance de dominance _que les animaux _purs. Bien que la

partition

de la variance de dominance ait été

_possible

dans

_quelques-uns

des fichiers

_{considérés,}

l’estimation des nouvelles

composantes

quadratiques

a été difficile et n’a _pas fourni de preuve

flagrante

que la variance de dominance chez les

_homozygotes

doive être

distinguée

de la variance de dominance

classique.

Un modèle de dominance

_regroupée

est

proposé

dans

lequel

les effets de dominance chez les

_consanguins

ont la même variance _que sur l’ensemble de la

_population.

En ce

_qui

concerne le

_{poids vif,}

les résultats

_suggèrent

_que l’architecture

génétique implique

de nombreux loci avec des allèles récessifs délétères mais _que cela

ne semblait pas être le cas _pour le rapport des densités de follicules.

_©

_{Inra/Elsevier,}

Paris

dépression de _{consanguinité}

_/

variance de dominance

_/

maximum de vraisem-blance restreint

_/

composantes de variance

_/

mouton

1. INTRODUCTION

The

_genetic

_analysis

of

_{populations undergoing rapid}

_inbreeding

is of interest

because the

_opportunity

for

_protective

mutations or

_haplotypes

to accumulate and obscure our view of the

_genetic

mechanisms involved is minimized. The

principal

phenomena predicted

from

_inbreeding

are the reduction of

_genetic

variation within families and the

_{disappearance}

of

heterozygosity.

When

asso-ciated with dominant _{gene action} this results in

_{inbreeding depression.}

Inbreed-ing depression

and its

_{seeming inverse,}

the heterosis obtained

_through

_crossing

of

_lines,

have received much attention over the whole of this

century

[10, 31].

Nevertheless,

the

_{interpretation}

of these

_phenomena

in terms of

_genetic

vari-ances and covariances has remained a

_{thorny problem.}

Harris

_[9]

and Cocker-ham

_[3]

_developed

complete

mathematical models for the

_genetic

variance of non-random

_{mating populations.}

These models were used to

_predict

_gene

fre-quency

changes

in

populations undergoing

selection

[4],

and,

while some

at-tempts

were made to

_apply

them to

_{agricultural populations}

_[5],

the models have for the most

_part

remained

_{computationally}

too intensive or the

_testing

of

them

empirically

too

_demanding

to be of

_practical

use. In this _paper we

_report

the results of a

_complete

variance

_component

_analysis

of an

_experiment

car-ried out between 1958 and 1974 in Scotland.

_Thorough

_analysis

of

_inbreeding

depression

and heterosis was

_possible

_previously,

and

these

have

been

_reported

for fleece and skin data

_{[29, 30],}

_weight

_[25],

and measures of

body

_size,

re-production,

fertility

and

_{profitability}

_[26-28].

_However,

a variance

_component

2. MATERIALS AND METHODS

2.1.

_Design

and measurements

Details of the

breeding designs

and the methods

_employed

in this

_experiment

are

_given

_by

Wiener

[24]

and Woolliams and Wiener

[30].

In

brief,

the

breeding

scheme was as follows: six rams and

_{approximately}

72 ewes of each of three

hill breeds

_(Scottish

Blackface,

South

_Country

Cheviot and Welsh

_Mountain)

were obtained from a

_variety

of different flocks in 1955. These were used as the

foundation animals in the

_pedigrees.

All nine

_possible

_purebred

and

_reciprocally

crossbred

_matings

were made. These were denoted

_F

_I

for crossbred and

0

1

for

purebred matings.

These

_mating

combinations

_(e.g.

Blackface x

Cheviot)

are

referred to as _groups.

_{Subsequently,}

within each _group,

_F,

or

0

1

females were

mated to unrelated

males, producing

F

2

and

₀

₂

_offspring.

Inbred crosses were

then carried out within the nine _groups between

_offspring

and younger

parent

to

produce

as _many as 27 lines _{per group.} The

_pattern

of

offspring

with younger

parent

matings

was carried on whenever

possible

for 10 years,

resulting

in

coefficients of

_inbreeding

of dams as

_high

as 0.375 and coefficients of

inbreeding

of lambs as

high

as 0.59.

Finally,

the

_separate

lines that remained were crossed

within the

_purebred

and crossbred _groups. A

_subpedigree

consisting

of nine

lines from the

purebred

Blackface _group is

_presented

in

_figure

1.

In this

_study,

we examine three traits: the fleece trait

N,,INp,

the ratio of

the

_secondary

follicle

_density

to the

_primary

follicle

density,

and

_weights

at

24 and 78 weeks. Of the many traits measured

during

this

experiment,

the

Ns/ Np

trait was chosen because of the

nearly

linear

relationship

previously

observed between its mean and

_inbreeding

coefficient. In contrast, the

_weight

traits tended to show less

inbreeding depression

for

_high

levels of

inbreeding

than for moderate levels. The

_weight

traits were chosen because of the

large

number of lambs measured for them

₍₇₃₀

_purebreds

in three _groups and 1480 crossbreds in three

_groups).

The

_N

_S/

_Np

measurements were made on all lambs

for the

_F

_Z/

₀

₂

_generation

onwards until the third inbred

generation

(F

=

0.5)

using

estimation

_techniques

described

_by

Carter and Clarke

_[2].

The

_weight

data were

_analyzed

for female lambs

_only,

but these data were run for the full

length

of the

_experiment

and included lambs with the

_highest

inbreeding

level

(F

=

0.59)

as well as the line cross lambs. 2.2. Statistical model and method

The

mixed linear

_model,

is made _up of fixed

effects, (

3,

and random effects

_including

additive allelic effects _ai, a dominance effect for the interaction between the alleles i and

j,

d

ij

where the

_expectation

is taken over all alleles

_segregating

at a

_single

locus. From these it follows that

_E(aid2!)

= 0. We can therefore write

_E(y)

=

_X

₁

₃

and

where the first term on the

_right-hand

side is

_commonly

denoted

_V

_A

and the last

_V

_d

_.

If _y

_represents

a vector of related but not inbred individuals in a

population,

we can write the covariance between _any two individuals _{in y} as a linear combination of these two variance

_components

where the coefficients

are based on

_{probabilities}

that the individuals share alleles or combinations of

alleles at a

_given

locus

_[8].

If the vector _y contains individuals that are

_inbred,

i.e. individuals with non-zero

_{probabilities}

of

_carrying

two identical alleles descended from a common ancestor at a

_{given locus,}

the situation becomes more

_complicated.

We must

now account for the

_{non-vanishing}

_presence of the term

_d

_ii

in our

_equations

since

_homozygotes

increase at the _expense of

_{heterozygotes.}

_Thus,

for an

individual

_randomly

chosen with an

_inbreeding

coefficient of F with

_respect

to the base

_population,

where F is the

_{inbreeding coefficient,}

and

The variance of _y must now be

_partitioned

into three more

_components

of

variance

_beyond

those

already

mentioned

_{[9, 23].}

These include the

_complete

homozygous

dominance

_variance,

and the

_expectation

of the

_squared

_inbreeding

_{depression effects,}

The covariance between additive effects and their associated

_homozygous

dominance effects is non-zero

(E(a

i

d

ii

) i-

0)

and _upon

_inbreeding

there is a

need to account for this covariance

Again,

the

_{expectations involving}

_homozygotes

are taken over all alleles

seg-regating

at a

single

locus

_using

the distribution of alleles in the base

_generation.

The

terminology

used here is taken from Cockerham and Weir

_[4].

To

_emphasize

the difference between

_homozygous

dominance effects and dominance effects in the context of random

_mating

with no

_inbreeding,

we name variance of the

the sum of the

_squared

_per locus

_inbreeding

_depressions.

If the _per locus

inbreeding

depressions

are all of similar small values and there are _{very many}

loci,

H* can be

_vanishingly

small even when the

_inbreeding

depression

is

_large.

It is also noted that if the trait is controlled

_by

a

_{single locus,}

H* will be the square of the

inbreeding

depression

as calculated

by

_regression

of the

phenotype

on the

_inbreeding

coefficient.

The variances of and covariances between individuals of known

_pedigree

are

_expressed

as linear combinations of these five variance

_components

with

coefficients based on the

appropriate

probabilities

of

identity

of alleles

by

descent. The

_probability

measures involve combinations of four alleles in two

individuals

_[3]

of which there are 16 if maternal and

_paternal

_gametes

are

distinguished

and nine if

_they

are not. In the case of the

present

analyses,

loci

_affecting

the trait are assumed to be

_autosomal,

so the nine

probability

measures are sufficient. Cockerham

_[3]

and Smith and Maki-Tanila

[20]

_gave

elegant

recursive

_algorithms

for

finding

these

_{probabilities}

and in the latter

case,

_{finding directly}

the inverse of the covariance matrix of an

_expanded

list

of allelic additive and dominance effects.

_Here,

we used Cockerham’s

_approach

to write the matrix V or

phenotypic

covariance

_matrix,

where the matrices

A, D,

M

D

, ,

_M

_D2

and

M

H

.

are the

appropriate relationship

matrices. In the case of the

_{weight data,}

a maternal environmental variance

component

and an

_appropriate

incidence matrix were also added.

The restricted

_log

likelihood

function,

where 0

is the

generalized

least squares solution for the fixed

effects,

was

maximized in the

_components

of variance

using

the Fisher

scoring algorithm

[14,

18].

The

_regression

of the

_phenotype

on the

inbreeding

coefficient F is also

included as a covariate

_which,

in the absence of selection

_bias,

will

_predict

the

inbreeding

depression.

In the Fisher

_scoring

_algorithm,

the inversion of V cannot be avoided and this restricts its use to

_relatively

small or

_felicitously

structured data sets such

as those here. Variance

_component

_{analysis, using}

the

approach

of Smith and

Maki-Tanila

_[20]

for the inversion of the mixed model

_equation

coefficient matrix C

_along

with

_recently

_developed

likelihood maximization

_algorithms

[15,

16],

is

_plausible

for

_larger

data

_sets,

_although

the dimension of C

_might

become _very

_large

_[20].

All of the data were

_analyzed

with _year of birth included both as a random

and a fixed effect. The results of these

analyses

were not

_{qualitatively different,}

with

_significant

_year variation but no

_long-term

trend.

_Reported

results in all cases are from

_analyses

in which year of birth was included as a fixed effect.

Separate

variance

_component

_analyses

were run on the six different

purebred

and crossbred combinations. To detect more

general

behavior and to boost

sam-ple

sizes,

the three crossbred combinations were combined into one crossbred

these combined

analyses,

a different covariate was fitted for

_inbreeding

depres-sion on each

purebred

and on each crossbred combination. Different levels for

the other fixed effects were also included so that the

_only

constraint

_present

in

the combined

_analyses

that was absent from the

_separate

analyses

was that all

groups were assumed to have the same

_genetic,

maternal environmental and

residual variances. Likelihood ratio tests were used to evaluate the

_significance

of this constraint. Whenever fixed effect estimates

_(e.g.

_inbreeding

depression

estimates)

from different

_analyses

were

_compared,

the

_analyses

were assumed to be

_independent.

As well as

_pooling

the

_purebreds

and the

_crossbreds,

the

_potential

_power

of the

analysis

was also increased

_by

combining

V!,.

and

_D2

_into

an

_agregate

dominance

_component,

_V

_D

_,

associated with the combined

_relationship

matrix

D +

_M

_D2

_.

Since in this model the dominance variance is not

_partitioned

into

_separate

_homozygous

and random

components,

we call it the

_’pooled

dominance’ model. It

_includes,

_along

with

V

A

and

_V

_D

_,

the covariance between additive and

homozygous

dominance effects

D

l

.

In all

_analyses,

_significance

levels for

_components

were tested

_by

a likelihood

ratio test in the

_following

order:

_V

_e

_,

_V

_A

_,

_V

_m

_(for

_weight),

_V

_d

_,,

_D2

_and

_D

_I

_.

Standard errors increased as more

_components

were added to the model. The standard errors

_reported

are those

_{corresponding}

to when all

_components

are

fitted,

so that

_they

do not reflect the levels of

_significance

attributed

_by

the likelihood ratio test that was used to test for a

particular component’s

_presence.

Likelihood ratios were

_compared

to the

_appropriate

_x2

_{distributions,}

i.e. a 50:50 mixture of

_X

_’(0)

and

_x

₂

₍₁₎

for null

hypotheses

of a

_single

variance

_component

on the

_boundary

of the

_parameter

_space and

x

2

(p)

for _{p variance}

_components

constrained in the interior

_[19].

The decision to

_{analyze separately purebred}

and crossbred data was made because a combined

_analysis

would constrain the variance

_components

from

very different

populations

to be the same. The constraint that this would be the

case in the combined

_purebred

data alone was found to be

_{highly significant}

_(see

Results).

An

_{analysis including}

all animals would be feasible

_{computationally,}

though

difficult.

Recently,

several studies have addressed the

_problem

of

_analyzing

crossbred data

_[11,

12,

22].

The methods which have been

_developed

use the variance

components

associated with the constituent

purebred parental populations

and,

in the case of

_dominance,

variance

_components

associated with the

crossbreds,

to

_predict

_genetic

values

_[11].

Estimation of the 26 covariance

components

associated with a

_general

two breed crossbreed

_pedigree

has not

yet

been

_{attempted; however,}

the

_theory

is

_{fully developed}

and methods such

as those

employed

here would suffice. In this _paper,

however,

we do not include

purebred

and crossbred

_genotypes

in the same data set and thus have no need to calculate

purebred

by

crossbred

_genotypic

covariances. Crossbred groups

sharing

a

_{purebred parental}

_origin

are included in a

_{single analysis,}

but the

covariances between individuals in different _groups would be _very small since

no

_mating

takes

_place

between the _groups in the _many

_generations

after

_they

are established. We therefore assume the _groups to be

_independent

and take the

_F

_2/

₀

₂

_generation

to

_represent

the base

_population

for each _{group. In} so

doing,

we reduce the number of covariance

_components

for two

_purebred

_groups

2.3. Simulation

_study

Although

strict attention was

_paid

that no artificial selection should take

place during

the

_experiment,

it was unavoidable that as the levels of

_inbreeding

increased,

many of the individual lines died out

_{(figure 1).}

This natural selection

clearly

favors lines that exhibit less

_inbreeding

_depression

for fitness traits. A simulation

_study

was carried out to assess the affect on variance

_component

estimation of loss of lines due to natural selection.

Populations

of

_potentially

500 were simulated based on ten

_{seven-generation}

pedigrees

similar to those found in the

_experiment

and shown in

_figure

1. Each

pedigree

consisted of

eight

unrelated founders mated in two _groups of one sire

and three dams. The second

_generation

consisted of six

_pairs

of full-sibs in two half-sib _groups. The half-sib _groups were then crossed to

_produce

six

non-inbred _{progeny in} the third

_generation.

These

_{third-generation}

individuals were

crossed with one of their

_parents

to

_produce

the first inbred

_generation

_(fourth

generation).

This

_crossing

was followed

by

three more

_generations

of

offspring

by

youngest parent

mating.

After the first

_{generation, then,}

each individual was

associated

_with,

and crucial to the continued

_propagation

of one of six different lines.

Each founder was

_assigned

two

_unique

alleles at each of 30 loci

₍₄₈₀

independent

alleles in each of ten

_pedigrees

_per

_replicate

data

_set).

Since the alleles

_assigned

to each founder were

_unique,

_homozygosity

at a locus could

_only

occur when the alleles were identical

_by

descent. Under the full

_{genetic model,}

correlated values for additive

_(a

_i

₎

and

_homozygous

dominance

_(d

_ii

₎

effects were

sampled

for each allele from

where nloc = 30 is the number of

_loci,

and id = -0.5 is the

inbreeding

depression.

For these

_simulations,

_V

_A

= 0.2 and

D2

= 0.5. Each non-identical

combination of alleles within a locus was

_given

a random dominance effect

(d

ij

for alleles i and

_j)

drawn from a normal distribution with mean zero and

variance

_V

_dr

= 0.2. Transmission of alleles _at each locus from one

_generation

to the next was simulated

_by

Mendelian

_segregation

and free recombination

into

_gametes.

_Phenotypes

were calculated as the sum of the

_genetic

values

from a combined

_pair

of

_gametes

_(the

_genotypic

_value)

to which was added an

independent

environmental effect with mean zero and variance

_V

_E

= 0.3.

Beginning

in the second

_generation,

individuals

_(and

_consequently

the lines derived from

_them)

were culled based on a linear function of

_phenotypic

value. Four such selection schemes were simulated. In the first scheme

_(I),

no selection was

_imposed.

In the second scheme

(II),

the lowest trait value

in a

_{given generation}

was culled with

_{probability 0.15,}

the

_highest

trait value with

_probability

_0.125,

and the intermediate trait values with

_probability

based

on a linear combination of these two. Schemes III and IV were similar

_with,

respectively,

0.2 and 0.25

_probability

of

_culling

of the lowest trait value for each

3. RESULTS

3.1. Simulations

The results of the simulation

_study

are

presented

in table I. The

inbreeding

depression

estimate is biased

_upwards

as selection becomes more intense and more lines with lower mean

phenotypic

values are lost. The variance

components

appear to be little affected

_except

in that the standard deviations

on the mean _{estimates grow} with data sets

_reflecting

_higher

levels of selection. These data sets are not

_only

smaller but

_they

lack information on animals at

high

levels of

_inbreeding.

3.2. Fixed effects for

_N

_s/

_Np

In earlier

_analyses,

when all observations were included in a

_single

data

_set,

the fixed effects found to be

_affecting

this trait were dam and lamb

inbreeding

depression

[29]

and _year of birth. The same fixed effects were fitted in this

analysis.

The estimates for

inbreeding

tend in the

_expected

directions

_(decline

in value with additional

_inbreeding

of the dam and the

_lamb).

The estimate for

inbreeding

depression

of the lamb

_{(table If)}

for each of the six _groups was

_rarely

more than a

single

standard deviation from _zero,

probably

due to the small

sample

sizes of the

_single

_group data sets

_(purebreds

had 145 to 189 observations

per group; crossbreds had 300 to 361 per

group)

and to the lack of

high

levels of

inbreeding

either _among the observed lambs or _among their mothers.

_However,

(results

not

_shown)

was less

pronounced.

Pooled over _groups, dam

_inbreeding

of

25

%

resulted in a trait mean decline from 3.88 ± 0.06 to 3.74 ± 0.10. Estimates of

_inbreeding

_depression

from

_purebred

_groups were

larger

in

magnitude

than

for crossbred groups but the effect was not

_significant

_(difference

0.57 ±

_0.42).

While effects of the _year of birth

_(estimates

not

_shown)

were

_significant,

there

was no evidence of a consistent

long-term

trend. In combined _group

_analyses,

separate

levels of fixed effects

_(including

_inbreeding

_depression)

were estimated

for each _group.

3.3. Variance

_components

for

_N

_s/

_Np

Variance

_components

results in table II are from a reduced model

including

only V

A

,

V

e

and

_,.

_d

_V

In most cases it was not

_possible

to estimate variances

due to

_{homozygous dominance,}

i.e.

_D

_l

_,

_D2

_and

H*. These were difficult to

estimate because

_large

_negative

_sampling

correlations between

_D2

_and

both

V

A

and V

e

[5]

resulted in infeasible estimates at

_best,

and

_instability

of the Fisher

_scoring

maximization

_algorithm

at worst.

3.3.1. Additive variance

Additive variance was detected

(P

<

_0.05)

in all of the _groups

_except

for Welsh and Cheviot-Welsh

_(table

_77).

_Heritability

estimates

_ranged

from 0 to

0.51.

There was no evidence from the likelihood tests for differences in the additive

component

within the

_purebred

and crossbred _groups nor between

purebred

(h

2

=

_0.31)

and crossbred

_(h

₂

=

_0.35)

_{combined data} sets

_{(table III).}

3.3.2. Dominance variance

In

_Welsh,

Blackface-Cheviot and Cheviot-Welsh

_breeds,

there was

(table

III),

there was evidence of random dominance in the crossbred _groups but not in the

_purebred

_groups,

_although

the difference in

_magnitude

between them was not

_significant.

_Homozygous

dominance variance

_components

either could not be

_estimated,

or did not differ

_{significantly}

from zero for these data

sets.

When the two forms of dominance variance

_V

_dT

and

_D2

_{were combined} in

the

pooled

dominance

_model,

the

_resulting

dominance

_component

was

_again

significant

for the crossbreds but not for the

_purebreds.

The

_pooled

dominance model resulted in a

higher

likelihood in the

_purebreds

over the model with

D*

=

D

I

=

_0,

but _no

change

in likelihood in the crossbreds. Since these models

are not

_nested,

a test was not

_possible.

3.4. Fixed effects for

_weight

The fixed effects

_{significantly affecting}

the

_weight

measurement data were,

for

_dams,

_{inbreeding coefficient,}

_age and

_parity,

and for

_lambs,

_year of

_birth,

inbreeding

coefficient and

_{birth/rearing}

_type

_{(born single/reared}

_single,

born

as

twin/reared

as

_twin,

_etc.).

The influences of the dam’s attributes and the

birth/rearing

type

were much less in the 78-week

_weight

than in the 24-week

weight.

Inbreeding

depression

for the lamb

_{(table IV)}

was observed within all breed

types

and was much

_greater

for the 78-week

_weight

than for the 24-week

_weight.

As in the

_NS/NP

_{trait, inbreeding depression}

for

_weight

in the crossbred _groups tended to be lower in

_magnitude

than

_expected

if the

_{inbreeding depressions}

of the constituent

_purebred

_groups were

_averaged;

but

_again

this was not

3.5. Variance

_components

for

_weight

3.5.1. Additive variance

A substantial amount of additive variation was evident in all _groups at both

ages

(table

IV).

Heritabilities

ranged

from 0.20 to 0.46 for 24-week

weight

and

from 0.12 to 0.69 for 78-week

_weight.

3.5.2. Dominance and maternal variance

Much of the information in the data used to calculate random dominance

variance comes from full-sib _groups and thus we find substantial

sampling

correlations between estimates for

_V

_dT

and

_{V&dquo;, .}

_Significant

_V

_dr

could not be detected in any of the

_purebred

_groups for 24-week

_{weight though}

it is

_present

(P

<

_0.05)

in two of the three crossbred groups. For 78-week

weight

there was

no evidence of random dominance in any of the

separate

group data sets

(see

Maternal variance in the

_separate

_group

_analyses

_{(table IVJ}

was

_generally

higher

for 24-week

_weight

than it was for 78-week

_weight,

_confirming

the observation of reduced influence of the dam from the fixed effects.

_Among

the

_purebred

_groups,

_significant

maternal variance was found in all but the Cheviot group for 24-week

weight

but could not be detected for 78-week

_weight.

The crossbred _groups, with the

_exception

of the Cheviot-Welsh

_breed,

showed

significant

maternal variance for both 24- and 78-week

_weight.

Sample

size and

_pedigree

structure were never

_adequate

to estimate

com-ponents

of variance

_{involving homozygous}

dominance as

_{distinguished}

from random dominance within each _group. When the

_pooled

dominance model was

fitted,

the combined dominance

_component

was

_present

in the Blackface and Blackface-Cheviot groups for both 24- and 78-week

weight

(results

not

_shown).

Correlations between

_homozygous

dominance and additive effects

_(D

_i

₎

were

negative

for both the Blackface and the Blackface-Cheviot breeds but was

_only

significantly

different from zero

(P

<

_0.05)

in the Blackface breed.

The results of the

_analysis

of the combined

purebreds

and combined

cross-breds are

complicated

since there was evidence of differences in the

_components,

among the combined groups, that were

_{statistically}

_significant

and not

_strictly

related to the mean

weight.

Therefore,

the constraint that the three _groups in

each combined data set share a

_single

set of variance

_components

_may have introduced bias into the combined results and table V must be viewed in this

context. The

_only

_{significant homozygous}

dominance variance in these

com-bined

_analyses

was in the combined

_purebred

data set. The

_magnitude

of this

variance was not

_{significantly}

different between 24 and 78 weeks

although

the

higher

additive variance at 78 weeks

_implies

a smaller

_negative

correlation be-tween additive and

_homozygous

dominance effects at the later _age

_(&mdash;0.91

at 24

_weeks,

-0.80 at 78

_weeks).

The combined crossbred line data sets showed very little evidence of ho-mozygous dominance variance. In the case of the 78-week

_weights,

where the

combining

of the constituent crossbred data sets

_required

the least

_constraint,

the inclusion of

_homozygous

dominance variance

_components

in the model

in-creased the

_log

likelihood

_by

a mere 0.22.

_Although

the estimates for the two

dominance variance

_components

in the combined crossbred data set are _very

nearly

the same, the constraint that these two were the same

(pooled

domi-nance

_model)

results in a

_slightly

_poorer fit than that

given

by

the model in

which the

_homozygous

dominance

_components

are constrained to zero. Since these models are not

_nested,

a test was not

_possible.

This

indicates, however,

that our _{power to} detect

_homozygous

dominance variance

_components

is

in-adequate

even

_though

random dominance can be

_{quite accurately}

estimated.

Homozygous

dominance variance is also not to be found in the 24-week data

which, nevertheless,

show

_{highly significant}

random dominance variance. The

pooled

dominance model

_again

in this case results in a

_negligible

likelihood

difference.

4. DISCUSSION

The structure of the data

_analyzed,

and their

_size,

are

_unique

in livestock and

unusual in

_mammals,

with

_rapid

_inbreeding

conducted up to and

beyond

50

%

of the inbred lines within a breed

_type.

Despite

this structure the estimation

of the variance

_{components pertaining}

to

_homozygous

dominance effects was

found to be difficult. For the ratio of

_secondary

to

_primary

follicle

_numbers,

random dominance variance was estimated to be of a similar

_magnitude

to

additive and to environmental

_variance,

_depression

to

_{complete inbreeding}

was of the order of 1 environmental standard

deviation,

but

_parameters

pertaining

to inbred dominance effects were

negligible.

For live

weight,

random

dominance variance was

_usually

smaller in

_magnitude

than additive variance

and

_depression

to

_{complete inbreeding}

was five or six times the environmental standard

deviation,

with estimates of inbred dominance variance

_components

rarely

obtainable and even more

_{rarely significant.}

At all times

_H

_*

_,

the sum of

the

_{squared homozygous deviations,}

was small or was estimated to be

_negative

and

_consequently

constrained to zero. The

_{interpretation}

of such results is

difficult.

For live

weight,

it is easiest to

_postulate

_many

_loci,

each

_having

alleles of small effect:

_thus,

the total sum of the

homozygous

dominance effects _may be

substantial

_(hence,

a

_{large inbreeding}

depression)

but H* is small. If a biallelic

model is considered at each

_locus,

then a model of rare recessives of

_large

effect

could be considered to fit the

_pattern

observed. The

_inbreeding

depression

per

locus would be

approximately p

r

d

r

,.

(where

r denotes the recessive

allele)

for

each locus and could be of

_{significant magnitude}

when summed over loci if each

term were

O(1/N)

where N is the number of loci. H* would be

_{approximately}

(p

r

d

rr

)

approximately

r

,

4p

)

2

r

d

(

r

which is not

_necessarily

small. For the follicle ratio this

_explanation

of _many loci with rare recessives is less

_satisfactory

since the

inbreeding depression

is weak while

V

dr

is

_large.

This tentative conclusion for live

_weight

appears to _agree with those of Caballero and

Keightley

[1]

for the

_quantitative

trait of bristle number in

Drosophila,

melanogaster.

From an

_analysis

of the distributions of P-element

mutagenesis,

they

concluded that a

_large

_proportion

of the additive variance and

_practically

all of the dominance variance arose from recessive or

_partially

recessive mutants. The relative

_magnitude

of the

_partition

between the additive and dominance variance found in their

_study

was 8 to

_1,

which has some

similarity

to the relative

_magnitude

observed here. Davis et al.

_[6]

observed that a few _genes of

_large

effect _may

_explain

some 75

%

of the additive

_genetic

variation for birth

_weight

in a crossbred

population.

Such _genes are

_predicted

to

be at least

_partially

recessive in the distributions of Caballero and

_Keightley

_[1].

There was no indication in the

analysis by

Davis et al.

_[6)

of whether dominance

was found with these

loci,

_although

the

_experimental

_design

should

_give

some

information on this.

Another

_{complication requiring}

consideration is the

_sensitivity

of dominance

components

to

_{epistasis. Epistasis}

has been observed for loci of

_large

effect on

bristle number

_[13]

and the presence of

epistasis

in

previous analyses

of this

data set were

significant

for live

_weight

[25]

but not for

_N

_S/

_Np

_[30].

Such

a

_sensitivity

to at least some form of

_epistasis

(which

is a _very

_general

term

covering

all interactions _among

_loci)

_might

be

_expected.

The

_possibility

of errors in estimation can be discounted since all the estimates from the simulation

study

were consistent with the true

_parameters

in the absence of selection. Recent treatments in the animal

_breeding

literature of the

_problem

of dom-inance variance with

_inbreeding

have

_proposed

that the

quadratic

components

of variance which involve

_homozygous

dominance variance

_{might safely}

be

ig-nored when

_genetic

evaluation is the

_goal

of the

_analysis

_[7,

_21].

The current

analysis

adds credence to this contention. Not

_only

are the

_homozygous

domi-nance

_components

_very difficult to

_calculate,

but

_they

_appear to be absent or

indistinguishable

from random dominance in the

_largest

and most

thorough

analyses

yet

completed

despite

the _presence of substantial random dominance and

_{inbreeding depression. However,}

the

_development

of finite locus models

[17]

would

prevent

the need for this choice between

incompleteness

and the

complexity

of the

_components.

ACKNOWLEDGEMENTS

J.A.W.

_{gratefully acknowledges funding by}

the

_Ministry

of

_Agriculture,

Fisheries and Food

_(UK).

F.H.S.

_{gratefully acknowledges}

_support from Pioneer Hibred

Inter-national, Inc.,

to

_develop

the _computer programs used in this paper, and a _grant from

the Underwood fund of the

Biotechology

and

_Biological

Sciences Research Council

(UK)

to support a

prolonged sojourn

at the Roslin

Institute, Edinburgh.

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