Genotypic covariance matrices and their inverses for models allowing dominance and inbreeding

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Genotypic covariance matrices and their inverses for

models allowing dominance and inbreeding

Sp Smith, A Mäki-Tanila

To cite this version:

Original

article

Genotypic

covariance matrices

and their

inverses for

models

allowing

dominance

and

_inbreeding

SP Smith

A Mäki-Tanila*

Animal Genetics and _{Breeding Unit, University}

_of

New

_England,

_Armidade, NSW _2351, Australia

(Received

11 June _{1988; accepted} 3 _July

₁₉₈₉₎

Summary - Dominance models are _{parameterized} under conditions of

_inbreeding.

The

properties of an infinitesimal dominance model are reconsidered. It is shown that mixed-model

_methodology

is _justifiable as _{normality assumptions} can be met. Tabular methods for _{calculating genotypic covariances among inbred relatives} are described. These methods

employ 5 parameters required to accommodate _additivity, dominance and

_inbreeding.

Rules for _calculating inverse _genotypic covariance matrices are _presented. These inverse matrices can be used _directly to set up the mixed-model equations. The mixed-model

methodology allowing

for dominance and

_inbreeding

_provides a powerful framework to

better _explain and utilize the observed variation in _quantitative traits.

dominance _/ inbreeding / infinitesimal models _/ inverse _/ mixed _{model / recursion}

Résumé - Matrices de covariances génotypiques et leurs inverses dans les modèles incluant dominance et consanguinité. Les modèles de _{génétique quantitative} incluant la dominance sont considérés dans des conditions de _{consanguinité.}

_Après

une discussion des _propriétés du modèle _{infinitésimal,} on _{montre que} la _{méthodologie} des modèles mixtes

peut être _appliquée à cette _situation, dans la mesure où les _hypothèses de normalité

peuvent être _satisfaites. On décrit des méthodes tabulaires pour calculer les covariances

génotypiques parmi des _{apparentés consanguins,} dont _l’emploi nécessite l’introduction de 5 composantes de variances. On

_présente

les

_règles

du calcul direct de l’inverse de ces matrices de covariances

_{génotypiques,}

connaissant la _{généalogie,} et ces 5 composantes de la variance. Ces matrices inverse peuvent être utilisées directement pour établir les équations

du modèle mixte. La _{méthodologie} du modèle _{mixte, prenant} en _compte les interactions

de dominance et la

_{consanguinité, fournit}

un cadre _pour une meilleure _explication et une meilleure utilisation de la variabilité des caractères

_{quantitatifs.}

dominance _{/ consanguinité /} modèle infinitésimal _/ inverse _/ modèle mixte /

algorithme récursif

*

Permanent address: Department of Animal Breeding, Agricultural Research Centre _(MTTK), 31600 Jokioinen, Finland.

**

INTRODUCTION

The mixed linear model has

_{enjoyed widespread}

_acceptance in animal

_breeding.

Most

_applications

have been restricted to models which

_depict

additive _{gene action.}

However,

there is also concern with non-additive effects within and between breeds and crosses

_(eg,

_Hill,

_1969;

_{Kinghorn, 1987;}

Miki-Tanila and

_Kennedy,

_1986).

Henderson

₍₁₉₈₅₎

_provided

a

statistical framework

for

_modelling

additive and non-additive

_genetic

effects when there is no

_inbreeding.

With

_inbreeding,

the mixed model allows statistical

_{analysis, however,}

considerable

_{developmental}

work

remains.

_Inbreeding

_complicates

covariance structures

_{(Harris, 1964).}

_Moreover,

inbreeding

depression

is a manifestation of interactions like dominance and

_epistasis.

Models which include

_only

additive effects and covariates for

_inbreeding

_(eg,

Hudson and Van

_Vleck,

₁₉₈₄₎

are

rough

_{approximations.}

The proper treatment of

_inbreeding

and dominance involves 6

_genetic

_parameters

(Gillois,

1964; Harris,

1964).

These _parameters define the first and second moments

of

_genotypic

values in the absence of

epistasis.

A

_{genetic analysis}

is

_{possible by}

repetitive

sampling

of lines derived from one

_population

_through

a fixed

_pedigree

(eg,

Chevalet and

_Gillois,

_1977).

_However,

we should like to

_perform

an

_analysis

where the

_pedigrees

are realized with selection

_and/or

random

_mating.

This could be done if an infinitesimal model was feasible and we could

_apply

normal

_theory

and the mixed model.

_Furthermore,

it would be useful to build covariance matrices

and inverse structures

_easily,

to enable use of Henderson’s

₍₁₉₇₃₎

mixed-model

equations.

This _paper shows how to

_justify

and

_implement

these activities. It is

an extension of Smith’s

₍₁₉₈₄₎

_attempt to

_generalize

models with dominance and

inbreeding.

DOMINANCE MODELS Finite loci

In this section we introduce the 6

_genetic

_parameters needed to model

_additivity,

dominance and

_{inbreeding depression.}

These _parameters are functions of _gene

frequency

(p

i

for the

i

th

_allele)

in much the same way that

heritability depends

on _gene

_frequency

for

_purely

additive traits.

First,

consider the

_{genotypic effect,}

_gij for 1 locus

_{represented by}

where p

is the _{mean, a}i and aj are the additive effects for the

i

th

and

_j

_th

_allele,

and

d

ij

is the

_{corresponding}

dominance deviation.

_Equation

₍₁₎

_represents a _system of

r(r

+

_1)/2

equations

in r + 1 +

_r(r

+

_1)/2

unknows

_(ie,

_{!, ai, aj,}

_d

_ij

₎

where r is

the number of alleles. To

_uniquely

determine _p, aiand

_d

_ij

requires

additional r + 1 constraints

_given

as:

These constraints are derived from effectual definitions

_applied

to

_populations

in

It follows that in

_{populations undergoing}

random

_mating,

the additive variance is:

and the dominance variance is:

To accommodate

_inbreeding

_requires

3 additional _parameters:

_(i)

the

_complete

inbreeding

depression:

(ii)

the dominance variance among

homozygotes:

and

_(iii)

the covariance between additive and dominance effects among

homozy-gotes:

It is convenient to work with the

_{parameter 6}

₂

=

06 + 6

which

is a second moment.

The

_symbol

&dquo;&dquo;’&dquo;

is a reminder that the associated

_{parameter refers}

to 1 locus. When there are n

_loci,

_{then the}

_parameters of

_interest,

_{say v,}

are

the

_single

locus

terms v

_{= (8fl, Qa, u6, 7.2}

82

_{-2 -}

&dquo;summed&dquo; over loci. All _parameters

terms

fol2,

fa2 a,

2, a2 - d, Ub, 2, U67 1 62

62

or

a2 bi

aa6l

&dquo;summed&dquo; over loci.

All parameters

in v =

{!a,

a-d,

U6

,

U

6

6

2

or

a-6,

_aa

₆₎

are formal sums. The column vector of

inbreeding

depression

(u

6

)

which is defined as a list of

E

6

for loci

1, 2, ... , n,

is

also

very

_useful.

Among

the

parameters,

we have the

dependancies

ua

=

u u

6

and

or 6 2 = 62 _ U2 6*

The _{parameters v} describe a

_{hypothetical population}

of infinite size

_undergoing

random

_mating

and

_inlinkage

_equilibrium.

This

_population

is sometimes referred

to as the base

population,

but we find this usage

misleading.

In the

spirit

of

Bulmer

_(1971),

let us introduce

_segregation

effects

defined as

deviations

from

mid-parent

values. In

_fact,

both additive and dominance effects have

_mid-parents

values,

as will be seen later. Now we can define v as _parameters that determine the

stochastic

_properties

of

_segregation

effects for an observed

_sample

of animals from a

known

_pedigree.

Whether or not these

_segregation

effects are

_{representative}

of some

ancestral

_population

_(perhaps

several

_generations

_old)

_is,

of course,

questionable.

Indeed,

ancestral effects associated with a

_sample

of animals can be treated as fixed

(Graser

et

_al,

₁₉₈₇₎

_{and, hence, segregation}

effects and estimates of v can be far

removed from the ancestral base. This

_{interpretation}

is robust under

_selection,

with the added

_assumption

that

_linkage

_{disequilibrium}

in one

_generation

influences the

next

_{generation only}

_through

the

_mid-parent

values. Our

_assumption

need

_only

be

approximately

correct over a few

_generations

_(perhaps

far removed from the

base).

It is

_important

to

_point

out that these views are definitional and no method of

The

_disruptive

forces of

_genetic

drift on our _usage of v are

_probably

of

_negligible

importance;

a small

_population

is

_just

another

_repetition

of a fixed

pedigree sampled

from the base

_population.

Infinite loci

It is feasible to define an infinitesimal model with dominance

(Fisher,

1918).

When

there is directional

_dominance,

we

_might

observe

1

U6

I

_going

to

_infinity

or

U2

_and

a!d

_d2

going

to zero

_(Robertson

and

Hill,

1983).

However,

it is our belief that this

_problem

is characteristic of

particular

infinitesimal

models,

not all infinitesimal models. To

show

_this,

we have constructed a counter

_example.

Because

_{o, 2,a}

₂

_,

_a2,

_aa6 and _U6 are formal _{sums, it is necessary}

(but

not

_sufficient)

for the contributions from

_single

loci to be of the order

n-

1

where n is the number

of

_{loci; ie,}

if the limit of v is finite.

_Whereas,

it

_might

seem reasonable to

_require

location effects like

_U

₆

to

_approach

0 at a rate of

_n-

₁

_/

₁

_,

this is not necessary and it _may result in infinite

_inbreeding

_depression.

Now let us

_imagine

an infinite number of

_loci,

each with 4

_possible

_alleles,

that

could be

sampled

with

equal

likelihood.

_Assuming

that the dominance deviations

for each locus are as

_given

in Table

I,

these deviations are consistent with constraints

(2).

In this

_example

it is

_possible

to use _any additive effects also consistent with

_(2),

where

_a2

_is

_proportional

to n-1. For a

_{particular locus,}

the

inbreeding

depression

and dominance variances are:

_U

₆

=

-1/(2n);

a’

=

1/(4n

2

)

₊

3/(8n);

a2

=

1/(4n

2

)

+

_1/(2n).

Summed over n loci these

become:

u6 =

-1/2;

Qd

=

1/(4n)

+

_3/8;

_a

₆

=

_1/(4n)

_{+ 1/2.}

_Letting

n drift to

_infinity

_gives

the

_following

non-trivial _parameters: u6 =

-1/2;

a

§

=

3/8;

or2

=

1/2.

This

provides

our counter

example.

There does not not seem to be an

_analogous

_example

_involving

_only

two alleles.

_However,

the biallelic situation is

_{uninteresting}

because it

_implies

a

singularity:

-2

= a2a2

smgu arlty:

Uab - uau8’ 6* >

The above demonstration may seem artificial because it is

spoiled by

global

changes

in _gene

_frequency

_(WG

_{Hill, 1988,}

_personal

_{communication).}

_However,

we can construct other more elaborate counter

examples.

For

_instance,

let loci _{vary in} their contribution to the _parameters. Let there be infinite loci indexed

1, 2, ... ,

n,

where

_Qd

_,

_{!6 !}

_0,

and there is no directional

_dominance;

_ie,

u6 = 0.

Among

where k

2

< n

<

_(k

+

₁₎

₂

_.

_By

_redefining

the contributions from

_single

loci to

E

6

=

-n-

l

/

2

₊

u

s

and

a2 a

2

<

n-

1

/

2

_(perhaps

_{by altering}

gene

frequency),

we notice that ub =

-1,

and

Qd

and

!6 are

non-zero at the limit when n _goes to

infinity.

We can create _yet another

_subsequence

with indices

2, 5, ... ,

k2

_{+ 1,}

where

k

2

+1 <_

n <

_{(k+ 1)}

₂

_{+ 1.}

_Taking

the

_previous

alterations and

_assuming

for the latter

sequence

E

6

=

1/2n-

1

/

2

+ E

6

,

the limit value

of u

6

is

-1/2.

It is

possible

_to select _a

finite number of

_subsequences

and make similar alterations. Each of these sequences

becomes

_{infinitely long}

as n

_{approaches infinity. Hence,}

infinitesimal

_models,

where

0 <

_J

_U61

< _00,

U2

q 0

0 and

_{Qa 5!}

_0,

are feasible.

With an infinitesimal

_model,

v is a function of summary statistics that involves gene

frequency.

Individual gene

frequencies

have little or no effect on v.

Moreover,

genotypic

effects summed over loci follow a normal distribution. This

_implies

that selection and

_genetic

drift can be accommodated

_by

the mixed

_model,

as

suggested by

Bayesian

arguments

(eg,

Gianola and

_Fernando,

_1986).

In

_particular,

the

_assumption

about the influence of

_linkage

_{disequilibrium,}

discussed

_earlier,

is valid under the infinitesimal model.

The real issue is not whether

_[u

₆

_is

infinite or dominance variances are zero, but

whether

_normality

and

_linearity

are

_appropriate

_assumptions

_given

a finite number

of loci. If

_[u

₆

is estimated from real

data,

it will be found to be

_infinite,

_although

it

may be very

large.

Furthermore,

if dominance variances are

_found

to be non-zero,

and if _many loci are

_involved,

then it would seem that a contrived infinitesimal

model

_(like

the ones

_above)

is

_appropriate.

Normal

_{approximations}

are

_adequate

under most realistic models for

_genetic

_variation;

there

_being

a small number of

major

loci and a

_large

number of minor loci

_(Robertson,

_1967).

_However,

with a very small number of

loci,

these

approximations

become less

adequate

with each

additional

_generation

of selection.

GENOTYPIC COVARIANCE STRUCTURES

Harris

₍₁₉₆₄₎

_developed

recursion formulae for

_evaluating

the

_identity

coefficients needed to determine covariances _among inbred relatives. In a later paper,

Cocker-ham

₍₁₉₇₁₎

elaborated on these methods.

_{Using zygotic}

_networks,

Gillois

(1964)

also devised a scheme to evaluate

_{identity coefficients,}

and Nadot and

_Vaysseix

(1973)

published

an

_algorithm

for

_implementing

Gillois’s

_procedure.

In this _paper, tabular methods for

_evaluating

second moments are

presented.

These

_techniques

allow the exact evaluation of

_genotypic

covariances without

cal-culating

individual

_identity

coefficients. The first class of methods are

_conceptually

easy and are modelled after the

_genomic

table described

by

Smith and Allaire

(1985).

The second class

_(those

based on

compression)

are

conceptually

more

Methods based on

_gametes

Each animal in a

_pedigree

receives 1

_genomic

half or _gamete from each of its _parents.

Thus,

every animal has 2

genomic

halves and the total number of such halves is

r =

_2s,

where _s is the number of animals.

Let ai be a column vector of additive

_effects,

such that the

It

h

element of ai

equals

the additive contribution of the

l

th

locus in the

i

th

_gamete. If there are n

loci,

then _aihas

_length

n. Under an infinitesimal

_model,

aiis

infinitely long.

Define

d

ij

as a vector of dominance

_{deviations, typical}

of the union of

_gametes

i and

_j.

The

it

h

element of

_d

_ij

_equals

the dominance contribution of the

E!h

locus. If i and

j

are

_genomic

halves from different

_animals,

the vector

_d

_ij

_depicts

the dominance deviations for a

_phantom

animal.

Like

_animals,

_gametes have a

_{pedigree; genomic}

halves in one animal form a

parental pair

for

_producing

_gametes. Let us assume the _gametes are ordered such

that i >

_j,

if _gamete i is a descendant of _gamete

_j.

_Furthermore,

let us assume

i

_{> j}

_implies

that

_{gamete j}

is a base

_population

_gamete

if i is.

_Next,

_imagine

the ordered _sequence:

where I is an

_identity

matrix of order n. This is a _very

_long

list

_comprising

of

(r+1)(r+2)/2

arrays.

Fortunately,

we need

only

select a much smaller

subsequence,

G =

{I,

_{gl, g2,...,}

gp}

from this

list;

_ie,

the arrays that _are

_actually

needed for

recursive calculations. An

_algorithm

for

_extracting

G is

_presented

in

_Appendix

A. The elements of G are used as row and column

_headings

in a table

depicting

the

second moments

_E{G’G}

which is

_represented

_by:

This table is referred to as the extended

_genomic

table

_(cf

Smith and

_Allaire,

1985),

and is denoted

_by

E.

Elements of E are

_{computed by}

recursion.

_Starting

with the first _row, elements

are evaluated from left to

_right.

When the first row is

_completed,

the first column is

filled in

_using

_symmetry. The

_remaining

elements in the second row are evaluated

from left to

_right

and the second column is then filled in

_using

_symmetry.

This

process is continued for each additional row and column. The recursions used to

compute E are listed

_below,

where B is defined as the index set of all base

_gametes,

i >

_{j, k,}

_m, k > _m, and _{parent gametes} of i are x and y. The

proofs

of these formulae

are due to

_properties

_involving

sums of

_expectations

and conditional

_{expectations.}

where i - x _{or y represents} the event that the

t

th

locus of

_gamete

i is identical

_by

descent to that x or _y,

_{respectively.}

The

_product

_gig

_w

is intended to involve the

gametes

i, j, k

and m

(v

and w are used to

_identify

the associated columns of

_G).

(i)

First n rows:

(ii)

Subsequent

rows:

(a)

Additive and additive

(c)

Dominance and dominance

the recursive formulae in

_(c)

_appears in Smith

_(1984).

When

_ie0,

the above recursions are initialized

_assuming

that

_gametes

are

_sampled

at random from a

_{single population.}

For this case, we have additional

_{simplification}

for all values of i:

Now that the recursive structure of E has been

_shown,

it is

_possible

to describe the

al

f

orithm

of

_Appendix

A. Define

_{f (v)}

as the _youngest

_gamete

associated with

the vtcolumn of

_G,

_say g&dquo;. The matrix or list G is said to be closed under

_gametic

recursion if the terms used to

_expand

any gv

by

parent gametes of

_{f (v)}

are also

of G. More

_formally,

_(g

_v

_{I f (v) =}

_x)

and

_(g

_v

_{I f (v)}

=

y)

_are columns of G when

f(v);(),

and has _parent

_gametes

x and _y. The

_algorithm

in

_Appendix

A is called

a

_depth-first

search and it

_produces

sets of vectors closed under recursion.

_Any

element needed to evaluate any recursion can

_always

be

_found

in E. The

_algorithm

of

_Appendix

A can also be used to define the

_subsequence

G

introduced below.

It is

_possible

to combine additive and dominance effects into

_{genotypic effects,}

say

i

ij

= _{ai +} a

j+

_d

_ij

_,

and use these as row and

column

headings

of a new table.

The

_headings

are ordered as some

subsequence,

_say

G,

of

The recursions for

E{G’G}

are

_exactly

as

_they

are for

Efd

ij

}

and

Eld

’

3d

k

,,,},

After

_building

a matrix of second moments, the

_(co)variance

matrix

_(for

_genetic

effects summed over

_loci)

is obtained

by

absorbing

the first n rows. The

_resulting

array is a function

of u

6

only through

u2

=

u6 U{j.

The

vw

th

element of the absorbed

array E is:

which reflects the

_assumption

that _genotypes are additive over

_{independently}

segregating

loci. This

_assumption

can be

relaxed,

as

linkage disequilibrium

can

sometimes be accommodated via conditional

_{(ie, Bayesian)}

_analyses.

In

_practice,

we never evaluate the entire _array E or

E{G’G}.

In

_particular,

the

first n rows and columns can be

represented

_implicitly

_by

one row and column: rows of:

are

simple multiple

of each other. Our purpose is to show structural

_properties

that allow inversion rules.

_{Nevertheless,}

the above recursions are

_helpful

in

_evaluating

particular

moments; eg, those needed to _compute the inverse. This can be

accom-plished by

adapting

Tier’s

₍₁₉₉₀₎

recursive

_{pedigree algorithm:}

one calculates

only

needed moments and avoids redundant calculation. _{We may} add to our

_recursions,

shortcuts for

_particular

_degenerate

cases:

These remarkable results do not

_depend

on i _>

_{j, k,}

m or

k >

m.

They

are

due

to the

_principle

of conditional

_independence

and to the rule that

_{probabilities}

are

additive for

_mutually

exclusive events. The first rule

_appeared

in Maki-Tanila and

Kennedy

(1986).

It is similar to a rule in Crow and Kimura

(1970,

p

134)

based

on additive

relationship,

_although

rule

(i)

is more robust under

_inbreeding.

We also

where 77

=

QabQa

2r

’

Y=

82U;

;

2,

a=

o-!o-!!

and

_p=

ubQ! 2.

Because

_E,

_excluding

the first n rows and

_columns,

is at most of the order

_r

₂

_/2

by

r

2

/2,

where r is the number of _gametes, one

_{might incorrectly}

conclude that

proposed

calculations are

_prohibitive

_(of

the order

_r

₄

_/8)

and of no

practical

value.

Recursive

_algorithms,

like the

_depth-first

search in

_{Appendix A,}

can be

_surprisingly

fast. The value

_{of r}

₄

_/8

should be

_regarded

as an _upper

_boundary

that _protects the

algorithm

from combinatorial

_{explosion -}

the kind of

_explosion

that

_might

occur

when

_{enumerating genetic pathways}

in a

_{pathological pedigree.}

Compressed

tables

The

_genomic

table

_{given by}

Smith and Allaire

₍₁₉₈₅₎

can be

_compressed.

We may

add

_together

elements in 2

_by

2 blocks

_{corresponding}

to

_animals,

and

_multiply

this matrix

_by

_1/2,

to

_give

the numerator

_relationship

matrix.

_Compression

_{of E}

is

also

_feasible,

and has

_already

been demonstrated above for a case

_involving

G. In

general,

E is

_{compressed by}

_combining

columns of G to create a new matrix C. To

be

_useful,

C should be smaller than G and contain

_pertinent

effects.

It is

_possible

to devise recursive methods for

_evaluating

_E{C’C},

when C

is not closed under recursion.

However,

methods become more meticulous. For

example,

since the vector of additive merits for animals is not closed under

_gametic

recursion,

we need to add

_inbreeding

coefficients to the

_diagonals

when

_calculating

the numerator

_relationship

matrix.

Whereas,

when

_compression

is defined as the addition of all G

_columns,

it is

possible

to

do

this

_{stochastically,}

as Harris

(1964)

has done. For

_{example, Harris,}

by

preferring

a

_{zygotic analysis}

over a

_{gametic analysis,}

devised a scheme where entities were created

_by

a random

_sampling

of _genes from

_existing

_genotypes.

Compression

is an

_important

area and it needs to be

_developed

further. Some

concepts will be illustrated later

_by

an

_example.

INVERSE STRUCTURES General rule

Matrix E contains second moments and not

_{(co)variances}

as

_{required by}

the

mixed model

_{equations. However,}

_deleting

the first n rows and columns of E-1

gives precisely

an inverse matrix of

_{(co)variances.}

The extended

_genomic

table

is characterized

_by

blocks

_along

the

_diagonal.

_{By inspecting}

labels attached to vectors in

_G,

it is seen that

_they

come _{in groups.} For

_example,

the _group associated

with _gamete i is a

_subsequence

of ai,

,

d

li

d2it ...

d

ii

.

Likewise,

when

considering

E =

E{G’G}

_we find blocks

_along

the

_diagonal

_{associated with gametes. Recursions}

above the

_diagonal

blocks are functions of column indices and not of row indices.

Now consider a submatrix

A

k

where

for some L and

_A

_k

contains the first k + 1 blocks. The matrix L is a

_simple

matrix defined

_by

column indices. If k = 1 note that:

for some

_L

_o

_,

where

A

o

corresponds

to the base

_assignments,

and

B

o

is the second block.

Let us assume that

Ao

is given

(perhaps

without the first n rows and

columns)

and note that:

-

-With

_(B

_o

_&mdash;

_LoA

_o

_L

_o

_)-

₁

_evaluated,

we find that

_All

is

a

simple

function of

AÕ

1

.

Given

_Ao

₁

_,

it is

_possible

to _compute

_A2

₁

_,

where

and

_B

₁

is the third block. In

_general,

_given

_A-’

we

can evaluate

A-1

1

,

where

and

B

k

is block k + 2. The

_general

inversion formula is:

To evaluate

_E-

₁

_,

_apply

this rule

_{recursively starting}

with k = 0.

It is

_hoped

that

B

k

-

_k

_L

_k

_L[A

will be

sufficiently

small or

sufficiently

sparse

so

that its inversion is feasible

_(eg,

Tier and

_Smith,

_1989).

For

_evaluating

_E-

₁

_,

the

worst scenario is that the order of

_B

_k

_-

_LkA!L,!

is r +1.

_However,

this occurrence

is

_unlikely.

Note that Henderson’s

₍₁₉₇₅₎

rule for

_calculating

the inverse numerator

There are some notable

_{simplifications}

when E-1 is to be evaluated.

_First,

evaluation of

_Ao

is

best done

by

absorbing

the first n rows and then

_deleting

the first n rows and columns. The

resulting

matrix is some

_permutation

of a block

diagonal

matrix

_involving

2

_by

2 matrices:

and 1

_by

1 matrices

_{a§ and}

_ad.

This is a trivial matrix to invert.

Second,

k

-

B

_k

_L

_k

_A

_k

_L

has a

_peculiar

structure that can be identified

_by

examining

the recursive definition in Section IIIA. If block

B

k

corresponds

to

gamete i which has _{parent gametes} x and _y, then

L

k

is a matrix that

&dquo;picks&dquo;

appropriate

terms from

A

k

that involve x and _y.

_{Moreover, B}

_k

is also defined

by

terms that involve x and _y. Assume that the column

headings

for Bk are:

It

_might

be that i =

j

m

.

Now define the column

headings

where

_x

₌

_H

_(Fi

_I

i =

_{x) =}

_{{a!, d}

_d

_x

_l’

_j

_Xj

_2’

_....

_d!,,.}

_}

and

_{Hy = (Fi}

_I

-

_{y) _}

_{ay,

_dy

_Ù’

_d

_Y

_j

₂

_{7 ...}

_dyj&dquo;,

_I

In the definition of

_H

_x

and

_H!,

it is understood that

_d

_Xj

_&dquo;,

=

d!!

and

d!j&dquo;,

=

dyy,

if i =

j&dquo;,.

Select elements from

A

k

and build the

matrix,

where

_M!!

=

E{HxH!}, M

_x

_y

=

M!!

=

E{H

x

Hy},

and

Myy

=

EIH’Hy}.

A direct

_application

of the recursions

_gives:

Futhermore,

as

L

k

is a matrix that

_{&dquo;picks&dquo;}

terms under

_headings

H!

and

_H!:

and thus:

Equation

(5)

can also be derived

if B,!-L!A,!Lk

is

recognized

as the

(co)variance

matrix for the

_segregation

effects due to recombination of

_{gametes x}

and y in the

formation of _gamete i. The

_mid-parent

values of

_F

_i

are the column vectors of

1/2H!

+

_1/2Hy.

As the

_{segregation effects,}

S =

F! - 1/2H! - 1/2H!,

have a mean

of _{zero, the}

_(co)variance

matrix is:

where S =

1/2H! -

Finally,

in rule

_{(3) L}

_k

_(B

_k

_-

_{L!A,!Lk)-1L!,}

_k

_-

_k

_(B

_-L

_L’A

_k

_L

_k

_)-

_l

and

(B! - Lj!A!L!) !

are contributions added under H

_by

H

_headings,

H

_{by F}

_i

head-ings,

and

_F

_i

_by

_F

_i

_headings,

_{respectively.}

Existence of inverses

When there is no

_inbreeding,

E-1 can be shown to exist.

_First,

we _present the

following

Lemma:

Lemma 1: In the absence of

_inbreeding,

_there

exists a matrix

_M

_k

_,

which is a

submatrix of

_A

_k

_,

and there exists a matrix

_X

_k

of full column rank such that:

where

_B!,,L,!

and

_A

_k

are associated with E.

Proof.

Because

_equ(5)

is

_given,

we

_only

_prove that such

M!

and

X

k

can be found

where XkM

k

X

k

=

1/4(M!! -

M!! - M!!

+

_Myy).

The matrix

_M

_k

_,

defined

_by

eqn(4),

will be ’a submatrix of

_A

_k

if there are no indices

_j

_m

= i,

j

v

= x

and j

w

_{= y}

used in the

_definition

of

_F

_i

_.

The

_{algorithm presented}

in

_Appendix

A will not create

indices

_j&dquo;,,

_{= i, j}

_v

= x

and j

w

=

y when there is no

_inbreeding.

For this _case, we can take

_M

_k

=

M

k

,

and ’

=

1/2{I, -I},

where the

identity

matrix I has order

m + 1. Matrix

_X

_k

has full column rank

_((a.E.D).

Theorem 1: If E is constructed

_{by applying}

the recursion rules to some finite and non-inbred

_pedigree,

then E-1

exists,

provided:

Proof.

The matrix

A

o

is

_non-singular

when the condition of the theorem holds. Now assume that

_A-’

_exists,

then

_by

the inversion rule

₍₃₎

_A!+1

exists if

_(B

_k

_-

_L’A

_k

_L

_k

_)-’

exists.

_By

the above

_Lemma,

_B

_k

_-

_LkA

_k

_L

_k

=

3CkM

k

Xk-Because

_M

_k

is a submatrix of

_A

_k

_,

it is

_{non-singular. Therefore,}

(X!M!X!)-1

exists because

_X

_k

has full column rank. We conclude that the existence of

_Ak

1

implies

the existence

_{of A!+1.}

As

_AÕ1

_exists,

the theorem follows

_by

mathematical induction

_(Q.E.D).

The reader

_might

think that the concurrence of identical twins would contradict

Theorem 1.

_However,

this is not the _case, as the

_theory

assumes that _gametes are

distinct and can be ordered

_using

indices.

_Thus,

for identical

_twins,

the recursive

formulae

_presented

earlier are

incomplete.

This is not a

_practical

_problem,

as

identical

_gametes

can be

_{represented only}

once in G.

Henderson

₍₁₉₈₅₎

_considered

a non-inbred

population

and studied a dominance

relationship

matrix D. He used D-1_{in many} formulae without

_proof

of its existence.

However,

as D is a submatrix of

_E,

the theorem

_implies

that D-1 exists.

When there is

_inbreeding,

the

_algorithm

in

_Appendix

A will

_produce

labels like

dii,

di.

and

_diy,

where

_i!9

and has _{parent gametes} x and _y. In

general,

E is

_singular

Even with

_inbreeding,

4 labels like those in

_eqn(6)

need not occur

_together

as

headings

of E.

_However,

it is

_possible

to _merge the

_identity

₍₆₎

with the recursion

and remove

d

ii

for

I(0

from the _sequence G to _get

G -

see

_Appendix

A. The vector

G

is closed under recursion if

_eqn(6)

is used to redefine

_d

_ii

for

_i18.

The matrix of second moments

E

=

Eli7x4U}

has no row and column

_heading

_d

_ii

for

_I(0

and it is

_{non-singular.}

To prove

this,

we introduce Lemma 2 for

E.

_{First, let}

us

_imagine

that

B

k

-

_LkAkL!

corresponds

to a

_particular

absorbed block of

E.

Lemma 2: There exists a matrix

M

k

,

which is a submatrix

_{of A}

_k

_,

and there exists

a matrix

X!

of full-column

rank,

such that

B

k

-

_k

_k

_L

_L’A

=

-

-

4

where

B

k

,

L

k

and

_A

_k

are associated with

E.

Proof.-

Because the

_pedigree

is

inbred,

we _expect

_{indices j,}

= x and

_j

_w

= _y

(x,

y parents of

_i)

in the definition of

F

i

- otherwise,

follow the

_proof

of Lemma 1.

By

construction,

there is no index

_j

_m

= i. This

implies

that there will be vectors

d

x

y,

d

xx

=

dxx, +dxx,,-dx!x&dquo;

and

_dyy

_{= d!yx+d!yy-dyzyv}

in

_H,

_{where x}

_x

_,

_{xy and}

y x

, yy are _{parent gametes} of x and _y,

_{respectively.}

With no loss in

_generality,

switch

the vectors aiwith

d

ix

,

and

d

i

y

with

di!&dquo;,

in

F

i

.

So as to maintain

_consistency

with the definition H =

{H

x

=

(Fi I

i =

x),

_{Hy = (F}

i

i

=

y)},

columns of H _may be altered

_by

_switching:

These

_permutations

preserve the

identity

where

_X

_k

=

1/2{I, -I}

and

M

k

=

E{H’H}

if _we further

stipulate

that selected

columns of

G

have also been

_rearranged.

Note that columns and rows m + 1 and

m+2 2 in

_M

_k

are

_redundant,

as

_they

are both

_represented

_by

_d

_x

_y

=

dy

x

.

Therefore,

we _may delete row and column m + 2 to create

_M

_k

_(delete

column m + 2 in

_H)

and note that

where

and I has order m &mdash; 2.

_Clearly,

i

k

is of

full-column

rank and is a candidate for

X

k

.

When x and _{y are} base

_gametes

(ie,

, xx_x!, Yx

and yy

are

unknown),

we are finished as we can take

M!

=

R

k

,

which is a submatrix of

_A

_k

_.

_{Unfortunately,}

when at

least one of the

_zygotes

_(x

_x

_,xy)

and

_,yy)

_Yx

₍

are

known,

M

k

is not a submatrix of

A

k

because of the

_composite

vectors

d!x

_{and/or dy}

.

.

We assume that both

(x

x

,

_yy)

and

_(y

_x

_{, y}

_x

₎

are known in the remainder of the

_proof

_(when

_only

1

_zygote

is

_known,

It

_might

be that

_d

_xx

_&dquo;

and

_d!!&dquo;

exist

_already

between columns 2 and m in the

redefined H.

_Likewise,

_d

_.

_y

_.

and

_d!!&dquo;

_may exist

_beyond

_position

m. If _any of the

labels _{xx!, xxy7yyx} and _yyy do not

_exist,

we introduce them as dominance vectors

in H.

_Further,

we may redefine the first and last columns to _represent labels _zzzy

and _yxyy,

_{respectively.}

This

_gives

us a modified matrix

Because

G

is

_{c_losed}

under

recursion,

all columns of

H

are in

G.

_Moreover,

all columns of

H

are

_unique;

_XxX

_{y =1=}

_yxy!, because animals cannot mate with

themselves. We conclude that the matrix

is a submatrix of

A

k

.

Now define the matrix

/.. . I ... r. B.

where m and f are null _{vectors, except} for ones at

_positions

_{corresponding}

to zz

z

, Xxyl yyx and yyy; M and F are like

_{identity matrices,}

_except that rows

corresponding

to _{zzz, xxy, YYx} and _yyy would be deleted if the

_{corresponding}

columns were introduced into H. Now

X

k

has full-column rank. Since

the Lemma is

_proved

_(Q.E.D).

Theorem 2: If

E

is constructed

_{by applying}

recursion modified

_by

_equ(6)

to some

finite

_pedigree,

then

E

exists,

_provided

Proof.

With the first n rows

_{absorbed, A}

_o

is

_non-singular

when the condition of the theorem holds.

_Therefore,

A

o

is

_non-singular

when the first n rows are intact.

The remainder of the

_proof

is identical to that of Theorem

_1,

_except that reference

is made to Lemma

2,

rather than Lemma 1

_(Q.E.D).

We should _expect

_{singularities}

when

_inbreeding

coefficients

_approach

_unity,

as

occurs with the numerator

_relationship

matrix.

_Indeed,

the matrix

S

=

1/2H

x

-1/2Hy

becomes a null matrix when _{gametes x}

_{and y are identically}

_equal.

As

E{S’S}

equals

eqn(5),

the absorbed block for

_gamete

i is a matrix of zeros and

E

is

_singular.

Our construction of

E

assumes that the base

_population

is a random set of unrelated _gametes which have been

_sampled

from some infinite

_population;

that is even

_though

the base

_population

is

_finite,

_inbreeding

in the base does not

exist. Under

_{diploidy, inbreeding}

coefficients cannot be

_unity

with finite

_pedigrees.

Feasibility

It is

_beyond

the _scope of the _{present paper} to demonstrate the calculation of E-1

pedigree

borrowed from a beef cattle selection

_experiment

conducted at the

Agri-cultural Research

Centre,

Trangie,

NSW

_(PF

Parnell

_(1988),

_personal

communica-tion).

The

_pedigree

involved 1 122 animals with records and 625 ancestors without

records. The _{average number} of

_generations

on the female side was 9 and the maxi-mum was 16. On the male

side,

the average was 8

_generations,

with a maximum of

13. There were

_only

55 base _gametes, and the average

inbreeding

of the

population

was 0.1

_(the

maximum was

0.26).

The order of E and

E

was about 190 000. Matrix

A

o

was of the order 1595.

However,

excluding

A

o

,

the maximum block sizes were 321 and

323,

respectively.

The distribution of block sizes is

_presented

in Table II. Most of the blocks were of

the order 2 and 3. Given the absorbed

blocks,

and

_{ignoring possible singularities,}

matrices E-1 and

E

can

be evaluated. The

computations

are not

_trivial,

but are

feasible.

There are other

_approaches

to

_modelling

dominance that do not involve E-1

or

E-

1

.

For

_example,

we

_might

extract from E

_only

those elements that involve animals or animals with records.

_{If the}

extracted matrix is put into a matrix

_D,

then we could attempt to evaluate

D-

1

_using

sparse matrix

absorption

(Tier

and

Smith,

1989).

Then

U-

1

could be used in the mixed-model

_equations.

_{Alternatively,}

D

could be used

_directly

in the same way that Henderson

(1985)

used D. Because

E is an

_enlarged

matrix with unrealized combinations of _gametes

_(the

so-called

phantom

animals),

alternatives to

E-

1

are attractive.

_However,

breeding

schemes that utilize dominance variation are bound to

_require

the

_prediction

of dominance effects that

_correspond

to untried

_{gamete pairs}

_(say,

among gametes of _parents that contribute _genes to the next

_generation).

Further research is needed to evaluate all

ILLUSTRATION

In this section we demonstrate our

methods, using

the

_simple

pedigree

displayed

in

_Fig.

1. This

_pedigree

involves 4 animals or 8

_genomic

halves. For

_simplicity,

we assume

or2

₌

or =

_1;

U2

₌

a

2

₌ 0. Given that _gametes

_{pairs 12, 35,}

64 and 78 represent dominance vectors for animals and are to be included in

_G,

the

_algorithm

of

_Appendix

A creates 14 additional

_{pairs -}

_ignoring

first _array I and additive vec-tors. The matrix G is

_given

as row and column labels of Table III. Second moments

of Table III are derived

_{by applying}

the recursion of formulae. For

_example,

the

element

_Eld’2ld5il

was calculated as

_1/2EId’

_l

_d

_ll

_}

+

_1/2EId’

_l

_d

_2ll

_{= 0 + 1/2}

= 1/2.

To evaluate E-1

_requires

the determination of

_(A

₂

_-

_LiBiL!)-1

for i =

_{0,1, 2, 3.}

The absorbed blocks can be evaluated

_recursively,

but for _now, the reader _may

obtain these

_{by applying}

Gaussian elimination

_directly

to Table III. The inverse blocks are:

The matrices

_L!

_for

i =

0,1,2,3,

_are

displayed

below the

diagonal

in Table IV. In accordance with formulae

_(3),

the elements of

E-

1

are

_given

above the

_diagonal.

We can now _{compress E}

_{by defining}

_d

_b

=

d

u

₊

d

22

(with

labels,

this is notated

as b: 11 +

_22),

_d, =

d

31

₊

,

32

d

_d

_d

= _{d41 +}

_d

₄₂

_,

_d

f =

dsi

+

d

52

,

di

=

d

74

₊

d

76

.

These

_definitions,

as well as other

_{symbolic definitions,}

are

_given

as row labels for

the

_compressed

matrix of Table V. As an

_exercise,

the

_compressed

table can be

evaluated

_{by inspecting}

Table III and

_adding

_together

_appropriate

elements. For

The real

_challenge

is to find

_simplifying

recursions that allow evaluation of

com-pressed

tables. In order to

_apply

inversion formulae

_(3),

recursions to the

_right

of blocks

_along

the

_diagonal

should be maintained

_exclusively

as a function of column

index,

as in the _present

_example.

These recursions are

_{given implicitly}

below the

diagonal

in Table VI as matrices

L

i

for i =

_{0,1, 2, 3.}

For

_example,

the element

E(d )di )can

also be obtained

_{by adding}

The blocks

_(A

_i

_-

_LiB

_i

_L

_il

_-

₁

for i =

0,1, 2,

are:

Inverse elements of the

_compressed

matrix are

_given

above the

_diagonal

in

Table VI.

DISCUSSION

Although

we have not

_emphasized

the evaluation of various

_identity

coefficients,

gametic

recursion of such coefficients is feasible.

_{Moreover, gametic recursions,}

like

those used to define

_E,

are _{very easy to} understand and _may

_provide

an

approach

useful in

_teaching,

whilst there may be more

_complicated

(but

_{perhaps numerically}

more

efficient)

alternatives. But

_gametic

recursion need not be

_numerically

ineffi-cient and is

_certainly

not

_subject

to combinatorial

_{explosion. Depth-first searches,}

like the

_algorithm

of

_{Appendix A,}

can be used to

_implement

_gametic

recursion in an

efficient _way. For

example,

to evaluate

_inbreeding

coefficients via

_gametic

_recursion,

one would first conceive of a

_symmetric

table with

_r

₂

_/2

elements.

However,

r

2

/2

operations

are not

required

to evaluate

_inbreeding

coeflicients via

_gametic

recur-sion. All that is needed is to

_identify

labels

_{i j}

of the dominance vectors

_d

_ij

in G.

By moving

down a list of such

_labels,

it is

_possible

to evaluate the coefficients

_using

one work vector. We need

_{only modify}

this scenario for

_{general identity}

coefficients. The

_{simplest explanation}

of the two

_commonly

found and

_{complementary}

phe-nomena

(heterosis

and

_inbreeding

depression)

is that there is dominance of alleles

at many loci.

_Therefore,

dominance should be

_regarded

as an essential feature of

genetic

models for loci

_affecting

_quantitative

traits. Characteristics

_closely

con-nected with

_fitness,

such as

_{reproduction,}

would

_mostly

have

_only

non-additive variation

_left,

as the additive has been exhausted

_(Robertson,

_1955).

There are also

suggestions

that dominance of alleles that _{maintain normal enzyme}

_activity,

is a

universal biochemical _property

_(Kacser

and

_Burns,

_1981).

It is clear that

_additivity,

dominance and

_inbreeding

can be modelled

_by

applying

mixed-model

_methodology.

The ramifications of such a

development

are

(i)

Determination of

_optimum

and

_{dynamic mating}

structures which

_capitalize

on additive and dominance variance while

_providing

for

_inbreeding.

Some of these

breeding strategies

can be studied

_by

the use of

_{moment-generating}

matrices for

regular mating

systems. Cockerham

_{( 1971)}

has

_given

an

_example

of such a transition matrix for full-sib

_mating.

Possible

_application

areas are:

(a)

mate selection

_(Jansen

and

_Wilton,

_1985;

Smith and

_Allaire,

_1985);

(b)

group selection

(Jansen,

1985;

Smith and

Hammond,

1987)

when the

selection of a random

_mating

_gene

_pool

is created

_by

a finite number of _parents.

The

_objective

of _group selection is to

_improve

both additive merit and the _average

specific

combining ability

of _{genes in} the

_pool;

(c)

crossbreeding plans

to utilize between breed additive and heterotic effects

(Kinghorn,

1987);

(d)

selection of clones and animals created

_by

futuristic

_techniques.

(ii)

To allow dominance variance to be

_partitioned,

and

_thus,

remove some of

the

_confounding

that would otherwise _corrupt statistical

_analyses.

(iii)

To

_improve

our

_{understanding}

of traits which are

_likely

to show considerable

amounts of non-additive

_genetic

variation. This

_{understanding}

can be advanced

_by

simulation

_studies,

_eg,

_concerning

selection bias and finite loci. Other studies may

involve simulation of infinitesimal models with dominance via

_sampling

from normal

distributions.

The

_theory

we

_present

here is still _very

underdeveloped.

New methods are

needed for the estimation of

_genetic

_parameters, as _past

_approaches

have met

with _very limited success

(eg,

_Gallais,

_1977).

It is our belief that an extension of

the derivative-free

_algorithm

of Graser et al.

₍₁₉₈₇₎

is

_possible,

and

_{consequently,}

genetic

parameters could be estimated

_by

restricted maximum likelihood. Research

is needed in the area of

_{computing strategies}

(eg,

_compression,

inversion).

Further,

models which allow different _parameters for different

_populations

would be useful in

crossbreeding

studies. A multivariate extension of our work would also be desirable.

The

_forgotten

_papers of Harris

₍₁₉₆₄₎

and Gillois

₍₁₉₆₄₎

have been resurrected. While the methods are

_arguably

_{complicated, they}

_are,

_however,

feasible.

_Whereas,

classical

_quantitative

_genetics

is unable to

_fully

utilize information on dominance

and

_inbreeding

in

_predictions,

the

_appropriate

mixed model under a wide _range of

assumptions,

including

selection and environmental

_noise,

does that. ACKNOWLEDGEMENTS

The authors thank Keith Hammond for his _{encouragement.} This work was

sup-ported

by

the Australian Meat and Livestock Research and

_Development