Restricted maximum likelihood to estimate variance components for animal models with several random effects using a derivative-free algorithm

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Restricted maximum likelihood to estimate variance

components for animal models with several random

effects using a derivative-free algorithm

K. Meyer

To cite this version:

Original

article

Restricted

maximum

likelihood

to

estimate variance

components

for

animal

models with

several

random

effects

_using

a

derivative-free

algorithm

K.

_Meyer

Edinburgh University,

Institute

_of

Animal

Genetics,

West Mains

_{Road, Edinburgh}

EH9

3JN, Scotland,

UK

(received

21 March

_{1988, accepted}

11

_January

₁₉₈₉₎

Summary - A method is described for the simultaneous estimation of variance

compo-nents due to several

_genetic

and environmental effects from unbalanced data

_by

restricted maximum likelihood

_(REML).

Estimates are obtained

by

evaluating

the likelihood

explic-itly

and

_using

_standard,

derivative-free

_{optimization procedures}

to locate its maximum. The model of

_analysis

considered is the so-called Animal Model which includes the ad-ditive

_genetic

merit of animals as a random

effect,

and

_incorporates

all information on

relationships

between animals.

_Furthermore,

random effects in addition to animals’ ad-ditive

_{genetic effects,}

such as maternal

_genetic,

dominance or _permanent environmental

effects are taken into account.

Emphasis

is

_{placed entirely}

upon univariate

analyses.

Sim-ulation is

_employed

to

_investigate

the

_efficacy

of three different maximization

_techniques

and the scope for

approximation

of

_sampling

errors.

_Computations

are illustrated with a

numerical

_example.

variance _{components -} restricted maximum likelihood - animal model - additional

random effects - derivative - free _approach

Résumé - Utilisation du maximum de vraisemblance restreint et d’un _algorithme

sans _{dérivation, pour} estimer les composantes de variance d’un caractère, selon un

modèle animal avec _plusieurs effets aléatoires. On décrit une méthode _{pour estimer}

simultanément les _composantes de la uartance "d’un

seul

caractère,

dues au milieu ou

plusieurs

effets

génétiques.

La méthode admet des données

_{non équilibrées}

et se

fonde

sur le maximum de vraisemblance restreint

_(«

_REML»).

Les

_composantes

estimées sont obtenues par l’évaluation

_explicite

de la

_fonction

de

_{vraisemblance,}

dont on recherche le maximum par des

techniques générales d’optimisation,

ne nécessitant _pas le calcul des dérivées. Le modèle

_d’analyse

est un «modèle

animal»,

où l’on considère la valeur

_génétique

individuelle des animaux comme un

effet

_aléatoire,

et tient

_compte

de toute

_{l’information}

généalogique disponible.

Des

_effets

aléatoires

_{complémentaires}

_(effets

maternels

_{génétiques,}

effets

de

_dominance,

_effets

de milieu

_permanent)

sont aussi

_pris

en compte. La simulation

est utilisée pour évaluer

l’efficacité

de trois

_techniques

de

maximisation,

et pour déterminer

approximativement les distributions des estimateurs. Les calculs sont illustrés par un

exemple numérique.

INTRODUCTION

Over the last

_decade,

restricted maximum likelihood

_(REML)

has become the

method of choice for

_estimating

variance

_components

in animal

_breeding

and related

_{disciplines trying}

to

_partition

the

_phenotypic

variation into

_genetic

and

other

_components.

This has been facilitated not

_{only by}

an increase in the

_general

level of

_{computational}

resources

_available,

but

_by

the

_development

of numerous

specialized

algorithms, exploiting specific

features of the data structure or model

of

_analysis

as well as

_utilizing

a

_variety

of numerical

_techniques.

So

far,

REML has found most

_practical

use in the

_analysis

of

_dairy

cattle data

under a &dquo;sire model&dquo;. For this

_model,

records of _progeny are used

_only

to obtain

information on half of their sires’

_{breeding value,}

while dams and

_{relationships}

between females are

_{ignored. Recently,}

interest has increased in more detailed

models,

in

_particular

the

_{conceptually simplest}

_breeding

value or

&dquo;Animal Model&dquo;

(AM)

where each record is taken to

_provide

information on the additive

_genetic

merit of the animal measured.

_{By including}

animals which do not have records themselves but are

_parents,

this allows for all information on

_{relationships}

to be

taken into account.

A

_large

_proportion

of REML

_applications

have been restricted to models with

one random factor

_{(e.g. sires)}

_apart from random residual _errors,

_estimating

two variance

_components

_only

in a univariate

_analysis,

or _p

_(p

_-f-1)

for a multivariate

analysis

of p traits. While

algorithms

for more

_complicated

models have been

described, they

are

_by

and

_{large computationally demanding.}

Often

_they

involve

inversion of a matrix of size

_equal

to the total number of levels

of all

random effects

fitted. This can be

_prohibitive

for

_practically

sized data sets. Thus REML has found

comparatively

little use so far for models

_fitting

several random effects.

Maximum likelihood estimation

_involves,

_{by definition,}

location of the maximum

of the likelihood function for a

_given

set of

data,

model of

_analysis

and

_parameters

to be estimated.

_Estimating

variance

_components

for unbalanced data

_generally

requires

iterative schemes. Standard textbooks on numerical

_{analysis classify}

pro-cedures to find the

_optimum

_(minimum

or

maximum)

of a function

_according

to

the amount of information

_required

from derivatives of the function. The so-called

Newton methods utilize both first and second

_derivatives,

i.e.

_{geometrically}

speak-ing slope

and

_curvature,

and are thus

_quickest

to _converge. Methods

_relying

on first

derivatives

_only

include

_steepest

_descent,

_{conjugate gradient}

and

_Quasi-Newton

pro-cedures

_{approximating}

second derivatives.

_Finally,

there are derivative-free methods

involving

direct search

_strategies

or numerical

_{approximation}

of derivatives

_(see

for

example

Gill et

_al.,

_1981).

In the

_main,

REML

_{algorithms currently employed}

in animal

_breeding

fall

into the first two

_categories.

Fisher’s Method of

_Scoring

is a

_special

case of the

Newton

_{procedures, requiring expected}

values of second derivatives of the

_log

likelihood function

_(G)

to be evaluated. As these are often difficult to

_obtain,

Expection-Maximization

(EM) type

algorithms

(Dempster

et

_al.,

_1977),

_exploiting

first derivative

_information,

are used more

_widely.

A derivative-free REML

_algorithm

has been

_{suggested by}

Graser et al.

₍₁₉₈₇₎

for univariate

_analyses

to estimate the additive

_genetic

and error variance under

procedure

was suitable for data from

_large

selection

_experiments

_involving

several

thousand animals.

This _paper describes the use of a derivative-free

_approach

to estimate variance

components

by

REML for AMs which include not

_only

animals’ additive

_genetic

merit but also additional random

effects,

and thus cover a wide range of

mod-els suitable for the

_analysis

of animal

_breeding

data. Univariate

_{analyses only}

are

considered at

_present,

extensions to multivariate situations will be discussed else-where.

CALCULATING THE LIKELIHOOD

The Model Let:

denote the linear model of

_analysis

with: Y the vector of N

_{observations,}

b the vector of NF fixed effects

_(including

_any linear of

_higher

order

_covariables)

X the NxNF incidence or

_design

matrix for fixed effects with column rank

NF

*

,

u the vector of all NR random effects

_fitted,

Z the NxNR incidence matrix for random

_effects,

and

e the vector of N random residual errors.

Assume:

&dquo;’{TI--B r&dquo;I

which

_gives:

The mixed model

equations

(MME)

pertaining

to

_[1]

are then

(Henderson, 1973):

or C

F

= _r. If C is _not of full

rank,

_as it is often the

case, estimates for b are not

unique.

The Likelihood

REML

_operates

on the likelihood of linear functions of the data vector with

expectations

zero, so-called error

_contrasts,

or,

equivalently,

on the

_part

of the likelihood

_(of

the data

_vector)

which is

_independent

of fixed effects. This results

in the loss in

_degres

of freedom due to

_fitting

of fixed effects

_being

taken into account

_(Patterson

&

_Thompson,

_1971).

For

_{Y !N(Xb, V),}

the

_log

likelihood is

(e.g.

Harville

_1977):

-

-where X*

_(of

order

_NxNF

_*

₎

is a full rank submatrix of X.

_Using

matrix

equalities

where C* is the coefficient’matrix in

_[2]

with X

_{replaced by}

_X

_*

_,

and P is a matrix:

Calculation of the first two terms

_required

in

_[4]

_depends

on the

specific

structure

of R and G in a

_{given analysis.}

The latter

_two,

_however,

can be determined in a

general fashion,

as

_{suggested by}

Graser et al.

_(1987),

_by

Gaussian Elimination

_(as

described in most Numerical

_Analysis

textbooks,

or

by

Smith & Graser

(1986))

applied

to _{the mixed model array: the} coefficient matrix in

_[2]

_{augmented by}

the

right

hand side and a

_quadratic

in the data vector.

Calculation

_ofy’Py

and

_log

_C*!

The mixed model _array for

_[1]

is:

&dquo;Absorbing&dquo;

rows and columns

_pertaining

to random effects into the rest to the

matrix then

_gives:

and

_eliminating

rows and columns for fixed effects

_{correspondingly,}

_yields

y!Py,

the

_weighted

sum of

_squared

residuals

_required

to evaluate

_log

.C.

_Absorption

is

most

_easily

carried out

_by

Gaussian elimination:

_{repeated absorption}

of one row

and column at a time. This will also allow

_log

_{I C}

_*

_to

be determined

_{simultaneously.}

Subdivide M of size KxK

_(K=NF

₊

_NR

₊

₁₎

with elements _m2! and column vectors mi into rows 1 to

_K—1,

and row K:

Partitioned matrix results then

give

with

Mx_

1

*

=

M

K

-

1

-M

KM

’K/M

KK

=

I

M

ij-

M

i

K

mjk!mKK} _ fm

ij

*

}

the matrix

resulting

when

_{&dquo;absorbing&dquo;}

row and column

K,

or

&dquo;pivoting&dquo;

on

_mK

_x

_.

_Repeated

use of this result shows that the

_required

determinant is then

_simply

the sum of the

log

of

_pivots

_log

mii*,

i =

_2,...,

_K)

_arising

when

_absorbing

all rows and columns of

M into the first _row, as

_required

to evaluate

_jrpy.

If X is not of full

_rank,

M has

to be set _up

_replacing

X

_by

X* _or,

_{equivalently, absorptions}

have to be carried out

Univariate

_analyses

Results

_presented

so far hold for any model of form

(1!.

Consider

now a univariate

analysis

with

_identically

and

_{independently}

distributed errors, i.e.

For

_given

values of the other variance

_components,

the error variance can be

estimated

_directly

in this _case, from the residual sums of squares as

(see

Harville,

1977;

or Graser et

_al.,

₁₉₈₇₎

Let the other

_parameters

to be

estimated,

i.e.

_{(co)variances}

of the random effects

fitted,

be denoted

_by

oi with i =

1, ..., p - l,

and p the total number

of

components

with up =

0-2 E*

As discussed

_by

Harville & Callanan

_(1988),

a function of REML estimates of a set of

_parameters

is also the REML estimate of this function.

_Hence,

instead of

maximizing

log

G with

_respect

to the _p

_components

Qi , we can

reparameterize

to

9

and

_{p—1 functions fi}

_(!i,

_(TÐ

of the other

_components

and the error variance.

An obvious choice is to _express the Qi as a

_proportion

_(À

_i

i/u2 )

of the

latter,

so

that

_having

found REML estimates of

_u

_§

_and

the

_,

_a

_i

we

can

estimate

6

i

=

62 E’

Furthermore,

for fixed values of

_A

_i

_,

_log

G attains its maximum with

respect

to

u§ at

the REML estimate of

_{U2 E.}

This allows estimation to be conducted in two

steps:

Firstly,

a &dquo;concentrated&dquo;

_likelihood

is maximized with

respect

to the

_A

_i

_only

which

_yields

REML estimates

!i.

_Secondly,

_&2

_is

obtained

_(from

_[9])

for the

iz

(Harville

&

_Callanan,

_1988).

The

_advantage

of this

_approach

is that it reduces the dimension of the numerical search for the maximum of

_{log L}

_by

one. As the number

of iterates and likelihoods to be evaluated to find the maximum of

_{log L}

_usually

increases

_{substantially}

with the number of

_parameters

to be

_estimated,

this can

lead to a considerable

_saving

in

computational

resources

_required.

From

_[8]

it follows

_immediately

that:

Log

!G!

depends

on the random effects fitted. For the

_simplest

model with

animals as the

_only

random

effect,

as considered

by

Graser et al.

(1987):

where

_QA

is the additive

_genetic

variance,

A the numerator

_relationship

matrix

between

_animals,

a the vector of

_(direct)

_genetic

effects for

_animals,

and NA denotes

the number of animals. Since

_log

_IAI

does not

_depend

on the

parameters

to be

estimated,

it is a constant and does not need to be calculated in order to maximize

log

G. The inverse of A is

_required

in

_[6]

_(for

_G-

₁

₎

_though,

but this can be set up

efficiently

from a list of

pedigree information, following

rules

described,

for

instance,

by Quaas

(1976).

Let c of

length

NC denote a vector of such effects to be included in the model of

analysis,

with .

This

_gives:

In other _cases, the model of

_analysis

_may involve two random effects for each animal. Let m, of

length NA,

denote the second animal effect and assume each

element has variance

_a

_’

_.

If there are

_repeated

records _per animal for a

_trait,

m

represents

the

_permanent

effects due to

_animals,

_excluding

additive

_genetic

effects.

These are

usually

assumed to be uncorrelated with any other effects in the

model,

so that

If m had variance

0&dquo;!1

D,

[13]

would be

augmented by log

!D!.

As with

log

IAI,

this term is constant and does not need to be evaluated. Note

_though

that

G-

1

and

consequently

D-

1

is

_required

in

_(6!.

A

_{typical example}

for this kind of structure is

a model where m stands for dominance

effects,

u m 2

for the

respective

variance and

D for the dominance covariance matrix among animals.

For other

traits,

for

_example

measures of

reproductive performance,

we

distin-guish

between a direct and a maternal

(or paternal)

additive-genetic

_component,

allowing

for a covariance between the two. In that

_situation,

there may not be a

record

_supplying

information on m for each

animal,

but information is

acquired

indirectly

via links

arising

from the

_genetic

covariance and

_{relationships.}

With UAM

denoting

the covariance between a and m and rAM the

corresponding

correlation,

and

_partitioned

matrix results

_give

For all models discussed so

far,

computational

requirements

to determine the

part

of

_log

_!G!,

which

_depends

on the

_parameters

to be

_estimated,

are trivial. This

results from random effects

_being

either

_{uncorrelated,}

so that G is

blockdiagonal

(!12!

and

_!13!),

or G

being

the direct

product

of a matrix of

_parameters

and a matrix

describing

correlations

_amongst

levels of random effects as in

_[14].

Extensions to

other models are

straightforward,

as

_long

as G can be

partitioned

into blocks of

such structure. For

_example,

fitting

permanent

environmental effects

_(c)

as well as

direct and maternal additive

_genetic

effects,

[14]

would be

_augmented

simply

by

(NC

log

Œð),

provided

c was uncorrelated to a and m. Table I summarizes

log

,C

for 10 models which may arise in the

analysis

of animal

breeding

data,

with up to 3 random effects and

involving

up to 5

(co)

variance

components.

Otherwise,

G

(or

using techniques

as described above for

log

!C*!.

For

instance,

if G contained a

block of form

the contribution to

_log

_IGI

would be

Assume

_{V(a )}

=

QA

A,

Cov(a,

c’)

=

Cov(m,

c’)

= 0 and

V(c)

=

o- c

I

for all models.

Terms are assumed to be the result of Gaussian Eliminations

performed

for M with

_aE

2 factored out.

Computational

Considerations

Typically,

the

_augmented

coefficient matrix M _{is very}

_large

but also very sparse. Hence use of sparse matrix

techniques, storing

the non-zero elements of M

_only,

is

_advantageous

and allows

_matrices

of order of thousands to be handled. Since M

is

symmetric,

only

the lower

_(or

_{upper) triangle}

is

_required.

One form of _sparse

matrix

_storage,

described in standard text books such as Knuth

(1973),

is a

so-called &dquo;linked list&dquo; . Such linked

_lists,

one list for each row of M in

_conjunction

with a vector

_pointing

to the first element in each _row, are well

_suited,

and allow

the Gaussian Elimination

steps

required

to evaluate

_{y!Py and}

_log

_{!C* !}

to be carried

out

_efficiently.

In

_setting

up

M,

the order of

_equations

can be of vital

_importance

as it affects the &dquo;fill-in&dquo;

_during

the

_absorption

_process, i.e. the number of additional non-zero

off-diagonal

elements

_arising.

For

_{computational efficiency}

this should be

kept

as small as

_possible.

There is extensive literature concerned with numerical

operations

on

sparse matrices. Tewarson

(1973),

for

example,

discusses

techniques

for the choice

of

_pivot

in each Gaussian Elimination

_step

which

_yields

the least local

_fill-in,

and also considers the _scope of a

priori

column

permutations.

A number of

strategies

for

re-ordering

matrices

_exists,

often

_{utilizing graph theory;}

see for instance Duff et al.

(1986).

Such

_{general techniques, making}

little or no

_assumptions

about the matrix

structure can be

_{computationally}

_expensive.

This _may be

_prohibitive

for situations

where the direct solution of a

_large

sparse

system

of

_equations

is

_required

a few

times

_only,

but _may be worthwhile for our

_application

where numerous likelihood

evaluations are to be

_performed.

Future research should consider this

_topic.

In the

_meantime,

critical

_inspection

of the data and

_relationship

structure with their

_implications

for the

_pattern

of

_off-diagonal

elements in the mixed model array,

and

_{judicious ordering}

of effects _may achieve a

_large

_proportion

of the

_potential

benefits from

_{general reordering algorithms.}

A standard

_strategy

in

_attempting

to minimize fill-in is to process rows with the fewest

_{off-diagonals}

first. Graser et al.

(1987)

therefore

_suggested

selection of

_pivots

_{corresponding}

to the

_youngest

animals first. For the models with several random

_effects

for each

animal,

these should be

assigned

to successive rows. In other _cases, it may be

_possible

to

_exploit

additional features of the data structure. For data from a

_{multi-generation}

selection

_experiment

with selection within

_families,

for

_{example, grouping}

of animals

_according

to female &dquo;founders&dquo; _appears

_preferable

to a

_grouping

_according

to

_generation.

On the

other

_hand,

if animals are nested within

_contemporary

_(fixed)

_{groups, it may} be

advantageous

to order

_equations

so that animals

_directly

follow their group effects.

For R of form

_(8],

_QE!’

_{is usually}

factored from

_(6].

In

this

case, calculations to

determine

_Y

_Py

and

_log

_[C

_*

as described

_above,

do not

_yield

the terms

_required

in

_[4]

_directly,

but

_(y

_p

_y

_!E)

and

_{(log IC}

_I

_*

+

_(NF

_*

_+NR)

_log

_U2

_),

which has to be

born in mind when

_assembling

the likelihood.

MAXIMIZING THE LIKELIHOOD

Choice of a

_strategy

to locate the maximum of the likelihood

function,

or

equiva-lently

the minimum of -2

_log

_G,

is determined

_by

several considerations.

_Firstly,

more

_demanding

than _any calculations

_{required by}

the

_{optimization procedure}

as

such.

_Hence,

a method which

_requires

a small number of function evaluations in

each iterate is desirable.

_Secondly,

the

_procedure

should be

_robust,

i.e. cope with

starting

values for

_parameters

_considerably

different from the eventual

_estimates,

and should be little affected

by problems

of numerical accuracy,

yielding sufficiently

precise

estimates of the minimum even for _very flat functions.

Thirdly,

constraints on the

_parameter

_space should be accommodated

_{and, preferably,}

not

_require

extra

function values or reduce the

speed

of convergence.

The

_suitability

of three different

_approaches

was examined

_using

simulated data

for models

_{1, 2,}

4 and 8 as

_specified

in Table I. Records were

sampled according

to

the model of

_analysis

for one or several

generations

(up

to

_four),

each

_comprising

a

_given

number of full-sib families

(ranging

from 25 to

₈₀₀₎

of variable size

₍₂

to

10),

with dams nested within sires and each sire mated to a

specified

number of

dams

₍₁

to

_5).

Error variances were estimated

_directly,

while all other

_components

were

_expressed

as a

_proportion

of the

phenotypic

variance, i.e.,

8

A

,

Om,

Oc

and

O

AM

for

_{a 2, 0fi, 02}

and QpM,

respectively. Obviously, B

A

is the

heritability

and

Oc

what is

_commonly

referred to as

&dquo;c

2

effect&dquo; . As described

above,

this reduced

the dimension of search to

_{1, 2,}

3 and 4 for Models

1,

2,

4 and

_8,

_{respectively.}

This

_{parameterization}

rather than

_expressing

components

as a

_proportion

of the error variance

(À¡)

was chosen since it allowed checks for

_parameter

estimates out

of bounds more

_{readily and,}

for the limited cases

_examined,

as it

appeared

to be

more robust

_against

bad

_starting

values.

Quadratic approximation

For a model with animals as the

only

random

effect,

Graser et

al.

(1987)

fitted a

quadratic

function in r =

0,2!la2 E to

the

log

likelihood,

predicting

the

maximum of

log G

as the maximum of this function. For one

parameter,

this

_required

function values for 3 different r values _per

_{approximation. Having}

calculated

log £

for 3 initial

points,

each iterate then involved one function

_evaluation,

for r* which maximized

the

_quadratic

function of the

_previous

step.

This value and those

_pertaining

to the two r values either side closest to r* were then utilized in

determining

the next

quadratic

approximation

to

_log

G. As

_{reported by}

Graser et al.

_(1987),

simulations for this model showed

_rapid

convergence. A bad initial guess for r

_generally

did not

affect the estimation

_{procedure greatly,}

as

_long

as the three

points

in the initial

approximation spanned

a

sufficiently large

range.

Though

the number of iterates

and likelihood evaluations

_required

tended to

_increase,

the same maximum of

log

G as for

_{&dquo;good&dquo;}

starting

values was attained without

_increasing

computational

demands

_excessively.

This

_approach

extends to the case of

multiple

_parameters.

For

_t,

with elements

Bi,

denoting

the vector of

_parameters

with

_respect

to which

_log

_G

is to be

_maximized,

and

_{log G (t)}

the

_{corresponding log likelihood,}

the

_{quadratic approximation}

is:

For p

parameters,

a total of z = 1 +

p(p +3)/2

different values of t and

log

G

_(t)

are

required

in each iterate to set _up and solve a

_system

of z

_equations

for the

_intercept

_q, the vector of linear coefficients _q and the

_symmetric

matrix

of

_quadratic

coefficients

_Q.

This number increases

_rapidly

with the number of

paramaters,

e.g, z =

_{6, 10,}

15 and 21 for

p =

2, 3,

4 and

5,

respectively.

For one

_parameter,

choice of the

_point

to be

_replaced

in each iterate was

straightforward.

In the multi-dimensional case,

however,

it was less obvious. Two

strategies

were

_explored.

After z initial

_points

had been

obtained,

the first

involved,

as for _p

=1,

in the

regular

case one function evaluation _per

iterate,

i.e. calculation

of

_log

G

_(t

_*

₎

for t* from the last iterate. This new

_point

was added to the set of z

points

which formed the basis to

_predict

t* in the

_previous

_step.

The worst of the

resulting

set of z + 1

_points

was then

eliminated,

and a new vector t* determined. If the

_quadratic

approximation failed,

i. e. if

_{log £}

_(t

_*

₎

was lower than all z function

values in the

set,

t* was

_{replaced by}

(t

*

+

) /

t

m

2,

where

t

nt

was the

_parameter

vector with

_highest

function value in the set. If _{necessary, this} was

repeated

until

the

_replacement

was successful.

_Hence,

each iterate increased the average likelihood

of the z current

_points.

The second

_strategy

_comprised

z function evaluations per iterate. Given a vector

of

_starting

values

to

(t

*

from the

previous

iterate),

np vectors

t

i

were derived

by

multiplying

the i-th element of

to

by

a factor

reflecting

a chosen

_step

_size,

1.10 for

steps

of

10%

in this case.

_Following

a scheme described

_by

Nelder & Mead

(1965),

further

_parameter

vectors were then determined as

(ti

+

_{tj )}

_/

2 for

_{i < j =}

0,

..., p.

This

_yielded

the

_required

total of z

grid

_points

and

subsequent

estimate t*. For

both

_strategies,

all vectors t were checked for elements out of the

parameter

space,

and if necessary these were set to their

_respective

bounds.

The

_quadratic

_{approximation}

_performed

well for Model

_2,

_though,

for the limited number of

_examples

_considered,

it was not

_consistently

better than the

two alternative

_procedures

studied,

in terms of the number of likelihood evaluations

required.

For Models 4 and

_{8, however,}

where the data structure was such that

only

a small

_proportion

of animals had direct information on the second

genetic effect,

problems

of numerical accuracy occurred. Often the

system

of z

_equations

to be solved was indeterminate or almost so.

Typically

this

yielded

non-positive

definite

estimates of

_Q

and useless

_predictions

of t*. For the second

_strategy,

an alternative

approach, slightly

more

robust,

was tried. This consisted of

_estimating

elements of

q and

Q by

numerical

differentiation,

i.e. as forward-difference

_{approximations}

to

the first and second derivatives of

_log

G,

respectively.

On the

whole,

quadratic

approximation

of the likelihood function

_involving

multiple

parameters

appeared

to be unsuitable as a

_general

search

procedure.

For a one-dimensional

search, however,

it

performed consistently

best among the 3

strategies

examined.

Quasi-Newton

Procedures which do not

_require

second derivatives of the function to be

_minimized,

but

approximate

the Hessian matrix

₍₌

matrix of second

_derivatives)

are referred

to as

Quasi-Newton

methods. This

approximation

is

usually performed iteratively,

vectors of

_changes

in

_gradients

₍₌

first

_derivatives)

and estimates between iterates. While most

_Quasi-Newton

_{procedures require}

first

_derivatives,

some are

derivative-free, approximating

the vector of

_{gradients using}

finite differences. These have been. found to show

_quadratic

convergence and are recommended as the derivative-free

methods to be used for smoth functions with continuous derivatives. Further details

are

beyond

the _scope of this _paper and can be found in standard

textbooks,

for

instance Gill et al.

_(1981).

Statistical

_library

subroutines to find the minimum of a function

_using

a

Quasi-Newton

_{algorithm, namely}

NAG routine E04JBF and IMSL routine

_ZXMIN,

have been

_applied

to -2

_log

₁

_:-.

These routines

require

the user to

_supply

a subroutine to

evaluate the function to be

_{minimized, passing}

the number and vector of

_parameters

as

arguments

and

_returning

the function value. In

_{addition, starting}

values for the

parameters,

the maximum number of iterates or function calls allowed and some

criteria of _accuracy of evaluation

_required

and

_rounding

errors tolerated have to be

specified.

E04JBF

_provided

the

_facility

to constrain

_parameters

between fixed _upper

and lower bounds

_(the

IMSL

_equivalent

ZXMWD was not

_tested),

i.e. 0 to 1

for

₀

_A

_{, 0}

_M

and

0c,

and -1 to 1 for

B

AM

.

However,

to

_impose

these

_constraints,

the routine

_required

function

values,

setting

all

_parameters

_{simultaneously}

to

their upper or lower limits.

Obviously,

this violated other restrictions on the

parameter

space in the

genetic

context,

i.e. that the sum of

_components

is bounded

correspondingly

(0 <

8

A

+

8

M

+

9

c

+

9

n

n,t <_

1),

and that the absolute value of

the

_genetic

correlation has a maximum of

_unity

(9A

M

<_

0

A

0

M

).

Consequently,

log G

could often not be determined for the

_required

constellation since M became

negative

definite or

Y

Py

assumed a

_negative

value. Hence minimization was carried

out unconstrained.

_Techniques

to

_implement

more

complicated

constraints

exist,

and further research should

_investigate

their

_suitabililty

for the kind of models which are

of

interest in animal

_breeding.

Unless a

_parameter

vector was encountered for which -2

_log

G could not be

evaluated,

the

_{Quasi-Newton algorithms performed}

well for all models examined. Each iterate

_required

_p

function

evaluations to

approximate

the vector of first derivatives. The number of iterates

_{performed depended}

on

_{user-specified}

criteria of accuracy and maximum number of function evaluations allowed. If likelihood functions were _wery

flat,

routines would

_stop

before the minimum of -2

_log

G was

determined as

_accurately

as

_{desired, flagging problems}

of numerical accuracy.

Figure

1 illustrates the

_typical

pattern

of

_changes

in likelihood and estimates

observed for an

_analysis

under Model 4 for a

_{&dquo;good&dquo;}

and a &dquo;bad&dquo; initial guess of

parameter

values. The simulated data for this

_{example comprised}

2

_generations

with 100 full-sib families of size 4 to 8 and 25 half-sib families each. Records were

sampled

for

_population

values of

_o-P

₌ 100

_{(phenotypic variance),}

9

A

=

0.50, Om

=

0.20 and IT AM = _-0.05.

Starting

values used were the

_population

values

(Set I)

and

ITA =

_{0.30, o-}

_M

= 0.15 and _ITAM = _0.05

_{(Set II),}

_{respectively.}

For Set

_I,

ZXMIN

required

93 likelihood

_evaluations,

for a

_given

_significance

level of 6

_significant

digits.

For Set

II,

however,

the routine used 204 function calls before it considered the maximum of -2

_log

G to be

_{found, although Figure}

1

_suggests

that likelihood

Simplex

The

_Simplex

method of Nelder & Mead

₍₁₉₆₅₎

is

_generally

advocated as the

derivative-free

_procedure

to use if the multivariate function to be minimized is

discontinuous,

though, initially,

it has been

_developed

with the maximization

of a likelihood function in mind. It relies on a

comparison

of function values

without

_attempting

to utilize any statistics related to derivatives of the function.

Such

_{optimization techniques}

are

generally

refered to as direct search

_procedures.

While

they

often have

been developed by

heuristic

_approaches

without

_proof

of

convergence,

they

have found to be effective in

_practice

_{(Swann, 1972).}

The

_Simplex

or

Polytope

method,

as some authors

prefer

to call it to avoid confusion with the

_Simplex

technique

used in Linear

_Programming,

was

_initially

suggested

by Spendley

et al.

_(1962).

It

_operates

on a set of

parameter

vectors and their

_pertaining

function

values,

which form the vertices of a

_simplex

in the

parameter

space, hence its name. As reviewed

_by

Swann

(1972),

it is based on

the

_conceptes

of

_{&dquo;evolutionary operations&dquo; , developed}

to

_optimize

_productivity

of industrial

_plants,

in which the

parameter

space is searched

following

some

_geometric

configuration.

The

_design

which

_requires

the least number of

_points

and hence

makes most efficient use of the function values

calculated,

is a

regular simplex.

This is defined

_simply

as a set of

_{mutually equidistant}

points,

n + for a

simplex

of

dimension n. For two

_dimensions,

for

_example,

the

_{regular simplex}

is an

equilateral

triangle.

A useful

_property

of a

_regular

is that a new

simplex

can be formed the

existing simplex by

addition of a

_single

new

_point.

The search

_proceeds

as follows. To

begin,

a

_simplex

of

specified

size is _{set up,}

including

the

_point

_representing

an initial guess for the

_optimum,

and

corresponding

function values are obtained. The aim in each iterate then is to

_replace

the worst

point,

i.e. for a minimization

problem

the

point

with the

highest

function. The new

_point,

defining

the next

simplex,

is chosen so as to preserve the

_geometric

shape

in a direction away from the discarded

point,

but

passing through

the center

repeated

until the

_simplex

straddles the

optimum.

Reducing

the size of the

simplex,

search then recommences with the new, smaller

design,

or terminates if the

simplex

becomes smaller than a

_specified

limit

(Swann 1972).

Subsequently,

the

_Simplex

method has been modified

_by

Nelder & Mead

_(1965).

Abandoning

the

_regularity

of

_design,

their

_procedure

allows the

_simplex

to rescale

itself

_{automatically}

in each

_iterate,

_changing

_shape

and size

_according

to the

landscape

of the surface searched. This

_adaptability

is achieved

_by

a combination

of so-called

_reflection,

_expansion

and contraction steps.

Consider a function

_F=f(t)

to be minimized with

respect

to _p

_independent

variables,

and let

_to

_{denot the guess} for the

_optimum

to start with. The initial

simplex

then

_comprises

to

and _p other

_points

t

i

(i

=

1,

..., p)

obtained

by

modifying

one co-ordinate of

_to

at a time

_by

a chosen

_step

size. Each iterate

_{begins by ordering}

and

_renumbering

the

_points

in the

_{simplex according}

to the

_pertaining

function values. The

_following

three

_points

are identified:

_to

with function value

_F

_o

is now

the

_currently

best

_point

in the

_simplex

_(F

_o

<

_F.t,

i =

1,

..., p),

tp

with function value

Fp

is the worst

_point

_(Fp

>

_Fi,

i =

_{0, ..., p-1)}

which is to be

_replaced

in this

_iterate,

and

_tp-,

is the next to worst

_point

_{(Fp_}

₁

<

_Fp

and

_{F!,_1}

>

Fi,

i =

_{0, ..., p -}

_2).

Next,

the center of the

_points

to remain in the

_simplex

is obtained as:

i.e.

_{by averaging}

each coordinate over the set of

_points.

A trial

_point

is then

generated by

&dquo;

reflecting&dquo; tp

towards the center:

where the reflection coefficient a is a

_positive

constant. The

_{corresponding}

function

value

_F,.

is

_compared

with

_F

_o

to

_Fp

to determine further

steps

in the iterate. Three

possible

outcomes are

_{distinguished.}

Firstly,

t

r

can be better than the worst

_point

but not better than the best

_point

(F

o

<

_F

_r

<

_Fp).

In this _case,

_t

_r

_{replaces tp}

and a new iterate is started.

_Secondly,

if

t,.

is the new best

_point

(F,.

>

_F

_o

_),

it is assumed that the direction of search has

been a

_good

one and search is

_{&dquo;expanded&dquo;}

in this direction

_by

_examining

a

_point

with

_{corresponding}

function value

_F

_e

_.

The

_expansion

coefficient ₇ is

_positive

constant. If

_F

_e

is less than

_F

_r

_,

the

_expansion

has been successful and

_t

_e

_replace

tp.

Otherwise

t

e

is discarded and

t,.

is substituted for

_tp

to

_complete

the iterate.

Thirdly,

t

r

may be worse than _any of the

_remaining

_p

_points

(F

r

>

_{Fp_}

₁

_).

This leads to &dquo;contraction&dquo; of the

_{simplex, obtaining}

a new

_point

_t

_e

with

_pertaining

function value

F,,

as:

if

_F

_T

is less than

_Fp,

and as

otherwise. The contraction

_{coefficient 0}

is a constant in the _range from 0 to 1. If

F

e

is less than the smaller of

_F

_r

and

_F

_N

_,

the contraction has been successful and

tc.

replaces tp

to end the iterate. If

_not,

the

_complete

simplex

is shrunk

_{by moving}

for i =

1, ..., p,

before _{a new} iterate is started.

Suggested

values for

a,

(

3

and 7

are

1.0,

0.5 and

2.0,

respectively

(Nelder

& Mead

1965).

Full

computational

details

_together

with a flow

_diagram

can be found in Nelder & Mead

(1965),

and a

FORTRAN

_{implementation}

for

_example

in O’Neill

_(1971).

Since differences in function values are not utilized in

_establishing

the direction

of

search,

restrictions on the

_parameter

_space can be

imposed easily by assigning

a

very

large

positive

function value to

parameter

vectors out of bounds. As

_pointed

out by

Nelder & Mead

_(1965),

this will be followed

_{automatically by}

a contraction

step

which will

_{eventually keep}

estimates within their bounds. The convergence criterion

_{suggested by}

Nelder

&

Mead

₍₁₉₆₅₎

is the

_variance,

or standard

_deviation,

of function values in the

_simplex,

rather than the more

_{conventionally}

used

_change

in estimates between iterates. The rationale for this was that in statistical

_problems

such as maximum likelihood

_estimation,

the curvature of the surface near the

minimum reflects the amount of information available on the

_parameters.

If the

surface is

_flat,

_sampling

errors are

_large

and it is not worthwhile to determine the

minimum with

_great

accuracy.

For REML estimation of

_multiple

variance

components,

the

_Simplex

method

proved

robust and easy to use. For a

_given

vector of

_{starting values,}

_to,

the initial

simplex

was made up of

to

and p vectors

t

i

,

obtained

_{by multiplying}

the i-th

element of

_to

_by

1.20. A factor this

_large,

_{corresponding}

to a

_step

size of

20%

was chosen to ensure

quick

_convergence even for bad

_starting

values. This

implied,

however,

that for

_starting

values close to the

_estimates,

the

_procedure

would search

unnecessarily

in the wrong direction. A smaller

step

size _may be sufficient and

perhaps

more eflicient i.e.

_yield

estimates with less likelihood evaluations. For cases where

_parameter

vectors out of bounds were not a

_problem,

the

Quasi-Newton

algorithm performed generally

somewhat better than the

_Simplex

method. The variance of function values in the

_Simplex

had to be less than

10-

8

or even

10-

9

for

the

_Simplex

to find the minimum of -2

_{log £}

with the same _accuracy as ZXMIN

(for

an _accuracy level of 5 or 6

significant digits).

In terms of

_changes

in

_parameter

estimates,

however,

a limit of 10-5 to 10-6

_{generally appeared}

sufficient.

Figure

2 illustrates the convergence behaviour of the

Simplex

method for the

same

_analyses

as

depicted

in

_Figure

1. Oscillations in likelihood and estimates

clearly

reflect the &dquo;trial and error&dquo; mechanism of this

_approach.

For both sets of

_starting

values,

88 likelihood evaluations were

required

before the variance of

function values in the

_{Simplex dropped}

below the

_specified

limit of

10-

9

.

SAMPLING VARIANCES

The matrix of

_approximate,

_large

sample

covariances among variance

component

estimates is

_{given by}

the inverse of the information matrix. This in turn can

be

_{approximated by}

the inverse of the Hessian

_matrix,

i.e. the matrix of second derivatives of the

_log

likelihood function with

respect

to the

_parameters

to be estimated. A

_quadratic

_{approximation}

of

_{log £}

and

_{Quasi-Newton procedures}

provide

an

_approximate

Hessian matrix

_(Q)

as a

_{by-product. Using}

the

_Simplex

method this has to be obtained

_explicitly.

Nelder & Mead

₍₁₉₆₅₎

described an

using

forward differences

_(see

_e.g. Gill et

al.,

1981).

While the mechanics for

approximating

second derivatives are

_{straightforward,}

it can be

problematic

in

practice.

As noted when

_examining

the

_quadratic

_{approximation}

of

_{log G}

for

_multiple

parameters,

Q

was often not

_positive

_definite,

whether derived

_{by least-squares}

as

described

_by

Smith & Graser

₍₁₉₈₆₎

or

_using

finite

_differences,

and

consequently

yielded

inappropriate

predictions

of the

_parameters

_{maximizing log}

G.

_Similarly,

the

_approximate

Hessian

_produced

_{by Quasi-Newton}

routines did not

_necessarily

give

meaningful

estimates of

_sampling

variances.

Indeed,

a

_{large proportion}

of the

literature on

Quasi-Newton

_algorithms

is concerned with

procedures

to deal with

bad or non

_positive

definite

_{approximations}

to the matrix of second derivatives. Simulation was

employed

to examine the

_sampling

distribution of variance

component

estimates and their

_predicted

variances for models

_1,

2 and 4. Data

were

sampled

for one to four

_generations

_consisting

of 25 to 800 full-sib families of

size 2 to 10 each. Dams were nested within sires with each sire mated to 1 to 5

dams. Variance

component

estimates were obtained

_using

the

Simplex

method with

population

parameters

as

_starting

values and a _convergence criterion of

V(-2

log

G)

<

10-

8

.

Second derivatives were

_approximated

as described

by

Nelder & Mead

(1965)

using

a

_step

size of

0.1% of

the estimates.

Approximation

of

_sampling

variances worked well for

_{model 1,}

i.e.

_estimating

the additive

_genetic

and error variance

_only.

It was

_satisfactory

for model 2 which

included a additional environmental

_component

due to full-sib families

_(litters),

but it failed for

_analyses

under model

_4,

_including

a maternal

_genetic

effect as an

additional random effect for each animal.

Table II summarizes

_empirical

and

_{predicted sampling}

variances and

_empirical

correlations between estimates for a

_variety

of

design

for model 2. Results

clearly

emphasize

the need to ensure that the data

_provide

sufficient information to

high

negative

sampling

correlation between

_a

₂

_and

_{<7!. or,}

_{equivalently,}

between

heritability

and

c

2

effect. In terms of the likelihood

_surface,

this

_implied

a maximum

along

a flat

_ridge,

i.e. an area where for a constant sum of the two

_parameters

the value of the likelihood

_changed

_very little with

_changes

in the

parameter

values. If each sire was mated to

_only

one

_dam,

i.e. there were no half-sib

families,

there was

little scope to

_partition

the within

_family

variance into its

_genetic

and environmental

components.

This

_yielded

a likelihood surface of a

_shape

which did not allow

sampling

variances to be

_{approximated.}

Including

data for a second

_{generation provided}

the covariance between

_parents

and their

_offspring

as an additional source of

information,

thus

_allowing

a

consid-erably

better discrimination between

₀

_&dquo;1

and

_{o- c 2}

as evidenced

by

the decreased

(absolute value)

sampling

correlation between the two

_components.

While for one

generation

sampling

variances decreased with

increasing

number of dams per sire

the reverse

_generally

held for data

_spanning

several

_generations.

These

_{relationships}

are also illustrated in

_Figure

3 for

_analyses

under Model 4.

Considering

one

_generation

_only

_{(top row),}

no animal had records

_expressing

both

its direct and maternal

_genetic

effects. Hence the

_sampling

correlation between

0

&dquo;1

and

_am

₂

was

_large

and

_negative.

This was

_{accompanied by}

little variation in

estimates of QAM

depicting

that this was determined

_by

the

_genetic

covariance structure _{among animals}

_only.

For data

_including

3

_generations

_{(bottom row)}

sufficient

_comparisons

between and within

_generations

were available to

_yield

estimates of

_Q

_A

and

_{am 2}

_{virtually independent}

of each

other,

while estimates of 0

&dquo;

AM were

highly

variable and showed a

_strong

_negative

association with

or M, 2

NUMERICAL EXAMPLE

Table III contains simulated data for 282 animals in two

_generations.

Each genera-tion consists of 18 full-sib families of size 6 to

_10,

where each sire is mated to 3 dams. Parents of

_generation

one did not have

records,

which introduced 24 base

_animals,

yielding

306 animals in total. Records were

_{sampled according}

to Model 8

_(see

Ta-ble

_I)

for

_population

values of

_u

_£

=

_40,

u

%

=

_15,0&dquo;

_AM

=

_-5,

u$

= 10 and

U2

=

50,

a

_phenotypic

mean of 200 and a fixed effect of 20 for

_generation

2.

Data were

_analysed

under Models

_1,

_{2, 3, 4,}

7 and

_8,

as described in Table I.

Analyses

were carried out

_using

the

_Simplex

method to find the maximum of the likelihood. Two sets of

_starting

values were

_chosen,

_namely

the

population

values as a

_good

_(Set

I:

_B

_A

_{=0.40, 0Nj}

=

0.15,

O

AM

= -0.05 and

Oc

=

0.10)

and

B

A

=

0.10,

om

=

_{0.30, O}

_AM

=

_{0.10, Oc}

= 0.20

(=

Set

II)

_{as a} bad initial guess. For models other

than

8,

the

_appropriate

subset of

_parameters

was used. Estimates for two values for

the minimum variance of function values in the

_Simplex

are summarized in Table

IV. Characteristics of the

_augmented

coefficient matrix M and intermediate results for the first likelihood evaluated in each run are

_given

to

_help

with the validation of

_computer

_programs. Gaussian elimination

_steps

are not

_dependent

on the model of

_analysis,

hence the

_{example given by}

Graser et al.

₍₁₉₈₇₎

should suffice as an

illustration. Numerical

_examples

to be used in

_checking

the

_building

of the mixed model _array for the various models can be found elsewhere in the literature. In