Computing approximate monogenic model likelihoods in large pedigrees with loops

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Submitted on 1 Jan 1995

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Computing approximate monogenic model likelihoods in

large pedigrees with loops

Llg Janss, Jam van Arendonk, Jhj van der Werf

To cite this version:

Original

article

Computing approximate monogenic

model likelihoods in

_large

_pedigrees

with

_loops

LLG Janss

JAM

Van

_Arendonk,

JHJ Van der Werf

Wageningen Agricultural University, Department of

Animal

_Breeding,

PO Box

_338,

6700 AH

_Wageningen,

The Netherlands

(Received

30

January

1995;

accepted

11

September

1995)

Summary - In this

_study

’iterative

_peeling’

is

_introduced,

a method

_equivalent

to

the traditional recursive

_peeling

method for

computing

exact likelihoods in

_nonlooped

pedigrees,

but which can also be used to obtain

approximate

likelihoods in

_looped

pedigrees.

Iterative

peeling

is an

_interesting

tool for animal

_breeding,

where exact recursive

peeling

is

_generally

unfeasible due to the abundant number of

loops

in animal

pedigrees.

In

_simulations,

_{hypothesis testing}

and _parameter estimation were

compared

based on

approximate

likelihoods in

_{looped pedigrees}

and exact likelihoods in

_nonlooped

_pedigrees,

showing

no biases introduced

_by

the

_{approximation}

in

_{looped pedigrees.}

likelihood

_/

pedigree peeling

/

major gene

/

looped pedigree

Résumé - Calcul approximatif de vraisemblance pour un modèle monogénique dans de grands pedigrees à boucles. Dans cette étude on introduit une

_procédure

itérative

de condensation de

_{l’information}

contenue dans un

pedigree, appelée « épluchage »,

qui

est

équivalente

à

_{l’épluchage récursif pour}

le calcul des vraisemblances exactes dans des

pedigrees

sans

_boucles,

mais qui est

_également

utilisable pour le calcul de vraisemblances

ap-proximatives dans les

_pedigrees

à boucles.

L’épluchage itératif

est une méthode intéressante en

_génétique

animale où la méthode récursive exacte est

généralement inapplicable

à cause du

grand

nombre de boucles dans les

pedigrees

animaux.

À

l’aide de

simulations,

on a

comparé

des tests

_{d’hypothèse}

et l’estimation de

paramètres

basés sur des vraisemblances approximatives dans des

_pedigrees

à boucles et des vraisemblances exactes dans des

pedi-grees sans

_boucles,

montrant

qu’il n’y

a _pas de biais introduit _{par le} calcul

approximatif

dans des

pedigrees

à boucles.

INTRODUCTION

Research into the use of

_major

_gene models in animal

_breeding

has been aimed

mainly

at

_{approximations}

to a mixed inheritance

_model,

_{including polygenes,}

in

one

_generation

half-sib structures

_(Hoeschele,

_1988;

Le

_Roy

et

_{al, 1989;}

Knott et

_al,

1992).

Because of the

_{pedigree loops}

that arise in animal

_{breeding situations,}

ex-tension to

_{multigeneration}

_pedigrees

is difficult. A

_{pedigree loop}

arises when 2 individuals are connected

_by

more than one

_path

of descent or

_marriage

relation-ships.

Lange

and Elston

₍₁₉₇₅₎

described various

types

of

loops,

among which are

inbreeding

loops, marriage rings

and

_{marriage loops.}

In animal

_breeding

_pedigrees

these kinds of

_loops

are _very common. In

_{particular, multiple matings}

which are

generally

applied

to males and often to

_females,

result _{in many}

_marriage

_loops

and

marriage rings.

For

_genotype

_probability

and likelihood

_{computation, loops}

can

only

be dealt

with in an exact manner in

_pedigrees

with a few

_{simple non-overlapping loops using}

the traditional recursive

_peeling

method

_(Elston

and

_{Stewart, 1971;}

_Cannings

et

_al,

1976;

Cannings

et

_al,

_1978).

_However,

in

_{highly looped}

_pedigrees,

common in animal

breeding,

exact recursive

_peeling

is too

_demanding

_{computationally}

and recursive

peeling

is not flexible

_enough

to allow for

_approximate

_{computations.}

In this

_study

we introduce ’iterative

_peeling’.

Iterative

_peeling

is

_developed

as

an exact method for

application

in

_{nonlooped pedigrees, equivalent}

to recursive

peeling,

but

_which,

unlike the

_original

recursive

_variant,

can be used without

modifications in

_looped

_pedigrees

to obtain

_approximate

likelihoods. The main

objective

of this paper is to introduce iterative

_peeling

for such

_{approximations}

in

_looped

_pedigrees,

_allowing

for a more

_{general application}

of

_major

gene models in animal

_breeding.

_{Using simulations,}

the usefulness of the

_{approximation}

for likelihood-based

_{hypothesis testing}

and

_parameter

estimation in

_looped

_pedigrees

is

_{investigated.}

A

_monogenic

model will be

_considered,

which can be extended to a

mixed inheritance

model,

as will be discussed.

RECURSIVE AND ITERATIVE PEELING

In the first

_section,

recursive

_peeling

is described for

_{obtaining monogenic}

model likelihoods in

_{nonlooped pedigrees.}

In the second

_section,

’iterative

_peeling’

is introduced as an

_equivalent

method for exact

_computations

in

_nonlooped

_pedigrees.

The

_equivalent

exact method in

_nonlooped

_pedigrees

can be used as an

_approximate

method in

_{looped pedigrees.}

Recursive

_peeling

Probability

and likelihood

_computations

in

_nonlooped

_pedigrees

can be done

_by

recursive

_peeling

_(Elston

and

_{Stewart, 1971;}

_Cannings

et

_al,

_1976;

_Cannings

et

_al,

1978)

using

2 basic

_{peeling operations}

of

_{’peeling up’}

and

_’peeling

down’.

Roughly,

considering

a

_{single family,}

a

_peel-up

_operation

_represents

the information in a

family

in

_{probabilities}

for the

_genotype

G

i

of a

_parent

_i,

and a

_peel-down

_operation

of the

peel-up operation

is denoted

_by

_prog(G

_i

₎

and the result of the

_peel-down

operation

is denoted

_by

_prior(G

_k

_).

The

_{corresponding}

notation in

_Cannings

et al

(1976, 1978)

is the

_R

_*

_(..;G

_i

₎

function for

_peeling

up and the

R+(..;G!)

function for

peeling

down.

Peeling operations

are used

_recursively,

_eg, the

_computation

of a _prog term for

a

_parent

based on _progeny data _may include

_{previously computed}

_{prog terms} of

those progeny,

representing

information from

_{grand-progeny.}

The aim of

_peeling

is

to condense all information from a

_pedigree

into a

_prior

and _prog term for a

single

individual

1, obtaining

the likelihood L for all data in the

_pedigree

as:

where

_{f (y, I G}

_i

₎

is the

_penetrance

_function,

which is the

_probability

for the observed data _yl on

_{individual l, given}

that it has

_{genotype G}

_i

_.

The individual

_may

be an

individual from the base

_population,

in which case the

_{base-population}

_genotype

frequency

P(G!)

is used in

_place

of

_prior(G

_i

_).

Individual

_{l may}

also have no own

data or no _{progeny, in} which case the

_{corresponding}

_penetrance

term or _prog term

is removed.

_{Computationally}

this is

_{implemented using}

a

_penetrance

or _prog term

containing

l’s.

Peeling equations

A

_peeling

_equation

for an individual is obtained

_by

_considering

the collection of

possible base-population

genotype

frequencies,

genotype

transmission

_{probabilities,}

penetrance

probabilities

and other

_peeling

terms

_pertaining

to the individuals in its

family

and

_summing

over all

_possible

_genotypes

of the

_family

members. The terms

thus

_entering

a

_{peeling equation}

are difficult to

_give

in

_{general. Here, equations}

will be

_given

to use

peeling

in a

_pedigree

structure with dams nested within sires. In this structure a

_family

is a half-sib

_family

of one sire with several

_mates,

_containing

groups of full sibs which are, across _groups,

_paternal

half-subs. Three different

peeling equations

are considered: 2 for

_peeling

_up,

_dependent

on whether this is

done for a sire or a

_dam,

and 1 for

_peeling

down. In the

_peeling

_equations,

_prior,

prog and

penetrance

functions on

family

members are

specified

in all

_places

where

they

can enter. When these are not

_relevant,

_eg, when a _progeny does not have

progeny of its own, these are removed _or,

computationally,

terms

_containing

l’s are used. Prior terms for individuals in the base

_populations

are substituted with

base-population

genotype

frequencies.

To condense all information in a _prog term for a sire i the

_following

_expression

is used:

prog(Gi) = r

l

jrgj

pr2or(Gj) fly7lC-T7)Hk!Gk

Pl!’klGi,

Gj)

f(

y

kl

-G

k

)

!r!9(CTk) [2]

where j

=

1 to ni are mates of

_i,

each mate

_having

k = 1 to

n

ij progeny, and

P(Gk !Gi, G! )

is the

_genotype

transmission

_probability

of sire i and a

_{dam j}

_to

offspring

k. To condense all the information from a half-sib

family

into a _{prog term}

where i is the sire of the

_family,

_prog-j.

₎

_i

_(G

is like in

_equation

_[2],

but

_excluding

dam

_j

_*

and k =

1,

n2!* are _progeny of dam

j

*

.

To condense all the information in

a

_prior

term for 1

_particular

_progeny with dam *k

_j

_*

_,

the

_following

_expression

is used:

where i is the sire of the

_family,

_phs(G

_i

₎

is a term that includes information on

the

_paternal

half-sibs of

_k

_*

_,

which is a function of the

genotype

of its sire i and is

computed

as:

Iterative

_peeling

Iterative

_peeling

is

_equivalent

to recursive

_peeling

used in

_{nonlooped pedigrees.}

Iterative

_peeling

is based on

_{algebraic partitioning}

of the likelihood and on

_repeated

computation

of

_peeling

_equations,

based on the idea of iterative

_computation

of

genotype

probabilities

(Van

Arendonk et

_al,

_1989).

Partitioning

of likelihood

The aim of

_obtaining

the likelihood of all data

_{using equation}

_[1]

_requires

families

to be handled in a certain order and

_requires

_peeling,

within each

_family,

to be

in a certain direction.

_{Peeling operations}

can be used to

_partition

the likelihood

pertaining

to

_parts

of the

_pedigree.

This

_partitioning

is continued until

_parts

are

obtained

_pertaining

to

_single

families. This allows a

_family-wise

evaluation of the

likelihood,

and the

_requirement

of

_peeling

to have a direction within each

_family

becomes obsolete.

Consider the

_pedigree

with 5 individuals in

_figure

1. In this

_pedigree

2 families

are

_present,

one

_family

with individuals

_1,

2 and

_3,

and a second with individuals

_3,

4 and 5.

_Here,

one

_partitioning

above and below individual 3 divides the

_pedigree

in

2

_families,

with individual 3

_being

in both families. Individual 3 is called a

_linking

individual. The likelihood for a

_{monogenic model, assuming}

data is available on all

5

_individuals,

is

_computed

as:

Now,

L is

_multiplied

and divided

by Li =!1!2!03

P(Gi) P(G

2

)

P(G

3

) Gi , G

2

)

!(

Y1

IG

1

where the

part !01!02

P(G

i

)

P(G

2

)

P(G

3

) Gi ,

)

G

2

!(

Y1

IG

d

f(y

2

) G2 )

has been

iso-lated. This

_part

is

_prior(G

₃

_).

The term defined as

_L

₁

can be rewritten as

E

G3

I;

G1

I;c

2

P(G

1

) P(G

2

) P(G

3I

G

1

, G

2

) !(

Y1I

G

1

) !(

Y2I

G

2

),

which is

_{I;c3 prior(G3).}

This

_simplifies

L to:

where

_prio&dquo;

_sC

_{( G}

₃

₎

stands for a

scaled,

or

normalised, prior

term. Now the likelihood can be written as L =

L

1

L

2

,

or

ln(L)

=

ln(L

i

)

₊

ln(L

2

),

with one likelihood term

per

family.

This is a

partitioning using

a

_prior

term for the

_linking

individual. It shows that for this

_type

_partitioning

_(i)

in the

_family

where the

_linking

individual is

a _progeny, after the

partitioning,

information on the

linking individual,

ie own data and progeny

data,

is

ignored;

and

(ii)

in the

family

where the

linking

individual is

a

_parent,

a scaled

_prior

term is used for the

_linking

individual. This term is used

in a manner like a

_{base-population}

_genotype

_frequency

for base individuals. The

scaled

_prior

term for a

linking

individual

1,

is

computed

in

general

as:

Although

the

_partioning

is shown

only

for 1

_example,

the

_partitioning

is _very

general.

The term

_L

₁

above is in

_general

the sum of the

_prior

term for a

_linking

individual

1,

which is the collection of all

_probability

terms

_pertaining

to anterior individuals of and the transmission

_probability

to

_l,

summed over all

_possible

genotypes

of l and of its anterior individuals. At the same time this term

_represents

the likelihood of the entire anterior

part

of the

_pedigree

and

l, excluding

data on

l. The

_remaining

_part

after the

partitioning,

L

2

in the

_example,

is the likelihood of the

posterior

part

of the

_pedigree

of

l, including

l with a scaled

prior

term. In

larger pedigrees

this

partitioning

is

_repeated

to

_yield

_parts

_{corresponding}

to

_single

families. When

_repeating

the

_{partitionings,}

results of earlier

partitionings

must be taken into

_account,

_eg, the result that after a

_partitioning

information on a

_linking

The likelihood of a

pedigree

can be

_{partitioned entirely using prior}

terms.

However,

the iterative

_computation,

as will be introduced

_hereafter,

can be

_speeded

up

by

also

using

a

_partitioning

of the likelihood

using

a _{prog term.}

_Showing

this based on the

example,

the likelihood L is

_multiplied

and divided

_by

a term

representing

the likelihood of

_family

_2,

_ignoring

data on individual

_3,

_L2

=

E

G3

E

G4

E

G5

P

(Ga) P(G

5I

G

3

,

Ga) !

(

Y3I

G

3

)

/(!!G4),

which leads to:

2

Here a term

_E

_G4

_E

_G5

_P(G

₄

₎

_P(G

_5l

_G

₃

_,

₎

_G

₄

_!(

_Y4I

_G

₄

₎

_!(

_Y5I

_G

₅

₎

has been

_isolated,

which is

_prog(G

₃

_).

The division

_by

_L2

_scales

this

_term,

_L2

_being

_E

_G3

_!rog(G3).

Hence,

L is written as:

where

_prot

_C

_{( G}

₃

₎

denotes the scaled or normalised _prog term. For a

_partitioning

using

a _prog term it is seen that

(i)

in the

_family

where the

_linking

individual is

a _progeny, a

_prog

_s

_’

term is added as information for the

_individual;

and

_(ii)

in the

family

where the

linking

individual is a

_parent,

all information from observations

and from

_prior

terms is

_ignored.

The scaled _prog term for a

_linking

individual l is

generally

computed

as:

Partitioning

in a nested

_design

In a nested

_{design, partitionings}

are carried

_through

until

_parts

are obtained

corresponding

to sire families. In such

_families,

several female

parents

can be

present.

The

_linking

individuals are all the sires and dams of the

families,

_except

when

_they

are in the base

_population.

In this

_design

we consider a

_partitioning

using

a _prog term for each male and a

_prior

term for each female that is a

_linking

individual. When all

_parents

of a

family

are in the base

_population,

the

_part

of the

likelihood

_pertaining

to such a

_family

is

_computed

as:

where i indicates the sire of

_family

_{s, j}

sums over the dams of the

_family,

k indicates

male _progeny that are

_linking

_individuals,

1 indicates female _progeny that are

_linking

individuals and m indicates all other _progeny. When the sire of the

_family

is not in

the base

_population,

the term

_P(G

_i

₎

_{f (y2!Gi)}

on the first line of

[5]

is removed and

of

_[5]

is

_replaced

with

_{priof’c (G j)}

_’

The considered

_{partitionings using}

_prog terms

for all male

_linking

individuals lead to this removal of information from sires on the

first line of

_[5]

when sires are not in the base

population

and lead to the inclusion of the

prog

sc

for males on the third line of

_equation

!5!.

The considered

_{partitionings}

using prior

terms for all female

linking individuals,

lead to the inclusion of a

_priors!

term on the second line of

[5]

when dams are not in the base

_population

and the removal of all information of females on the fourth line of

_equation

!5!.

Based on

the results from the

_previous

_paragraph,

the likelihood of the entire

_pedigree

after the

_{partitionings}

is:

Repeated computation

of

_peeling

_equations

Iterative

peeling

uses

_repeated

_computation

of

_{peeling equations.}

The

_repeated

computation

is a method to establish the order in which

_equations

should be handled.

Therefore,

iterative

_peeling

does not

_{require knowledge}

of such an order

beforehand,

as is

required

for recursive

peeling.

For each individual a

_prior

and a _prog term is

_computed

and remains stored because results of

_peeling

terms can be

_required

as

_input

for the

_computation

of

other

peeling

terms. Iterative

_peeling

_computes

a series of solutions

_{priorlo, rtorlil,}

etc,

for these terms.

_Starting

values are taken for individual i as

pr!or!(Gt) =

P(G

i

),

the

genotype

frequencies

in the base

population

and

_I

_l

_°

_prog

_(G

_i

₎

_equals

1 for all

_G

_i

_.

Iterative

_computation

starts

_{by computing}

prior

[l

] (G

i

)

for each individual

i,

in order of

descending

age. Evaluation of these

prior

terms is based on

prior!l!

terms

of

_parents,

which are available because older individuals are

_updated

before _younger

individuals,

and on

prog

lol

terms of sibs.

_{Subsequently,}

_{proglo (Gi)}

is

_computed

for each individual

_i,

in order of

_ascending

_age. Evaluation of these prog terms is

based on

prior!l!

terms of

_mates,

on

prog[l]

terms of _progeny, which are available

because now _younger individuals are

updated

before older

individuals,

and for

female

parents,

on a

prog

iol

or

prog

[l]

term of their male mate. Whether this last

term is

_{already updated}

as

prog

[l]

depends

on the order in which _prog terms are

computed.

After

_computation

of all

_prior!l!

and

_prog

_[l]

terms is

_completed,

a new

iteration starts

_computing

prior

[2

]

and

_prog!2!,

etc.

Starting

values are such that

prior

lol

terms are correct for all individuals in the base

_populations,

and

_prog

_lol

terms are correct for all individuals without progeny.

Terms that can be correct after the first

_cycle

of

_computations

are, for

instance,

prior

l

1]

terms of individuals

_descending

from 2 base individuals and

_prog

_[l]

terms

of

_parents

without

_{grandprogeny.}

Correct

_computation

of a term is shown when in

the next

_{cycle recomputed}

terms are

_equal

to old terms. Once it is found that a

term is

_{correctly computed, recomputation}

can be omitted in

_following

iterations

of the

_algorithm.

The order in which terms are found correct

_gives

information on

the order in which recursive

_peeling

could be used.

_Generally,

in each

_iteration,

reasonably large

groups of terms _appear correct,

keeping

the number of

_cycles

required

to

_compute

all terms

_correctly

_{reasonably small, typically}

about the number of

_generations

in the data set. When all terms are found

_correctly

_computed,

Application

in

_looped

_pedigrees

The series of solutions

_prior

_f

_°

_]

_,

_prio!ll,

_etc,

obtained with iterative

_peeling

can be

considered as

_temporary

solutions for the

_required

terms,

corresponding

to solutions based on a not

_yet

_fully

determined

_peeling

order.

_{’Temporary’}

likelihoods can

also be

_{computed using}

_[5]

and

_[6]

based on a not

_yet

_fully

determined order. In

_nonlooped

_pedigrees,

a

_peeling

order can

_eventually

be found and

_temporary

solutions become exact. In

_looped

_pedigrees,

a

_peeling

order for recursive

_peeling

cannot be determined. In the iterative

_{peeling algorithm}

the

_{impossibility}

of

_finding

a

_peeling

order in

_{looped pedigrees}

is shown

_by

_continuing

_changes

in

_peeling

terms. In

_{looped pedigrees,}

these

_changes

were found to decrease in size

quickly

and

temporary

likelihoods were found to

_{stabilise, supplying}

an

_{approximation.}

Because

in iterative

_peeling

_every

_following

_update

of terms includes information from

50%

less related

_individuals,

a

_geometric

rate of _convergence is

_plausible.

As a

_stopping

rule to use the

_{approximation}

in

_looped

_pedigrees,

we used the _average absolute

difference between

subsequent

normalised

_{heterozygote probabilities,}

based on

computed

peeling

terms. For

_convenience,

_only

the

_heterozygote

_probability,

which

changed

the

_most,

was monitored.

SIMULATION STUDY

Application

of iterative

_peeling

to obtain

_approximate

likelihoods in

_looped

pedi-grees was the aim of this

_study.

Simulations were therefore

_performed

to

_investigate

the usefulness of this

_{approximation.}

Because exact

_computations

are unfeasible in

large looped

pedigrees,

approximate

likelihoods could not be

_compared

with exact

ones.

_Hence,

an indirect way to

study

the

approximation

was found

_{by studying}

the distribution of test statistics and of

_parameter

estimates over a number of

replicated

analyses

in

_looped

as well as in

_{nonlooped pedigrees.}

In

_{nonlooped pedigrees}

exact

likelihoods could be

_{computed, serving}

as a reference. Simulations and

_analysis

are

based on a biallelic autosomal locus and a normal

_penetrance

function. Simulated data

Data sets had a nested structure each

_generation,

with full-sibs nested within

paternal

half-sibs. Three different data structures were used

_{(table I),}

1 structure

without

_loops

and 2 structures with

_loops.

The data structures were

_designed

to

contain

_{approximately}

the same number of

_{observations,}

the same number of base

individuals

_(structure

1 vs

₂₎

and the same

family

sizes

(1

vs

3).

In structures 2 and

3,

the third

_generation

was

_{produced by taking}

1 son from each sire and 1

_daughter

from each

_{dam, maintaining}

the same

_breeding

structure across

_generations.

No directional selection was

_practised,

and

_breeding

females for a male were each

taken from a different

_sire-family.

Half- and full-sib

_matings

were

_avoided,

so that

inbreeding

was absent within the 3

_generations

considered. The additional third

generation

in structures 2 and 3 caused _many

_{pedigree loops}

in the form of

_marriage

loops.

All individuals used for

_breeding

the last

_generation,

ie 120 for structure 2

and 60 individuals for structure

_3,

were involved in 1 or more such

_loops,

often

Genotype

G

i

of an individual

_equals

_1,

2 or 3

_{corresponding}

to

_genotypes

_A

₁

_A

_1’

A

l

A

2

and

A

2

A

2

at an autosomal locus.

_Genotypes

for individuals in the base

population

were

_{randomly sampled}

_using

_genotype

_{frequencies according}

to

Hardy-Weinberg

proportions,

after which

_genotypes

of other individuals were

_randomly

sampled

based on realised

parental

_genotypes

_assuming

Mendelian transmission

probabilities.

For each individual a random

normally

distributed environmental

component

was

_sampled

and added to a

_{pre-determined}

effect of each

_genotype

to

obtain a

phenotypic

observation. Random numbers were

_generated

_using

GGUBFS

and

_GGNQF

_{(IMSL, 1984).}

Details on the

_parameters

used for these simulations

are

_given

in the

_following

sections. Model and model

_fitting

The statistical model can be

_{specified by}

the

_probability

terms in

_!2!,

_[3]

and

_[4]

which are

P(G

i

),

the

_genotype

_frequency

in the base

population

for individual

_i,

P(GiIG

s

,

G

d

),

the transmission

_probability

for individual i

_given

the

_genotypes

of its sire s and dam

d,

and the

_penetrance

function

/(<

/

t!G,),

the

_probability

for

the data _y2 on individual i

_given

the

_genotype

_G

_i

of individual i. From these 3

terms,

transmission

_{probabilities}

are assumed known to be Mendelian.

_Genotype

frequencies

in the base

_{population depend}

on the unknown

_{frequency f}

of the

A

1

allele, assuming Hardy-Weinberg proportions

of

_genotypes.

The

_penetrance

function for an individual i is taken as:

This

_penetrance

function is a normal

_{probability density}

function with variance

a-2

around

the mean _JiGi for

_{genotype G}

_i

_.

No dominance is assumed. For

_analysis,

means attributed to the

genotypes

are

expressed

as _Ji1 =

p -

1/2t,

!2 = tc and A’3 =

J

i +

1/2t,

where t is the difference between

homozygotes,

referred to as the

gene effect. The unknown

parameters

in the model are then

_f,

_p.,

_t,

and

Q2

.

Likelihoods were

computed using

iterative

peeling.

For structure

_1,

without

loops,

computations

were done

_exactly

_by

_repeating

the

_computations

until no

further

_changes

_occurred,

_having

found the order for recursive

_computation.

For the

looped

pedigrees

of structures 2 and

_3,

iterative

_peeling

was used to obtain

approximate

likelihoods. The

_stopping

rule was a

change

less than

10-

8

for the average absolute

heterozygote probabilities

of all individuals. The maximum of the

likelihood was

_{sought using}

the downhill

simplex algorithm

(Nelder

and

Mead,

1965),

using

as _{convergence criteria} the variance of likelihood values of

_points

in

Comparisons

Looped

and

nonlooped

pedigrees

were

compared

in

hypothesis

tests and

_parameter

estimation. In

_hypothesis

_testing,

a null

_{hypothesis postulating}

the absence of a

major

gene is

used,

described

by

a model with

parameters it

and

_a

₂

_,

and an

alternative

_{hypothesis postulating}

the presence of a

_major

_{gene is}

used,

described

by

a model with

_parameters

_f,

_/1,

t and

a2.

Tests are based on the likelihood ratio

(LR)

test

_statistic,

which is twice the natural

_logarithm

of the ratio of maximum likelihoods under each

_{hypothesis. Type}

I error and _power, the

complement

of

_type

II _error, were

_investigated

at their nominal

_level,

ie

_assuming

the

_expected

classical

asymptotic

X2

distribution

for the LR test statistic under the null

hypothesis

(Wilks, 1938).

Using

the classical

_{rules, rejection}

thresholds were obtained from a

X2

2

distribution with 2

_degrees

of

_freedom,

ie the difference in number of

_parameters

between the null and alternative

_hypothesis.

It should be noted that for

testing

mixtures,

these classical rules do not lead

_exactly

to the nominal

type

I errors

(Titterington

et

_al,

_1985),

but this is not of

_importance

for the

_comparisons

between

looped

and

_nonlooped

_pedigrees

to be made here. The likelihood

_L

_o

for the null

hypothesis

is

_computed

as:

where y

i

are observations with i =

1, ... , N,

the total number of

observations,

assumed

_normally

and

_{independently}

distributed. Under the null

hypothesis,

the maximum likelihood estimate for the mean is

_íi

=

&dquo;

BYi/

N

and for the variance is

(;

2

=

_&dquo;B(

_Yi

_-

_íiO)

₂

_/N.

Type

I error of the test for a

_major

_gene was

_investigated

_by

_simulating

1 000

data sets of each structure

_{(table I),}

_generating

for each individual

_only

a

_randomly

distributed error term with

U

2

= 100 as

phenotype.

Likelihoods for the null

hypothesis

and the alternative

_hypothesis

were

_computed

in each of these

_replicated

data

_sets,

and the likelihood ratio test statistic was obtained. The number of

significant

tests in these 1 000 data sets was counted

_{using rejection}

thresholds

of 4.605 and

_5.991,

_{corresponding}

to nominal

_type

I errors of 10 and

5%.

Power to

detect a

_major

_gene was

_{investigated by}

_simulating

100 data sets of each structure

(table I)

for 3 different gene effects t =

_5,

t = 7.5 and t = 10 and

_using

allele

frequency f

= 0.5 and residual variance

U

2

= 100.

Hence,

relative _gene effects

t

l

a

were

_0.5,

0.75 and 1. Power was based on a nominal

_type

I error of

5%,

_using

a

rejection

threshold of 5.991. Parameter estimates were

compared using

the 100 data

sets of each structure

_(table

_I)

used to

_investigate

_power with t = 10.

RESULTS

Type

I errors were

significantly

lower than their

_nominal,

ie

_{asymptotically}

ex-pected, level,

but

_comparison

of

_type

I errors between

_looped

and

_nonlooped

struc-tures did not show

_significant

differences

_(table

_II).

This indicates that absolute values of

_approximate

likelihoods obtained are on _average close to

_expected

and that the distribution of the test statistic over a number of

_replicates

is not

estimates for _gene effect under the alternative

_hypothesis

are biased in

_general,

but estimates for gene effects as well as allele

frequency

do not differ between

_looped

and

_nonlooped

structures

_{(table IV).}

This indicates that location of the maximum

is,

on _average over

_replicates,

not altered for

_approximate

likelihoods.

DISCUSSION AND CONCLUSIONS

An alternative

_peeling

_algorithm,

called iterative

_peeling,

has been

_presented.

The iterative

_{peeling algorithm}

includes an

_algorithm

to find an order for

_evaluating

peeling

equations.

When an order cannot be

found,

as in

_looped

_pedigrees,

an

approximate

likelihood is

supplied.

In this _case, use of a

_{partitioned computation}

of the likelihood is also crucial. Traditional recursive

_peeling

does not involve such

approximations,

because this method

_only

_computes

the exact likelihood once a

peeling

order is found and

computes

the likelihood

_{by representing}

all

_pedigree

information in terms for a

_single

individual. Usefulness of iterative

_peeling

as

an

_approximate

method in

_looped

_pedigrees

was

_{investigated by}

simulations. At an

_aggregate

level,

ie

_compared

on _average over a number of

_replicated

data

sets,

no differences were found between

_looped

and

_{nonlooped pedigrees.}

Exact

computations

were unfeasible due to the

_large

number of

_loops

in the

_typical

animal

breeding pedigrees

we

considered,

and

_properties

of iterative

_peeling

could not be studied

_comparing

exact and

_approximated

likelihoods in individual data sets.

The iterative

_peeling

method _may be of interest for

_application

in animal

breeding.

In human

_populations,

_pedigrees

are

_generally

small and

_loops

are not

abundant so that exact

_computations

can be considered

_using

more

_complicated

forms of

_peeling

_(see

_Cannings

et

al,

1978).

These more

_complicated

forms of

peeling

consider

_genotypes

on sets of individuals

_jointly.

_{Larger pedigrees}

and

more abundant

_looping

in animal

_{breeding, however,}

makes the sets of

_genotypes

considered

_jointly

too

_large

for exact

_computations

to be feasible.

_Therefore,

approximate

methods are

_required

for

_application

in animal

_breeding.

Iterative

peeling

seems _very

suited,

_being

exact without

_loops,

and

_{automatically supplying}

approximate

likelihoods when

_loops

are

_present.

Note

_that,

due to the

_partitioned

computation

of

_likelihood,

iterative

_peeling

also

_{automatically}

handles

_pedigrees

consisting

of

_{independent families,}

ie data

_{traditionally}

handled with sire or

sire-and-dam models. The

_equations

and

_{partitionings given}

here could be extended to

allow for more

general pedigrees.

In

particular,

allowance could be made for females

being

mated with several males. In this _case,

_{partitionings}

should accommodate for

’linking

individuals’

_being

_parents

in several

_families,

rather than

_just

one. The

monogenic

model used could also be extended to a mixed inheritance

model,

the

model

_{usually required}

for

_analysis

of animal

_breeding

data. In iterative

_{peeling only}

uni- and bivariate functions of

_genotypes

are considered on

_single

families. This can

be combined with for instance a Hermitian

_integration

_(Le

_Roy

et

al, 1989;

Knott

et

_al,

₁₉₉₂₎

to include a

_polygenic

_component.

ACKNOWLEDGMENTS

This research was

supported financially by

the Dutch Product Board for Livestock and

Meat, the Dutch

Pig

Herdbook

Society,

Bovar

_BV,

VOC Nieuw-Dalland BV, Euribrid BV

REFERENCES

Cannings C, Thompson EA,

Skolnick MH

₍₁₉₇₆₎

The recursive derivation of likelihoods

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Appl

Prob _8, 622-625

Cannings C, Thompson EA,

Skolnick MH

₍₁₉₇₈₎

_Probability

functions on

_complex

pedigrees.

Adv

_Appl

Prob

_10,

26-61

Elston

RC,

Stewart J

₍₁₉₇₁₎

A

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model for the

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Hoeschele I

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Genetic evaluation with data

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gene

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Genet

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Methods of

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Hum Hered

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Roy P,

Elsen JM, Knott SA

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Comparison

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JA,

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DM, Smith

AFM,

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_York,

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_JAM,

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_{C, Kennedy}

BW

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Method to estimate _genotype

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Genet

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735-740 Wilks SS

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