Breeding value estimation with incomplete marker data

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Breeding value estimation with incomplete marker data

Marco C.A.M. Bink, Johan A.M. van Arendonk, Richard L. Quaas

To cite this version:

Original

article

Breeding

value estimation with

incomplete

marker data

Marco

C.A.M.

Bink

Johan

A.M.

Van

Arendonk

a

Richard

L.

_Quaas

a

Animal

_Breeding

and Genetics

_{Group, Wageningen}

Institute of Animal

_Sciences,

Wageningen

Agricultural University,

PO Box 338, 6700 AH

_Wageningen,

the Netherlands b

Department

of Animal

Science,

Cornell

University, Ithaca,

NY _14853, USA

(Received

20

_{January 1997; accepted}

17 November

₁₉₉₇₎

Abstract -

_Incomplete

marker data _prevent

_application

of marker-assisted

_breeding

value estimation

_using

animal model BLUP. We describe a Gibbs

_{sampling approach}

for

Bayesian

estimation of

breeding values, allowing incomplete

information on a

single

marker that is linked to a

quantitative

trait locus. Derivation of

_sampling

densities for

marker _genotypes is

emphasized,

because reconsideration of the

gametic relationship

matrix structure for a marked

quantitative

trait locus leads to

_simple

conditional densities. A small numerical

_example

is used to validate estimates obtained from Gibbs

sampling.

Extension and

_application

of the

_{presented approach}

in livestock

_populations

is discussed.

©

Inra/Elsevier,

Paris

breeding values

_/

quantitative trait locus

_/

_incomplete marker data

_/

Gibbs _sampling

Résumé - Estimation des valeurs génétiques avec information incomplète sur les marqueurs. Un typage

incomplet

pour les marqueurs

empêche

l’estimation des valeurs

génétiques

de _type BLUP utilisant l’information sur les marqueurs. On décrit une

procédure d’échantillonnage

de Gibbs pour l’estimation

bayésienne

des valeurs

_génétiques

permettant une information

_incomplète

_pour un _marqueur

_unique

lié à un locus

_quantitatif.

On

_développe

le calcul des densités de

_{probabilités}

des

_génotypes

au _{marqueur parce}

que la reconsidération de la structure de la matrice des corrélations

_gamétiques

_pour

un locus

quantitatif marqué

conduit à des densités conditionnelles

_simples.

Un

_petit

exemple numérique

est donné pour valider les estimées obtenues par

échantillonnage

de Gibbs.

L’application

de

_l’approche

aux

_populations

d’animaux

_domestiques

est discutée.

©

Inra/Elsevier,

Paris

valeur _génétique

_/

locus _quantitatif

_/

marqueurs incomplets

/

échantillonnage de

Gibbs

*

1. INTRODUCTION

Identification of a

_genetic

marker

closely

linked to a _gene

(or

a cluster of

genes)

affecting

a

_quantitative

_trait,

allows more accurate selection for that trait

_[5].

The

_possible

_advantages

of marker-assisted

_genetic

evaluation have been described

extensively

(e.g. [13,

16,

17]).

Fernando and Grossman

_[1]

demonstrated how best linear unbiased

_prediction

(BLUP)

can be

performed

when data are available on a

single

marker linked to

quantitative

trait locus

_((aTL).

The method of Fernando and Grossman has been modified for

_including

multiple

unlinked marked

QTL

[23],

a different method of

assigning

QTL

effects within animals

_[26];

and marker brackets

_[5].

These methods

are efficient when marker data are

_{complete. However,}

in

_practice,

incompleteness

of marker data is very

likely

because it is

expensive

and often

impossible

(when

no DNA is

available)

to obtain marker

genotypes

for all animals in a

pedigree.

For _every unmarked

_animal,

several marker

_genotypes

can be

_fitted,

each

_resulting

in a different marker

_genotype

configuration.

When the

proportion

or number of

unmarked animals

_increases,

identification of each

_possible

marker

_genotype

con-figuration

becomes tedious and

analytical computation

of likelihood of occurrence

of these

_{configurations}

becomes

impossible.

Gibbs

_sampling

_[3]

is a numerical

_integration

method which

_provides

opportuni-ties to solve

_analytically

intractable

_problems.

_Applications

of this

technique

have

recently

been

_published

in statistics

_(e.g.

_{[2, 3])}

as well as animal

_breeding

(e.g.

[18,

25]).

Janss et al.

_[10]

_{successfully applied}

Gibbs

_sampling

to

_sample

_genotypes

for a

bi-allelic

_major

gene, in the absence of markers.

Sampling

genotypes

for multiallelic

loci,

e.g.

genetic markers,

may lead to reducible Gibbs chains

[15, 20].

Thompson

[21]

summarizes

_approaches

to resolve this

potential

reducibility

and concludes that

a

_sampler

can be constructed that

_efficiently

samples

multiallelic

_genotypes

on a

large pedigree.

The

_objective

of this _paper is to describe the Gibbs

_sampler

for marker-assisted

breeding

value estimation for situations where

genotypes

for a

_single

marker locus

are unknown for some individuals in the

pedigree.

Derivation of the

conditional,

discrete,

sampling

distributions for

_genotypes

at the marker is

_emphasized.

A small numerical

_example

is used to _{compare estimates} from Gibbs

_sampling

to true

posterior

mean estimates. Extension and

_application

of our method are discussed.

2. METHODOLOGY

2.1. Model and

_priors

We consider inferences about model

_parameters

for a mixed inheritance model

of the form

where _y and e are n-vectors

_representing

observations and residual _errors,

₍

₃

is a

p-vector

of ’fixed

effects’,

u and v are _q and

2q-vectors

of random

polygenic

and

consider three random

_genetic

_effects,

i.e. two additive effects at a marked

QTL

(v!

and

_v2,

see

_figure

1)

and a residual

_polygenic

effect

(u;).

Here e is assumed to have the distribution

_N

_n

_{(O, 1}

₀

_&dquo;;),

independently

of

(3,

u and v. Also u is taken to

be

_{Nq(0, A}

_O

_,

₂

_),

where A is the well-known numerator

_relationship

matrix.

Finally,

v is taken to be

_N2q(OGQ!),

where G is the

_{gametic relationship}

matrix

(2q

x

2q)

_computed

from

_pedigrees,

a full set of marker

_genotypes

and the known map distance between marker and

QTL

[26].

In case of

incomplete

marker

data,

we

_{augment genotypes}

for

_ungenotyped

individuals. We then denote

_f

_fi(

_k

₎

and

G(

k

)

as the marker

_genotype

_{configuration}

k and as the

_{corresponding}

_gametic

relationship

matrix.

_{Further, /3,}

_{u, v,} and

_missing

marker

_genotypes

are assumed

to be

_independent,

a

_priori.

We assume

_{complete knowledge}

on variance

_components

and _map distance between marker and

_QTL.

2.2. Joint

_{posterior density}

and full conditional distributions for location

parameters

The conditional

_density

of _y

_{given /3,}

_u, and v for the model

_given

in

_equation

(1)

is

_proportional

to

_{exp{ -1/2a;}

₂

_{(y -}

_{X,3 -}

Zu -

_{Wv)’(y -}

_{X/3 -}

Zu -

_Wv},

The

joint posterior

density

includes a summation

_(n

_c

₎

over all consistent marker

genotype

configurations

(

M

(

k

))-

In the derivation of the

_sampling

densities for marked

_{QTL effects,}

_however,

one

_particular

marker

_genotype

_{configuration,}

_m(

_k

_),

is fixed. The summation needs to be considered

_only

when the

_sampling

of marker

genotypes

is concerned.

To

_implement

the Gibbs

_{sampling algorithm,}

we

_require

the conditional

_posterior

distributions of each of

(3,

u, and v

_given

the

_remaining

_parameters,

the so-called full conditional

_{distributions,}

which are as follows

and

_gametic

covariances in the

_{pedigree, respectively.}

Note that the means of the

distributions

_{(3), (4)}

and

₍₅₎

_correspond

to the

_updates

obtained when mixed model

equations

are solved

_by

Gauss-Seidel iteration. Methods for

_sampling

from these distributions are well known

_{(e.g. [24,}

_25]).

2.3.

_Sampling

densities for marker

_genotypes

Suppose

m is the current vector of marker

_genotypes,

some observed and some

of which were

_augmented

(e.g.

_sampled

_by

the Gibbs

_sampler).

Let m-i denote

a

particular

_genotype

for the marker locus. Then the

_posterior

distribution of

genotype

gm

is

the

product

of two factors

with,

where

₁

_G-

_corresponds

to marker

_genotype

set

_,

_Mi

_M

_i

_-

_I

=

g

m

).

Thus, equation

(7)

shows that

_phenotypic

information needed for

sampling

new

_genotypes

for the

marker is

_present

in the vector of

_QTL

effects

_(v).

Now,

it suffices to

_compute

_equation

₍₆₎

for all

_possible

values of gm, and then

randomly

select one from that multinomial distribution

_[20].

In

_practice

consid-ering

only

those _gm that are consistent with _m-i and Mendelian inheritance can

minimize

_{the, computations. Furthermore, computations}

can be

simplified

because

&dquo;transmission of genes from

parents

to

_offspring

are

conditionally independent

_given

the

_genotypes

of the

_{parents&dquo;}

_[15].

_Adapting

notation from Sheehan and Thomas

[15],

let

_S

_j

denote the set of mates

_(spouses)

of individual i and

_0;,!

be the set of

offspring

of the

_pair

i and

_{j. Furthermore,}

the

_parents

of individual i are denoted

by

s

(sire)

and d

_(dam).

_{Then, equation}

₍₆₎

can be more

_specifically

written as

p(mi = gm, m-i IV, oV 2 ,Mobs, r)

When

_parents

of individual i are not

known,

then the first two terms on the

right-hand

side of

_equation

₍₈₎

are

_{replaced by}

x(m;),

which

_represents

frequen-cies of marker

_genotypes

in a

_population.

The

_probability

_p(m;

=

9

.

1

M

.,

Md

).

cor-responds

to Mendelian inheritance rules for

_obtaining

marker

genotype

gi

given

parental

genotypes

ms and md, similar for

p(m

1

Im¡

=

gm, m!).

The

computation of

p{v

i

lv

d

,m¡,m

s

,m

d

,r}

(and

,r})

1

,m

j

,m

i

,m

Vj

Iv¡,

1

p{v

can

efficiently

be

performed

by

utilizing special

characteristics of the matrix

G-

1

.

Let

_Q

_i

denote a

_gametic

contribution matrix

_relating

the

_QTL

effects of individual i to the

_QTL

effects of its

_parents.

The matrix

_Q

_i

is

_{2(i —}

₁₎

x 2. For founder

animals,

matrix

_Qi

is

_simply

zero. The recursive

algorithm

to

_{compute G-}

1

of

_Wang

et al.

_(1995,

_equation

_{[18] )}

can be rewritten as

where

_D¡1

=

_{(C; -}

_Q;G¡-

_1Q

_¡)-

₁

_(which

reduces to

D¡

1

=

_(C

_i

_-

_QfG

_i

_-,Q

_i

_)-’

with no

inbreeding),

_O

_i

is a

2(q—i)

x 2 null matrix. The

_{off-diagonals}

in

_{C; equal}

the

Henderson’s rules for

A-

1

_[6].

The nonzero elements of

G-

1

pertaining

to an animal

arise from its own contribution

_plus

those of its

_offspring.

So,

when

_sampling

the ith animal’s marker

_genotype,

_only

those contribution matrices need be considered that contain elements

_pertaining

to animal i. These are the individual’s own

contributions and those of its progeny when i appears as a

_parent.

where Vk is the vector of animal k’s two marked

_{QTL effects,}

and

_Qp

_denotes

the

rows of

_Q

_k

_pertaining

to

_P,

one of k’s

_parents.

_Again,

we

_recognize

each term in

the sum is the kernel of a

_(bivariate)

normal which is

_pfv

_i

_Iv

_s

_,

_{, vdm¡, ms, md,}

_r}

or

p{v1Iv¡, Vj, m¡, mj,m1, r}.

2.4.

_Running

the Gibbs

_sampling

The Gibbs

sampler

is used to obtain a

_sample

of a

_parameter

from the

_posterior

distribution and can be seen as a chained data

_augmentation

_algorithm

_[19].

_So,

one

_augments

data

_(y

and

_mobs)

with

parameters

(0)

to

_obtain,

for

_example,

p(e

1

Ie

2

, ... ,

O

d

,

y).

For the purpose of

breeding

value

_estimation,

Gibbs

_sampling

works as follows:

1)

set

_arbitrary

initial values for

_9!°!,

we use zeros for fixed

and

_genetic

effects and for each unmarked

_animal,

we

_augment

a

_genotype

that is consistent with

pedigree,

Mendelian

_inheritance,

and observed marker

_data;

2)

sample

01’

+ll

from

[3],

i =

1, 2, ..,

p; for fixed

effects,

[4],

i =

p +

1,

p +

2, ..,

p + q; for

polygenic effects,

[5],

i =

p + q +

1,

p + q +

2, ..,

p + q +

2q;

for marked

QTL effects,

or

[6],

i =

p +

3q

+

_1,

_p +

3q

+

2,..,

p +

3q

+ t;

for marker

genotypes,

and

replace

6!T!

with

ei

T+1

]

;

.

3)

repeat

2)

N

_(length

of

_chain)

times.

For any individual

parameter,

the collection of n values can be viewed as a

simulated

_sample

from the

_appropriate

_marginal

distribution. This

_sample

can be

used to calculate a

_marginal

_posterior

mean or to estimate the

_{marginal posterior}

distribution. For small

_pedigrees

with

_only

a few animals

_missing

observed marker

where B* is a

_fixed,

_polygenic

or marked

QTL

effect. This

provides

a criterion to

compare the estimates obtained from Gibbs

sampling.

3. NUMERICAL EXAMPLE

A small numerical

_example

is used to

_verify

the use of the Gibbs

_sampler

to obtain

_posterior

mean estimates and illustrate the effect of the data on the estimates

obtained from two different

_estimators,

i.e. a

_posterior

mean and the well-known

BLUP estimator

_(by

_solving

the MME

_given

in the

_Appendix).

Pedigree

and

data of the

_example

are in

_figure

2. Both sire

(01)

and dam

(02)

have observed

marker

_genotypes,

AB and

_{CD, respectively,}

but do not have

_phenotypes

observed.

Three full sibs have a marker

_genotype

BC and a

phenotype

+20

(denoted

FS

_03,

04,

05);

three other full sibs have a marker

_genotype

AD and a

_phenotype

-20

(denoted

FS

06, 07,

08).

Both animals 09 and 10 have no marker

genotypes

but have a

_phenotype

+20 and

_-20,

_{respectively. Complete knowledge}

was assumed on

variance

_components

and recombination rate between marker and

_MQTL

_{(table I).}

The

_thinning

factor in Gibbs

_sampling

chain was 50

cycles

and the burn in

period

was twice the

_{thinning factor,}

and 20 000 thinned

_samples

were used for

_analysis.

3.1. Estimates for

_genetic

effects

The

_posterior

estimates obtained from Gibbs

_sampling

were similar to the TRUE

posterior

estimates,

as shown in table 11. The

_posterior

estimates of

_MQTL

effects of

animals 09 and 10

_(f0.70)

were much less

divergent

than those of their full sibs that

had their marker

_genotypes

observed

_(f2.48).

These less

_divergent

values reflect

the

uncertainty

on marker

genotypes

of animals 09 and 10. The TRUE and GIBBS

posterior

densities for an

MQTL

effect of animal 09 were also _very similar

(figure 3).

The

_posterior

variance was

_52.3,

which was

_larger

than the

_prior

variance

(ufl =

50)

and reveals that the data are not

_decreasing

the

prior uncertainty

on

MQTL

effects

for animals 09 and 10 in this situation. For the other full

_sibs,

the

_posterior

variance

was

_47.02,

which was lower than the

_prior

variance because

_segregation

of

_MQTL

effects was known with

_{higher certainty,}

i.e. marker

_genotypes

were known. The

BLUP estimates for

_MQTL

effects of animal 09 and 10 were

_equal

to

_1/6

of the

3.2. Marker

_genotype

_{probabilities}

In the

_following

marker

genotype

AB

_represents

both AB and BA. In the latter case, alleles for both marker and

MQTL

are

_{reordered, maintaining linkage}

between

marker and

_MQTL

alleles within an animal.

_So,

four marker

_genotypes

were

_possible

for animals 09 and 10

_{(table III).}

Based on

pedigree

and marker data

solely,

each of

these four

_genotypes

was

_{equally likely}

(prior

_probability

=

0.25).

After

including

phenotypic

data,

(posterior)

probabilities changed:

marker

_genotype

BC and AD for animal 09 became more and less

probable, respectively.

The reverse holds for

animal 10. The estimates from the Gibbs

_sampler

were

_again

_very similar to the TRUE

posterior probabilities.

Complete phenotypic

and marker information on six full sibs gave the

MQTL

effects linked to marker alleles B and C

_positive

values and marker alleles A and D

_negative

values. Note that

_{probabilities}

_(TRUE)

for marker

genotypes

AC and BD also

_(slightly)

_changed

after

_considering

the

_phenotypic

data.

4. DISCUSSION

Marker-assisted

_breeding

value estimation in livestock has been

hampered by

incomplete

marker data.

_Previously

described methods

_[1,

_23,

_26]

can accommodate

ungenotyped

individuals which do not have

_offspring

themselves as was shown

by

Hoeschele

_[7].

_However,

_they

do not

_provide

the

_flexibility

to

_incorporate

parents

with unknown

_genotypes

which results in the loss of information for

estimating

marker linked effects. The described Gibbs

_{sampling algorithm}

now

provides

this

_required

_flexibility.

The innovative

_step

in our

_approach

is the

_sampling

of

genotypes

for a marker locus that is

closely

linked to

_QTL

with

normally

distributed allelic effects.

_Normality

of

QTL

effects is a robust

_assumption

to allow

segregation

of _many alleles

_throughout

a

_population

and allow

_changes

in allelic

effects over

_generations,

_e.g. due to mutations and interactions with environments

phenotypes

of animals in the

_pedigree

are used. Jansen et al.

_[9]

indicate

that,

as a result of the use of

_{phenotypic information,}

unbiased estimates of effects at the

QTL

can be obtained in situations where animals have been

_selectively

_genotyped.

In this paper we have concentrated on the use of information from a

single

marker locus.

_Using

information from

_multiple

linked markers can increase _accuracy of

_predicting

_genetic

effects at the

_QTL.

The

_{principles applied}

here have been extended to situations where

_genotypes

for all the linked markers are known for all

individuals

_{[5, 22].}

In order to

_incorporate

individuals with unknown

_genotypes,

the method

_presented

in this _paper needs to be extended to a

multiple

marker situation.

In

_extending

the method to

_multiple

_markers,

the

_problem

of

_reducibility

deserves

special

attention.

Reducibility

of Gibbs chains can arise when

sampling

_genotypes

for a

polymor-phic

locus with more than two alleles

_[20].

The

_{reducibility problems}

will become

more severe when

sampling

_genotypes

for

multiple

linked markers.

Thompson

[21]

suggested

several, workable,

approaches

to

_guarantee

_{irreducibility}

of the Gibbs

chain. These

_approaches

make use of

Metropolis-coupled

samplers

[11],

importance

sampling,

with

_0/1

_weights

_[15],

and

_{’heating’}

in the

Metropolis-Hastings

steps

[12].

Alternatively,

Jansen et al.

_[9]

_sampled

IBD values for all marker loci

_indicating

parental origin

of alleles instead of actual alleles to avoid the

_{reducibility problem.}

In

_extending

the method to

_multiple

linked

markers,

attention also needs to be

paid

to an efficient scheme for

_haplotypes

or

_genotypes

at the linked loci.

_Updating

of

genotypes

at

_closely

linked loci will be more efficient when

_genotypes

at the linked loci are

_updated

_together

(’in blocks’)

in order to reduce auto-correlation in the Gibbs

_sampler

_[10].

For

_posterior

inferences on the

_breeding

value of an animal a minimum of

100 effective

_samples

is needed. In the numerical

_example

this minimum would

correspond

to a chain of 5 000

_cycles

which

required

8 s of CPU at a HP9000

K260 server. It has been found that

_{computing requirements}

increase more or

less

_linearly

with the number of animals

_[10].

The

presented

method can be

applied

to data

_originating

from nucleus herds which

_comprise

the

_relatively

small number of

_genetically

_superior

animals from the

_population.

In a marker-assisted

selection scheme marker

_genotypes

will be collected

_largely

on these

animals,

with

sufficient animals

_having

marker

_genotypes

observed to

_improve

selection of

_superior

individuals.

Straightforward application

in

_large

commercial

_populations

with thousands of marker

_genotypes

_missing,

is not a valid

_option

because of

_{computational}

requirements

of Markov chain Monte Carlo

_(MCMC)

_algorithms

such as Gibbs

sampling. Hybrid

schemes will need to be

_developed

to

_incorporate

information from the commercial

_population

into the marker-assisted

_prediction

of

_breeding

values of nucleus animals. Similar schemes have been

_implemented

to

_incorporate

foreign

information into national evaluations in

_dairy

cattle.

Our

_{Bayesian approach}

can also be considered as a first

_step

towards a MCMC

algorithm,

not

_necessarily

Gibbs

_sampling,

that can also estimate

_hyper

parame-ters,

which were held constant in this

_study.

The next

_step,

_{therefore, comprises}

MCMC

_algorithm

can then be used for the

_analysis

in

QTL mapping experiments

with

complex pedigree

structures,

such as

(grand-)

_{daughter designs,}

in outbred

populations.

ACKNOWLEDGEMENTS

Valuable

_suggestions

by

S. van der Beek and anonymous reviewers are

_gratefully

acknowledged.

The financial support of Holland Genetics is

_{highly appreciated.}

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Al. APPENDIX

A1.1.

_Computation

of _average G with

_incomplete

marker data

Wang

et al.

_[26]

_{suggested computing}

an _average

G,

here denoted

G,

as

where

_G(

_k

₎

is the

_gametic

_relationship

matrix

_given

a

_particular

marker

_genotype

configuration

k

);

m(

and

)lM

,)

ob

k

(

M

p(

is the

probability

of

m(

k

)

given

mobs. This

equation

is not conditioned on

_phenotypic

information.

Al.2. Marker-assisted best linear unbiased

_prediction

of

_breeding

values

where a&dquo; =

Qe !Qu,

_a&dquo; =

Qe !Q!

and G are all known. Solutions can be obtained

by

iteration on the data

[14].

These

equations

can be used in three situations.

First,

G

is

_unique

_(complete

marker

_data).

Second,

with

_{missing markers,}

a linear estimator

is obtained

_{by taking}

G = G.

Third,

with G =

G!!!,

they

are used to

compute