Bayesian analysis of genetic change due to selection using Gibbs sampling

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Bayesian analysis of genetic change due to selection

using Gibbs sampling

Da Sorensen, Cs Wang, J Jensen, D Gianola

To cite this version:

Original

article

Bayesian

analysis

of

_genetic

_change

due

to

selection

_using

Gibbs

_sampling

DA Sorensen

_{CS Wang}

J Jensen

D

Gianola

1

National

Institute

of

Animal

Science,

Research Center

_Foulum,

PB _39, DK8830

Tjele, Denmark;

2

Department of Meat

and Animal

_{Science, University of Wisconsin-Madison,}

Madison, WI53706-128.!,

USA

(Received

7 June 1993;

accepted

10

_February

₁₉₉₄₎

Summary - A method of

_analysing

_response to selection

using

a

_Bayesian

_perspective is

presented.

The

_following

measures of response to selection were

_analysed:

₁₎

total _response in terms of the difference in additive

_genetic

means between last and first

_generations;

2)

the

_slope

_(through

the

_origin)

of the

_regression

of mean additive

_genetic

value on

generation;

3)

the linear

_{regression slope}

of mean additive

_genetic

value on

_generation.

Inferences are based on

_{marginal posterior}

distributions of the above-defined measures of

genetic

response, and uncertainties about fixed effects and variance components are taken into account. The

marginal

posterior distributions were estimated

using

the Gibbs

sampler.

Two simulated data sets with

_heritability

levels 0.2 and 0.5

_having

5

_cycles

of selection were used to illustrate the method. Two

analyses

were carried out for each data

set, with

_partial

data

_{(generations 0-2)}

and with the whole data. The

_{Bayesian analysis}

differed from a traditional

_analysis

based on best linear unbiased

_predictors

(BLUP)

with an animal

model,

when the amount of information in the data was small. Inferences about selection response were similar with both methods at

_{high heritability}

values and

using

all the data for the

_analysis.

The

_{Bayesian approach correctly}

assessed the

_degree

of

uncertainty

associated with insufficient information in the data. A

_{Bayesian analysis using}

2 different sets of _prior distributions for the variance _components showed that inferences differed

only

when the relative amount of information contributed

_by

the data was small. response to selection

/

Bayesian analysis

/

Gibbs sampling

/

animal model

Résumé - _{Analyse bayésienne} de la réponse génétique à la sélection à l’aide de

l’échantillonnage de Gibbs. Cet article

présente

une méthode

d’analyse

des

expériences

de sélection dans une _perspective

_bayésienne.

Les mesures suivantes des

réponses

à la sélection ont été

analysées:

i)

la

réponse totale,

soit la

différence

des valeurs

génétiques

additives moyennes entre la dernière et la

_{première génération;}

_ii)

la _pente

_(passant

par

l’origine)

de la

_régression

de la valeur

_génétique

additive moyenne en

_fonction

de la

génération;

iii)

la pente de la

_régression

linéaire de la valeur

_génétique

additive _moyenne en

fonction

de la

génération.

Les

inférences

sont basées sur les distributions

marginales

a

incertitudes sur les

effets fixés

et les _composantes de variance. Les distributions

marginales

a

_posteriori

ont été estimées à l’aide de

l’échantillonnage

de Gibbs. Deux ensembles de données simulées avec des héritabilités de _0,2 et 0,5 et 5

_cycles

de sélection ont été utilisés pour illustrer la méthode. Deux

analyses

ont été

_faites

sur

_chaque

ensemble de

_données,

avec des données

incomplètes

(génération 0-!!

et avec les données

_{complètes. L’analyse}

bayésienne différait

de

l’analyse traditionnelle,

basée sur le BL UP avec un modèle

animal,

quand

la

_{quantité d’information}

utilisée était réduite. Les

_inférences

sur la

_réponse

à lt sélection étaient similaires avec les 2 méthodes

_quand

l’héritabilité était élevée et que toutes les données étaient prises en _compte dans

l’analyse. L’approche bayésienne

évaluait correctement le

degré

d’incertitude lié à une

information insuffisante

dans les données. Une

_{analyse bayésienne}

avec 2 distributions a

_priori

des composantes de variance a montre que les

inférences

ne

difj&dquo;éraient

_que si la _part

d’information fournie

_par les données était

faible.

réponse à la sélection

_/

analyse bayésienne

/

échantillonnage de Gibbs

_/

modèle animal

INTRODUCTION

Many

selection _{programs in} farm animals

_rely

on best linear unbiased

_predictors

(BLUP)

using

Henderson’s

₍₁₉₇₃₎

mixed-model

_equations

as a

_{computing device,}

in order to

_{predict breeding}

values and rank candidates for selection. With the

increasing computing

power available and with the

development

of efficient

algo-rithms for

_writing

the inverse of the additive

_relationship

matrix

_(Henderson,

_1976;

(auaas,

1976),

’animal’ models have been

_{gradually replacing}

the

_originally

used sire

models. The

_appeal

of ’animal’ models is

_{that, given}

the

_model,

use is made of the information

_provided

_by

all known additive

_{genetic relationships}

_among individuals. This is

_important

to obtain more

_precise

_predictors

and to account for the effects of certain forms of selection on

_prediction

and estimation of

_genetic

_parameters.

A natural

application

of ’animal’ models has been

_prediction

of the

_genetic

means

of

_cohorts,

for

_example,

_groups of individuals born in a

_given

time interval such

as a _year or a

_generation.

These

_predicted

_genetic

means are

typically computed

as the average of the BLUP of the

genetic

values of the

_appropriate

individuals. From

_{these, genetic}

_change

can be

expressed

as, for

example,

the

regression

of the mean

_predicted

additive

_genetic

value on time or on

_appropriate

cumulative

selection differentials

_(Blair

and

Pollak,

1984).

In common with selection

index,

it is assumed in BLUP that the variances of the random effects or ratios thereof

are

_known,

so the

predictions

of

breeding

values and

_genetic

means

depend

on such ratios.

This,

in turn, causes a

dependency

of the estimators of

_{genetic change}

derived from ’animal’ models on the ratios of the variances of the random effects

used as

_’priors’

for

_solving

the mixed-model

_equations.

This

_point

was first noted

by Thompson

(1986),

who showed in

_{simple settings,}

that an estimator of realized

heritability

given

by

the ratio between the BLUP of total _response and the total selection differential leads to estimates that are

_{highly dependent}

on the value of

to

_expect

that the statistical

_properties

of the BLUP estimator of _response will

depend

on the method with which the

’prior’

heritability

is estimated.

In the absence of

selection,

Kackar and Harville

₍₁₉₈₁₎

showed that when the estimators of variance in the mixed-model

_equations

are obtained with even

estimators that are translation invariant and functions of the

data,

unbiased

predictors

of the

_breeding

value are obtained. No other

properties

are known and

these would be difficult to derive because of the

_nonlinearity

of the

_predictor.

In selected

_populations,

_{frequentist properties}

of

_predictors

of

_breeding

value based

on estimated variances have not been derived

_analytically

_using

classical statistical

theory.

Further,

there are no results from classical

theory

indicating

which estimator of

_{heritability ought}

to be

used,

even

_though

the restricted maximum likelihood

(REML)

estimator is an

intuitively appealing

candidate. This is so because the

likelihood function is the same with or without

selection,

provided

that certain

conditions are met

_(Gianola

et

_{at, 1989;}

Im et

at, 1989;

Fernando and

Gianola,

1990).

However,

frequentist properties

of likelihood-based methods under selection have not been

_{unambiguously}

characterized. For

example,

it is not known whether the

maximum likelihood estimator is

_always

consistent under selection. Some

_properties

of BLUP-like estimators of _response

_{computed by replacing}

unknown variances

_by

likelihood-type

estimates were examined

by

Sorensen and

Kennedy

(1986)

using

computer

simulation,

and the

_methodology

has been

_{applied recently}

to

_analyse

designed

selection

_experiments

_(Meyer

and

_Hill,

_1991).

_{Unfortunately, sampling}

distributions of estimators of _response are difficult to derive

_analytically

and one

must resort to

_{approximate results,}

whose

_validity

is difficult to assess. In _summary, the

_problem

of exact inferences about

_genetic

_change

when variances are unknown has not been solved via classical statistical methods.

However,

this

_problem

has a

_{conceptually simple}

solution when framed in a

Bayesian

setting,

as

_{suggested by}

Sorensen and Johansson

(1992),

drawing

from

results in Gianola et at

₍₁₉₈₆₎

and Fernando and Gianola

_(1990).

The

_{starting point}

is that if the

_history

of the selection process is contained in the data

employed

in the

_analysis,

then the

_posterior

distribution has the same mathematical form with or without selection

(Gianola

and

Fernando,

1986).

Inferences about

breeding

values

_(or

functions

_thereof,

such as selection

response)

are made

using

the

marginal

posterior

distribution of the vector of

_breeding

values or from the

_marginal

_posterior

distribution of selection response. All other unknown

parameters,

such as ’fixed effects’ and variance

components

or

heritability,

are viewed as nuisance

_parameters

and must be

_integrated

out of the

_{joint posterior}

distribution

_(Gianola

et

_at,

_1986).

The mean of the

posterior

distribution of additive

genetic

values can be viewed as a

weighted

average of BLUP

predictions

where the

weighting

function is the

marginal

posterior density

of

_{heritability.}

Estimating

selection response

by

giving

all

weight

to an REML estimate of

posterior

distribution.

_However,

this

_{approximation}

_may be poor if the

experiment

has little informational content on

heritability.

A full

_{implementation}

of the

_Bayesian

_approach

to inferences about selection

response relies on

having

the means of

_carrying

out the _necessary

_integrations

of

joint

posterior

densities with

_respect

to the nuisance

_parameters.

A Monte-Carlo

procedure

to _{carry out} these

_integrations

_numerically

known as Gibbs

_sampling

is

now available

(Geman

and

_{Geman, 1984;}

Gelfand and

_Smith,

_1990).

The

procedure

has been

_implemented

in an animal

_breeding

context

_{by Wang}

et al

_{(1993a; 1994a,b)}

in a

study

of variance

_component

inferences

_using

simulated sire

_models,

and in

analyses

of litter size in

_pigs.

_Application

of the

_Bayesian

approach

to the

_analysis

of selection

_experiments

_yields

the

_marginal

_posterior

distribution of _{response to}

selection,

from which inferences about it can be

made, irrespective

of whether

variances are unknown. In this paper, we describe

_Bayesian

inference about selection

response

using

animal models where the

_{marginalizations}

are achieved

_by

means of Gibbs

_sampling.

MATERIALS AND METHODS Gibbs

_sampling

The Gibbs

sampler

is a

technique

for

_generating

random vectors from a

_joint

distribution

_{by successively sampling}

from conditional distributions of all random variables involved in the model. A

_component

of the above vector is a random

sample

from the

_appropriate

_marginal

distribution. This numerical

_technique

was

introduced in the context of

_{image processing}

_by

Geman and Geman

₍₁₉₈₄₎

and

since then has received much attention in the recent statistical literature

_(Gelfand

and

Smith,

1990;

Gelfand et

_al,

_1990;

Gelfand et

_al,

_1992;

Casella and

_George,

1992).

To illustrate the

_procedure,

let us _suppose there are 3 random

_variables,

_X,

Y and

_Z,

with

_joint

_density

_{p(x, y, z);}

we are interested in

_obtaining

the

_marginal

distributions of

_X,

Y and

_Z,

with densities

_p(x),

_p(y)

and

_p(z),

_{respectively.}

In _{many cases,} the _necessary

_integrations

are difficult or

impossible

to

perform

algebraically

and the Gibbs

sampler provides

a means of

sampling

from such a

marginal

distribution. Our interest is

_typically

in the

_marginal

distributions and the Gibbs

_{sampler proceeds}

as follows. Let

)

o

, z

o

, y

o

(x

_represent

an

arbitrary

set of

starting

values for the 3 random variables of interest. The first

sample

is:

where

_p(xl

_y

=

y o

, Z =

z

o

)

is the conditional distribution of X

_given

Y =

y o and

Z = zo. The second

sample

is:

where x, is

updated

from the first

sampling.

The third

sample

is:

repeated

k

times,

where k is known as the

length

of the Gibbs _sequence. As k -> oo, the

points

of the kth iteration

₍

_Xk

_,

_Yk,

_Zk

₎

constitute 1

_sample

_point

from

_p(x,

_y,

_z)

when viewed

_jointly,

or from

_{p(x), p(y)}

and

_p(z)

when viewed

_marginally.

In order to obtain m

samples,

Gelfand and Smith

(1990)

_suggest

_generating

m

independent

’Gibbs

_sequences’

from m

_arbitrary

_starting

values and

_using

the final value at the kth iterate from each sequence as the

sample

values. This method of Gibbs

_sampling

is known as a

_{multiple-start sampling}

_scheme,

or

_multiple

chains.

_{Alternatively,}

a

single

long

Gibbs sequence

(initialized

therefore once

only)

can be

_generated

and

every dth observation is extracted

_(eg,

_Geyer

₁₉₉₂₎

with

the total number of

_samples

saved

_being

m. Animal

_{breeding applications}

of the short- and

long-chain

methods

are

_given

_{by Wang}

et al

_(1993,

_1994a,b).

Having

obtained

the m

_samples

from the

_marginal

distributions,

possibly

corre-lated,

features of the

_marginal

distribution of interest can be obtained

_appealing

to the

_ergodic

theorem

_(Geyer,

_1992;

Smith and

_Roberts,

_1993):

where

_x

_i

_(i

=

_{1, ... ,}

m)

are the

samples

from the

_marginal

distribution of _{x, u}’ _{is any}

feature of the

_marginal

_{distribution,}

eg, mean, variance or, in

general,

any feature

of a function of _x, and

_g(.)

is an

_{appropriate operator.}

For

_example,

if u is the mean

of the

_marginal

_{distribution,}

_g(!)

is the

_identity

_operator

and a consistent estimator

of the variance of the

_marginal

distribution is

(Geyer, 1992).

Note that

_S

_m

is a Monte-Carlo estimate of _u, and the error of this

estimator can be made

_arbitrarily

small

_by

_increasing

m. Another _way of

_obtaining

features of a

marginal

distribution is first to estimate the

_density

_using

_[11,

and then

compute

summary statistics

(features)

of that distribution from the estimated

density.

There are at least 2 ways to estimate a

density

from the Gibbs

samples.

One is to

use random

_samples

_(x

_i

₎

to estimate

_p(x).

We consider the normal kernel estimator

(eg,

Silverman,

1986).

where

_p(x)

is the estimated

_density

_{at x,} and h is a fixed constant

_(called

the window

_width)

_given

_by

the user. The window width determines the smoothness of

the estimated curve.

Another way of

estimating

a

density

is based on

_averaging

conditional densities

and

_given

_knowledge

of the full conditional

distribution,

an estimate of

p(x)

is

obtained from:

«

where _{t/i,2:i;...,t/!,2:nt} are the realized values of final

_Y,

Z

_samples

from each

Gibbs sequence. Each

pair

y2, zi constitutes 1

sample,

and there are m such

_pairs.

Notice that the xi are not used to estimate

_p(x).

Estimation of the

_density

of a

function of the

_original

variables is

_accomplished

either

_{by applying}

_[2]

_directly

to the

samples

of the

_function,

or

by applying

the

theory

of transformation of random variables in an

_appropriate

conditional

_density,

and then

_using

!3!.

In this case, the Gibbs

_sampling

scheme does not need to be rerun,

provided

the needed

samples

are

saved. Model

In the

_present

_paper we consider a univariate mixed model with 2 variance

components

for ease of

_exposition.

Extensions to more

_general

mixed models are

given

by Wang

et al

_{(1993b; 1994)}

and

_by

Jensen et al

_(1994).

The model is:

where _{y is} the data vector of order n

_by

_1;

X is a known incidence matrix of order n

by

p; Z is a known incidence matrix of order n

by

_q; b is a _p

by

1 vector of

uniquely

defined ’fixed effects’

_(so

that X has full column

_rank);

a is a _q

by

1 ’random’

vector

_representing

individual additive

_genetic

values of animals and e is a vector of random residuals of order n

by

1.

The conditional distribution that

_pertains

to the realization of y is assumed to

be:

where H is a known n

by n

_matrix,

which here will be assumed to be the

_identity

matrix,

and

_aj

_(a

_scalar)

is the unknown variance of the random residuals. We assume a

_genetic

model in which _genes act

_additively

within and between

loci,

and that there is

_effectively

an infinite number of loci. Under this infinitesimal

model,

and

_assuming

further initial

_{Hardy-Weinberg}

and

_{linkage equilibrium,}

the distribution of additive

_genetic

values conditional on the additive

_genetic

covariance is multivariate normal:

In

_[6],

A is the known q

by

q matrix

of additive

_genetic

_{relationships}

among

animals,

and

_Q

_a

_(a

_scalar)

is the unknown additive

_genetic

variance in the

_conceptual

base

_population,

before selection took

_place.

The vector b will be assumed to have a _proper

_prior

uniform distribution:

p(b)

oc _constant,

To

_complete

the

_description

of the

model,

the

_prior

distributions of the variance

components

need to be

specified.

In order to

_study

the effect of different

_priors

on inferences about

_heritability

and _response to

_selection,

2 sets of

_prior

distributions will be assumed.

_Firstly,

_u2

_and

_{aj will}

be assumed to follow

independent

proper

prior

uniform distributions of the form:

p

(u?)

oc _constant,

where

_a

_2maX

and

_afl!!!

are the maximum values

_{which, according}

to

_prior

know-ledge,

a!

and

!2

can take. In the rest of the paper, all densities which are functions of

_b,

_Q

_a,

and

ae

will

_implicity

take the value zero if the bounds in

7

and

[8]

are

exceeded.

Secondly,

Q

a

and

_<7!

will be assumed to follow a

_priori

scaled inverted

_chi-square

distributions:

where vi and

Sf

_are

parameters

of the distribution. Note that a uniform

_prior

can

be obtained from

_[9]

_by

_{setting v}

_i

= -2 and

S2

= 0.

Posterior distribution of selection response

Let a

_represent

the vector of

parameters

associated with the model.

_Bayes

theorem

provides

a means of

_deriving

the

_posterior

distributions of a conditional on the data:

The first term in the numerator of the

_right-hand

side of

_[10]

is the conditional

den-sity

of the data

_given

the

_parameters,

and the second is the

_{prior joint}

distribution of the

_parameters

in the model. The denominator in

_[10]

is a

_normalizing

con-stant

_(marginal

distribution of the

_data)

that does not

_depend

on a.

Applying

!10!,

The

_{joint posterior}

under a model

_assuming

the _proper set of

_prior

uniform distributions

_[8]

for the variance

components

is

_simply

obtained

_by

_{setting v}

_i

= -2

and

_SZ

₌ 0

_(i

=

e, a)

in

!12!.

To make the notation less

burdensome,

we

drop

from

now onwards the

conditioning

on v and S.

Inferences about response to selection can be made

_working

with a function of the

_marginal

_posterior

distribution of a. The latter is obtained

_integrating

[12]

over the

_remaining

parameters:

where E =

(o, a 2,a e 2)

and the

_expectation

is taken over the

_joint

_posterior

distribution of the vector of ’fixed effects’ and variance

_components.

This

_density

cannot be

written in a closed form. In finite

samples,

the

_posterior

distribution of a should be neither normal nor

_symmetric.

Response

to selection is defined as a linear function of the vector of additive

genetic

values:

where K is an

_{appropriately}

defined transformation matrix and R can be a vector

_(or

a

scalar)

whose elements could be the mean additive

_genetic

values

of each

_generation,

or contrasts between these means, or

alternatively,

regression

coefficients

_representing

linear and

quadratic changes

of

_genetic

means with

_respect

to some measure of

_time,

such as

_{generations. By}

virtue of the central limit

theorem,

the

_posterior

distribution of R should be

_{approximately}

normal,

even if

[13]

is not.

Full conditional

_posterior

distributions

_(the

Gibbs

_sampler)

Using

results from

_Lindley

and Smith

_(1972),

the conditional distribution of 0

given

all other

_parameters

is multivariate normal:

where

and 0

satisfies:

which are the mixed-model

_equations

of Henderson

_(1973).

and

_using

standard multivariate normal

_theory,

it can be shown for _any such

partition

that:

where

0

1

is

_{given by:}

Expressions

[19]

and

_[20]

can be

_computed

in matrix or scalar form. Let

_b

_i

be

a scalar

_{corresponding}

to the ith element in the vector of ’fixed

effects’, b_

i

.

be b without its ith

_element,

xi be the ith column vector in

_X,

and

_{X_.L}

be that

_part

of matrix X with xi excluded.

Using

[19]

we find that the full conditional distribution

of

b

i

given

all other

_parameters

is normal.

where

_b

_i

satisfies:

For the ’random

_{effects’, again using}

_[19]

and

_letting

a-i be a without its ith

element,

we find that the full conditional distribution of the scalar _ai

_given

all the

other

_parameters

is also normal:

where zi is the ith column of

Z,

iic =

(Var(ai!a_,,))-1

Q

a

is the element in the ith row and column of

A-

1

,

and

a

i

satisfies:

The full conditional distribution of each of the variance

_components

is

_readily

obtained

_{by dividing}

_[12]

_by

_p(b,

a,

!-ily),

where

E-

i

is E with

_!2

_excluded.

Since this last distribution does not

_depend

on

a

2

,

this becomes

_(Wang

et

_al,

_1994a):

which has the form of an inverted

_{chi-square distribution,}

with

_parameters:

When the

_prior

distribution for the variance

_components

is assumed to be

uniform,

the full conditional distributions have the same form as in

_[25],

_except

that,

in

_[26]

and

subsequent formulae, v

i

= -2 and

S2

= 0

(i

=

e, a).

Generation of random

_samples

from

_marginal

posterior

distributions

using

Gibbs

_sampling

In this section we describe how random

samples

can be

generated indirectly

from the

_joint

distribution

_[12]

_{by sampling}

from the conditional distributions

_{!21!, [23]}

and

_!25!.

The Gibbs

sampler

works as follows:

(i)

set

_arbitrary

initial values for

_b,

a and

_E;

(ii)

sample

from

_!21!,

and

_{update bi,}

i =

_{1, ... ,}

_p;

(iii)

sample

from

_[23]

and

update

ai, i =

_{1, ... , q;}

(iv)

sample

from

_[25]

and

update

aa;

(v)

sample

from

_[25]

and

update

!e;

(vi)

repeat

(ii)

to

_(v)

k

_(length

of

_chain)

times.

As k - oo, this creates a Markov chain with an

_equilibrium

distribution

_having

[12]

as

_density.

If

_along

the

_{single path k,}

m

samples

are extracted at intervals of

length

d,

the

_algorithm

is called a

_{single-long-chain algorithm.}

If,

on the other

hand,

m

independent

chains are

implemented,

each of

length k,

and the kth iteration is

saved as a

_sample,

then this is known as the short-chain

_algorithm.

If the Gibbs

_sampler

reached convergence, for the m

samples

(b, a, E) 1, ... ,

(b, a, E)&dquo;,

we have:

where - means distributed with

marginal

_posterior

density

p(.ly).

In _any

particular

of the univariate

_marginal

distributions

_p(bi!y)

and

_p(a

_il

_y).

In order to estimate

marginal

posterior

densities of the variance

_components,

in addition to the above

quantities

the

_following

sums of _{squares must} be stored for each of the m

_samples:

For estimation of selection response, the

quantities

to be stored

_depend

on the structure of K in

_(14!.

Density

estimation

As indicated

_previously,

a

_{marginal density}

can be

_estimated,

for

_{example, using}

the

normal

_density

kernel estimator

_[2]

with the m Gibbs

samples

from the relevant

marginal distribution,

or

by

_averaging

over conditional densities

(equation

[3]).

Density

estimation

_by

the former method is

_{straightforward, by}

_applying

_(2!.

_Here,

we outline

_density

estimation

_using

_(3!.

The formulae for variance

_components

and functions thereof were

_{given by Wang}

et al

_{(1993, 1994b).}

For each of the 2 variance

_components,

the conditional

_density

estimators are:

and for the error variance

_component:

The estimated values of the

_density

are obtained

by fixing

Q

a

and

Qe

at a number

of

_points

and then

_evaluating

_[27]

and

_[28]

at each

_point

over the m

_samples.

Notice

that the realized values of

_Q

_a

and

_Q

_e

obtained in the Gibbs

sampler

are not used to estimate the

_marginal

_posterior

densities in

_[27]

and

_[28].

To estimate the

_marginal

_{posterior density}

of

_heritability

h

2

=

a£ /(a£ +!e),

the

point

of

_departure

is the full conditional distribution of

_{a£ .}

This distribution has

_!e

as a

_conditioning

_variable,

and therefore

Q

e

is treated as a constant. Since the inverse

transformation is

_a!

=

<T!/(1 -

h

2

),

and the Jacobian of the transformation from

Q

The estimation of the

_marginal

_posterior

_density

of each additive

_genetic

value follows the same

_principles:

where,

for the

_jth

sample:

Equally

spaced points

in the effective _range of ai are

chosen,

and for each

_point

an estimate of its

_density

is obtained

_by

_inserting,

for each of the m Gibbs

_samples,

the realized values for

_o,2,

_b,

_a-i, and k in

_[30]

and

_[31].

These

_quantities,

_together

with

_a

_i

_,

need to be stored in order to obtain an estimate of the

marginal

_posterior

density

of the ith additive

_genetic

value. This process is

repeated

for each of the

equally

spaced points.

Estimation of the

_marginal

_posterior

_density

of response

depends

on the _way

it is

_expressed.

In

_general

R in

_[14]

can be a scalar

_(the

_genetic

mean of a

_given

generation,

or _{response per} time

_unit)

or it can be a

_vector,

whose elements could

describe,

for

_example,

linear and

_higher

order terms of

_changes

of additive

_genetic

values with time or unit of selection _pressure

_applied.

We first derive a

_general

expression

for estimation of the

_marginal

_posterior

_density

of R and then look at some

special

cases to illustrate the

_procedure.

Assume that R contains s elements and we wish to estimate

_p(Riy)

as:

In order to

_implement

_(32!,

the full conditional distribution on the

_right-hand

side is needed. This distribution is obtained

_applying

the

_theory

of transformations to

p(a

il

The s additive

_genetic

values in ai must be chosen so that the matrix of the transformation from ai to R is

non-singular.

We can then write

[14]

as:

so that we have

expressed

R in a

_part

which is a function of ai and another one which is not. The matrix

_K

_i

is

_non-singular

of order s

_by

_s, and from

_[33],

the inverse transformation is

Since the Jacobian of the transformation from ai to R is

_det(K!

₁

_),

_letting

Vi

=

Var(ai!b, a_i, E, y),

and

_using

standard

theory,

we obtain the result that

the conditional

_posterior

distribution of _response is normal:

where, using

[19]

and

_!20!,

it can be shown that:

and

In

_[35]

and

_[36],

_C

_i

is the s

_by

s block of the inverse of the additive

_genetic

relationship

matrix whose rows and columns

correspond

to the elements in ai, and

CZ!_2

is the s

_by

(q &mdash;

_s)

block

_associating

the elements of aiwith those of a-i. With

[34]

available,

the

_marginal

_posterior

distribution of R can be obtained from

!32!.

As a

_simple

_{illustration,}

consider the estimation of the

_marginal

_posterior

_density

of total response to

_selection,

defined as the difference in average additive

_genetic

values between the last

_generation

_{( f )}

and the first one:

where _afj and _a1j are additive

_genetic

values of individuals in the final and first

generations,

and _n

_and

n1 are the number of additive

_genetic

values in the final

and first

_{generation, respectively.}

We will

_arbitrarily

choose _af1 and _{carry out} a linear transformation from

_{p(afllb, a- 11,!, y) to p(Rlb, a- 11,!, y).}

We write

_[37]

in the form of

_!33):

and the

_marginal

_posterior

_density

is estimated as follows:

As a second illustration we consider the case where response is

expressed

as the linear

_regression

_(through

the

_origin)

of additive

_genetic

values on time units. Thus

R is a scalar. We assume as before that there are

f

time

units,

and the response

expressed

in the form of

_[33]

and

_[38]

is:

is their number. A little

_manipulation

shows that:

EXAMPLES

To

_illustrate,

the

_methodology

was used to

_analyse

2 simulated data sets.

_Genotypes

were

_{sampled using}

a Gaussian additive

_genetic

model. The

_phenotypic

variance was

always

_10,

and base

population

heritability

was 0.2 and 0.5 in data sets 1

(30

fixed effects in the

_complete

data

_set,

but these values are of no

_importance

here),

an additive

_genetic

effect and a residual term. Both additive

_genetic

and residual effects were assumed to follow normal distributions with null means and

variances based on the 2

_heritability

levels. In the

generation

of Mendelian

_sampling

effects,

parental inbreeding

was taken into account.

Five

_cycles

of

_single

trait selection based on a BLUP animal model were

practised.

Each

_generation

resulted from the

_mating

of 8 males and 20

_females,

and each female

_produced

3

_offspring

with records and 2 additional female

_offspring

without records which were also considered as female

replacements.

Males were selected

across

_generations

(the

best 8 each

_cycle)

and

_females,

which bred

_only

_once,

were selected from the

_highest

_{scoring predicted}

_breeding

values available each

generation.

The total number of animals at the end of the program was

_968,

of

which 720 had records. The total number of sires and dams in each data set was 42

and 241. Details of the simulation

_including

a

description

of the

_generation

of fixed

effects and additive

_genetic

values can be found in Sorensen

(1988).

Two

_analyses

were carried out for each data set: an

_analysis

based on all

_records;

and one

_using

data from

_generations

0-2

only.

The rationale for this is that it is

not uncommon in the literature to find

_reports

of selection

_experiments

_{in progress.}

Hence,

we can illustrate in this manner how inferences are modified as additional

information is collected.

For each

_analysis,

_marginal

_posterior

densities for the additive

_genetic

variance

a ,

residual variance

₍

_Q

_e),

_phenotypic

variance

_(aP)

and

heritability

(h

2

)

were estimated. In

_addition,

_marginal

_posterior

densities of 2

_expressions

for selection

response were estimated. These were:

1)

difference between final and initial _average

additive

_genetic

values or total _response

(TR);

and

2)

linear

_regression

through

the

_origin

of the linear

_regression

of additive

_genetic

values on

_generation.

Unless

otherwise

_stated,

the

_{analyses reported}

below were

performed assuming

that

variance

_components

follow a

_priori

_independent

uniform distributions as shown in

(8!.

The _upper bounds of the additive and environmental variances

_(a

₂

₂

were

arbitrarily

chosen to be 10 and

_20,

_{respectively.}

The Gibbs

_sampler

was

implemented

_using

a

single

chain of total

length

1 205 000,

with a _warm-up

(initial

iterations

_discarded)

of

_length

_{5 000,}

and a

sub-sampling

interval between

_samples

of d = 10.

Thus,

a total of m = 120 000

samples

were saved. Calculations were

programmed

in Fortran in double

_precision.

Random numbers were

_generated

_using

IMSL subroutines

_(IMSL

_Inc,

1989).

Plots of the estimated

_marginal

_posterior

densities of

_Q

_{a, Q}

_e,

_h

₂

_,

TR and 6 were obtained with both the _average of conditionals and with the normal kernel

density

estimator;

those for

_{a!, t50}

and

6

1

were

_generated

_using

the normal kernel

density

estimator

_only,

for technical reasons. The window used for the normal

kernel estimator was the _range of the

effective

domain of the

_parameter,

divided

by

75. The effective domain covered

99.9%

of the

_density

mass. Each of the

_plots

was

_{generated by dividing}

the effective domain of that variable into 100

_evenly

spaced

intervals.

_Summary

statistics of

_{distributions,}

such as the _mean, median

and variance were

computed

_by

Simpson’s integration rules, by

further

dividing

the effective domain into 1 000

_evenly

_spaced

intervals

_using

_{cubic-spline techniques}

(IMSL

Inc,

1989).

Modes were located

_{through grid}

search. For

illustration

and not

was carried out.

_First,

REML estimates of variance

_components

were obtained.

These REML estimates were then used in lieu of the true values in Henderson’s mixed-model

_equations

to obtain

estimates

of additive

_genetic

values.

_Finally,

estimated

_genetic

responses to selection were

_computed

based on

_appropriate

averages of the estimated additive

genetic

values. Results of this classical

analysis

are

reported

later under the

heading

ST in

figures

5-8. The estimated

marginal

posterior

densities of the variance

_components

and of

_heritability

are shown in

figures

1 and 2 for the

analysis

based on

partial

(PART)

and whole

(WHOLE)

data,

respectively,

when base

_population

_heritability

was 0.2.

_{Corresponding}

plots

For the data simulated with

h

2

=

_0.2,

the

_posterior

distributions reflected

considerable

_uncertainty

about the values of the

_parameters.

As

expected,

the variances of the

_posterior

distributions were smaller in WHOLE

_{(fig 2)}

than in

PART

_{(fig 1).}

In

_particular,

the PART

_analysis

of

_U2

_{and h2}

showed

_highly

skewed

_{distributions;}

the mean, the mode and the median differed from each

other. The REML estimates in the PART

_analysis

for the additive

_genetic

variance and

_heritability

were _very close to zero.

_Asymptotic

confidence intervals were not

computed,

but

_they

would

_probably

include a

_region

outside the

parameter

space.

The

_{Bayesian analysis}

indicated with considerable

_probability

that both

_parameters

was

_reduced,

but

_point

estimates in the

_{Bayesian analysis}

of

_genetic

variance and of

_heritability

were different from those in the REML estimates. For

example,

the

_posterior

mean, mode and median of

_heritability

were

_0.13,

0.08 and

0.12,

respectively,

in

_comparison

with the REML estimate of 0.07. The lack of

_symmetry

of the

_posterior

distribution of

_Q

_a

_{and h2}

_clearly

_suggests

that the simulated selection

_experiment

does not have a

_{high degree}

of resolution

_concerning

inferences

about

_genetic

_parameters.

This

_point

would not be as

_forcefully

illustrated in a standard REML

analysis,

where,

at

best,

one would

_get

_{joint approximate}

In the case of

h

2

= 0.5

(fig

3 and

4),

it should be noted that the distributions

were less skewed than for

h

2

=

_0.2,

_although

still skewed for PART

(fig 3).

Both

the

_Bayesian

and the REML

_{analyses suggested}

a moderate to

_{high heritability,}

but the fact that the

_posterior

distribution of

_heritability

for PART

_(fig

₃₎

was

very skewed indicates that more data should be collected for accurate inferences about

_{heritability.}

The WHOLE

_analysis

_{(fig 4)}

_{yielded nearly}

_{symmetric posterior}

distributions of

_heritability

with smaller

variance,

centered at 0.55

_(the

REML estimate was

_0.543).

It should be noted that the residual variance was

poorly

estimated in PART

_{(fig 3),}

but the

_{Bayesian analysis clearly depicted}

the extent of

_uncertainty

about this

_parameter.

This also illustrates the fact that a variance

contrary

to the common belief in animal

_breeding

that residual variance is

_always

well estimated.

The assessment of response to selection for the

_population

simulated with

h

2

= 0.2 is

_given

in

_figures

5 and 6 for the PART and WHOLE data

_analyses,

_{respectively.}

The true response in the simulated data at a

_{given generation, computed}

as the average of true

_genetic

values within the

_generaton

in

_question,

was TR = 0.643 units and TR = 1.783

units,

for PART and WHOLE. The

regressions

of true response on

_generation

were 0.321 and

_0.343,

for PART and

WHOLE,

respectively.

The

_hypothesis

of no _response to selection cannot be

_rejected

in the PART

_analysis

(fig 5).

The classical

_approach

_gave an estimated response of zero.

_However,

the

Bayesian analysis

documents the extent of

_uncertainty,

and

_assigns

considerable

posterior

probability

to the

_proposition

that selection _may be

_effective;

the

_posterior

probability

of response

(eg,

Ql > 0 or TR =

_t

₂

_-

_to

_>

0)

_was much

larger

than that of the

_complement.

The

_strength

of additional evidence

_brought

_{up in} the whole

_analysis

_{(fig 6)}

_supports

the

_hypothesis

that selection was effective. For

example,

the difference in

_genetic

mean between

_generation

0 and 5 was centered

around

_1.852,

_though

the variance of the

_posterior

distribution was as

_large

as 1.075. The classical

analysis

gave an estimated value of ST = 1.296

_units,

but _no exact measure of variation can be associated to this estimate.

_Approximate

results could be

_obtained,

for

_example,

_using

Monte-Carlo

_simulations,

but this was not

attempted

in the

present

work.

In the simulation with

h

2

=

_0.5,

the _{true response in} the simulated

data,

computed

as the _average of true

_genetic

values from the relevant

_generations,

was TR = 3.258 units and TR = 7.148 units for the PART and WHOLE data

sets,

respectively.

The associated

_regressions

on

_generation

were 1.629 and

1.436,

respectively.

The

_Bayesian

and classical

_analyses

indicated response to selection

_beyond

reasonable doubt

_(fig

7 and

_8).

Note, however,

that the

_posterior

distribution of TR =

_t

₂

_-

_to

in PART _was skewed

(fig 7).

In

figure

_8,

all the

posterior

distributions were

_nearly

_symmetric,

and the classical and

_Bayesian

analyses

were

_essentially

in

_agreement.

In

_spite

of this

_symmetry,

considerable

uncertainty

remained about the true rate of response to selection. For

example,

values of

t

5

-

to

ranging

from 3 to 11 units have

_appreciable

_posterior

_density

(fig 8).

In the

_light

of

_this,

results from a similar

_experiment

where realized

_genetic

change

is,

say,

50%

higher

or lower than our

_posterior

mean of about 6.7

_units,

could not be

_interpreted

as

yielding conflicting

evidence.

In order to

_study

the influence of different

_priors

on inferences about

_heritability

and _response to

_selection,

_part

of the above

_analyses

_using

the same

_data,

was

repeated assuming

that variance

_components

_followed,

a

_priori,

the inverse

chi-square distributions

(9!.

The

’degree

of belief’

_{parameters v}

_i

and the a

_priori

value

of the variance

_components

_S

₂

_,

were assessed

_by

the method of moments from

an

independently

simulated data set with base

_{population heritability}

of

_0.5,

and

consisting

of 3

_cycles

of selection. An REML estimate of

_heritability

from this data

was 0.46 with an

_asymptotic

variance of 2.94 x

10-

2

.

The method of moments

as described in

_Wang

et al

_(1994b),

_produced

the

_following

estimates: _va =

11;

v

e =

_464;

Sa

_{= 4.50;}

Se

= _{5.19. A}

reanalysis

of the PART and WHOLE

_replicate

(heritability = 0.5)

yielded

the results shown in table I. The

_figures

in the table

the

_input

contributed

_by

the

_prior.

Thus in the PART

_dataset,

the uniform

_prior

and the informative

_prior

_(which

_implies

a

_prior

value of

heritability

of

0.46)

lead to clear differences in the

_expected

value of the

_marginal

_posterior

distribution of

heritability

and selection _response.

_However,

in the WHOLE data

_set,

inferences

using

either

_prior

are _very similar. The

_figures

also illustrate how the variance of

marginal

posterior

distributions is reduced when an informative

_prior

is used. The

_examples

indicate the power of the

Bayesian

analysis

to reveal

_uncertainty

appropriate parameters

is small

_(eg,

PART

_analysis

at

h

2

=

_0.2).

Other

_plots,

such as

_marginal

_posterior

distribution of

_generation

_means, for

_example,

are

possible

and could illustrate the evolution of the drift variance as the

_experiment

progresses.

DISCUSSION

We have

_presented

a _way of

analysing

_response to selection in

_experimental

or

is the construction of a

_marginal

_posterior

distribution of a measure of selection

response

which,

in this

study,

was defined as a linear function of additive

_genetic

values. The

_marginal

distribution can be viewed as a

weighted

_average of an

infinite number of conditional

distributions,

where the

_weighting

function is the

marginal

posterior

distribution of variances and other

parameters,

which are not

necessarily

of interest. The

_approach

relies

_heavily

on the result

(Gianola

and

Fernando,

1986)

that if the data on which selection was based are included in the

that the

_{joint posterior}

or _any

_marginal

_posterior

distribution is the same with or without selection. This is a

_particular

case of what has been known as

_’ignorable

selection’

_(Little

and

Rubin, 1987;

Im et

_al,

₁₉₈₉₎

and it

_implies

that selection

can be

_ignored

if the information of _any data-based selection is contained in the

Bayesian

model.

_Furthermore,

inferences

_pertain

to

_parameters,

_eg,

_{heritability,}

of the base

population.

As shown with the simulated

data,

the

_marginal

posterior

distributions of

vari-ances and functions thereof are often not

_symmetric.

This fact is taken into account

when

_computing

the

_marginal

_posterior

distributions of measures of selection

re-sponse. In this way, the

uncertainty

associated with all nuisance

_parameters

in the model is taken into account when

_drawing

inferences about response to selection. This is in marked contrast with the estimation of _response that is obtained

_using

BLUP with an animal

model,

which assumes the variance ratio

_known,

and

_gives

100%

weight

to an estimate of this ratio.

The

_analysis

of the simulated data used uniform distributions for the fixed effects and variance

_components.

The choice of

_{appropriate prior}

distributions

is a contentious issue in

_{Bayesian inference,}

_especially

when these

_priors

are

supposed

to _{convey vague} initial

_{knowledge. Injudicious}

choice of noninformative

prior

distributions can lead to

_improper

_{posterior distributions,}

and this may not be

easy to

_recognize,

_especially

when the

_analysis

is based on numerical

_methods,

as in

the

_present

work. The

_subject

is still a matter of debate and an

_important

_concept

is that of reference

_priors

_(Bernardo,

_1979;

_Berger

and

_Bernardo,

_1992),

which have the

_property

that

they

contribute with minimum information to the

_posterior

distribution while the information

_arising

from the likelihood is maximized. In the

single

parameter

case, reference

priors

often

yield

the

Jeffreys

priors

(Jeffreys, 1961).

The uniform

_prior

distributions we have chosen for the fixed effects and the variance

components represent

a state of little

_prior

_knowlege,

but are

clearly

not invariant

under transformations and must be viewed as

_simple

ad iaoc

_{approximations}

to the

appropriate

reference

_prior

distributions. We have

illustrated,

however,

that when the amount of information contained in the data is

_adequate

inferences are affected

little

_by

the choice of

_priors.

It is not

_generally

a

_simple

matter to decide when

one is in a

_setting

of

_{adequate information,}

and it _may be therefore

revealing

to carry out

_analyses

with different

_priors

to

_study

how inferences are affected

(Berger,

1985).

If use of different

_priors

leads to _very different

results,

this indicates that the information in the likelihood is weak and more data

_ought

to be collected in order to draw firmer conclusions.

The richness of the information that can be extracted

using

the

Bayesian

approach

is at the cost of

computational

demands. The demand is in terms of

computer

time rather than in terms of

_{programming complexity. However,}

an

important

limitation of iterative simulation methods is that

_drawing

from the

appropriate posterior

density

takes

_{place only}

in the

limit,

as the number of draws becomes infinite. It is not _easy to check for convergence to the correct

_{distribution,}

and there is little

_{developed theory indicating}

how

_long

the Gibbs chain should be. The rate of convergence can be

exceedingly

slow,

especially

when random variables

are

highly correlated,

which is the case with animal models.

Further,

there is often a

_possibility

that the Gibbs _{sequence may}

_get

_{’trapped’}

near zero which is an