Use of covariances between predicted breeding values to assess the genetic correlation between expressions of a trait in 2 environments

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Use of covariances between predicted breeding values to

assess the genetic correlation between expressions of a

trait in 2 environments

Dr Notter, C Diaz

To cite this version:

Original

article

Use

of

covariances between

_predicted

breeding

values

to

assess

the

genetic

correlation between

_expressions

of

a

trait in

2

environments

DR

_Notter,

C Diaz

Virginia Polytechnic Institute and State University, Department of Animal Science, Blacksburg, VA _24061, USA

(Received

23 June 1992; accepted 23 March

₁₉₉₃₎

Summary - Procedures to _interpret correlation and _regression coefficients _involving

pre-dicted _breeding values

_(BV)

calculated for the same animals in different environments

have been _developed. Observed correlations are a function of the additive _genetic

cor-relation between _performances in the 2 environments but are also affected _by selection

of animals that _produce data in both environments, the accuracy of BV predictions in

each environment, relationships among animals within and across environments and

co-variances among BV predictions within an environment arising from estimation of fixed effects in best linear unbiased _prediction

_(BLUP)

of animal BV. Methods to account for effects of selection and variable accuracy and experimental designs to minimize effects of

relationships and covariances among BV predictions from estimation of fixed effects have been described. The _regression of predicted BV in environment ₂ on _predicted BV in

envi-ronment 1 is _generally not affected by selection in environment

1,

but both correlation and

regression coefficients are sensitive to covariances among

breeding

value predictions within environments. In _general, caution must be exercised in interpreting observed associations

between _{predicted breeding} values in different environments.

predicted breeding value _{/ genetic} correlation _{/ regression /} selection index _/ best

linear unbiased prediction

Résumé - Utilisation des covariances entre les valeurs _{génétiques prédites pour} estimer la corrélation _génétique entre les _expressions d’un caractère dans 2 milieux. Des _procédures sont établies pour interpréter les _coefficients de corrélation et de _répression

impliquant des valeurs _{génétiques prédites}

_(VG)

calculées pour les mêmes animaux dans

différents milieux. Les corrélations observées sont fonction de la corrélation _génétique

entre les _performances dans les 2 milieux, mais elles _dépendent aussi de la sélection des

animaux sur lesquels des données sont recueillies dans les _{2 milieux,} de la _précision des

prédictions de VG dans chaque milieu, des _parentés entre animaux intra-milieu et entre

milieux et des covariances entre les _prédictions de VG dans un milieu _qui résultent de

VG. L’article _présente des méthodes pour prendre en _compte les effets de la sélection et

de la _précision variable des _prédictions et des _{plans d’expérience} pour minimiser les effets

de la _parenté et des covariances entre les _prédictions de VG à _partir de l’estimation des

effets fixés. La _régression des VG _prédites dans le milieu 2 en _fonction des VG _prédites dans le milieu 1 n’est _{généralement} pas affectée par la sélection dans le milieu 1, mais

la corrélation et la _régression sont toutes deux _{influencées par} les covariances entre les

prédictions de VG intrn-milieu. D’une manière _générale, une _{grande prudence} est _requise

dans _{l’interprétation} d’associations entre des valeurs génétiques prédites dans _différents milieux.

prédiction de valeur _{génétique /} corrélation _{génétique / régression /} indice de sélection _/ meilleure _prédiction linéaire sans biais

INTRODUCTION

Procedures to estimate additive

_genetic

correlation

_(r

_G

₎

between

_expressions

of the

same trait in different environments were introduced

by

Falconer

(1952),

Robertson

(1959),

Dickerson

₍₁₉₆₂₎

and Yamada

_(1962).

The

procedures

are

analogoes

to those for estimation of

_genetic

correlation between 2 traits in the same _environment, but

_recognize

that

_performance

is

_normally

not measured on the same animal in

multiple

environments.

_Instead,

related animals

_{(often half-sibs)}

are

produced

in

each environment and rG is derived

by

comparing

the resemblance among relatives in different environments to that observed among relatives in the same environment. In

_{single-generation}

_experiments

_utilizing

_half-sibs,

sires can

_produce

_{progeny in}

pairs

of environments. If sires are evaluated in environment 1 before

_being

used

in environment _2,

_divergent

selection of sires can increase

_precision

of estimates of the

_{genetic regression}

of one trait on the other when a fixed number of _progeny

is measured

_(Hill,

_1970;

Hill and

_Thompson,

_1977).

This _strategy makes use of

the fact that ’selection of sires biases correlation between parent

predicted breeding

value

_(BV)

and

_{offspring performance}

but does not affect the

_regression

of progeny

performance

on _parent

_predicted

BV so

_long

as there is no selection of _progeny records.

Data from

industry

performance-recording

programs often include records of

relatives evaluated in different _{environments,} but the data structure is not under

experimental

control. Animals differ in the amount of information

_available,

and unknown

_non-genetic

sources of resemblance _{among relatives} can exist.

Likewise,

little information _{may exist} on

_procedures

used to select _parents in each environ-ment. Procedures to estimate additive

_genetic

covariances from these

_industry

data

sets exist

_{(Meyer, 1991)}

and have been used to estimate covariances between

ex-pressions

of the same trait in different environments

(eg,

Dijkstra

et

_al,

_1990).

These

analyses require

that the model includes all

_genetic

and

_nongenetic

sources of

re-semblance _among relatives and that information used to select _parents be included

in the data.

_Large

numbers of records often _exist, but

_only

a fraction of them _mey

If

_predicted

BV for the same animals in different environments were derived

using

only

data from within each _environment, correlations _among

_predicted

BV

across environments should

provide

information about rc. Observed correlations in such situations

_(Oldenbroek

and

_Meijering,

1986;

DeNise and

_Ray,

_1987;

Tilsch et

al,

1989a,b;

Mahrt et

_al,

₁₉₉₀₎

were

usually

< _1,

_but,

as noted

_by

Calo et al

₍₁₉₇₃₎

and Blanchard et al

_(1983),

the

_expected

value of the correlations is also < _1, even

if the

_underlying

_genetic

correlation is

_unity.

Thus correlations between

_predicted

BV in different environments must be

_interpreted

relative to their

_expected

value. This paper will consider the

expected

values of observed correlations and consider alternative

_{experimental designs. Expected}

values of correlation and

regression

coefficients under ideal conditions will first be reviewed. Effects of non-random

selection,

variation in the _accuracy of BV

_{predictions, relationships}

among animals within and across

_{environments,}

and covariances _among BV

_{predictions arising}

from estimation of fixed effects under best linear unbiased

_prediction

_(BLUP)

will then each be considered.

RELATIONSHIP BETWEEN PREDICTED BV IN 2 ENVIRONMENTS

Let a

population

in environment 1 have additive

_genetic

variance

Q

u

l

for some

trait. Predict BV

_(Mi)

in that _environment, and choose m sires to

_produce

progeny

in environment 2. Predict BV in environment 2

_(Û

₂

₎

_using

_only

data from that environment. Let the additive

_genetic

variance for the trait in environment 2 be

ol u

2

2and

the

_genetic

correlation between BV in environment 1

_(u

_l

₎

and 2

_(u

₂

₎

be rG. Let the accuracy of BV

prediction,

aland a2 for environments 1 and _2,

_{respectively,}

be the correlation between actual and

_predicted

BV and be constant within each environment.

Under certain

_conditions,

the

expected

correlation between

predicted

BV in the 2 environments

_(Tulu2!

is a,rGa2 and rG can be estimated as rG =

r&mdash; !

/a

l

a

2

-ul!!2)

Ul 2

The conditions include

_{(Taylor, 1983): 1)}

no environmental correlation between

performance

in the different _{environments;}

₂₎

no

_{relationships}

_{among parents} of

measured

_animals;

and

₃₎

no other _{covariances among}

_predicted

BV within either

environment. An additional _assumption

₍₄₎

is that sires are chosen at random. For sire evaluation with these

_assumptions,

_{a ij 2}

=

n

ij/

(n

ij

₊

A)

where n

ij is the number of _progeny for sire i in

_{environment j}

and A is the ratio of residual to sire variance.

_Assumption

₍₁₎

is

_normally

met if different animals are measured

in different environments.

_Assumptions

₍₂₎

and

₍₄₎

can be met

_through

choice of sires.

_Assumption

₍₃₎

will not

_normally

hold for

_BLUP,

but may

approximately

hold under some conditions.

The

_regression

_(b)

of

_u

₂

on

ill

has the expectation:

such that TO =

b!2u1

(Q!1 /a2!!z ).

Taylor’s

(1983)

assumptions

are

_required

for

this

_expectation,

but random selection of sires is not.

_Knowledge

of

_a!l

and

_Q

_u

₂

is

u 2

. Effects of

using

incorrect values of

_o,2

_on estimates of rG will not be considered

further,

but _may be

_important.

Expected

confidence limits for observed correlation

_(F)

and

_regression

coefficients

(b)

can be used to evaluate

_experimental

_designs

in terms of their

_ability

to detect

significant

departures

of

r and

from their

_{expectations.}

For correlation

_analysis,

Fischer’s z =

0.5[ln

(1 +r) -In

(1 - r)] (Snedecor

and

Cochran,

1967)

has variance of

!

(m-3)-

1

_where m is the number of sires in the

_sample.

_{For ) r)}

₆

_0.65, confidence bounds on F are of similar width at fixed m, whereas for

Irl

>

_0.65,

confidence bounds narrow with

_increasing

r.

_Large

numbers of sires are thus

_required

if

accuracies are low to avoid confidence limits that

_overlap

0. For

_regression

_analysis,

variance

of

_{b [V(b)]}

is:

(Snedecor

and

Cochran,

1967)

where

Q!2I!1

is the variance in

_Û

₂

at a fixed value of

!

U2 Ut

Û

l

_(ie,

the mean _square for deviations from

_regression)

and

SS(ul)

is the sum of

squares for

u

l

.

Given

_{T_aylor’s}

(1983)

assumptions

for m sires

_sampled

at random from environment _1,

V(b)

is:

Numbers of sires and progeny

required

to detect

_significant

_departures

of

6 from

its

_expected

value are

_given

in

_figure

1 for several values of _{rc, , , Oau&dquo;U2 and} al = _a2

or al = 0.95. When _al = _a2

(eg,

when sires are

_being

_proven

simultaneously

in 2

environments),

numbers of _progeny

_required

in each environment to

_reject

the null

hypothesis

that rG = 1 are minimized at _aj = 0.7 _to 0.8 for _rG between 0.5 and

0.8. For al = 0.95

(eg,

when

proven sires are chosen from environment

_1),

progeny

numbers in environment 2 are minimized with one _{progeny per sire,} but increase

little until a2exceeds 0.5 to 0.6. Thus

relatively

efficient

designs

at al = 0.95 would

include 35 to _{45 sires with 400-500 progeny at rG} = 0.6, but 250-400 sires with

1200-1300 progeny at _rG = 0.8.

Critical numbers

_required

for correlation

_analysis

were similar to those for

regression

analysis

at low accuracies and rG, but lower at

_higher

accuracies due to

_asymptotic

declines in the width of confidence intervals as

expected

r increased.

The ratio of the critical number of sires for correlation

_analysis

to critical number for

regression

analysis

(SRAT)

was

_predictable

(R

2

=

0.983)

as a function

_{of q =}

ala2 and resuch that SRAT = 1.115-0.101

_q-0.667

q

2

-0.161

_{1 rG .} This ratio

adjustment

can be

_{applied directly}

to values in

_figure

1 to

_approximate

critical sire and progeny numbers for correlation

_analysis.

The above derivations assume that accuracies are calculated

_correctly

in both

environments. Under

BLUP,

accuracies of

u

i

for non-inbred animals are

_given

_by

(1 &mdash;

Ci2/Q!)’S

where

_C

_ii

is the ith

_diagonal

element of

_C

₂₂

_,

the

_prediction

error

covariance matrix of u

_{(Henderson, 1973).}

If the model is

_complete

and

_properly

parameterized,

accuracies are

expected

to

_equal

correlations between actual and

predicted

BV. In most

_{applications,}

u is derived

_by

iterative solution of Henderson’s

elements of

_C

₂₂

are

_approximated

but

_off-diagonal

elements of

_C

₂₂

are

usually

not estimated. To

date,

no

_{completely satisfactory procedures}

to obtain

_diagonal

elements of

C

22

exist. Alternative methods have been

_{presented by}

Van Raden and Freeman

_(1985),

_Greenhalgh

et al

_(1986),

Robinson and Jones

_(1987),

_Meyer

₍₁₉₈₉₎

and Van Raden and

_Wiggans

_(1991).

Evaluation of

_procedures

to estimate accuracy

is

_beyond

the _scope of this

_study,

but the

_assumption

that accuracies are estimated

correctly

is critical to the discussion.

Effects of

_departures

from the ideal conditions described above will now be discussed.

Effects of non-random selection from environment 1

Let sires be

_non-randomly

selected based on

Û

l

and accuracies be constant within each environment. Let unselected

_population

_{variances, covariances,} correlations and

_regressions

be

_{symbolized by}

_Q

_2,

0&dquo;, r and

b,

respectively,

and let

V, Cov,

Corr and

_{Regr respectively}

_represent observed values for some

sample

from the

population.

For truncation selection on

u

l

,

where w = 1 -

V(Û¡)/

0

&dquo;1

(Robertson, 1966).

For directional truncation

selection,

m

w =

i(i &mdash; x)

and for

divergent

truncation selection _w = -ix where i is the standardized selection differential

_{(Becker, 1984)}

and x is the truncation

_point

on a standard normal curve

_(Snedecor

and

_Cochran,

₁₉₆₇₎

_{corresponding}

to random selection of sires from the upper or lower

fraction,

p, of the

u

l

distribution for directional selection or from the upper and lower fraction

p/2

for

divergent

selection.

Also:

(Hill,

1970;

Johnson and

_Kotz,

_1970;

_Robertson,

_1977).

The observed correlation is thus biased

_by

selection but the observed

_regression

is _not, and the deviation of

Regr

(u

2

u

1

)

from its

_expected

value

_provides

a test of the

_hypothesis

that rG = 1.0.

If selection is non-random but not

_clearly

directional or

_divergent

or not based on

truncation, additional

_{complications}

arise. To account for such

selection,

let

_{V(Ei )}

be calculated for the selected

_sample

and define w

_empirically

as the observed value

1-

_{V(iil)lo,!! .}

Use of this

_empirical

value of w to

_predict

rG

using equation

[3]

was

U¡ i

evaluated

_by

_computer simulation. Predicted BV for the ith sire in environment 1

where

61i

is a random normal deviate

(SAS, 1985),

0

&dquo;;B

= 315 and _al = 0.7.

Predicted BV in environment 2 were then simulated for _a2 = 0.7 and

Q!! oru2

as:

Three selection scenarios

_(SS)

were considered:

SS1. 80%

_divergent,

20% random: 80% of the bulls chosen such

_{that lxl}

>

_1.282(i

=

1.755)

and 20% chosen at

_random;

SS2. 50%

_high,

50% random: 50% of the bulls had x > 0.842

_(i

=

1.400);

SS3. 50%

_high,

50%

_{stabilizing: lxl}

< 0.5 for 50% of the bulls and x > 1.282 for

50% of the bulls.

Each scenario was

_repeated

for _rG = 1.0 or 0.5 and

replicated

10 times. Each

_replicate

contained 5 000 selected animals.

Agreement

between

_predicted

and simulated values of

_V(ii

₂

₎

and

_Corr(ic

_l

_u

₂

₎

(table I)

was within theoretical 95% confidence limits of the

expected

value

(Snedecor

and

Cochran,

1967).

Thus

_equation

_[3]

predicted

Corr(û

l

û

2

)

satisfac-torily

in bulls selected

_non-randomly

on

u

l

with fixed _accuracy al.

With selected _sires, the

expected

V(b)

is:

using

values from

_equations

_[1]

and

_[3].

The SD of is

_{inversely proportional}

to

vi --w and

varies from 48 to 243% of its value when w = 0 _{as w varies} from

- 3.39

_(divergent

selection from the top and bottom 5% of the

_population)

to 0.83

departures

of

_Regr(u

₂

_u1)

from its

_{expectation using}

selected sires can be derived

from

_figure

1

_by

_dividing

sire and total animal numbers

_by

1 - w.

Effects _{of variation among animals in} accuracy of

predicted

BV

Calo et at

₍₁₉₇₃₎

and Blanchard et at

₍₁₉₈₃₎

derived the

_expected

correlation between

_predicted

BV for 2 traits when BV for each trait were estimated in

separate

single-trait analyses

and individuals differed in _accuracy of BV

_prediction

as

C

’

OT’r(ui

M2

)

= _rGal2 for:

(see

Appendix)

and recommended

_using

this

_expression

to estimate rG from ob-served

_C

_O

_rr(iil

_i

_i

₂

_).

_Similarly:

Taylor

(1983)

criticized

_equation

_[5]

as

unstable, however,

_asserting

that it

may

yield

estimates of rG that are outside the _{parameter space,} and

presented

assumptions required

to allow estimation of rG with this

equation.

Taylor

(1983)

concluded

that,

if all _assumptions are _met,

_equation

_[5]

is

_appropriate

to estimate

r

G so

long

as the

a!

are derived from MME as

(1 -

C,,/

0

,2).

The

_equations

of Calo et al

₍₁₉₇₃₎

and Blanchard et al

₍₁₉₈₃₎

do not consider selection on

Ûl.

With

_{selection, equation}

[5]

_appears

_appropriate

to estimate

Corr(u

l

u

2

)

for the selected

sample,

but not within the unselected

_population.

Equations

[3]

and

_[5]

_could,

however,

perhaps

be combined to

_give

the

_expected

correlation in a selected

_sample

of animals with variable _accuracy as:

To evaluate

_equation

_[7],

several accuracy scenarios

(AS)

were considered

by

simulating

samples

from the

u

i

distribution.

AS1: 20 000 animals from the upper 10% of the

u

l

distribution. al varied

uniformly

over the interval 0.7 to 0.95.

AS2: 20 000 animals from the upper and lower 10% of the

ui

distribution. al varied

uniformly

over the interval 0.7 to 0.95.

AS3: 10 000 animals from the _upper 10% of the

_Û

_l

distribution with al

uniformly

distributed over the interval 0.7 to 0.95 and 10 000 animals selected from the lower 10% of the distribution with al

uniformly

distributed over the interval

0.5 to 0.7.

the bottom 80% of the distribution with al

uniformly

distributed over the

interval 0.5 to 0.7.

ASS: 15 000 animals from the _{upper 10%} and 5 000 animals from the lower 80%

of the

Û

l

distribution with al

uniformly

distributed over the interval 0.7 to

0.95.

AS6: 5 000 animals from the upper 10% and 15 000 animals from the bottom 80% of the

_Û

_l

distribution with al

uniformly

distributed over the interval 0.5 to 0.995.

AS7: 10 000 animals from the upper 10% of the

Û

l

distribution with al

uniformly

distributed over the interval 0.795 to 0.995 and 10 000 animals from the lower

80% of the distribution with al

uniformly

distributed over the interval 0 to

0.50. a

2 was

_uniformly

distributed over the interval 0.5 to 0.7 for all scenarios. Each scenario was

_repeated

for rc = 0.5 _or 1.0 and

replicated

(table II).

Empirical

calculation of w

_requires

use of

a2 ,

which varies with accuracy. Simulated values

! ul

of

_Mi

were thus standardized

by dividing by

_{aliaul and}

_empirical

w calculated as

1 &mdash;

_V(u

_l

₎

_using

standardized

_u

_l

_.

For all accuracy scenarios, observed

_Regr(u

₂

_u1)

_{agreed closely}

with

_predicted

values from

_equation

_[6]

_{(table II).}

Observed values of

_Corr(u

_l

_u

₂

₎

were

_usually

also close to

_expectations

from

equation

[7],

but with some

_systematic

_departures

Thus

_equation

_[7]

_produced

a small

_negative

bias under directional selection with

variable accuracy. This result was confirmed

_{by producing}

10 more

_replicates

at

r

G =

_1;

the results _were identical.

For

_divergent

selection,

differences between observed and

_predicted

correlations

were

_again

_small,

but sometimes

_significant

and now

_negative

for

AS2,

3,

5 and

6,

ranging

from -0.001 to -0.008

_(!0.002).

However,

with both

_{non-symmetrical}

selection from

_high

and low groups and different accuracy distributions between

groups

(AS4

and

AS7),

observed correlations were

_{considerably larger}

than

pre-dicted,

especially

for rG = 1

(table II).

The

appendix

shows _exact

_expectations

for

correlations and

_regressions

_{involving u}

_i

and

_Û

₂

under non-random selection from environment 1 and variable accuracies within environments. Correlation between

means and accuracies of

_divergently

selected _groups violate some of the

assump-tions used to derive _equation

_[7]

and

_presumably

account for the

_departures

from

predicted

values in AS4 and AS7.

Equation

[7]

thus

_{produced slightly}

biased

_predictors

of

_Corr(Ei

_E2

₎

but still

appears

useful,

especially

when exact selection rules are unknown.

However,

biases

in

_predicted

values of

_Corr(û

_l

_Û2

₎

in

_equation

_[7]

will be

_{multiplied by}

the inverse of the coefficient of rG in

equation

[7]

to estimate rG. Potential bias in re thus is

larger

with lower _aj or more directional selection. If

V(u

i

)

is

larger

than

expected

from random selection and _greater

_precision

than that

_provided

_by

_equation

_[7]

is

desired, Ap

P

endix

equations

can be used.

The

_expected

correlation between

Û

l

and

Û

2

thus

_depends

on the distribution of accuracies within each _environment, the selection

_applied

on

Û

l

(quantified

by

w).

and rG. To evaluate net effects of these

variables,

values of

Corr(û

l

û

2

)

from

equation

[7]

were calculated for rG = 1 and when _a12 varied from 0.10 _to 0.90 and w

varied from -3.3 to 0.9

_{(table III).}

For re = _1.0 and _a12 = _0.90,

Corr(u

l

u

2

)

varied

from 0.55 to 0.97 due to

_selection,

_although

_rG exceeded 0.79 so

_long

as directional

selection was not intense

_(w

6

_0.6):

For w = 0 and _rG = 1.0,

Corr(u

lu2

)

equalled

a l2

, but still varied from 0.10 to

0.90,

depending

on observed values of _al and _a2.

Effects of

relationships

Previous results assume

_independence

of

predicted

BV within each environment.

However,

relationships

among animals lead to covariances among

predicted

BV within and across environments. If

_ui

and

_U

₂

are

predicted

_by

BLUP,

covariances

among

predicted

BV within environments also arise from estimation of fixed

effects. These covariances affect

_expectations

of both

_Corr(u

_l

_u

₂

₎

and

_Regr(u

₂

_u1).

Their

_impact

is difficult to

_{generalize, depending}

_upon the extent and nature of

relationships

in the data and the distribution of records _among fixed effect classes. Covariances among

predicted

BV associated with

_{relationships}

and estimation of fixed effects arise simultaneous in BLUP solutions to

_MME,

but effects of

relationships

alone can be seen under selection

index,

or best linear

prediction

(BLP),

assumptions

of known mean and variance for both _ul and _u2. In that case:

where

_Û

_j

is a vector of

_breeding

value

_predictions

for

_{environment j}

_{and yj} is

matrix _{between yj} and

_u’..

For non-inbred

animals,

_H

_j

=

Z

j

G

=

Z

j

Ao, 2

where

Z

j

is the incidence matrix

_relating

_yj to _Uj, and G and A are additive covariance

and numerator

_relationship

_matrices,

_{respectively,}

for animals in _Uj. The covariance matrix of

_{Gj (Qj )}

and the covariance matrix between

_Û

_l

and

_u2

_(Q

_i2

₎

are thus:

Expectations

of

_sample

variances of

_U

_j

_{(s3- )}

and covariance between

M

i

and

_ Uj

u2(Sulu2)

are functions of elements of

_Q

_j

and

Q

12

:

where tr is the trace of the matrix and sum is them sum of all elements. If animals

recorded in the 2 environments resemble one another

only

because of

relationships

This

_assumption

is warranted if animals are evaluated

_using

unrelated

popula-tions of mates in the 2 environments but _may not be correct if mates are

poten-tially

related across environments. If selected animals are likewise

unrelated,

and

relationships

among recorded animals within each environment arise

only through

relationships

to selected

animals,

_Q

_j

will be

_diagonal

with elements

_{a? . 2}

for the ith animal in the

_jth

_environment, A is an

_identity

matrix of size m and

Q

lz

is

di-agonal

with elements

_{aiiaiirGaul!u2.}

_Sample

correlation and

_regression

coefficients then have

_expectations

_equal

to those

_previously

discussed.

In most

_{applications,}

_Q

_j

are not

_diagonal

and

_off-diagonal

elements are not calculated due to the nature and size of

_{V! 1.}

Thus

_explicit

consideration of

off-diagonal

elements of

_Q

_j

may not be

_possible

unless the data set is small or

_highly

structured.

However,

if the _accuracy of all

_Û

_ij

_approaches

_1,

_Q

_j

_approaches

_{Ao, &dquo;j 2}

(ie, Qu --!

j

)

or2 u

and

_Q

₁₂

_approaches

_utO

_u2

_&dquo;

_&dquo;

_rcO

_A

such that

_Corr(û

_l

_û

₂

₎

= _TG and

°/J ’

Regr(û

2

û¡)

=

r

C

O&dquo;U

2/

O

&dquo;

U

t’

A more realistic situation is one in which sires from environment 1 have a,

approaching

1 but are evaluated in environment 2 with a2 < 1. In this case,

Q

l

=

AQ!I,Qz

=

H2VZ

l

Hz,

and

Q

12

=

&dquo;

’

)

U2

(rc

Ut/O

&dquo;

O

2

Q

These

_quantities,

and

associated

_{expected sample}

variances and _covariances, could be obtained if the size

or structure of the data allows calculation of all elements of

_Q

₂

and would allow derivation of an exact

_predicted

value for

C’orr(uiU2)-A small

_example

will demonstrate the

impact

of

_{relationships}

on

_Corr(ûlû2)

and

_Regr(û

₂

_ûl),

which are

equal

in these

examples.

Let

h

z

= _{0.25, Qu,} =

O &dquo;U 2 = 1

and rG = _1. Let 3 sires

produce

8 _progeny each in each of 2 environments. If sires are unrelated and _progeny are related

_only

_through

the _sires,

_sample

variances and covariances

_{involving Û}

l

and

_Û

₂

_{equal expected population}

_values,

and

_Corr(û

_l

_Û

₂

₎

= _{. zala}If all 3 sires _are full

sibs,

Q

j

and

_Q

_lz

are no

longer diagonal

and al = _a2 = 0.643. The

expected

Corr(ûlû2)

is 0.211 versus _ala2 = 0.414. If sires are half

sibs,

Corr(û

l

Û

2

)

= 0.286 versus _ala2 = 0.365,

reducing

bias

by

one

half as

relationships

among

sires decline.

Still,

with _many close

relationships

_among

sires, ala2 may

considerably

overestimate the

_expected

correlation.

If

_only

2 of the 3 sires are full

sibs,

bias is reduced. al = _a2 = 0.621 for related sires and 0.590 for the unrelated _sire, and a12 = 0.374

(equation

[5]).

Now _a12

overestimates the observed correlation

_{by only}

3.9%

_(0.374

versus

0.360).

Thus if

sampled

animals _represent a reasonable number of unrelated

families,

Q

j

and

Q

12

are

correspondingly

_sparse and little bias in

Corr(u

i

U

2

)

is

expected.

Turning

to effects of

_{relationships}

within

_{environments,}

let the 3 sires be unrelated but cross-classified within each environment with

_only

2 dams. In that

case, al =az =0.566 and

Corr(ii

IU2

)

= 0.364 which is 13%

larger

than _ala2= 0.321.

However,

under the more realistic

_assumption

of cross-classification with 2 maternal

grandsires,

Corr(ûl Û2)

= 0.352 _{versus alaz} =

_0.338,

yielding

little bias. When sires

were cross-classified with 8

dams,

Corr(û

l

û

2

)

= 0.364 versus _ala2 =

0.349, again

yielding

little bias. These

_examples

suggest selection of

_widely

proven,

lowly

related

animals from environment 1 followed

_by

evaluation in environment 2

_using

a broad

Effects of covariances

_arising

from estimation of fixed effects

If fixed effects are estimated

_{simultaneously}

with BLUP of

_iij,

Mallinckrodt

₍₁₉₉₀₎

noted that

_off-diagonal

elements of

_Q

_j

and

_Q

₁₂

are not _zero, even in the absence

of

_{relationships. By}

BLUP:

f

or

# =

(X!V! 1X!)-1X!V! ly!

and

_{P_,}

=

X!(X!V! 1X!)-1X!

and

_wherep

is the vector of fixed effects with incidence matrix X.

The covariance matrix of

_Û

_j

is:

composed

of a term due to BLP

_{relationships}

minus a term due to estimation of fixed effects. If

_{relationships}

across environments arise

only through

sires

sampled

from environment 1 such that

_Cov(y

_l

_y2)

is

_given

_by

_equation

_(10],

_Q

₁₂

is still

_given

by

equation

[11],

but

_regardless

of the

_relationship

structure of the

_data,

_Q

_j

now

approaches

a

_diagonal

matrix

_only

as _aj

_approach

1.

_Also,

for a

_given

number of

progeny, aijwill be less for BLUP than for BLP and

depends

on the number of sires

and their distribution among fixed effect classes. The above solutions are identical

to those obtained from MME

_(Henderson,

_1963,

₁₉₈₄₎

such that:

The

_impact

of fixed effect estimation on

_Corr(û

_l

_û

₂

₎

can be seen most

_readily

using

MME for a sire model without

relationships

_among animals in u and where

P

includes

_only

_{contemporary group} effects. Note that in all

_remaining

_examples,

Corr(£i

E2

)

=

Regr(u

2

u

1

)

when _al = _a2. For such _a

model,

after

absorption

of

fixed effects into u

_equations

and

_factoring

of residual variance from both sides of the

_equation,

the coefficient matrix for u has:

(Do, 1991)

where A is the ratio of residual to sire variance and ni.,n.k and

n

ik are numbers of records for sire

_i,

_contemporary group k

(of g)

and sire x

contemporary

group subclass

ik,

respectively.

For balanced

data,

nik = _n for

off-diagonals

reduce to

_(-gn/m)

_[vs

0 for

_BLP].

The

_{corresponding}

inverse of the coefficient matrix has

diagonal

elements of

_(gn

+

_mA)/[m(gn

+

_A)]

and

off-diagonals

of

_gn/[mA(gn

+

_{A)] (Searle,}

_1966).

Q

will have

_{corresponding diagonal}

and

_off-diagonal

elements of

_{gn(m &mdash;}

1)/!m(gn

+

_A)]

=

a

2

and

-gn/[m(gn

₊

A)],

respectively.

s

2

is thus

_gn/(gn

+

_A)

= _a

!(m -

l)/

7

7t].

If

design

matrices _are the

same for both environments and rG = _1,

Q

12

will have

diagonal

elements of

9

2

n

2

(

M

_ 1)/[,rn(gn+A)

2

1

=

a

4

m/

(m-1)

and

_{off-diagonals}

_{of -g}

₂

_n

₂

_/[m(gn+À?

_]

₌

-a9m/(m

-

1)

to

_give

S12 =

g2

n

2/(g

n

₊

A)

2

=

a

4

m

2

/(m -

1)

2

and

Corr(ûlû2) =

gn/(gn

+

_A)

=

a

2

m/(m -

1).

With balanced

designs,

Corr(u

l

u

2

)

has the same

expectation

under both BLUP and

BLP,

but accuracies are lower under BLUP such that la2a from BLUP underestimates

_expected

2

).

l

û

Corr(û

The extent of bias is

proportional

to

_{m/(m&mdash;1)}

and decreases from 20% at m = 5 to 11% at m = 10 and

2.6% at m = 40. This

_expectation

is maintained if

design

matrices differ between

environments

_{provided designs}

are balanced within each environment.

The situation is more

complicated

for unbalanced

designs,

but

general

conclu-sions are similar in that the number of sires

_compared

as

_{contemporaries}

needs to

be

_{large enough}

to minimize

_confounding

between sire BV

_predictions

and fixed effects estimates.

_Otherwise,

BLUP accuracies are reduced and their

product

un-derestimates

_expected

_Corr(u

_l

_u

₂

_).

For

_example,

consider a block of 8 sires with

progeny distributed over 4 _{contemporary groups}

(eg,

4 yr of an

experimental

eval-uation in some

environment)

as shown in table IV. The size of the _{experiment may} be varied

_by

_increasing

the number of sire blocks

_(to

_16,

_24, etc,

_sires),

by

varying

the number of _{progeny per} sire and _{contemporary group}

_(n),

_{by replicating}

the sire block over additional _{contemporary groups}

(8,

_{12, 16,}

etc),

or

by

a combination of

these

_approaches.

The same

_design

is assumed for each environment.

Define bias

_(fig

₂₎

as the difference between the

expected

Corr(u

l

u

2

)

calculated

from

_equations

_[8]

and

_[9]

and the

_product

ala2 which is constant for all sires in this

_design.

With n = 6 and

only

8 sires, bias was

_{relatively large}

with

Corr(£1 £2 )

= 0.41 _{vs ala2}= 0.35. Bias decreased _as number of sire blocks increased and was < 0.03 with 24 sires

_{(12/contemporary}

_group).

_Doubling

n or

_replicating

pattern of bias.

_Expected

values of

_Corr(ûlû2)

were also

_compared

to the

_product

of BLP accuracies calculated

_ignoring

_{contemporary group} effects. The

_product

of BLP accuracies overestimated

_Corr(u

_l

_u

₂

₎

as shown

_{by negative}

bias in

_figure

_2,

but the

_product

of BLP accuracies was

_superior

to the

_product

of BLUP accuracies

as an estimator of

_Corr(u

_l

_u

₂

_).

Design

like that in table IV can be used under

experimental conditions,

but are less feasible when sires are

compared

on _cooperator farms. It

_particular,

use

of

_large

numbers of

experimental

sires on individual farms _may not be feasible.

Instead,

sires from environment _{1 may} be tested

_together

on several farms in a

loosely

connected

_design

but with

_only

a few sires

represented

on _{any one} farm.

If

_only

data from introduced sires are used in the

_evaluation,

considerable bias in

Corr(u

l

u

2

)

may result.

However,

this bias can be reduced if introduced sires are

evaluated with sires

_{represented only}

in environment 2 and data from all sires are

included in the evaluation. To demonstrate this

_effect,

two additional sires were

added to each _contemporary group in table IV to

_give

16

_sires/sire

block with

resulting

values of

_Corr(ul

_u2

₎

for introduced sires was reduced

_{(fig 2).}

Thus if

sires from one environment are introduced into another and if evaluation occurs

primarily

in _cooperator

_herds,

sires should be evaluated in _{contemporary groups}

containing

reasonably large

numbers of sires

_(either

introduced or

native)

to increase

precision

of estimates of _{contemporary group}

_effects,

and data from all sires should be included in the evaluation.

CONCLUSIONS

Interpretation

of correlations between

_predicted

BV in different environments is

not

_{straightforward. Expectations}

of such correlations are influenced

_by

_accuracy

of evaluation of animals in both _{environments,}

_by

selection of animals chosen for

evaluation,

by relationships

among chosen animals and

by

the

design

of the evalua-tion in both environments. If animals are chosen from environment 1 for evaluation

in environment _2,

_{Regr(£2£1 )}

_may be a more useful statistic than

Corr(u

l

u

2

)

be-cause it is unbiased

_by

selection on

iij.

However,

both

Regr(û

2

û¡)

and

Corr(£

1£

2 )

are biased

by

covariances among

predicted

BV within environments.

Also,

use of

proven sires

(a

-!

1.0)

from environment 1

simplifies

interpretations

and reduces

the number of sires

_required

to attain a

specific

level of

significance

for measures of

association.

Equations

in this paper allow calculation or

_{approximation}

of

expected

values of

Corr(£

1

£

2

)

under various sorts of selection and with variable accuracies in each environment

_(equation

_[7]).

Evidence for selection can be obtained

empirically

if _necessary

_by

_comparing

observed

_V(Û¡)

to its _expectation from Blanchard et

al

_{(1983) (see Appendix).}

_Expected

values of both

_Corr(£

₁

_£

₂

₎

and

_Regr(û

₂

_û¡)

may involve

off-diagonal

elements of

Q

j

matrices which are often not available for BLUP BV

_predictions.

Effects of

_{off-diagonals}

_may be minimized

_by

_ensuring

that a

number of families are

_represented

in both the

_experimental

animals and their mates and

_by

using

several

_{(eg 8-16)}

sires _{per contemporary group. Use} of small numbers of sires per contemporary group can lead to considerable underestimation of the

expected

value of

_{Corr(£1 £2 )}

if

_off-diagonal

elements are not considered. Prediction of

_expected

values of

_Corr(£

₁

_{£2 )}

and

₂

_iii)

_g

_r(ii

_Re

from observed accuracies _may be

superior

when selection index

_(BLP)

rather than BLUP accuracies are used.

A number of

_potential

difficulties in

_deriving

and

_interpreting

_Corr(û

_l

_Û

₂

₎

have not been

_explicitly

considered. Accuracies are assumed to be

_properly

calculated,

even

though

_{approximations}

are

normally

used and are

probably

not

_completely

satisfactory.

Additive

_genetic

variances must be known for both environments in

or-der to calculate

u

l

and

U

2

correctly

and to

_interpret

_Regr(u

₂

_u

₁

_).

If sires introduced into environment 2 for evaluation are a selected

_sample

from environment 1 and BLUP evaluations are

_{used, grouping strategies}

and(or)

_adjustment

of covariances may be

required

to derive unbiased

ii

2

-

Values of

_{o,2 U2}

for animals selected from environment 1 will

_depend

both on selection

_applied

and on _rG. Thus results

_given

for correlations

_involving

BLUP

_predictions

of

_Û

_j

are

probably striclty

correct

_only

for random selection of sires. See Diaz

₍₁₉₉₂₎

for additional discussion of effects of selection and

_grouping

on

Corr(£

1

£

2

)

with BLUP

_{predictions. Despite}

these

environments will often be

_relatively

_easy to obtain

and,

if

_{properly interpreted,}

can

provide

information on _rG.

APPENDIX

This

_appendix

addresses

_general

_expectations

for variances and covariances of

ill

and

_M

₂

with non-random selection on

Û

l

and variable accuracies in both

environments.

_Sample

m animals from environment 1. If

sampling

is non-random

and selection rules are not

_specified

or if accuracies

differ,

the

resulting

distributions

of

_u

_j

and

_M

_z

are mixtures of m

_unique

distributions. The

sample

sum of _squares

(SS)

of

_u

_i

is:

The

_expected

value

_(E)

and variance of

_Û

_2i

with accuracy a2i are:

The

_{resulting expected}

_SS(u

₂

₎

and sum of

cross-products

(SCP)

are:

Expected

correlation and

_regression

coefficients can be derived from these SS

and SCP. If animals are chosen at

_random,

_l

_u

and

_U

₂

have mean 0 and

_expected

Corr(£

1£2

)

is

_given

_by

text

_equation

_[5]

_(Blanchard

et

al,

1983).

If selection is non-random but accuracies are constant within _{environments,} formulae for

_SS(u

₂

₎

and

_SCP(u

_l

_u

₂

₎

become consistent with text formulae 2 and 3. For directional selection with constant _accuracies, w can be

_{replaced by}

i(i &mdash; x)

without loss of

_generality.

For

_divergent

selection,

let fraction wH of the selected

standardized mean and truncation

_{point of i}

H

and xH,

respectively.

Let

fraction W

L come from the lower fraction

_ø

_L

_.

Let accuracies within

_high

and low _{groups, and} within environment

_2,

be constant at alH, alL and a2,

respectively.

V(û¡)

within selected _groups are

[1 -

_i

_H

_(i

_H

_-

x

H

)]aî

H

O&dquo;î

and

[1 -

i!(iL -

x

L

)]aî

L

O&dquo;I.

The

expected

value and variance of

_u

_l

and

u

2

,

and their

_expected

covariance are:

The

_resulting

_Corr(û

_l

_û

₂

₎

reduces to text

_equation

_[4]

_only

if alH = _alL and selection is

_symmetrical

_(ie,

wH = _WL

_{and i}

_H

=

-i

L

).

ACKNOWLEDGMENT

This research was conducted while C Diaz was _recipient of an INIA

_(Spain)

graduate fellowship.

REFERENCES

Becker WA

₍₁₉₈₄₎

Manual

_{of Quantitative}

Genetics. Academic

_Enterprises,

Pull-man,

WA,

4th edn

’

Blanchard

JP,

Everett

RW,

Searle SR

₍₁₉₈₃₎

Estimation of

_genetic

trends and correlations for

_Jersey

cattle. J

Dairy

Sci _66, 1947-1954

Calo

_LL,

McDowell

_RE,

VanVIeck

_LD,

Miller PD

₍₁₉₇₃₎

Genetic _aspects of beef

production

among Holstein-Friesians

pedigree

selected for milk

production.

J Anim

Sci

_37,

676-682

DeNise

_{RSK, Ray}

DE

₍₁₉₈₇₎

_{Postweaning weights}

and

_gains

of cattle raised under

range and

gain

test environments. J Anim Sci 64, 969-976

Diaz C

₍₁₉₉₂₎

The effect of selection on the covariance between direct and

maternal

_breeding

value

_predictions

in different environments. Ph D

_Diss,

_Virginia

Polytechnic

Institute and State

_Univ,

_{Blacksburg, Virginia}

Dickerson GE

₍₁₉₆₂₎

_Implications

of

_{genetic-environmental}

interaction in animal

breeding.

Anim Prod

_4,

47-64

Dijkstra

J,

Oldenbroek

_JK,

Korver

_S,

VanderWerf JHJ

₍₁₉₉₀₎

_Breeding

for veal and beef

_production

in Dutch Red and White cattle. Livest Prod Sci _25, 183-198 Do C

₍₁₉₉₁₎

Improvement

in _accuracy

_using

records

_lacking

sire information in

the animal model. Ph D

Diss,

Virginia Polytechnic

Institute and State

Univ,

Blacksburg, Virginia

Greenhalgh

SA,

Quaas

RL,

VanVIeck LD

₍₁₉₈₆₎

_{Approximating prediction}

error

variances for

_multiple

trait sire evaluations. J

_Dairy

Sci _69, 2877-2883

Henderson CR

₍₁₉₆₃₎

Selection Index And

_Expected

Genetic Advance. Natl Acad Sci Natl Res Council Publ _982,

_Washington,

DC

Henderson CR

₍₁₉₇₃₎

Sire evaluation and

_genetic

trands. In: Proc Anim Breed Genet

_Symp

in Honor

_of

Dr

_Jay

L Lush. Am Soc Anim

Sci, Champaign, IL,

10-41 Henderson CR

₍₁₉₈₄₎

_Applications

of linear models in animal

_{breeding. University}

of

_{Guelph, Guelph,}

Ontario

Hill WG

₍₁₉₇₀₎

_Design

of

_experiments

to estimate

_{heritability by}

_regression

of

offspring

on selected _parents. Biometrics _26, 566-571

Hill

WG,

Thompson

R

₍₁₉₇₇₎

_Design

of

_experiments

to estimate

_offspring

_parent

regression using

selected _parents. Anim Prod

_24,

163-168

Johnson

_NL,

Kotz S

₍₁₉₇₀₎

Distributions in Statistics: Continuous Univariate Distributions-1. John

_Wiley

and

_Sons,

New York

Mahrt

_GS,

Notter

_DR,

Beal

_WE,

McClure

_WH,

Bettison LG

₍₁₉₉₀₎

Growth of crossbred progeny of Polled Hereford sires

divergently

selected for

yearling weight

and maternal

_ability.

J Anim Sci

68,

1889-1898

Mallinckrodt CH

₍₁₉₉₀₎

_Relationship

of milk

_expected

_progeny differences to actual milk

_production

and calf

_weaning

_weight.

M Sc

thesis,

Colorado State

_Univ,

Fort

Collins,

CO

Meyer

K

₍₁₉₈₉₎

_Approximate

accuracy of

genetic

evaluation under an animal

model,

Livest Prod Sci _{21, 87-100}

Meyer

K

₍₁₉₉₁₎

_Estimating

variances and covariances for multivariate animal models

_by

restricted maximum likelihood. Genet Sel

Evol 23,

67-83

Oldenbroek

_JK,

_Meijering

A

₍₁₉₈₆₎

_Breeding

for veal

_production

in Black and White

_dairy

cattle. Livest Prod Sci _15, 325-336

Robertson A

₍₁₉₅₉₎

The

_sampling

variance of the

_genetic

correlation coefficient. Biometrics _15, 469-485

Robertson A

₍₁₉₆₆₎

A mathematical model of the

_culling

process in

dairy

cattle.

Anim Prod _8, 95-108

Robertson A

₍₁₉₇₇₎

The effect of selection on the estimation of

_genetic

_parameters.

Z

_{Tierxuchtg Zuchtgsbiol}

94, 131-135

Robinson

GK,

Jones LP

₍₁₉₈₇₎

_{Approximations}

for

_prediction

error variances.

J

_Dairy

Sci

_70,

1623-1632

SAS

₍₁₉₈₅₎

SAS User’s Guide: Statistics. SAS Institute

Inc,

Cary,

NC

Searle SR

₍₁₉₆₆₎

Matrix

_{Algebra for}

the

_Biological

Sciences. John

_Wiley

and

_Sons,

NY

Snedecor

GW,

Cochran WG

₍₁₉₆₇₎

Statistical Methods. The Iowa State Univ

_Press,

Ames,

IA,

6th edn

Taylor

J

₍₁₉₈₃₎

_{Assumptions required}

to

_approximate

unbiased estimates of

_genetic

(co)variance

by

the method of Calo et al

_(1973).

In: Genetics Research 1982-1983.

Rep

Eastern Artificial Insemination

_{Coop, Dept}

Anim Sci Cornell

Univ, Ithaca,

NY,

256-261

Tilsch

_K,

Wollert

J,

Nurnberg

G

_(1989b)

Studies on sire x

_{sex/environment}

interactions and their effect of _response to selection in beef sires

_{progeny-tested}

for

_fattening

_performance

and carcass

yield.

Livest Prod Sci

_21,

287-302

VanRaden

PM,

Freeman AE

₍₁₉₈₅₎

_Rapid

method to obtain bounds on accuracies

and

_prediction

error variances in mixed models. J

_Dairy

Sci

_68,

2123-2133

VanRaden

_PM,

_Wiggans

GR

₍₁₉₉₁₎

Derivation,

calculation,

and use of animal

model information. J

_Dairy

Sci,

74, 2737-2746