Estimation of covariance components between one continuous and one binary trait

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Estimation of covariance components between one

continuous and one binary trait

H. Simianer, L.R. Schaeffer

To cite this version:

Original

article

Estimation of

covariance

_components

between

one

continuous

and

one

binary

trait

H.

Simianer

L.R. Schaeffer

2

Justus

_{Liebig University, Department of}

Animal

_Breeding

and

_Genetics,

Bismarckstr.

16 D-6300

_Giessen,

FRG

2

University of Guelph, Department of Animal

and

_{Poultry Science, Guelph,}

Ontario N1G 2W1 Canada

(received

17 December

_{1987, accepted}

14 October

₁₉₈₈₎

Summary - A method is described to estimate variance and covariance _components in a

multiple

trait situation with one continuous and one

_binary

trait. An

_underlying

bivariate normal distribution is assumed with one variable dichotomized on the observable scale

through

a fixed threshold. A mixed linear model is

_applied

to the

_{underlying scale,}

and

Bayesian

arguments are

_employed

to derive estimation

_procedures

for both location and

dispersion

parameters. This leads to a nonlinear _system of

_equations

similar to the mixed model

_equations

for observations that have been transformed

_by

a

_Cholesky

_{decomposition}

of the residual variance-covariance matrix so that the residual covariance between the two

transformed traits is zero,

thereby simplifying

construction of the

_multiple

trait mixed model

_equations.

The

_procedures

for

_estimating

_genetic

variances and covariances and the residual variance for the continuous trait are

_equivalent

to restricted maximum likelihood

in the multivariate normal case. The residual correlation is estimated

_using

a maximum likelihood

_approach.

Suitable

_{computing strategies}

are indicated and a simulation

_study

is

given

to illustrate the use of the method. The

_impact

of small subclass size on the estimates

is seen to be a serious drawback to the

_proposed

method. Possible

_{generalizations}

of the method and

_{potential problems}

in its

_{practical application}

are discussed.

variance - covariance - estimation - _binary trait methods

Résumé - Estimation des _composantes de la covariance entre un caractère continu et

un caractère binaire. On décrit une méthode pour estimer les composantes de variance et de

_covariance,

dans le cas de deu!

caractères,

l’un continu et l’autre binaire. On _suppose

l’existence d’une distribution

_{sous-jacente binormale,}

où l’une des variables ne

_présente

que deux états observables en

_fonction

de sa valeur _{par rapport} à un seuil

_fixe.

On écrit

un modèle linéaire mixte pour les variables continues

_{sous-jacentes,}

et l’on

_s’appuie

sur une

_{approche bayésienne}

_{pour construire} des estimateurs des

_paramètres

de

_position

et de

dispersion.

On obtient un

_{système d’équations}

non linéaires semblable aux

_équations

d’un

modèle

_mixte;

une

_{simplification}

des

_équations

est obtenue si la covariance résiduelle entre

les deux caractères est

_nulle,

ce

_qui

s’obtient en réalisant au

_préalable

une

_{décomposition}

de

corrélation résiduelle est estimée selon le

_principe

du maximum de vraisemblance. On

indique

des

_{stratégies adaptées}

au

_calcul,

et une simulation illustre l’utilisation de la

méthode. Cette méthode s’avère sensible à l’existence de

_petits

nombres d’observations dans certaines cellules. On discute les

_{généralisations possibles}

de la

méthode,

ainsi que

les

_{problèmes potentiels}

de son

_application

_pratique.

composantes de la variance - composantes de la covariance - estimation - caractère binaire

INTRODUCTION

Discrete traits are

_predominant

in certain areas of animal

_production,

_e.g.,

_fertility,

prolificacy, viability

and animal health. These traits are of

_increasing

_importance

since _many

_producers

are under

quota systems

and a means of

_increasing

income

is to

_improve

_{efficiency through}

these

non-production

traits.

_However,

traditional

traits like milk

_yield,

milk

_composition,

_growth

and feed

_efficiency

are still

impor-tant, and so are the

_{relationships}

between the discrete and the traditional

_traits,

most of which are continuous. Non-linear

_procedures

based on the threshold

con-cept

(Dempster

&

_Lerner,

₁₉₅₀₎

and

_Bayesian

_arguments

have become available to

analyze categorical

observations

_(Gianola

&

_Foulley,

_1983;

Harville &

_Mee,

_1984).

Bayesian

methodology

has been established as a

general

framework for the

analysis

of _any

_type

of observation

_arising

in animal

_production

_(Foulley

&

_Gianola,

_1984;

Gianola et

al., 1986;

H6schele et

_{al., 1986, 1987;}

_Foulley

et

al., 1987a,

1987b).

Observation structures that include both continuous and

_binary

traits have

been discussed as far as the estimation of

genetic

and environmental effects is

concerned

_(Foulley

et

_ad.,

_1983).

The

_objective

of this paper is to

_present

a method

for the estimation of

_dispersion

_parameters

of the

_joint

distribution of continuous

and

_binary

traits. A simulation

_study

was conducted to

_study

the effects of small

subclass size on

_parameter

estimates.

MODEL AND DATA STRUCTURE

Each animal is recorded for two different traits. Let _y.il be the continuous trait 1

(e.g.,

milk

_yield)

for the i-th

animal,

and let ci be the response for

binary

trait 2

(e.g.,

mastitis),

which is either 1 or 0.

_Categories

of response for the

binary

trait

are assumed to be

_mutually

exclusive and exhaustive. Observations for trait 1 form

the s x _{1 vector y}i, and the observations for the

binary

trait can be

_represented

as an s x 1 _{vector, c’=}

_(c

_l

_,

_c2, ...,

c,)’,

where ci = 0 if no disease is

observed,

otherwise ci = 1.

A non-observable

_underlying

continuous

variate,

Yi2, is

assumed,

and Yil and y

22 follow a multivariate normal distribution.

According

to the threshold

_concept

(Dempster

&

_Lerner,

_1950),

_y22 can be

_thought

of as

_liability

to contract the disease.

Only

if the value of y22exceeds a fixed threshold on the

_underlying

_scale,

does animal

i show the disease.

where :

p

j

= _p x 1 vector of fixed effects for trait

_j,

u j

= q x 1 vector of additive

_genetic

effects for trait

_j,

ej= s x

1 vector of residuals for trait

_j,

X = s x _p incidence matrix

_{corresponding}

to

_P

_j

_,

Z = _{s x}

q incidence

matrix

_{corresponding}

to _uj,

p = the number of levels of all fixed

effects,

and

q = the number of additive

genetic effects,

which could be

_greater

than or

equal

to s.

The

_expectations

and variance-covariance matrices of the random variables are:

where:

G = 2 x 2 additive

_genetic

variance-covariance

matrix,

A =

q x q additive

genetic relationship

matrix,

R = 2 x 2 residual variance-covariance

matrix,

I = s x

_{s identity}

_matrix,

*

= direct

_(Kronecker-)

_product,

gjj= additive

genetic

variance of trait

j,

r!! = residual variance of trait

j,

912

= additive

genetic

covariance between traits 1 and

2,

and

p e

= residual correlation between traits 1 and 2. The unknown

_parameters

are the location

_parameters,

and the

_dispersion

_parameters

The residual variance structure is a function of

_only

two

_parameters,

because _Yi2 is

a

conceptual

variable

and, therefore,

an

arbitrary

value for r22 can be chosen.

Note that this model assumes that every animal is observed for both traits and

that the same model

_applies

to each trait.

METHODS OF INFERENCE

Assume

_{that P}

has a flat

_prior

distribution

(i.e.,

nothing

is known a

_priori

about the

elements

_{of !),}

and that u has _a multivariate normal distribution with

expectation

null and variance-covariance matrix

_equal

to G * _{A. The}

priori

assumed to be

_independent.

The

_{joint posterior}

distribution of all unknowns

can be written as:

where

_{f (}

_Y

₎

is the

_{joint prior}

_density

of the

_dispersion

_parameters

_(Foulley

et

_al.,

1987a).

Estimation of location

_parameters

Integrating

T

out

of

₍₂₎

would

_hardly

be

_possible,

as was

_pointed

out

_by

Gianola

et al.

_(1986).

An alternative is to

_replace

_!y

_by

some

_value,

-y!t!,

so that

as elaborated

_{by Foulley}

et al.

_(1983).

_Ignoring

the

_superscript

_[t]

and

_assuming

that there is

_only

one continuous and one

_binary

trait and that both traits are

described

by

the same

_model,

then

_following

the

arguments

of Foulley

et al.

_(1983),

the residual variance of the

conceptual

variable is chosen to be:

and therefore:

For

_{lp, <}

_1.

R is

_positive

definite and a lower

_triangular

matrix T

_exists,

so that

TT’ = _R.

T-

1

is used to

_perform

a

Cholesky

transformation to remove the residual

covariance between transformed variables where

Let the tilde

_symbol,

-, above a variable indicate that same variable on the

transformed

_scale,

then

for i = _{1 to}

s, and

similarly

for elements

ofp,

u, and e. On the transformed

_scale,

variances and covariances are

Foulley

et al.

₍₁₉₈₃₎

show that under multivariate

_normality

and the residuals on the transformed scale are

_{uncorrelated,}

one can

proceed

as

in a

_{single binary}

trait

_analysis,

a

_simplication

of

_(Gianola

&

_Foulley,

₁₉₈₃₎

from

several

_categories

to

_two,

_computing

where

_0(.)

and

_4)(.)

are the

density

and distribution function of a standard normal

variate, respectively.

Let:

then from (4),

The mode of the

posterior

density

(3)

can be calculated

_{by applying}

the

Newton-Raphson algorithm

to the

_log

of

_(3),

which leads to the nonlinear

system

of

equations

for the

_[k

+

_l]th

round of iteration

_(Foulley

et

al.,

1983)

where

an s x s

_diagonal

matrix with wi in the

diagonal

and

Note,

that W and _{Y2 in} round

_[k

+

_1]

are

_computed

based on the solutions from

the

_preceding

round

_!k!.

Let

and :

be transformed back to the

_original

scale

_(the

_superscript

_[t]

_indicating

the set of

dispersion

parameters

used):

Estimation

_{of genetic}

variances and covariances

Foulley

et al.

₍₁₉₈₃₎

discuss

_briefly

the case of the variance-covariance structure

being

unknown and

_suggest

the

_application

of the restricted maximum likelihood

(RE1VIL)

algorithm

described

_by

Schaeffer et al.

₍₁₉₇₈₎

to estimate the entire

_genetic

variance-covariance structure as well as the residual variance of the continuous trait

only,

and the residual covariance between continuous and

_binary

traits.

Methods to estimate

_dispersion

_parameters

have been

developed

from a

Bayesian

background

for different data structures. Procedures for

_{single polychotomous}

traits have been described

_by

Harville & Mee

₍₁₉₈₄₎

and

_by

H6schele et al.

_(1987).

Methods for multivariate

_binary

traits have been

_{suggested by Foulley}

et al.

(1987a),

and for multivariate continuous traits

_by

Gianola et al.

_(1986).

_Essentially,

all authors recommend an

_{algorithm analogous}

to the

_{expectation-maximization}

(EM)

algorithm

(Dempster

et

_al.,

₁₉₇₇₎

for REML and its multitrait

_extension,

respectively.

Let

which

_{requires integration}

of

₍₂₎

with

_respect

to 0.

_Foulley

et al.

_(1987a)

show that

(7)

can be written as

where

E

e

is the

expectation

taken with

_respect

to

_{f (u ! Y,}

_yl,

_T

_).

_Furthermore,

Foulley

et al.

_(1987a)

show that whenever a flat

_prior

for

_{g is}

used,

the

_joint

_posterior

distribution

_of

_T

can be maximized with

_respect

to

_{g by maximizing}

_{E!{ln f( u I g) I}

at each iterate. The

resulting

iterative scheme for the

_[t

+

_1]th

round is

where tr

_(.)

is the trace

_operator

and

These terms cannot be derived

_explicitly,

but

_{approximations}

have been

_suggested

(Harville

&

_{Mee, 1984;}

Stiratelli et

_al.,

_1984).

_Making

use of these

_{approximations}

and

C

ii

,

by

the

_U

_t

x

_,

_i

_u

_part

of the inverted left hand side of

_(6).

The new estimates are on the transformed scale and need to be transformed back:

Foulley

et al.

_(1987a)

have shown that if the initial G is

_{positive definite,}

then the

subsequent

estimates of G are also

_positive

definite,

which holds in combination

with the

_applied

_Cholesky

transformation.

Estimation of the residual variance of the continuous trait

Maximum a

_posteriori

_predictors

tend to _converge to trivial results when flat

_priors

for the

_dispersion

_parameters

are used

_(Lindley

&

_Smith,

_1972).

As an alternative

Gianola et al.

₍₁₉₈₆₎

suggest

estimators

_arising

from an

_approximate

integration

of the variances. For g these are

equivalent

to the

_algorithms

described in section B as

_long

as flat

_priors

for the variances are used. For the

residual

variance of the

continuous trait estimator is

which can be shown to lead to the same results as

_iterating

on

assuming

full column rank of X. Estimators on the

_original

scale are obtained

by:

which

_yields

Estimation of the residual correlation between traits

Foulley

et al.

₍₁₉₈₃₎

_suggest

the use of restricted maximum likelihood estimation

as described

_by

Schaeffer et al.

₍₁₉₇₈₎

for multivariate normal data.

However,

this

requires

that the residual correlations between continuous and

_binary

traits be known.

_Foulley

et al.

₍₁₉₈₃₎

show that

_{proportionately}

_except

for a constant.

The

_problem

is to find

_y

_[t]

and 8

_(which

is a function of

T

14

)

which maximize this

log

posterior density.

Estimates for

_0,

_g, and rll can be obtained under the

_pretense

that _pe =

pe

similar _to the _one dimensional

_grid

search

technique proposed by

Smith

&

Graser

_(1986).

In this case the

_strategy

would be:

-

-

2)

for ith value of pe in the

set,

iterate on

(6)

_using

the

_dispersion

_parameters

(g,

rll,

Pe

i)

until convergence of

0,

then do one

_step

of

₍₉₎

and

₍₁₀₎

and continue

to iterate until _convergence of

_(g

and

_r

_ll

₎

is achieved. Then

compute

the value of

(11)

defined as h

_(p

_ei

_).

_Repeat

for all p,i in the

set;

-

3)

to find _{the p,i}that maximizes the

_{log likelihood,}

fit a second

_{degree polynomial}

through

the

_points

_(p

_ei

_,

h

_)).

_e2

_(p

Add the mode of this curve to the set of

_possible

values of pe and

repeat

the second

_step

with this new value. Continue this iterative

process until convergence of pe is achieved.

This

_strategy

is

_{computationally}

very

demanding

as _{convergence is very slow.} An

alternative

_{empirical approach}

_using

a maximum likelihood

_procedure

to estimate

p

e was

_{suggested by}

_equation

4.4 of Tate

₍₁₉₅₅₎

for the biserial correlation

_problem.

The

_[t

+

_l]th

improved

estimate of pe is

computed

as:

where

v = s x 1 vector with elements vi, i =

_1,

_s,

1 = s x 1 vector of

I’s,

Yi

(

y

i-

1 *

)

I

!

*

l/(

d

tl)

l

/

2

,

!

I

= phenotypic

mean of _y21 for all

_animals,

and

OW

]

is

the estimated value of the threshold which can be calculated from

!lk’]

analogously

to a least _squares mean. Note that this

_{procedure yields}

a new estimate of _pe on the

_{original scale,}

and

_therefore,

no backtransformation is needed.

Computing algorithm

Combining

all the different

_steps

_described,

an obvious

_computing

_strategy

would be:

-

1)

choose a set of

_starting

values

_yl’],

set

_(t]

=

_1;

-

2)

iterate on

(6)

until _convergence to

_get

_{9 ! Y}

=

T

14

;

-

3)

do one

_step

each of

_{(9), (10)}

and

₍₁₂₎

to

_get

a new set of

_dispersion

_parameters

T

l’

+

’],

set

_[t]

=

[t

+

1]

and continue with the

_precedent

step.

Iterate on the above scheme until the estimates

_{of y are converged. Empirically,}

(10)

and

₍₁₂₎

were found to converge more

_quickly

than

_(9),

and the estimates in r seemed to be

_{relatively independent}

of those in _g. This

_suggests

the use of a

different

_strategy:

-

1)

choose a set of

_starting

values

_g

_l]

and

_],

_rI

_l

set

_[t]

= 1 and

[s]

=

1;

-

2)

do one round of

(6)

to

_get

an

_improved

estimate of

0

g =

gI

t

],

r =

r1

4

;

-

3)

do one

step

each of

₍₁₀₎

and

₍₁₂₎

to

_get

a new set of

_parameters

_r

_Is+1]

_,

set

[s]

=

[s

₊

1]

and continue with the

_precedent

_stage

-

4)

if convergence in the second

_step

is

_reached,

do one

_step

of

(9)

to

_get

a new

estimate of

_gI

_t+1]

_,

set

_[t]

=

[t

₊

1]

and continue with the second

_step.

Iteration is

stopped

when the estimates in g are

converged.

This

algorithm

was

et al.

_(1987a)

for the

_multiple

_binary

trait case.

_Iterating

on r

_only

for some rounds

with _every new value of g

improved

the

procedure

even

further,

as the estimates of

r _converge

_quite

_rapidly.

SIMULATION STUDY

Model and methods

A simulation

_study

was done to assess the

_sampling

_properties

of the

proposed

estimators. The assumed true model was:

where:

-

Y ij

kl =

phenotypic

record

(i

=

1)

_or

underlying liability

value

(i

=

2)

of animal I

in

_{herd j}

with sire

_k,

-

pz = mean of trait

_i,

-

hij

= effect of

_{herd j on}

trait

_i,

-

sz! = effect of sire k on trait

_i,

and

-

e2!! = residual term.

Pairs of

_herd,

sire and residual effects were

_generated

for each animal as random

samples

from a bivariate normal distribution. Trait 2 was then

dichotomized,

applying

a threshold at 1.282 standard

_deviations,

which

_assigns

90 and 10 per cent of the observations to the

_phenotypic

classes 0 and

_1,

_{respectively.}

Records

were

_generated

for

10,000

individuals per

replicate,

which were

_{randomly assigned}

to sires

_(of

which there were

50)

and to herds

_(of

which there were either 100 or

1000).

With

_only

100 herds there were

13.5% empty

sire

_by

herd

subclasses,

and with 1 000 herds there were

81% empty

sire

_by

herd subclasses. The

_objective

of

having

two data sets was to

_study

the effect of small sire

_by

herd subclass size on

the

_parameter

estimates from the above methods. The true

_dispersion

parameters

were:

The model of

_analysis

was

_equal

to

_(13),

_except

that herd effects were treated as

fixed and the non observable y2 was

_{replaced by}

the

_phenotypic

observation in the

binary

trait. The

_starting

values of iteration were

and the

_stopping

criterion was

considering

the relative

_change

to account for the different

_magnitude

of the true

Results

The mean estimates and

95%

confidence intervals are

_given

in Table I. Fisher’s

x-transformation for correlations was

_applied

to

_{hi, h’,}

_{r9 and pe,} and a t-test was

applied

to examine differences between true and estimated

_parameters.

Estimates

from data set I with 100 herds

_agreed

_quite

_closely

with the true

_parameters.

The

only significant

bias was observed for the residual

correlation,

which is

partly

due

to the small confidence interval for this estimate. The

_{genetic correlation}

had a _very

large

confidence interval

_compared

with the other estimates.

The

results for data set II

_{(1000 herds)}

were similar with the

_exception

of a

50%

overestimation of the

_heritability

of the

_binary

trait. H6schele et al.

₍₁₉₈₇₎

_reported

similar results in a simulation

study considering

estimation of

heritability

for

binary

traits. The bias was observed when the average subclass size was below

_2,

which

agrees with the

present

results,

as the mean subclass sizes were

_{approximately}

2.3 and 1.1 in data set I and

_{II, respectively.}

H8schele et al.

₍₁₉₈₇₎

considered

the

_{approximation}

of the conditional distribution in the derivation of

₍₉₎

as the main cause of the

bias,

as this

_{approximation}

is based on a

normality

_assumption

DISCUSSION

A method has been

_presented

for the estimation of variances and covariances for the

simplest

case of one continuous and one

_binary

trait.

_{Theoretically,}

_{generalization}

to

more

_complex

observation structures is

_{straightforward.}

For the case of n continuous

and one

_binary

_trait,

the method to estimate location

_parameters

is as

_{given by}

Foulley

et al.

_(1983),

and the

_procedures

to estimate

_dispersion

_parameters

can be

readily

extended, using

results of Hannan & Tate

₍₁₉₆₅₎

for the maximum likelihood estimation of residual correlations. Further

_{generalizations, including}

the use of

informative

_priors,

_may be derived

_{through Bayesian}

_arguments

as

_{employed by}

Gianola et al.

₍₁₉₈₆₎

and

_Foulley

et al.

_(1987a).

In

_{practice, however,}

an

_analysis

of even the

_simplest

case poses a formidable

computational

task with a

_{large body}

of

_data,

in which case the

optimization

of

computing strategies

is of crucial

_importance.

_Actually,

the

_{methodology presented}

was

_developed

for a

_joint

_analysis

of more than

_200,000

records on

_production

and

disease data in

_dairy

cattle.

Observations on both traits for all animals were assumed

_complete

_throughout

the

_data,

which seldom is the case in

_practice,

where selection and hence

_missing

data due to selection

_play

an

_important

role. This has been addressed

_by

Henderson

(1975)

for continuous traits and

_by

_Foulley

& Gianola

₍₁₉₈₆₎

for

_categorical

traits. The

_multiple

trait

_methodology

allows for

_sequential

selection where the decision whether an animal is

_given

the

_opportunity

to be observed for the discrete trait is made on the basis of its

_performance

for the continuous

_trait,

but not vice _versa,

as was indicated

_by

Foulley

et al.

_(1983).

General

_approaches

to the more

_complex

types

of selection are

needed,

as most of the reasons for inevitable natural selection

(diseases, fertility)

are of

_categorical

nature.

The simulation

_study

was an illustration

of

the

_methodology

for two

_special

cases.

A

_general

assessment of the

_sampling

properties

of the estimates would

_require

a series of simulations over a wide range of

parameter

combinations and more

realistic

_population

structures as was done

_{by Hbschele}

et al.

_(1987),

for the

_single

categorical

trait case.

Since

the

_general

framework

of the methods is the _same, a

similar behaviour of the estimates can be

_expected,

which was shown for the biased

estimate of

_heritability

for the

_binary

trait when the average size of the smallest

subclass was less than two. This is a serious

_problem

in

_{practical applications,}

_e.g. when

_dairy

_progeny test data are

_{analyzed by}

a model in which sire and herd x

year x season effects are cross-classified.

The

_complexity

of the

_{Bayesian approach}

in the context of the

_given

model and observation structure makes _{it necessary} to use certain

_assumptions

and restrictions as discussed

_by

Gianola et al.

₍₁₉₈₆₎

for the multivariate normal case.

Although

the

_approach

of

_basing

the inference about 0 on the conditional distribution

ACKNOWLEDGEMENT

This

_study

was carried out while H. Simianer was

_visiting

the

_University

of

_Guelph,

supported by

the Deutsche

_{Forschungsgemeinschaft.}

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