Genetic evaluation of horses based on ranks in competitions

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Genetic evaluation of horses based on ranks in

competitions

A Tavernier

To cite this version:

Original

article

Genetic evaluation of horses based

on

ranks in

competitions

A Tavernier

Institut National de la Recherche

_Agronomique,

Station de

_{Génétique Quantitative}

et

_Appliquée,

Centre de Recherches de

_{Jouy-en-Josas,}

7835!

Jouy-en-Josas Cedex,

France

(Received

14 October

_{1988; accepted}

9

_January

₁₉₉₁₎

Summary - A method is presented for

analysing

horse

performance

recorded as a series

of ranks obtained in races or _{competitions.} The model is based on the assumption of the existence of an

_underlying

normal variable. Then the rank of an animal is

_merely

the

phenotypic expression

of the value of this

_underlying

variable relative to that of the other horses

_entering

the same

_competition.

The

_breeding

values of the animals are estimated

as the mode of the a

_{posteriori density}

of the data in a

_Bayesian

context. Calculation of this mode entails

_solving

a non-linear system

by

iteration. An

_{example involving}

the results of races of

_{2 .yr-old}

French trotters in 1986 is

_given.

Practical

_computing

methods

are

_presented

and discussed.

horse

_/

_ranking

_/

order statistics

_/

Bayesian methods

Résumé - _{Évaluation génétique} des chevaux à _partir de leurs classements en

compéti-tion. Cet article

_présente

une méthode

_d’analyse

de

_{performances enregistrées}

sous la

forme

de classements obtenus dans des

_{confrontations}

restreintes et variables

_(courses

ou

concours).

Le modèle

postule

d’existence d’une variable normale

sous-jacente.

Le

classe-ment d’un cheval est alors

_{simplement d’expression phénotypique}

de la valeur de cette

va-riable

_sous-jacente

relativement à celles des autres animaux

participant

à la même

épreuve.

Les valeurs

_génétiques

des animaux sont estimées à

_partir

du mode de la densité a

poste-riori des données dans un contexte

_bayésien.

Le calcul de ce mode amène ic la résolution

d’un

_système

non linéaire par itérations. Un

exemple

d’application

est réalisé sur les

résul-tats des courses des chevaux Trotteurs

_Français

de 2 ans en 1986. Des méthodes de calculs

pratiques

sont

proposées

et discutées.

cheval

_/

classement

_/

_statistiques d’ordre

_/

méthodes bayésiennes INTRODUCTION

Choosing

a

good

selection criterion is one of the

major

problems

in

genetic

evaluation of horses. The

_breeding

_objective

is the

_ability

to succeed in

_riding

competitions

(jumping,

dressage,

3-day-event)

or in races

_(trot

and

_gallop).

But

how should success be measured?

The "career" of a horse is made _up of a series of ranks obtained in races or

competitions.

A

_"physical"

measure of

performance

is not

_always

available. Such a

These data are not

_always

collected

and, furthermore,

they

may

give

a _poor

indication of the real level of the

_performance:

a

_racing

horse must be fast but it

_must,

above

_{all, adapt}

to

_particular

conditions

_prevailing

in each event. This may

explain

the

relatively

low

heritability

of time

performance

of

thoroughbreds

(Hintz,

1980;

Langlois,

1980a).

In the case of

_{riding horses,}

it is difficult to assess

the technical level of a

_jumping

event. It

_depends

not

_only

on the

_height

of the

obstacles

_but,

to a

_greater

_extent, on the difficulties encountered when

_approaching

the obstacles and on the distance between obstacles. None of these variables can be

easily

quantified.

Therefore,

information

_{provided by}

the

_ranking

of horses in each event deserves attention.

_Ranking

allows horses

_entering

the same event to be

_compared

to the others.

However,

the level of the event has to be determined too. The most

frequently

used criterion related to

_ranking

is transformed

earnings.

Each horse that is

_{&dquo;placed&dquo;}

in an event,

ie,

ranked among the first ones, receives a certain

amount of money.

Prize-money

in a race is allocated in an

exponential

_way: for

instance,

the second horse earns half the amount

_given

to the

_first,

the third half of that

_given

to the second and so on... If the rate of decrease is not

_50%,

it often

equals

a fixed

_percentage,

for instance

75%

in horse shows. The

earnings

of a horse in a race can then be

expressed

as G =

ax( k-

l

) D

with _a

being

the

_proportion

of

the total endowment

_given

to the winner

_(constant),

x

_being

the rate of decrease of

earning

with rank

_(constant),

k the rank of the horse in the race and D the total

endowment of the race. The constants a and x must

_satisfy

_{(axK-1-!+(1-a)}

=

0)

with K the total number of horses

_{&dquo;placed&dquo;. So,}

a

_logarithmic

transformation

_gives

Log(G)

=

Log(a)

₊

Log(D)

₊

(k - 1) Log(x).

This is _a linear function of the rank

of the horse. To use it as a function of the

_ability

of the

_horse,

Log(D)

should be

assumed to be a linear function of the level of the race. The total amount of _money

given

in a race or a

_competition

should

_depend

on the technical

_difficulty

or the

level of the

_{competitors. Hence,}

with

_{adequate competition}

_programmes

_(Langlois,

1983),

the

_logarithm

of

_earnings

of a horse may be a

_good

scale for

_measuring

horse

_performance

and it has been

_widely

used

_(Langlois,

_{1980b, 1989;}

Meinardus and

_{Bruns, 1987; Tavernier, 1988, 1989;}

Arnason et

_{al, 1989; Klemetsdal, 1989;}

Minkema,

1989).

However,

this criterion

_strongly

depends

on the _{way money is} distributed. The choice of the amount of money

given

in

_jumping

_competitions

does not follow strict technical rules in France and does not

_{directly depend}

on

the scale of technical difficulties but on the choice of the

_organizing

committee.

Therefore,

it _appears that ranks should be taken into account without reference to

earnings.

The purpose of this article is to

_present

a method for

_estimating

the

_breeding

value of an animal

_using

a series of ranks obtained in events where it

_competed

against

a

sample

of the

population.

In order to

_interpret

these

data,

the notion of

underlying

variable will be used as in Gianola and

_Foulley

(1983)

for estimation

of

_breeding

value with

_{categorical data,}

and in

_Henery

₍₁₉₈₁₎

for

_constructing

the likelihood of outcomes of a race. The horse’s &dquo;real&dquo;

_performance,

which cannot be

measured,

is viewed as a normal

_variable;

this is a reasonable

_assumption

for traits

with

_polygenic

determination.

_Only

the location or

_ranking

of this

_performance

is recorded instead of a

performance.

Practical

computational

aspects

as well as an

application

to trotters are

presented.

METHOD

Data

The data

_(Y)

consist of the ranks of all the animals in all the events. The total number of observations is therefore

_equal

to the sum of the number of animals

per event. It is assumed that the ranks are related to an

underlying

unobserved

continuous variable. The rank

_depends

on the realized value of this

underlying

unobserved variable

_{(&dquo;real&dquo;}

animal

_performance)

relative to that of the other animals

_entering

the same event. The

_genetic

model is the same as for usual traits

with

_polygenic

determinism. The

_{underlying performance}

_yjk follows a normal

distribution with residual standard deviation (Fe and

expected

value !,2!. The model is:

where:

- y2!! _ &dquo;real&dquo;

performance

of

horse j under

environmental conditions i in the kth race of

_j;

-

b

i

= environmental effect

_{i (eg}

_age, sex,

rider...);

- u

j = additive

breeding

value of horse

j;

-

p

j = environmental effect common to the different

_performances

of horse

_j,

as

it may

participate

in several

events;

- eij

k = residual effect in kth race.

The vector of

parameters

to be estimated is 0 =

(b’,

u’,

p’)

where b =

{b

i

},

u =

(uj )

and

p =

{

Pj

}.

Inference is based on

_Bayes

theorem. Since the

marginal

density

of Y does not _vary with 0:

where

_pee)

is the

_prior

_density

of

_0,

_g(Y/6)

is the likelihood function and

_{f (9/Y)}

is the

posterior

density

of the

_parameters.

Prior

_density

The vectors

_b,

u, p and e are assumed to be

mutually independent

and to follow

the normal distributions:

_N(13,

_{V), N(O, G),}

_{N(O, H), N(O, R),}

respectively.

Prior information about b is assumed to be vague, which

implies

that the

diagonals

of V tend to +00.

Then,

the

_prior

_density

of b is uniform and the

_posterior

_density

of e does not

_depend

on

_{!3 !}

G =

Ao,’

where

A is the

relationship

matrix and

0

-; is

the additive

_genetic

variance. H is a

_diagonal

matrix with

diagonal

elements

equal

to the variance of

_p

_{(u p 2).}

The variances

₀

_-;

_and

_a

_P2

are assumed to be

known,

0

-; is

Likelihood function

Given ai, the

performances

y2!! are

_{conditionally independent.}

Let _y(l),

_!(2),...,

Yen)

be the ordered

underlying performances

of the n horses which

competed

in an event

_(for

_notation,

see for

_example

David, 1981,

p

4).

Then,

the likelihood

of

_obtaining

the observed

_ranking

in that event can be written as

(Henery,

_1981;

Dansie,

1986):

where:

-

yis the standard normal

density.

-

J1(t)

is the location

parameter

of the horse ranked &dquo;t&dquo; in that event.

This

_probability

can be

_interpreted

in the

_following

way: the

performance

of the

last animal may

vary

between -oo and +00, the

_performance

of the next to last varies from that of the last to +oo and so on.

_Thus,

the

_performance

of a horse varies

from that of the horse ranked

_just

behind it to +00, hence

_leading

to the bounds of each

_integral

in

P

k

.

Each

_integration

variable

_(t)

follows a normal distribution

with mean

_J1(

_t

₎

and standard deviation ue = 1.

Given 1L

(

t

),

these distributions are

independent

for all animals in the same

_competition.

This

_probability

may be

expressed

in terms of a multivariate normal

_integral

with thresholds

_independent

of

_integration

variables

_(Godwin,

_{1949; David,}

_1981):

where the distribution of

_{(xl, ... , !t, ... , !n-1 )}

is normal with mean

₍

_/1

₍

₁

₎

-!(2!, ... , ,!(t) - /1(t+1) , ... ,/1(n-1) -

/t(

n

))

and variance V =

{v

ml

}

with

Vmm =

_2,

Vm,m-1 = v m , m+1

= -1 and all other Vml = 0. Then:

Results of races are

_likely

to be correlated.

_However,

if the model is

_appropriate,

this correlation would

depend only

on

_genetic

or environmental effects ie

_given

the

J

.

L

ij’

S

,

the races are

_independent.

The likelihood function is

equal

to

the

_product

of the

_{probabilities}

of each event:

Estimation of

_parameters

The

_posterior

_density

of the

_parameters

is:

The best selection criterion is known to be the mean of the

posterior

distribution

(Fernando

and

_Gianola,

_1984;

Gof&net and

Elsen,

1984).

As

_expressing

it

analyti-cally

is not

_possible

for the model used

here,

we will take as estimator of 0 the

mode of the

_{posterior distribution,}

which can be viewed as an

_{approximation}

to the

optimum

selection criterion.

_Finding

this mode is

_{computationaly equivalent}

to the maximisation of a

_{joint probability}

mass

density

function as calculated

_by

Harville and Mee

₍₁₉₈₄₎

for

_categorical

data

_(Foulley,

_1987).

It is more convenient to use

the

_logarithm

of the

_{posterior density:}

/C=1

where m is the number of events.

The

system

which satisfies the first-order condition is not linear and must

be solved

_iteratively,

for

_{example using}

a

Newton-Raphson

_type

algorithm.

This

algorithm

iterates with:

where

9

is the solution for 0 at the

_qth

round of iteration and

AM

= 9!q!-e!q 1!.

Iterations are

_stopped

when a _convergence

_criterion,

a function of

0,

is less than an

arbitrarily

small number.

The first and second derivatives of

_L(O)

with

_respect

to

_b,

u, p are

reported

in

Appendix

1.

The

_system

can be written in the

following

way:

m

where

A, B, C,

D are sub-matrices of minus the second derivatives of

_L

_Log(P

_k

₎

k=l

m

with

_respect

to 0 and w, z are the vectors of first derivatives of

_E

_Log(P,!)

with

k=l

The numerical solution of

_system

_(I)

raises the

_problem

of the calculation of the

_{corresponding integrals.}

Multivariate normal

_integrals

_may be calculated with numerical methods such as’that of Dutt

_(1973),

described and

_{programmed by}

Ducrocq

and Colleau

_(1986).

A second method consists of

_using

a

Taylor’s

series

expansion

about zero which seems to

_{give good}

results

_(Henery,

_{1981; Dansie,}

1986; Pettitt,

1982).

This

requires

that animals

_{participating}

in a

_given

event have

relatively

close means _Itij, which is a reasonable

_assumption

in the

_present

context

of horse

_{competitions.}

This

expansion

involves moments of normal order

_statistics,

as

explained

in

Appendix

2.

Example

In order to illustrate these

_{computations,}

a

_simple

_example

was constructed. This

example

involves 5 unrelated horses. There are no fixed

_effects,

_{hence a}

=

(u

₊

p)

is estimated. The variance-covariance matrix

_{of p}

is

_diagonal

with each term

_being

9/11.

Two races with 4 runners are considered. _{The first gave the}

_{following ranking:}

No

_1,

No

_2,

No

_3,

No 4 and the second: No

3,

No

_2,

No

_5,

No 4. The

_starting

value for all

_A

_’s

was 0. The

system

to be solved at the first iteration of the

_{Newton-Raphson}

algorithms

as well as the

corresponding

solution are the

_following:

The

_{algorithm converged}

at the 5th iteration:

_{(A’ A )°.}

₅

= _{6 x}

10-

17

.

The

correspon-ding

values as well as the solutions and the coefficient of determination

(CD)

with

CD

=

(1 —

ciilo, u 2)

where _c2i is the

diagonal

element of the inverse of the matrix of

second derivatives of the

_logarithm

of

_posterior

_density

are:

---solution:

_[

_Al

_{p2 P3 !4}

_P5]

₌

_[0.621

0.237 0.271 - 0.902 -

_0.226]

accuracy:

[0.242

0.434 0.404 0.348

0.293]

It should be

noted that

the value of the first derivative for a horse in a

_given

race is

equal

to the

_expectation

of the normal order statistic

_{(normal score)}

_{corresponding}

to its rank.

_Similarly,

second derivatives for a

_given

race are functions of the variance

of,

and covariances

_between,

normal order statistics. This is the

_logical

_consequence of the choice of 0 for _JL as

_starting

value: all distributions of

_performances

are the same with a mean of 0 and all

_{integrals correspond}

to

_expectations

of normal order statistics. The accumulated values for all races are the sum of these.

At _convergence, these values have

_changed

and the final solution differs from the estimates obtained from the

_expectation

of normal order statistics. The

interpreta-tion of a rank

_depends

not

_only

on the number of

_competitors,

which is taken into

account

_through

the normal order

_statistics,

but also on the level of the

competi-tion.

At

convergence, the first derivative of the

_log

of a

_posteriori

_density

is set to

0.

_So,

estimates of horses are

_equal

to the first derivatives of the

_log

of likelihood function divided

_by

the variance term. These derivatives are different for the same

rank in different races.

_{They depend}

on the level of the race estimated a

_posteriori

by

the estimates of the horses

_{participating}

this

particular

race,

taking

into account

all races. In the

example,

for the winners of the 2 races, the first derivatives of the likelihood function were much lower than the

_expected

values of order statistics.

This is because the

_competitors

of these races have much lower estimates than the

winners:

0.237, 0.271,

-0.902 for horses No

_2,

No 3 and No 4

_against

0.621 for horse No 1 winner of the first race and

_{0.237, 0.226,}

-0.902 for horses No

2,

No 5 and No 4

_against

0.271 for horse No 3 winner of the second race.

_Therefore,

the first race

for No 1 and the second race for No 3 was easier than if

they

had

_{competed against}

3 horses of

_{equal ability}

to

_themselves,

ie with the same _ui, as

_implied

with the normal order statistics. The values of the first derivatives were 0.7589 and

_0.8475,

respectively,

compared

to 1.0294 for the

expectation

of the normal order statistics of the first out of 4. In the same _way, in the first _race, horse No 3

(0.27)

was beaten

by

a horse of lesser

_ability

_(No

2

_(0.24)),

_and,

therefore was more

_penalized

than if

it had been defeated

_by

a horse of

_equal

_ability.

The first derivative was

_-0.5165,

compared

to -0.2970 for the

_expectation

of the normal order statistics of the third

out of 4.

APPLICATION Data

horses in these races were recorded in the file. Ten races

(38

horses)

were discarded

because

_they

involved

_only

horses that did not

_compete

more than _once, and

which,

therefore,

were

_totally

disconnected from the rest of the file. We had to limit the

analysis

to

_{&dquo;placed&dquo;}

horses in each _race,

_ie,

horses ranked _among the best 4 or

_5,

because the

_ranking

of other

_participants

were not available. This does not

_prevent

us from

_testing

and

_comparing

our method to usual

_earning

criteria

_assuming

that these races involved

_only

4 or 5 horses.

_Indeed,

this is _neccessary for a fair

comparison

since

_earnings

also involve

_{only &dquo;placed&dquo;}

horses. With our

_approach,

&dquo;non

_placed&dquo;

horses

_could,

of _course, be treated as the others

_provided

that

_they

are filed.

The data set was made up of 251 races

(211

with 4 horses ranked and 40 with

5 horses

_ranked),

_involving

490 different horses. The total number of

_performances

was 1044

_places,

ie 2.1 _per horse on _average, with a maximum of 9 and a minimum

of

1. A

horse

_competed

_against

3.3 horses on _average. The model used was:

where: -

y!! _ &dquo;real&dquo;

performance

of

horse j

in the kth race of

_j;

- u j

= additive

breeding

value of horse

j;

-

p! = environmental effect common to the different

_performances

of horse

_j;

-

e jk

= residual effect in kth race about

_{&dquo;expected&dquo; performance}

_lLj.

No fixed effect was considered because

particular

conditions of each race

(dis-tance, type

of

_{ground, season...)}

are the same for all horses in the race and so have no effect on the result and because trainer and driver effects cannot be used on a

small data set

_(only

one horse for the

majority

of trainers or

_drivers).

The

expectations

and variance-covariance matrices are:

where h

2

=

0

,2/

U2

is the

_heritability

and r =

(

U2

₊

a;)/a;

is the

repeatability

of the trait. Values of

h

2

= 0.25 and r = 0.45 _were chosen _as

they correspond

_to usual

estimates of these

_parameters

obtained from

_{competitions.}

RESULTS

The elements of

_system

_(I)

were recalculated at each

Newton-Raphson

iteration with Dutt’s

!1973)

method for

_{integrals. Convergence}

was reached after 5 iterations

(with

(ð.’ ð.) .

5

/490

= 2 x

10-

15

).

The accuracies of these solutions were measured

by

coefficient of determination

_(CD).

If cii is a

_diagonal

element of the matrix of second

_derivatives,

CD =

(1 -

c

ii/

o

u

).

Breeding

value estimates had a mean of

_0,

a standard deviation of

_0.30,

with a

maximum of 0.94 and a minimum of -0.82. The mean _accuracy was

_0.23,

with a

standard deviation of

_0.08,

a maximum of 0.43 and a minimum of 0.12.

These values were

_compared

to criteria

_{usually employed}

in trotters

_(Thery,

0.73 with

_{Log(yearly earning),}

0.88 with

_Log(yearly

_earning

per

&dquo;place&dquo;),

0.79

with

_Log(yearly

_earning

per

start).

The correlation with a selection index

_using

as

_performance

the mean of the

_logarithm

of

_earnings

in each race

(with

_parameter

values

h

2

= 0.25 and r =

_0.45)

was 0.94. Correlations with criteria related to

racing

time were

_lower,

as were correlations between

_earnings

and

_racing

time. The correlation was -0.43 between our estimate and the best time per kilometer and - 0.47 between our criterion and a selection index

_using

as

_performance

_{the average}

racing

time

_(with

_parameter

values

h

2

= 0.25 and r =

0.45).

These

figures

also

suggest

that the best

_racing

time is not a

_good

measure of success in a race for

2-yr-old

horses.

This

_application

_suggests

some

_{peculiarities}

of our method. The first one relates

to the

_spread

of _{accuracy values. These}

_depend

not

_only

on the number of

&dquo;places&dquo;

but also on the

_{&dquo;place&dquo;}

of the horse in the race. Accuracies

_ranged

from 0.25 to 0.33 and from 0.20 to 0.28 for horses

_having

3 and 2

_{&dquo;places&dquo;, respectively.}

The minimal

accuracy

corresponding

to a

_single

_{&dquo;place&dquo;}

(0.12)

was smaller than the

_heritability

(0.25).

This is the result of the loss of information because ranks are used instead of continuous

_{performances.}

_{The average &dquo;loss&dquo; of accuracy}

_ranged

from 0.10

_points

for horses ranked once to 0.05 for those ranked more than 7 times.

The second

_point

of interest is the relative

_importance

of the number of horses

per event and the level of the horses

_{participating}

in the event. At _convergence, the first derivative of the

_logarithm

of

_{posterior density}

is

_equal

to

_0,

so estimates

are

_equal

to the

_part

of the first derivative without variance terms divided

_by

these variance terms

_(see

_Appendix

_I).

When all horses

_{participating}

in an event are of

the same level

_(ie,

have the same real

_racing

ability)

this derivative is

equal

to

expectations

of normal statistics. These

_expectations

_depend

only

on the number of animals per event. In our method the first derivative also

depends

on the real

racing

abilities of the

_competitors.

So the same rank in different events does not

give

the same derivative.

_Figure

1 shows the distribution of the derivatives in all

the races with 5 horses

&dquo;placed&dquo;

for the different ranks. For a

_{given rank,}

these

derivatives are different in each race and so,

being

first in a race sometimes

_gives

a

lower estimate than

_being

second in a race of a

_higher

level.

Our method can be used as a tool to

_improve

the

_{correspondence}

between the level of the race and the

_prize

_money to be distributed. The average

competitive

&dquo;level&dquo; of the race can be

approximated

as the mean of the estimates of real

producing ability

(

Jij

)

of each horse. In

_practice,

the correlation between such a measure and the

_logarithm

of total endowment of the race was 0.30 for races with 4

horses

&dquo;place&dquo;,

and 0.65 for races with 5

_{&dquo;placed&dquo;.}

Races with 5 horses

&dquo;placed&dquo;

have

the

_greatest

_prize-money,

and endowment seemed to be a

_good

indicator of the value

of

_{participating}

horses. It is also

_possible

to calculate a

_posteriori

the

probabilities

of

_obtaining

the observed

_ranking

in each race - or even of fictitious races -

_using

the estimates for each horse. These

_{probabilities}

were

_directly

calculated from the

formula for

_P

_k

and do not take into account the _accuracy of the estimates. The average

probability

of

obtaining

the observed ranks was

11%

and

3%

in races with

4 and 5

_horses,

_{respectively.}

If all horses had the same real

_producing

_ability,

this

probability

would be

4%

in races with 4 horses

(24 possibilities)

and

0.8%

in races

DISCUSSION

In the

_light

of the results obtained with

_2-yr-old

trotters,

the

_proposed

method seemed

_{satisfactory:}

the estimated values are consistent with other criteria.

In

practice,

solving

a much

_larger

_system

of

_equations

_presents

difficulties. Two

numerical

_problems

arise,

namely

the calculation of the

_integrals

P!

and their

derivatives

and the dimensions of the whole

system.

Two methods for

_computing

the necessary

integrals

have been

suggested,

the first

being

a numerical calculation

of multivariate normal

_integrals

and the second an

_{approximation}

by Taylor’s

series.

Beyond

certain

_dimensions,

it takes a _very

long

time to

_compute

multiple integrals

of the normal distribution. For each iteration of

_{Newton-Raphson}

and for each race

of n

horses,

it is _necessary to calculate one

integral

of order

(n - 1),

n

integrals

of order

_{(n - 2)}

and

_[n(n

+

_1)/2!

_integrals

of order

_{(n -}

_3).

_Therefore,

the time needed to

_accomplish

this becomes

_prohibitive

for a number of horses per race

> 5 or 6. On the other

hand,

our _purpose is to be able to

_apply

this

technique

to all

_types

of horse

competitions

(for

example

show

_jumping)

that sometimes involve more than 100

_{participants. Then,}

it is necessary to turn to

_{approximations}

like those

_{proposed by Henery}

₍₁₉₈₁₎

using

Taylor’s

series. The _accuracy of these

approximations

is difficult to test. In

_{particular, approximate}

formulae for the

moments of order statistics

_superior

to 2

_(Pearson

and

_Hartley,

_1972;

David and

Johnson,

1954)

need to be tested and

_compared

to

_integral

calculations of

_high

order.

Such

an

_{approximation}

reduces calculation times

considerably.

The moments

of order statistics not

_given

in tables can be calculated once and for all.

Then,

each

derivative

_only

consists of a linear combination of the

producing

abilities of the

The overall dimension of the

system

constitutes a second

problem. Using

an

animal

model with

repeated records,

this dimension is

_equal

to the number of horses to be evaluated

_plus

the number of

_performing

horses and fixed effects. At the

_{present time,}

in

_France,

100 000 horses are evaluated in

_jumping

with an

animal model

_(BLUP

_method)

based on

yearly earnings

and 70 000 are evaluated

in

_{trotting-races}

_(Tavernier,

_198(,)b,

_1990).

For each

Newton-Raphson

iteration of the

_proposed

method,

an iterative solution such as Gauss-Seidel will be needed.

This method has been

_developed

to include all horses in every race

_including

&dquo;non-placed&dquo;

horses.

However,

they

will have to be treated in a

slightly

different manner: the _purpose is to consider the horses

_{&dquo;placed&dquo;}

as better than the

&dquo;non-placed&dquo;,

but detailed

_ranking

of

&dquo;non-placed&dquo;

horses is of little interest. The

competitor

which no

_longer

has a chance of

finishing &dquo;placed&dquo;

is not

_going

to

_try

to

improve

its rank

_{and, therefore,}

its rank relative to the other

_{&dquo;non-placed&dquo;}

horses does not

_accurately

reflect its real

_ability.

Therefore the

_{&dquo;non-placed&dquo;}

should be treated as

_having

a

performance

below that of the last

&dquo;placed&dquo;. Then,

the likelihood

of the outcome of a race can be written as:

where there are n’ horses in the race and n horses

&dquo;placed&dquo;.

This

integral

can be

used in this form or

_equivalently

as the sum of all the

integrals

over all

_possible

rank combinations between

_{&dquo;non-placed&dquo;}

horses,

which allows a

simplified

application

of

the calculation

_{by Taylor’s}

approximation.

Another

_difficulty

is the estimation of the

_genetic

parameters.

The estimation of variance _components could

_probably

be made

_using

a

_marginal

maximum likelihood

approach

which

_requires

the inversion of the matrix of second

derivatives,

as

discussed

_by

Gianola et al

₍₁₉₈₆₎

and

_applied

_by

_Foulley

et al

_{(1987a, b).}

In

_practice,

this method can be

_{applied only}

on a reduced data file or with a &dquo;sire&dquo; model.

The

_heritability

of a

_single

_performance

is lower than that of

yearly

earning

criteria.

_Yearly

criteria are

_compound

functions of the number of events and of success in each event. For

_instance,

for

_{single performance,}

Meinardus and Bruns

(1987)

reported h

2

= 0.18 and r = 0.48 for the

logarithm

of

earnings

in

jumping

shows,

Klemetsdal

₍₁₉₈₉₎

_reported

h

2

= 0.18 and r = 0.65 for time in

trotting-races,

Thery

(1981)

found

h

2

= 0.23 and r = 0.52 for the _same criterion and

h

z

= 0.07

and r = 0.13 for the

logarithm

of

earnings. However,

the number of

elementary

performances

during

the lifetime of a horse is sufficient to

_expect

_good

accuracies of estimations.

_Taking

the

_previous

_examples

and a number of

yearly

starts

_equal

to 12

_(the

_{average number of}

_yearly

starts for an adult horse is 12 in

_{trotting-races}

and 14 in

_{riding competitions),}

the accuracies of

_breeding

value estimation

_ranged

from 0.27 to 0.41 after one _year of

_performance.

With a loss of 0.10

_point

due to

the use of

ranks,

accuracies of evaluations based on ranks would _range from 0.17

to

_0.31,

which is reasonable.

This model

_requires

a

sufficiently large

amount of

_comparisons

between horses

to allow a proper classification. The presence of isolated events which do not

the

_necessity

of

_good

connections between races, which is the

_only

_guarantee

of a

reliable result.

CONCLUSION

This article describes a method of evaluation of the

_breeding

value of an animal

from its rank relative to those of other

_competitors

in a

_given

_event, without

_using

a

direct measure of

_performance.

It is

_interesting

that the method

_suggests

a solution based on a conventional

_genetic

model. It can be

_applied

to an &dquo;individual animal&dquo;

model as well as to a &dquo;sire&dquo; model. It takes into account the level of the

_competition

which is the main factor

_influencing

a rank’s

value,

_together

with the number of

participants

in the event.

_Although

use of ranks _may seem to lead to a loss of information

_compared

to a

physical

_measure, it is sometimes more reliable. In the case of horse races,

ranking

is

_absolutely

_necessary as a real

_physical

measure is

not identifiable. It _may also be useful in the case of a distorted scale of measure or

when the usual

_physical

measure is

_nothing

but the

_{transcription}

of a rank.

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B

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H,

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₍₁₉₈₇₎

BLUP

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₍₁₉₈₉₎

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82-94.

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₍₁₉₇₂₎

Biometrika Tables

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₍₁₉₈₂₎

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_{(1988) Advantages}

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149-160

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₍₁₉₈₉₎

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41-54

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_(1989b)

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leur estimation

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_par un BLUP modèle animal. Ann Zootech

38,

145-155 Tavernier A

₍₁₉₉₀₎

Caract6risation des chevaux de concours

_hippique

_fran!ais

d’apr6s

leur estimation

_g6n6tique

_par un BLUP modèle animal. Ann Zootech

_39,

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C

₍₁₉₈₁₎

_{Analyse g6n6tique}

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pp 80

APPENDIX 1

Calculation of the first and second derivatives of the

_logarithm

of the a

posteriori density

Let

_y(

_l

_{), Y(2),...,Y(n)}

be the ordered

_{underlying performances}

of the n horses

which have

_participated

in race k

(see,

for

_{example, David, 1981,}

_p

_4).

_Further,

let

Q(t),k, R(t)(t),k, R(t)(z),k

be:

We have:

where: -

(k, (t))

E j

indicates the set of events k in which the

_{animal j competed}

and obtained the rank

t;

-

where: -

(t)

E i indicates the horses ranked at the

_place

t in the event k and with associated fixed effect i

and,

if the

_{horses j and}

have

_participated

in the same event:

I I _ --

!-The second derivatives with

_respect

to u and _p, or p and p are built in the same

way. The

only

value that

changes

is the covariance which is

equal

to 0 between u

p and is

equal

to

_1/(J!

on the

_diagonal

of the second derivatives with

respect

to p

and _p. p

where: -

!i(t)

= 0 if fixed effect i does _{not influence the horse ranked t}

-

!i(t) =

1 if fixed effect i influences the horse ranked t

APPENDIX 2

Approximation

of first and second derivatives of

_log

of the a

posteriori

density

using

Taylor’s

series

_expansion

These

_expansions

are drawn from those used

_by

_Henery

₍₁₉₈₁₎

and Dansie

₍₁₉₈₆₎

who

_approximate

the

_probability

_P

_k

_.

An

_example

of these

_{decompositions}

is

_given

for

_{(Q!t!,K/P!;):}

where,

for n

independent

normal distributions:

-

e t:n

:

expectation

of the tth order statistic -

o-tt:

, : variance of the tth order statistic -

<!tp:n : covariance between the tth order statistic and the

pth

order statistic -

Pt p z:n