A Tobit-Model for breeding value evaluation in German trotters

Transcription

A Tobit-Model for breeding value
evaluation in German trotters
A.-E. Bugislaus1, N. Reinsch2
1Agrar-
und Umweltwissenschaftliche Fakultät der
Universität Rostock,
2Leibniz-Institut
16.07.2012
für Nutztierbiologie Dummerstorf (FBN)
UNIVERSITÄT ROSTOCK | FAKULTÄT AGRAR- UND UMWELTWISSENSCHAFTEN
Breeding goal in German trotters
The goal in German trotter
breeding is a fast, sound, welltempered and good-gaited
trotter with precocity and a
correct exterior.
Actual breeding value evaluation in
German trotters
¾ Model:
BLUP animal model utilizing individual race results from all
starting trotters
¾ Traits:
Earnings per race
Rank at finish
Racing time per km
¾ Question:
Do only trotters with earnings show their real racing potential
whereas trotters without earnings are not driven to their limit?
Censored race results could be responsible for a potentially severe
bias in actual genetic evaluation in German trotters
16.07.2012
3
Objectives of the study
¾ Definition of censoring in race results of German trotters
¾ Tobit-like-Threshold-Model of racing performances
¾ Use of a real Tobit-Model for the censored trait racing time per km
¾ Comparison with linear model that treated all individual racing times
per km as uncensored
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4
Data for genetic estimations
¾ Total data set consisted of 105,981 race performances from 6,504 trotters
(mean of racing time per km = 79.7 s/km)
¾ Data set involved 14,148 races with 7.5 participants in average
¾ Starting method was in all races the auto start
¾ Pedigree back to the fourth generation was
included (20,703 animals)
16.07.2012
5
Censoring in race results of German trotters and
description of different used genetic models
Trait
Exemplary
results of
one race
sorted by
ranks at
finish
Uncensored racing Tobit-like-Threshold-Model Censored racing time per
time per km (y)
km for Tobit-Model (y*)
for placing status
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
5
4
3
2
1
0
0
0
0
0
y1
y2
y3
y4
y5
y6*
y7*
y8*
y9*
y10*
Model
Linear-Model:
Bayesian analysis
Threshold-Model:
Bayesian analysis
Tobit-Model: Bayesian
analysis with data
augmentation
Program
LMMG
(REINSCH, 1996)
LMMG_TH
(REINSCH, 1996)
LMMG_TOB
(REINSCH, 2011)
16.07.2012
6
Trait
Exemplary
results of
one race
sorted by
ranks at
finish
time per km (y)
for placing status
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
5
4
3
2
1
0
0
0
0
0
y1
y2
y3
y4
y5
y6*
y7*
y8*
y9*
y10*
Model
Linear-Model:
Bayesian analysis
Threshold-Model:
Bayesian analysis
analysis with data
augmentation
Program
LMMG
(REINSCH, 1996)
LMMG_TH
(REINSCH, 1996)
LMMG_TOB
(REINSCH, 2011)
16.07.2012
7
Trait
Exemplary
results of
one race
sorted by
ranks at
finish
time per km (y)
for placing status
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
5
4
3
2
1
0
0
0
0
0
y1
y2
y3
y4
y5
y6*
y7*
y8*
y9*
y10*
Model
Linear-Model:
Bayesian analysis
Threshold-Model:
Bayesian analysis
analysis with data
augmentation
Program
LMMG
(REINSCH, 1996)
LMMG_TH
(REINSCH, 1996)
LMMG_TOB
(REINSCH, 2011)
16.07.2012
8
Trait
Exemplary
results of
one race
sorted by
ranks at
finish
time per km (y)
for placing status
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
5
4
3
2
1
0
0
0
0
0
y1
y2
y3
y4
y5
y6*
y7*
y8*
y9*
y10*
Model
Linear-Model:
Bayesian analysis
Threshold-Model:
Bayesian analysis
analysis with data
augmentation
Program
LMMG
(REINSCH, 1996)
LMMG_TH
(REINSCH, 1996)
LMMG_TOB
(REINSCH, 2011)
16.07.2012
9
Tobit-Model for censored trait racing time per km:
Data augmentation
¾ For each y* > y5 the threshold is determined as standardized value:
t=
y 5 − xi ' β − zi ' u
σe
¾ A random variable ai < t is drawn from a truncated standard normal
distribution and is subsequently transformed to the original scale:
yi* = ai * σe + xi ' β + zi ' u
¾ For one iteration yi* is treated as observation, in the following round yi*
is again determined
16.07.2012
10
Univariate genetic-statistical model
y =
Xb + Z1a + Z2pe + e
y:
vector of observations containing either the uncensored or
censored trait racing time per km or the threshold trait placing
status of each trotter in each individual race
b:
fixed effects
a:
random animal effect
pe:
random permanent environmental effect
e:
residual effect
X, Z1, Z2: incidence matrices
¾ 1 million cycles were generated (Gibbs sampling algorithm)
¾ As burn-in period 250,000 rounds were considered
16.07.2012
11
Fixed effects
¾ sex (stallion, mare, gelding)
¾ age of trotter (12 classes)
¾ year-season of race (three months are one season)
¾ condition of race track (fast, good, medium, heavy, muddy)
¾ distance of race (10 distance classes)
¾ driver (1, …, 1572)
¾ each individual race (1, …, 14148)
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12
%
h² = 0.095 (0.016)
Uncensored racing time per km
(Linear model)
Placing status
(Tobit-like-Threshold model)
h² = 0.207 (0.025)
Censored racing time per km
(Tobit model)
h² = 0.208 (0.024)
0.034
16.07.2012
0.067
0.100
0.133
0.166
0.199
heritability
0.232
0.265
0.298
0.331
13
%
h² = 0.095 (0.016)
(Linear model)
Placing status
h² = 0.207 (0.025)
(Tobit model)
h² = 0.208 (0.024)
0.034
16.07.2012
0.067
0.100
0.133
0.166
0.199
heritability
0.232
0.265
0.298
0.331
14
%
h² = 0.095 (0.016)
(Linear model)
Placing status
h² = 0.207 (0.025)
(Tobit model)
h² = 0.208 (0.024)
0.034
16.07.2012
0.067
0.100
0.133
0.166
0.199
heritability
0.232
0.265
0.298
0.331
15
(seconds/km)²
Variance components estimated for
racing time per km with two different models
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Tobit model
Linear model
additive‐genetic
16.07.2012
permanent environment
residual
16
Boxplots for residuals estimated with a Linear Model for the
trait racing time per km over different placings
Plot of breeding values for the trait racing time per km
estimated either with a Linear- or with a Tobit-Model
Rank correlations (r) between breeding values estimated
with different genetic models as well as the percentage (%)
of incorrectly selected stallions
r
%
Tobit-Model vs. Linear Model
0.89
25.5
Tobit-Model vs. Tobit-like-Threshold-Model
0.96
16.1
Tobit-like-Threshold-Model vs. Linear Model
0.86
33.8
Conclusion for trotter breed
¾ Trotters without earnings didn‘t show their real racing potential and
should be regarded as censored observations.
¾ Tobit-(like-Threshold)-Models with censored race results represented
good suitability for genetic evaluation.
¾ Heritability estimates for the threshold trait placing status and the
censored trait racing time per km were almost identical.
¾ Also the high rank correlation between the breeding values of placing
status and the breeding values of censored racing time per km
showed great agreement.
Thank you for your attention!
Breeding goal in German thoroughbreds
The goal in German thoroughbred
breeding is a highly competitive and
sound horse with a correct exterior.
Previous developed genetic evaluation systems
for German thoroughbreds
¾ Model:
BLUP animal model using individual race results from all
starting thoroughbreds
¾ Traits:
Rank at finish
or
Distance to first placed horse
¾ Handicap:
Consideration of carried weights in the genetic model
either as fixed linear regression
or
creation of new performance traits independent of
carried weights
¾ Definition of censoring in race results of German thoroughbreds
¾ Derivation of individual racing times for each starting thoroughbred
using the racing time of the first placed horse and the stewards
decision for the further placed horses
¾ Proof if a real Tobit Model for the new created and censored trait
racing time per km is appropriate
16.07.2012
23
Data for first genetic estimations
¾ Total data set consisted of 62,412 race performances from 6,244
thoroughbreds
¾ Data set involved 6,524 races with 9.6 participants in average
¾ Four generations were conisdered in pedigree (18,766 animals)
Censoring in race results of German thoroughbreds
and description of different used genetic models
Trait
Exemplary
results of
one race
sorted by
ranks at
finish
Uncensored
Tobit-like-Threshold-Model
square root of rank
for placing status
at finish (y)
New created and
censored racing time for
Tobit-Model (y*)
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
4
3
2
1
0
0
0
0
0
0
y1
y2
y3
y4
y5*
y6*
y7*
y8*
y9*
y10*
Model
Linear-Model:
Bayesian analysis
Threshold-Model:
Bayesian analysis
analysis with data
augmentation
Program
LMMG
LMMG_TH
(REINSCH,
1996)
(REINSCH,
1996)
UNIVERSITÄT ROSTOCK
| FAKULTÄT AGRARUND UMWELTWISSENSCHAFTEN
16.07.2012
LMMG_TOB
(REINSCH, 2011)25
Average distances between different ranked horses
between
rank A and rank B
Distance in
body lengths
1 and 2
2,23
2 and 3
2,25
3 and 4
2,23
4 and 5
2,91
5 and 6
3,78
6 and 7
4,46
7 and 8
5,19
8 and 9
5,46
Univariate genetic-statistical model
y =
Xb + Z1a + Z2pe + e
y:
vector of observations containing either the uncensored trait
square root of rank at finish or the threshold trait placing status
of each thoroughbred in each individual race
b:
fixed effects
a:
random animal effect
pe:
random permanent environmental effect
e:
residual effect
X, Z1, Z2: incidence matrices
¾ 1 million cycles were generated (Gibbs sampling algorithm)
¾ As burn-in period 250,000 rounds were considered
16.07.2012
27
Fixed effects
¾ sex (stallion, mare, gelding)
¾ age of trotter (10 classes)
¾ year-season of race (12 seasons)
¾ distance of race (4 distance classes)
¾ trainer (1, …, 706)
¾ jockey (1, …, 686)
¾ each individual race (1, …, 6524)
¾ fixed linear regression of carried weights
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28
h² = 0.091 (0.014)
Uncensored square root
of rank at finish
(Linear Model)
Placing status
(Tobit-like-Threshold-Model)
h² = 0.189 (0.027)
0.0356
0.0740
0.1124
0.1508
0.1892
heritability
0.2276
0.2660
0.3044
¾ Definition of censoring in race results of German thoroughbreds
¾ Derivation of individual racing times for each starting thoroughbred
using the racing time of the first placed horse and the stewards
decision for the further placed horses
¾ Proof if a real Tobit Model for the new created and censored trait
racing time per km is appropriate
16.07.2012
30
Derivation of racing time for non-winning horses
(Mota et al., 2005)
¾ Only winning horses receive a racing time at finish
¾ Racing times for the other non-winning animals in the race were calculated by
multiplying the number of body lengths behind the winner per 2/10 of second
¾ The trait racing time was analysed at distances of 1000, 1100, 1200, 1300,
1400, 1500 and 1600 m
¾ Heritabilities for the trait racing time at finish varied from 0.29 to 0.05 for the
different distances in races
Creation of new performance traits independent of
carried weights (Bugislaus et al., 2004)
¾ Carried weights ranging from 47 kg to 74.5 kg
¾ Analysed trait was „Square root of distance to first ranked horses in races“
¾ Observed performances were biased because of different carried weights in
individual flat races
¾ Estimation of regression coefficients within thoroughbreds
¾
Total data set consisted of 33,223 performance observations from 2,507
thoroughbreds that started at least in six flat races
¾ A univariate random regression model for the transformed distance to first rank
was used for estimations of coefficients on carried weights
¾
¾
¾
Model included only a random animal effect and a residual effect
The space variable was the carried weight
A first order polynomial on carried weights for the animal effect was applied
Distribution of coefficients from the carried weight on the
square root of distance to first rank
350
300
Frequency
250
200
150
100
50
0
0,1825
0,1325
Regression coefficients over all horses
0,0575
New distance to first rank = ((10 – distance0.5) + (0.1325 * carried weight))
¾ Estimated heritability was h² = 0.145 (0.019)
First conclusion for the thoroughbred breed
¾ Thoroughbreds without earnings didn‘t also show their real racing
performance potential and should be regarded as censored observations
¾ A Tobit-like-Threshold-Model was suitable for genetic evaluation
¾ A real Tobit-Model is only appropriate when using new calculated racing
times for all placed thoroughbreds
16.07.2012
36

A Tobit-Model for breeding value evaluation in German trotters

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