Transcript Accounting for heterogeneous variances in beef cattle breeding

Slides available at http://www.msu.edu/~tempelma/nbcec1.pdf
Accounting for heterogeneous variances
(heteroskedasticity) in genetic evaluations
National Animal Breeding Seminar Series
Fall Semester 2004
Robert J. Tempelman
Michigan State University
A typical genetic evaluation model for
postweaning gain (PWG)
y = X1b1 + X2b2 + Z1u1 +Z2u2+ e
Fixed effects
Random effects
Random contemporary group
effects: u1
Non-genetic effects: b1 (age
of dam, length of PW
pd, calf sex)
Var (u1) -> autoregressive ys
within herds or NIID
Random additive genetic
effects: u2
Genetic effects: b2 (Breed and
dominance and
recombination loss
effects)
Var(u2) -> function of one or
more (multibreed)
components
y = Xb + Zu + e
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???????
2
Homoskedastic error models
e ~N (0,Ise
2)
Common s2e across environments,
factors, etc. may not be a suitable
assumption.
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Example of Heterogeneous
Variances
• Garrick et al. (1989)
– Separate genetic (s2g) and residual (s2e) variances estimated by
%Simmental and sex for postweaning gain.
Residual
s 2g
s2e
Genetic
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Structural (mixed effects) modeling
of variances (Foulley et al., 1992)
– model residual and genetic variances as a
function of fixed and random effects
2
– Example: Consider the residual variance s e jk
unique to fixed calf sex j and random CG k.
Log linear “mixed effects” model on log variance
 
log s e2jk  baseline  sex j  cgk ;
cgk ~ IID  0,  
Antilog both sides (Multiplicative model)
s
2
e jk
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e
baseline sex j
e
ecgk  s e2 j vk ; vk ~ IID 1, CV 2  f    
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First known application of structural
variance model to beef cattle data
• San Cristobal et al. (1993) analyzing
muscular development scores in French
Maine Anjou cattle
– Scored on 0 to 100 scale.
• Considered structural variance model on
both residual AND genetic variances.
– Effects considered:
• classifier (random), condition score (fixed), year
(random), month(random) for residual variance
• Sex for genetic variance
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Representative results from San
Cristobal (multiplicative scale)
Factor
Level
Estimate
Baseline
1
97.57
Classifier
1
1.17
2
1.07
3
1.06
1
1
2
0.74
3
0.65
1
0.98
2
1.14
1
0.94
2
1.02
3
1.00
Condition Score
Year
Month
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For example, an animal
evaluated by Classifier 2
with condition score 2 born
in year 1 and month 2 has a
residual variance of:
97.57
x1.07
x0.74
x0.98
x1.02
=77.23
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The underlying model for calving
ease (1-5 scale)
Colored
areas =
probability of
occurence
1= Unassisted
calving
5= Caesarean
Section
1
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2
3
4
5
(l)
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Heterogeneous variances for
calving ease (CE)?
• Genetic evaluations based on threshold mixed
effects model.
– Underlying liability (l) is typically modeled as a function
of fixed (e.g. calf sex) and random effects (herd-yearseason) + IID residual (e); i.e.
li  xi' β  zi' u  ei ;
var  ei   1, i  1,2..., n.
– Heteroskedastic theory provided by Foulley and
Gianola (1996)
• Demonstrated that statistically significant calf sex by age of
dam interactions for CE in homoskedastic error threshold
models may be an artifact of heterogeneous residual variances
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ALLOWING FOR HETEROGENEOUS RESIDUAL
VARIANCES IN THRESHOLD MODELS
1
2
3
4
5
Note how
probability of
extreme outcomes
particularly depend
on residual
variance
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Genetic evaluations accounting for
calving ease
• French Holstein, Normande, and Montbeliarde
breeds (Ducrocq, 2000)
– Heteroskedasticity is breed dependent:
– ~15% lower residual variance in winter versus
summer.
– Larger residual variance (1.07-1.18x) for male calves.
• Italian Holsteins (Canavesi et al., 2003)
– Larger residual variance (1.03) for males
– Regional differences for residual variance
• Both evaluations only consider fixed effects
models for residual variances
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Fixed and random effects for log residual variances in
threshold models for calving ease
Kizilkaya and Tempelman (2005; GSE)
First parity Italian Piedmontese cattle
Parameter
F
Linear Mixed Model Analysis
of Birth Weights
Threshold Mixed Model
Analysis of Calving Ease
Estimate ± SE
Estimate ± SE
Sire Variance
1.13  0.20
0.13  0.02
MGS Variance
0.50  0.11
0.02  0.01
Sire-MGS covariance
0.35  0.11
0.02  0.01
CG variance
1.68  0.19
0.13  0.02
Male residual variance
14.44  1.03
1.09  0.09
Female residual variance
10.19  0.73
0.71  0.06
Sex difference in residual
variances
4.26  0.53
0.38  0.05
CV for herd-specific
0.60  0.09
0.74  0.14
R variances
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Fixed effects and Random effects for Residual Heteroskedasticity
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•
Estimates ( )of and 95% credible sets ( ) for Herd
Specific Variances for CE Relative to Baseline (1.0)
Note: Because sire-mgs
model was used, residual
heteroskedasticity may be
partly genetic
CV = 0.74
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Impact on calving ease EPD’s?
Heteroskedastic vs. Homoskedastic Error
2
sˆ Sire
 0.34
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Impact of residual heteroskedasticity across CG on Sire
EPD’s for birthweights
(Kizilkaya and Tempelman, 2005)
CV = 0.60
Implications of
ranking herds for
product uniformity!
Herd 66
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sˆ sire
 1.06
Sire A
All of Sire’s A progeny were from Herd 66
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Multiple Breed Populations
• Might naturally expect heterogeneous
genetic variances (for different
breedgroups and different levels of
heterozygosity)
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Multibreed genetic modeling
• Additive model (Lo et al., 1993)
• For any individual j, its additive genetic
effect aj has variance:
B
 
Var a j  
b1
fb s A2b
j
B 1 B
2
b1 bb

sj sj
fb fb

dj dj
fb fb


s S2bb  0.5cov a sj , a dj
fb j Expected allelic contribution due to Breed b
in individual/parent j
s A2
Additive genetic variance of Breed b
s S2
Variance due to genetic segregation
between Breeds b and b’
b
bb '
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
Simple two breed example
Suppose
s S2  20
12
P1
s
2
g  P1 
s g2P   50 P2
 100
2
2
s
F1
g ( F )  75
1
2
F2 s g ( F2 )  95
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Theory used for QTL
mapping in pig breed
crosses: better power
than Haley-Knott
regression
(Perez-Enciso and
Varona, 2000)
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Application:Nelore-Hereford data
(Fernando Cardoso PhD)
 Data set:
 22,717 post-weaning gain (PWG) records
on Hereford and Nelore x Hereford calves
raised in Brazil (from 1974-2000)
 40,082 animals (including ancestors in
pedigree file)
 Breed compositions of animals with
records ranged from purebred Hereford to
7/8 Nelore
 Purebred Herefords and F1’s represent 90%
of the data
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But maybe the residual variances are
heterogeneous too!

Beef cattle performance is recorded across diverse
production systems and environments, with data
quality often compromised by, e.g.


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Recording error, preferential treatment,
disease, etc.
Hierarchical model constructions have been
independently used to address

heteroskedasticity (Foulley et al., 1992;
SanCristobal et al., 1993) and

robustness to outliers (Stranden and Gianola,
1998, 1999).
Important to discern outliers from highvariance subclasses
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First stage: Specify the Linear
Mixed Model
y = X1b1 + X2b2 + Z1u1 +Z2u2+ e
Fixed effects
Random effects
Non-genetic effects: b1 (age
of dam, length of PW
period, calf sex)
Random contemporary group
effects: u1
Genetic effects: b2 (Breed
additive, dominance and
recombination loss
effects)
Random additive genetic
effects: u2
y = Xb + Zu + e
OR
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y j  x'j β  z'j u  e j , i  1, 2..., n.
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Second stage: Structural
variance model

e j ~ N 0, s e2j
log
 
s e2j
 
o
 
 log s e   pm j log  m    log  k
2
baseline
EXAMPLES
m1
Regression
parameters
s e2j
k 1
 j
s
 
  log vl
l 1
 j
Calf
sex
 
 log wj 1
Fixed
Random
classification classification
effects
effects
Breed
proportion
Breed
heterozygosity
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r

Lack-offit term
with
mean 0
CG
o
r
s




p
2
mj
s e     m    k j    vl j  
 m1
 k 1
 l 1


wj
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Distributional assumptions on
random effects
• Location parameters:
– u includes 940 CG (uCG) and 40,082 additive genetic
effects (uA):
• uCG ~ N(0,Is2CG)
• uA ~ N(0,G(f)) where f includes breed specific variances and
segregation variances.
• Residual variance
– v = [v1 v2 v940] includes random relative variances
for 940 CG
• vi ~ IID Inverted-gamma with mean 1 and standard deviation
sv
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Need to consider one more thing

• Recall e j ~ N 0, s
2
ej

where
log
 
s e2j
 
o
r
 
 log s e   pm j log  m    log  k
2
m1
k 1
 j
s
 
  log vl
l 1
 j
 
 log wj 1
• What about wj?
– Lack-of-fit term
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• 1) If wj ~ Gamma(n/2, n/2) then this is
equivalent to specifying:

e j ~ t 0, s ,n
2
ej

i.e. Student t error
Demonstrated to be
resistant to outliers
Stranden and Gianola
(1998; 1999)
• 2) If wj = 1 for all j, then

e j ~ N 0, s
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ej

Many other options!!!
See Rosa et al. (2003)
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Now (At least) four distributional
possibilities!
• 2 × 2 factorial based on distribution
(normal versus Student t) and
homoskedastic versus heteroskedastic
residuals :
2
e
~
N
0,
s
 e
j
1. Homoskedastic normal
2
2. Homoskedastic Student t e j ~ t  0, s e ,n 
2
e
~
N
0,
s
3. Heteroskedastic normal
j
e
2
4. Heteroskedastic Student t e j ~ t 0, s e ,n

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
j
j


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Some results
•
•
Based on Pseudo Bayes Factors (PBF), the
Student t heteroskedastic model provided the
best data fit; the homoskedastic normal model
the worst data fit.
The heteroskedastic Student t error model was
the best fit:
– The posterior mean of the degrees of freedom
parameter (n) was 7.33 ± 0.48 indicating a heavier
tailed residual distribution than normal (n =∞) for
PWG data
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Heteroskedastic residual variance
results from
Fixed
effects
Parameter
EST.
SE
95%PPI
Gender (1)
1.13
0.09
(0.97, 1.31)
Nelore proportion (1)
1.15
0.45
(0.48, 2.20)
Heterozygosity (2)
0.70
0.16
(0.46, 1.06)
CG (sn)
0.72
0.06
(0.62,0.86)
Pr 1  1
 0.9378
Pr   2  1
 0.0449
Random
effects
Evidence of genetic homeostasis? (Lerner, 1954)
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What do these estimates mean
again?
• Example a male F1 calf in a herd (Herd 5)
with above average variability (vˆ 5  1.2 )
f1 j  0.50
– Nelore proportion
– Heterozygosity
f12j  1.00
• Estimated residual variability:
sˆ ˆ  ˆ1 
2
e 1
0.50
ˆ 2 
 sˆ 1.13 1.15 
2
e
 1.02sˆ
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2
e
1. 00
vˆ 5
0.50
 0.70 
1.00
1. 2
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Posterior densities of heritabilities under
homoskedastic normal error model
Posterior density
a) Gaussian homoskedastic model
Cardoso and Tempelman, 2004
30
25
20
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
Heritability
Nelore
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Hereford
F1
A38
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Posterior densities of heritabilities under
heteroskedastic normal error model
Posterior density
c) Gaussian heteroskedastic model
35
30
25
20
15
10
5
0
Some of Why
mostthe
variable
herdsfrom
were
“flip flop”
exclusively
homoskedastic
normal error?
Why the “flipHerefords
flop”
->Some of most variable herds
were exclusively Herefords
0
0.1
0.2
0.3
0.4
0.5
Heritability
Nelore
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Hereford
F1
A38
densities look very similar under Student t heteroskedastic
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Where do we go from here?
• Genetic evaluation for residual variability?
– Relevance: Uniformity of product premium.
– San Cristobal-Gaudy et al. (1998, 2001)
Sorensen and Waagepeterson (2003)


yi  xi' β  z i' u  ei ; ei ~N 0,s e2i , i  1, 2..., n,
 
ln s e2i  pi' δ  qi' v
 0   As u2
u  2 2
 v  | s u , s v , r ~ N  0 , 
 
    Ars us v
Ars us v  
2 
As v  
A: numerator relationship matrix
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r: genetic correlation between location and log variance effects
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Sire EPD for litter size
variability (v)
Litter size in sheep (San Cristobal
et al., 2003)
For litter size
in pigs, a
negative rˆ
was estimated
(Sorensen and
Waagespeterson,
2003)
rˆ  corr
r (u, v)  0.19
Sire EPD for litter size (u)
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Multiple trait analysis?
• The standard for genetic evaluations today
• Perhaps genetic covariances/correlations
between traits are heterogeneous across
environments too.
• Hopefully, these issues will be investigated
further.
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References
Cardoso, F.F., and R.J. Tempelman. 2004. Hierarchical Bayes multiple-breed inference with an application to genetic
evaluation of a Nelore-Hereford population. Journal of Animal Science 82:1589-1601.
Canavesi F., Biffani S., Samore A.B., Revising the genetic evaluation for calving ease in the Italian Holstein Friesian.
Interbull Bulletin 30 (2003) 82-85 http://www-interbull.slu.se/bulletins/framesida-pub.htm.
Ducrocq V., Calving ease evaluation of French dairy bulls with a heteroskedastic threshold model with direct and
maternal effects, Interbull Bulletin 30 (2000) 82-85 http://www-interbull.slu.se/bulletins/framesida-pub.htm.
Foulley, J.L. 1997. ECM approaches to heteroskedastic mixed models with constant variance ratios. Genetics,
Selection, Evolution 29:297-315.
Foulley, J. L., M. S. Cristobal, D. Gianola, and S. Im. 1992. Marginal likelihood and Bayesian approaches to the
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Analysis 13: 291-305.
Foulley J.L., Gianola D., Statistical analysis of ordered categorical data via a structural heteroskedastic threshold
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Garrick, D.J., E.J. Pollak, R.L. Quaas, and L.D. Van Vleck. 1989. Variance heterogeneity in direct and maternal weight
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Perez-Enciso, M., and L. Varona. 2000. Quantitative Trait Loci Mapping in F2 Crosses Between Outbred Lines.
Genetics 155:391-405.
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References (cont’d)
Robinson G.K., 1991. That BLUP is a good thing - the estimation of random effects, Statistical Science 6 15-51.
Robert-Granie, C., B. Bonati, D. Boichard, and A. Barbat. 1999. Accounting for variance heterogeneity in French dairy
cattle genetic evaluation. Livestock Production Science 60: 343-357.
Robert-Granie, C. B. Heude, and J.L. Foulley. 2002. Modeling the growth curve of Maine-Anjou beef cattle using
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Rodriguez-Almeida, F. A., L. D. Vanvleck, L. V. Cundiff, and S. D. Kachman. 1995. Heterogeneity of variance by sire
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Animal Science 73: 2579-2588.
Rosa, G. J. M., C. R. Padovani, and D. Gianola. 2003. Robust linear mixed models with normal/independent
distributions and Bayesian mcmc implementation. Biometrical Journal 45: 573-590.
San Cristobal, M., J. L. Foulley, and E. Manfredi. 1993. Inference about multiplicative heteroskedastic components of
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San Cristobal-Gaudy, M., Bodin, L., Elsen, J-.M., Chevalet, C. 2001. Genetic components of litter size variability in
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