6/16/2009
Genomic Prediction & Selection
TRAINING & VALIDATION
Agenda
• A Bioinformatics System to Implement B
Bayesian methods for Genomic Prediction i
th d f G
i P di ti
• Results from various methods applied to real and simulated data
• Training on EBVs, deregressed data and weighting information in training
weighting information in training
1
6/16/2009
Bioinformatics Infrastructure
• Research from analysis of high‐density genotypes t
to predict merit has several objectives
di t
it h
l bj ti
– Determine predictive ability of
• same‐density panels in validation/target populations closely related to the training population
• same‐density panels in validation/target populations less related or unrelated to the training population
related or unrelated to the training population
• low‐density panels in populations closely related to the training population
– Motivate other genomic selection research
Bioinformatics Infrastructure
• Identify informative regions for fine‐mapping and gene discovery
d
di
• Provide a platform for collaborating (beef) researchers to undertake genomic training
– eg US Meat Animal Research Center
• Provide
Provide a platform for delivering genomic a platform for delivering genomic
predictions to (the beef) industry
2
6/16/2009
Various Methods
y = Xb + ∑ Mi a i + e
estimate σ ai2 and σ e2
BayesA
y = Xb + ∑ Mi a iδ i + e
estimate δ i , σ ai2 and σ e2
BayesB
estimate δ i , σ a2 and σ e2
BayesC
estimate π , δ i , σ a2 and σ e2
BayesCPi
Various Methods
Markers in Model
Marker Effects
All (π=0)
Fraction (1‐π)
Random ‐ Individual Variance (Normal)
“Bayes A” (B0)
“Bayes B”
Random ‐ Constant Var (when in model)
Bayes C (C0)=“BLUP”
Bayes C
Random – Constant Var (when in model)
Fraction (1‐π) estimated from data=Bayes CPi
Categorical Variants (threshold models)
Other Variants (estimate scale, heavy tails)
Fixed (LS MSE )
Fixed (LS PRESS) Any π options (Bayes
Any π
options (Bayes K and Bayes
K and Bayes T)
StepWise
PRESS
Bayes A & B are not really truly Bayesian – Bayes C and Bayes Cpi are (Gianola) 3
6/16/2009
Pi influences convergence
Priors Influence Shrinkage
• Influence varies with method and prior degrees of freedom
4
Bayes A df=4
Bayes A df=3
Bayes A df==4
Bayes A df==3
6/16/2009
9
Bayes B df=4 ∏=0.99
10
5
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Bayes C0
11
Bayes C df=4 ∏=0.99
12
6
6/16/2009
BayesB then BayesA (100 markers)
“Heritability” for 100 markers chosen for trait in row, applied to trait in column
0.64
0.50
0.23
0.33
0.29
0.22
0.45
0.30
0.24
0.53
0.61
0.24
0.33
0.29
0.23
0.45
0.30
0.26
0.27
0.29
0.57
0.33
0.29
0.22
0.36
0.30
0.25
0.27
0.27
0.23
0.67
0.29
0.26
0.42
0.30
0.29
0.28
0.24
0.23
0.33
0.57
0.25
0.40
0.35
0.27
0.27
0.29
0.26
0.33
0.29
0.53
0.42
0.30
0.25
0.29
0.29
0.23
0.33
0.29
0.25
0.70
0.26
0.25
0.29
0.27
0.24
0.33
0.29
0.22
0.36
0.63
0.24
0.32
0.27
0.26
0.33
0.29
0.25
0.42
0.30
0.65
13
Bayes B then Bayes A (100 markers)
Correlation in training data chosen for trait in row applied to trait in column
0.79
0.68
0.37
0.41
0.42
0.33
0.56
0.46
0.39
0.69
0.76
0.38
0.4
0.44
0.34
0.54
0.42
0.41
0.39
0.41
0.77
0.4
0.39
0.35
0.5
0.4
0.39
0.36
0.36
0.35
0.78
0.41
0.41
0.53
0.45
0.43
0 41
0.41
04
0.4
0 38
0.38
0 36
0.36
0 79
0.79
0 39
0.39
0 51
0.51
0 51
0.51
0 41
0.41
0.39
0.4
0.39
0.45
0.41
0.72
0.55
0.41
0.38
0.41
0.4
0.35
0.45
0.4
0.41
0.87
0.4
0.41
0.43
0.41
0.37
0.4
0.48
0.37
0.5
0.79
0.37
0.44
0.4
0.39
0.44
0.38
0.37
0.5
0.45
0.78
14
7
6/16/2009
1st attempt Cross Validation
• Dataset 1 comprising 8 breeds
• Select best 100 markers in all data using BayesB
g y
✓
B1
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
B3
✓
✓
B4
✓
✓
✓
B5
✓
✓
✓
✓
B6
✓
✓
✓
✓
✓
B7
✓
✓
✓
✓
✓
✓
B8
✓
✓
✓
✓
✓
✓
✓
Validation
B1
B2
B3
B4
B5
B6
B7
Trraining
B2
✓
B8
Bayes B then Bayes A (100 markers)
markers in row chosen from Bayes B on all data, Bayes A trained in cross‐
validation for trait in column, predicting merit in omitted data
0.66
0.53
-0.02
0.09
0.02
-0.06
0.07
0.08
-0.03
0.53
0.65
0.01
0.03
0.1
-0.02
0.06
-0.02
0.06
0.01
0.03
0.68
0.02
-0.03
-0.02
-0.04
-0.01
-0.05
-0.05
-0.06
0.01
0.68
0.02
0.04
0.02
0.08
0.11
0.09
0.07
-0.02
0
0.68
0.04
0
0.2
0.04
-0.02
0.01
0.06
0.14
0.08
0.58
0.11
0.03
-0.03
-0.01
0.01
-0.04
0.14
0
0.1
0.74
-0.07
0.04
0.06
0.05
0.01
0.05
0.22
0.07
0.06
0.69
-0.05
0.08
-0.02
0.02
0.15
-0.08
-0.01
0.01
0.14
0.7
16
8
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StepWise then BayesA
17
StepWise then BayesA
Successive datasets have previously best markers removed
18
9
6/16/2009
StepWise and BayesA
19
Proper cross‐validation
• Marker subset selection and marker estimation are undertaken on each training ti ti
d t k
ht i i
data subset and used to predict “virgin” data • Correlation dropped to 0.18 (at best) when properly (100 marker subset chosen in training data) cross‐validated
g
)
20
10
6/16/2009
Training and Validation
Purebred (PB) Purebred (PB)
PB Î PB
50K SNP
Validation
• Almost always SNP that spuriously fit the data well
– Having a model that fits the training data well provides relatively little information about how good the prediction will be in new data
• Many world‐changing research discoveries are announced in news releases and then never‐to‐be‐
h d f
heard‐of‐again
i
• Training & Validation can be done together to quantify the likely confidence in predictions 11
6/16/2009
Cross Validation
Training
• Partition the dataset (by sire) into say three groups
G1
G2
✓
G3
✓
Validation
Derive g‐EPD
Compute the
Compute
the
correlation between
predicted genetic
merit from g‐EPD and observed performance
G1
Cross Validation
Training
• Every animal is in exactly one validation set
✓
G1
G2
✓
G3
✓
✓
G1
G2
Validation
✓
✓
G3
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6/16/2009
Cross‐Validation
• 1800 bulls with EPDs ‐ split into 3
– At random
– By sire ID ‐ sire of bulls nested in subset
– By sire ID ‐ sires also fitted as fixed effects
– By time ‐ oldest, middle‐aged, youngest 25
Results
41028m
Random
Sire
Sire+cg
Time
Bayes
y A (B0)
( )
0.745
0.726
0.646
0.732
Bayes B (.99)
0.722
0.700
0.618
0.712
Bayes C0
0.746
0.728
0.648
0.730
Bayes C(.50)
0.746
0.728
0.647
0.730
Bayes C(.99)
0.728
0.708
0.625
0.717
C.99/C100m
0.553
0.567
0.389
0.583
StepWise
0.547
0.558
0.393
0.542
PRESS
0.523
0.539
0.365
0.574
100m
26
13
6/16/2009
Simulated SNP Results ‐ 1184 QTL
52566 markers
Number of training animals
π=0.977
1000
2000
3000
4000
B(true)
0.65
0.76
0.82
0.84
C(true)
0.62
0.74
0.80
0.83
B(inflated)
0.63
0.75
0.80
0.83
C(inflated)
0.60
0.71
0.77
0.80
B(0.50)
0.62
0.74
0.79
0.82
C(0.50)
0.60
0.70
0.75
0.78
B(0)
0.64
0.74
0.79
0.81
C(0)
0.59
0.70
0.75
0.78
True=#QTL/#markers; inflated=0.9 true; heritability=0.5
(Christian Stricker for Swiss Cattle Breeders)
27
Simulated Results
2000 animals
Number of QTL
171
493
1184
B(true)
0.88
0.82
0.76
C(true)
0.88
0.81
0.74
B(inflated)
0.84
0.79
0.75
C(inflated)
0.70
0.74
0.71
B(0.50)
0.81
0.78
0.74
C(0 50)
C(0.50)
0 65
0.65
0 72
0.72
0 70
0.70
B(0)
0.82
0.77
0.74
C(0)
0.64
0.72
0.70
True=#QTL/#markers; inflated=0.9 true; heritability=0.5
(Christian Stricker for Swiss Cattle Breeders)
28
14
6/16/2009
50k within‐breed predictions
50k within‐breed predictions
• These predictions are characterized by correlations between genomic merit and realized
correlations between genomic merit and realized performance from 0.5 to 0.7
– They will account for 25 (0.52) to 50% (0.72) genetic variation
– Compared to a trait with heritability of 25%, the genomic predictions would be equivalent to observing 6 t 15 ff i i
6 to 15 offspring in a progeny test
t t
• Correlations of 0.7 are similar to the performance of genomic predictions in dairy cattle 15
6/16/2009
50k within‐breed predictions
• These predictions are not as highly accurate as can be achieved in a well designed and b
hi d i
ll d i d d
managed progeny test, say with 100 or more offspring
• However, for many traits they are much more reliable for animals of a young age (eg
y
g g ( gp
prior to first selection) than is currently achievable from individual performance
Across‐breed prediction
• Refers to the process of predicting performance for a breed or cross that was not in the training dataset
breed or cross that was not in the training dataset
• Critical interest to those selecting breeds that are not well represented in the training populations
• May not be as reliable as within‐breed predictions due to complexities associated with non‐additive genetic effects (dominance and epistasis)
• Potential can be assessed by simulating the effects of P t ti l
b
d b i l ti th ff t f
major genes using real SNP genotypes on various populations
16
6/16/2009
Introduction
• Toosi et al.,(2008) simulated genotypic and
phenotypic
p
yp data
– Training in crossbred and MB populations
– Successful selection of PB for MB performance
• Linkage Disequilibrium (LD)
– Simulated LD in pure and MB populations may
not accurately reflect real LD in beef cattle
populations
Objective
Training Populations Î
Validation Populations
Multi‐breed (MB) Purebred (PB)
MB Î PB
MB Î
50K SNP
Purebred (PB) Multi‐breed (MB)
PB Î MB
50K SNP
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6/16/2009
50K SNP Datasets
MB Population (N=924)
PB Population (N=1086)
Angus
Angus
239
Brahman
10
Charolais
183
Hereford
78
Limousin
45
1086
Maine-Anjou 137
Shorthorn
97
South Devon 135
Simulation of Additive Genetic Merit and
Phenotypic Performance
50K SNP
50K SNP
X1 X2 X3
Xk
X6
X50K
0, 1 or 2
SNP chosen at random
QTL 50, 100, 250, 500
QTL1 QTL2
QTLj
X6
Xk
xijα j
xi1α1 xi 2α 2
Additive Genetic Merit
N
Phenotypic performance
yi = ∑ xijα j + ei
j =1
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6/16/2009
Marker Panels
50K SNP
X1 X2 X3
X5 X6
Xk
LD=r2
50K w/o QTL
X50K
Xk
HLD1 HLD2
HLDj
X5
X1 X2 X3
Xm
QTLj
X6
HLD50, 100, 250, 500
X50K
LD=r2
QTL1 QTL2
QTL 50, 100, 250, 500
Xm
Xm
X5
N
Bayesian Analysis
y = 1μ + ∑ x jα jδ j + e
j =1
Simulated Phenotypes/real 50k Data
• Effect of number of available markers
50 QTL
Just QTL
QTL + Best markers
Q
QTL + 50k
Train in Multibreed
Validate in
Purebreed
0.953
0 931
0.931
0.766
Train in Purebreed
Validate in
Multibreed
0.962
0 938
0.938
0.842
19
6/16/2009
Simulated Phenotypes/real 50k Data
• Effect of number of available markers
50 QTL
Just QTL
QTL + Best markers
Q
QTL + 50k
Just Best markers
50k w/o QTL (real life)
Train in Multibreed
Train in Purebreed
Validate in Purebreed Validate in
Multibreed
0.953
0 931
0.931
0.766
0.570
0.388
0.962
0 938
0.938
0.842
0.489
0.422
Kizilkaya et al, ASAS, 2009
Effect of number of available markers
• Redundant markers reduce accuracy
– Increased type I errors
• Accuracy suffers greatly when QTL not on panel
– Not enough markers of sufficiently high LD to act as good proxies on a one‐for‐one basis
g
p
• Multibreed population generally inferior to purebred
20
6/16/2009
Purebred or Crossbred
Highest LD markers for random QTL with Training in Purebred
1.2
y=1
1.2018x
2018x - 0.371
0 371
R² = 0.48309
Means (r)
Purebred = 0.717
Multi-breed = 0.491
1
Multi-Breed (r)
0.8
Few QTL with
LD <0.4 in training
0.6
0.4
Many markers
erode in
validation
population
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Purebred (r)
Purebred or Crossbred
Highest LD markers for random QTL with Training in Crossbred
1.2
y = 0.8775x + 0.0406
R² = 0.39451
Means (r)
Multi-breed = 0.625
Purebred = 0.589
1
Purebred (r)
0.8
0.6
Many QTL with
LD <0.4 in
training
0.4
Most markers
still robust
still robust
in validation
population
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Multi-Breed (r)
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6/16/2009
Effect of number of available markers
• Easier
Easier to find high LD markers in purebreds than to find high LD markers in purebreds than
multibreed populations because average LD is higher
– Favors the use of purebred populations
– Necessitates higher density SNP panels in multibreeds • Markers chosen in purebreds may be less informative in multibreed populations as they will have less LD
• Markers that work well in multibreed populations seem p p
to work just as well in purebred populations
• Nice to have larger multibreed populations & denser panels Correlations between true and predicted genetic
merits in validation population
Panel: QTL
QTL
50
100
250
500
MBÎPB
0.953
0.938
0 840
0.840
0.720
PBÎMB
0.962
0.941
0 853
0.853
0.786
22
6/16/2009
Simulated Phenotypes/real 50k Data
• Effect of number of QTL
Effect of number of QTL
50k w/o QTL
50 QTL
Train in Multibreed
Validate in Purebreed
0.388
0.289
0.247
0.200
100 QTL
250 QTL
500 QTL
Train in Purebreed
Validate in Multibreed
0.422
0.308
0.276
0.299
• Identical trends when panel comprises QTL only
• These correlations a/c for < 20% variation at best
Correlations between true and predicted genetic
merits in validation population
Panel: HLD
QTL
MBÎPB
PBÎMB
50
0.570
0.486
100
0.513
0.480
250
0.510
0.429
500
0.372
0.391
23
6/16/2009
Average LD between QTL and HLD marker
in PB or MB populations
HLD to QTL
chosen from
PB
HLD-QTL LD
assessed in
PB
MB
0.549
0.322
MB
0.412
0.408
Conclusions
• MB population
– A good choice to carry out genomic selection
– Reasonably
R
bl accurate
t estimate
ti t off genetic
ti merits
it
of selection candidates in a PB population
• Accuracy of genetic merit in genomic selection
– Higher with fewer QTL
– Erodes when more uninformative SNPs added
• The extent of LD hence r2 are highly variable
– Lower average r2 in MB than PB populations
– No complete LD for all QTL with SNPs
– Denser markers are needed
24
6/16/2009
Training and Validation
Purebred (PB) Purebred (PB)
PB Î PB
Reduced
Panel
Reduced panel within‐breed selection
• Two‐stage Bayesian analysis
– Run all 50k markers
• in each of the three training sets (2&3, 1&3, 1&2)
– Select the best 600 markers on model frequency and genomic coverage
– Rerun the training and validation analyses using only the markers on the 600 marker panel
25
6/16/2009
50k versus 600 markers
384 SNP Panels
• Panels of 600 markers per trait for 8 traits would require a single panel of 4 800 markers
would require a single panel of 4,800 markers
• Technology is moving such that larger panels are costing the same as smaller panels used to, rather than reducing the cost of smaller panels
• Significantly cheaper panels are currently Si ifi tl h
l
tl
limited to 384 (or less) SNP
– Allow 100 or so of the best SNP for 3‐4 key traits
26
6/16/2009
Even Smaller Panels
Validation in New AI Bulls
27
6/16/2009
Summary – beef cattle in US
• 50k
50k within breed (like 5
within breed (like 5‐15
15 progeny)
progeny)
• 50k across breed (like 1 individual record or 5 progeny)
• Reduced panel within breed (
(varies up to 50k accuracy)
p
y)
Validation Statistics
• Proportion of additive variation accounted for by the genomic prediction
the genomic prediction
– Molecular BV used as an observation
1/Multivariate model using the MBV as a trait to estimate (eg ASREML) the genetic correlation
2/ Reduction in estimated sire variance when the MBV is included as a fixed effect in the model
3/ Regression of phenotype on MBV
Thallman et al, 2009 BIF
28
6/16/2009
Thallman et al, 2009 BIF
BVN
BVN
res cov estd res cov 0
heritability
rg
Data Simulated from Additive Model Only
0.1 0.04
0.11
0.08
0.1 0.16
0.21
0.23
0.1 0.36
0.38
0.44
0.1 0.64
0.54
0.64
0.3 0.04
0.06
0.05
0.3 0.16
0.17
0.19
0.3 0.36
0.35
0.40
0.3 0.64
0.64
0.68
0.5 0.04
0.05
0.05
0.5 0.16
0.16
0.18
0.5 0.36
0.35
0.39
0.5 0.64
0.63
0.66
Reduction Regression
0.02
0.17
1.40
0.29
0.04
0.15
0.35
0.66
0.04
0.16
0.36
0.63
0.05
0.21
6.62
-0.23
0.05
0.20
0.42
0.83
0.05
0.18
0.39
0.72
1,000 animals representing 100 sires
Training on EBVs
29
6/16/2009
Ideal Model (Equation)
g = 1μ + Ma
M +ε
g is (true) genetic merit (BV)
M is columns of covariates (genotypes)
a are substitution effects
ε is lack-of-fit (hopefully small)
Ideal Model
g = 1μ + Ma + ε
g is genetic merit (BV)
var(g) = A? or G?
var(Ma) = G genomic relationships
var(ε ) = Iσ ε2 ? or cA? for c=
σ ε2
σ g2
the fraction of var(g) unaccounted by markers
30
6/16/2009
Towards a Practical Model
g + e = 1μ + Ma
M + (ε + e )
g is (true) genetic merit (BV)
e is usual e
(g + e ) is phenotype (no fixed effects)
var(ε + e) = cA + Iσ e2 since cov(ε , e') = 0
Or include a random polygenic effects explicitly, with var(u)=A etc
Practical Model
y = Xb + Ma + (ε + e )
Xb are usual fixed effects
var(ε + e) = cA + Iσ e2
Not reasonable to assume
ass me var(
ar(ε + e) = Iσ e2
unless markers are fitting very well
But we do all the time ! And the results are fairly similar
31
6/16/2009
Heterogeneous Residual
If y has correlated or heterogeneous residual
y = Xb + Ma + (ε + e )
var(ε + e) = cA + R
R is the usual value that would be used in MME
(eg y is means of varying k numbers of observations)
y k = Xb + Ma
M + (ε + ek )
Repeatability model
σ e2
, var (ek ) = D (diagonal)
⎡⎣ k + (k − 1)rI ⎤⎦
with rI =intraclass correlation of residuals
var(ek ) =
Heterogeneous Residual (cont)
y k = Xb + Ma + (ε + e )
var(ε + e) = cI + D Simplify by ignoring polygenic
Then "weights" are the diagonals of R −1
1
c
or (arbitrarily)
c + var(e )
c + var(e )
so weights
i ht approach
h 1 as k → ∞ (if rI = 0)
r ii =
var(e ) =
σ e2
⎡⎣ k + (k − 1)rI ⎤⎦
with rI =intraclass correlation of residuals
32
6/16/2009
Family Data
When means are from relatives,
relatives
rather than the same individuals,
genetic relationships can contribute
to the intraclass correlation
EBVs as data
g = Ma + ε
g + (ĝ - g) = ĝ = Ma + ε + (ĝ - g)
with var( ĝ − g) = PEV > 0
p
Similar to previous
g + e = Ma + ε + e
where var(g+e) > var(g)
33
6/16/2009
EBVs as data
g = Ma + ε
g + (ĝ - g) = ĝ = Ma + ε + (ĝ - g)
with var( ĝ − g) = PEV > 0
Generally var(ĝ − g) = var(g) + var( ĝ) − 2 cov(ĝ, g)
p
shrinkage
g pproperties
p
But BLUP has special
cov(ĝ, g) = var (ĝ) so that var(ĝ − g) = var(g) − var( ĝ)
var( ĝ)
r2 =
≤ 1 so 0 ≤ var( ĝ) ≤ var(g)
var(g)
Other Relevant Properties of BLUP
cov( ĝ, ĝ − g) = var( ĝ) − cov( ĝ, g)
var( ĝ) − var(g)
var( ĝ) = 0
= var(g)
So prediction errors are uncorrelated with
estimated merit
ĝ
ĝ − g
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6/16/2009
Other Relevant Properties of BLUP
But cov(g, ĝ − g) = cov( ĝ, g) − var(g)
= var(g)
( ˆ ) − var(g)
( )<0
Really good animals are underestimated
Really bad animals are overestimated
g
ĝ − g
Genomic Prediction
usual linear regression of y on (fixed) x
cov(y, x)
,
var(x)
(random) regression of EBV on markers
ĝ on Ma involves cov(ĝ, Ma) ≈ cov(ĝ, g) for small c
But cov(ĝ, g)=var(ĝ) will differ for every animal
β y.x =
according to its accuracy r 2
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6/16/2009
Need to “inflate” observations
g = Ma + ε
g + (kĝ - g) = kĝ = Ma + ε + (kĝ - g)
Want to choose k so that
cov(g kg
cov(g,
kĝ - g) = 0
cov(kĝ, g) to be constant
Finding k
Want cov(g, kĝ - g) = 0
cov(g, kĝ - g) = cov(g, ĝ - g) = kvar(ĝ) - var(g)
so we want k =
var(g) 1
=
var(ĝ) r 2
Want cov(kĝ, g) to be constant
cov(kĝ, g)=k var (ĝ)=
var(g)
var (ĝ)= var(g)
var(ĝ)
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6/16/2009
Implications
g
ĝ
= d,
d a deregressed
d
d "observation"
" b
" with
h h2 = r 2
2
r
d
It is used in the right-hand side of Cĝ= 2
σe
where C is such that the PEV is C−1 = (1 − r 2 )σ g2
Proof d = σ e2 Cĝ=
σ e2
(1 − r 2 )σ g2
(1 − h ) ĝ= ĝ
2
ĝ=
(1 − r 2 )h 2
r2
More Implications
But deregressed observations have heterogeneous variance
ε + (kg-g
kĝ-g ) with k = r −2 so kr 2 = 1
var (ε +kĝ-g )= var (ε ) + var(kĝ-g)
= var (ε ) + k 2 var (ĝ ) + var (g ) − 2k var(ĝ)
= var (ε ) + k 2 r 2 var (g ) + var (g ) − 2kr 2 var(g)
1 − r2
r2
Therefore the weights representing diagonals of R -1 are
= var (ε ) + (k − 1)var (g ) and k − 1 =
w=
1
c + 1 − r2 / r2
(
)
or scaled to w ' =
c
c + 1 − r2 / r2
(
)
so w' → 1 as r 2 → 1
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6/16/2009
Removing Parent Average
During the deregression process, parent average effects
should be removed
Why?
Animals with own and/or progeny information are shrunk
towards the parent average
Imagine if many bulls had no own/progeny info
y should not contribute anything
y
g to training
g
They
Imagine if some parents were segregating a major effect
We dont want this effect shrunk in all the offspring
Deregression is no problem if deregressed information is derived
directly from animal models during evaluation
Removing Parent Average
Deregression and removal of parent average effects
can be approximately achieved using only the EBV and r 2
values from trios of the training animal, its sire and dam,
by setting up mixed model equations for the parent average
and offspring, reconstructing the left-hand side to obtain
th published
the
bli h d reliabilities,
li biliti before
b f reconstructing
t ti the
th implied
i li d
right-hand side to determine the deregressed observation
and its appropriate r 2 ignoring the parental contribution
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6/16/2009
Approximate Method
First solve for Z ' Z
⎡ Z ' Z PA + 4 λ
⎢
−2 λ
⎢⎣
−2 λ
Z ' Zindividual
⎤ ⎡ ĝPA
⎥⎢
+ 2 λ ⎥ ⎢ ĝindividual
⎦⎣
⎤ ⎡ yPA
⎥=⎢
⎥⎦ ⎢⎣ yindividual
⎤
⎥
⎥⎦
−1
⎤
⎡ Z ' Z PA + 4 λ
⎡ C PA
⎤
−2 λ
C off
⎢
⎥ = ⎢ off
⎥
−2 λ
Z ' Zindividual + 2 λ ⎥
C individual ⎥⎦
⎢⎣
⎢⎣ C
⎦
2
2
to compute rPA
= 0.5 − λC PA and rindividual
= 1.0 − λC individual
Then use [Z ' Zindividual + λ ][ĝindividual ] = [yindividual ]
2
so rderegressed
= 1.0 − λ / (Z ' Zindividual + λ )
Substitution Effects
Animal pperspective,
p
Falconer recognizes
g
a and d effects
as the allelic contributions to phenotypic performance
Parent perspective, the allelic contributions to breeding value
are α =a + d(q - p)
Mixed animal and offspring data will be estimating a
mixture of a and α effects
39