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Penalized maximum likelihood
estimates of genetic covariance
matrices with shrinkage towards
phenotypic dispersion
Karin Meyer1
, Mark Kirkpatrick2
, Daniel Gianola3
1
Animal Genetics and Breeding Unit, University of New England, Armidale
2
Section of Integrative Biology, University of Texas, Austin
3
University of Wisconsin-Madison, Madison
AAABG 2011
Penalized REML | Introduction
Motivation
Multivariate genetic analyses: more than 2-4 traits
→ desirable!
→ technically increasingly feasible
→ inherently problematic
SAMPLING VARIANCE ↑↑ with no. of traits
K. M. / M. K. / D. G. | | AAABG 2011 2 / 13
Penalized REML | Introduction
Motivation
Multivariate genetic analyses: more than 2-4 traits
→ desirable!
→ technically increasingly feasible
→ inherently problematic
SAMPLING VARIANCE ↑↑ with no. of traits
Measures to alleviate S.V.
large → gigantic data sets
parsimonious models → less parameters than covar.s
estimation → use additional information
Bayesian: Prior
REML: Impose penalty P on likelihood
→ P = f(parameters)
→ designed to reduce S.V.
K. M. / M. K. / D. G. | | AAABG 2011 2 / 13
Penalized REML | Introduction
Penalized REML
Maximize:
log LP = log L − 1
2
ψ P
Tuning factor
Penalty on parameters
Standard, unpenalized REML log likelihood
Objectives:
Introduce new type of P
→ prior: Σ ∼ IW
Compare efficacy of different P
K. M. / M. K. / D. G. | | AAABG 2011 3 / 13
Penalized REML | REML and penalties
Penalties to improve estimates of ΣG
ˆΣP estimated much more accurately than ˆΣG
Idea: ‘Borrow strength’ from ˆΣP
1 Shrink canonical eigenvalues towards their mean
→ ‘bending’ (Hayes & Hill 1981)
Pℓ
λ
∝ i(log λi − ¯λ)2
λi : eig.values of ˆΣ
−1
P
ˆΣG
K. M. / M. K. / D. G. | | AAABG 2011 4 / 13
Penalized REML | REML and penalties
Penalties to improve estimates of ΣG
ˆΣP estimated much more accurately than ˆΣG
Idea: ‘Borrow strength’ from ˆΣP
1 Shrink canonical eigenvalues towards their mean
→ ‘bending’ (Hayes & Hill 1981)
Pℓ
λ
∝ i(log λi − ¯λ)2
λi : eig.values of ˆΣ
−1
P
ˆΣG
2 a) Shrink ˆΣG towards ˆΣP
→ assume ΣG ∼ IW(Σ−1
P
, ψ)
→ obtain penalty as minus log density of IW
PΣ ∝ C log | ˆΣG| + tr ˆΣ−1
G
ˆΣ0
P
b) Shrink ˆRG towards ˆRP
PR ∝ C log |ˆRG| + tr ˆR−1
G
ˆR0
P
K. M. / M. K. / D. G. | | AAABG 2011 4 / 13
Penalized REML | Simulation study
Simulation
Paternal half-sib design: s = 100, n = 10
5 traits, h2
1
≥ . . . ≥ h2
5
, MVN
60 sets of population values
→ vary mean & spread of λi
1000 replicates per case
3 penalties
Pℓ
λ
regress log( ˆλi) towards their mean
PΣ shrink ˆΣG towards ˆΣP
PR shrink ˆRG towards ˆRP
Obtain ˆΣψ
G
and ˆΣψ
E
for values of ψ, range 0 − 1000
K. M. / M. K. / D. G. | | AAABG 2011 5 / 13
Penalized REML | Simulation study
Simulation - cont.
Estimate ψ using population values (V∞)
→ construct MSB & MSW → validation
→ ˆψ maximize log L in valid. data
Evaluate effect of penalty
→ Loss: deviation of ˆΣX from ΣX
L1(ΣX, ˆΣX) = tr(Σ−1
X
ˆΣX) − log |Σ−1
X
ˆΣX| − q
→ Percentage Reduction In Average Loss
PRIAL = 100






1 −
¯L1(ΣX, ˆΣ
ˆψ
X
)
¯L1(ΣX, ˆΣ0
X
)






K. M. / M. K. / D. G. | | AAABG 2011 6 / 13
penalized
unpenalized
Penalized REML | Results | PRIAL
PRIAL in estimated covariance matrices
Pλ
l
PΣ PR
40
60
80
100
q
q
q
q
q
q
q
Genetic
71.4 70.6 72.3
Pλ
l
PΣ PR
0
20
40
60
80
Residual
43.4 13.3 37.1
Pλ
l
PΣ PR
0
2
4
6
8
q
q
q
qq
q
qq
q
q
q
q
q
Phenotypic
1.2 1.2 2.2
K. M. / M. K. / D. G. | | AAABG 2011 7 / 13
Penalized REML | Results | PRIAL
PRIAL for ˆΣG – 60 individual cases
in order of PRIAL for Pℓ
λ
5K
5L
5J
5D
2K
5H
5F
2F
5E
2L
4F
4K
2H
4L
5I
2D
4H
3K
3F
2E
5C
1D
4D
3L
1C
3H
3D
2J
5G
4J
4E
5B
3E
5A
4I
1G
3C
3A
3B
1E
3G
3J
1F
4A
2I
4C
4B
1K
3I
4G
2C
1B
2G
1H
2B
2A
1L
1J
1A
1I
40
60
80
100
q q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q q q q q q
q q
q q q
q
q q
q q
q
q
q q
q
q
q
q q
q
q
q
q
q
Penalty
Pλ
l
PΣ PR
K. M. / M. K. / D. G. | | AAABG 2011 8 / 13
Penalized REML | Results | PRIAL
Extension: PRIAL “double” penalties
P so far: improve ˆΣG
“Double”
→ PΣ on ˆΣG & ˆΣE
PR on ˆRG & ˆRE
Pℓ
λ
on log λi &
log(1−λi)
→ 1: joint ψ
2: separate ψ
Pℓ
λ
PΣ PR
ˆΣG G 71 71 72
G+E 1 73 70 73
2 74 73 74
ˆΣE G 43 13 37
G+E 1 62 54 60
2 65 65 63
Little impact on PRIAL for ˆΣG
Can ↑↑ PRIAL for ˆΣE w/out ↓ PRIAL for ˆΣG
Separate ˆψ: extra effort, limited add. improvement
K. M. / M. K. / D. G. | | AAABG 2011 9 / 13
Penalized REML | Results | PRIAL
Summary: PRIAL
Substantial improvements in ˆΣG & ˆΣE feasible
→ Results shown ‘optimistic’ → ˆψ using pop.values
PRIAL heavily influenced by spread of can. eigenvalues
Pℓ
λ
best if λi ≈ ¯λ; overshrink if not
PΣ & PR worst if λi ≈ ¯λ; more robust than Pℓ
λ
if λi ≈ ¯λ
similar canonical eigenvalues unlikely in practice?
New type of P useful
K. M. / M. K. / D. G. | | AAABG 2011 10 / 13
Penalized REML | Results | Bias
Mean estimates of can. eigenvalues
Penalties act differently – illustrated for single case
λi = 0.5, 0.2, 0.15, 0.1, 0.05; ¯λ = 0.2
0
0.2
0.4
λ1 λ3 λ5
q
q
q
q
q
None
q
q
q
q
q
Pλ
l
λ1 λ3 λ5
q
q
q
q
q
PΣ
q
q
q
q
q
q
q
q
q
q
PR
q Pop.value
K. M. / M. K. / D. G. | | AAABG 2011 11 / 13
Penalized REML | Results | Bias
Mean relative bias (in %)
1 2 3 4 5
Can. eigenvalues
None 9 27 17 -20 -79
Pℓ
λ
-4 16 29 58 101
PΣ 8 25 25 39 75
PR 1 16 21 37 57
Heritabilities
None -1 4 5 7 12
Pℓ
λ
-7 5 12 23 45
PΣ 1 10 15 26 44
PR -2 2 5 9 17
K. M. / M. K. / D. G. | | AAABG 2011 12 / 13
Penalized REML | Finale
Conclusions
Can obtain ‘better’ estimates of genetic cov. matrices
→ borrow strength from phenotypic covariance matrix
→ easy to implement by penalty on log L
K. M. / M. K. / D. G. | | AAABG 2011 13 / 13
Penalized REML | Finale
Conclusions
Can obtain ‘better’ estimates of genetic cov. matrices
→ borrow strength from phenotypic covariance matrix
→ easy to implement by penalty on log L
Comparable improvement (PRIAL) from different P
None best overall
PR: Shrinking ˆRG towards ˆRP
less variable
less bias in estimates of genetic parameters
easier to implement than Pℓ
λ
K. M. / M. K. / D. G. | | AAABG 2011 13 / 13
Penalized REML | Finale
Conclusions
Can obtain ‘better’ estimates of genetic cov. matrices
→ borrow strength from phenotypic covariance matrix
→ easy to implement by penalty on log L
Comparable improvement (PRIAL) from different P
None best overall
PR: Shrinking ˆRG towards ˆRP
less variable
less bias in estimates of genetic parameters
easier to implement than Pℓ
λ
Penalized REML recommended!
Small to moderate data sets, ≥ 4 traits
K. M. / M. K. / D. G. | | AAABG 2011 13 / 13
P-REML
part of
everyday
toolkit

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Penalized maximum likelihood estimates of genetic covariance matrices with shrinkage towards phenotypic dispersion

  • 1. Penalized maximum likelihood estimates of genetic covariance matrices with shrinkage towards phenotypic dispersion Karin Meyer1 , Mark Kirkpatrick2 , Daniel Gianola3 1 Animal Genetics and Breeding Unit, University of New England, Armidale 2 Section of Integrative Biology, University of Texas, Austin 3 University of Wisconsin-Madison, Madison AAABG 2011
  • 2. Penalized REML | Introduction Motivation Multivariate genetic analyses: more than 2-4 traits → desirable! → technically increasingly feasible → inherently problematic SAMPLING VARIANCE ↑↑ with no. of traits K. M. / M. K. / D. G. | | AAABG 2011 2 / 13
  • 3. Penalized REML | Introduction Motivation Multivariate genetic analyses: more than 2-4 traits → desirable! → technically increasingly feasible → inherently problematic SAMPLING VARIANCE ↑↑ with no. of traits Measures to alleviate S.V. large → gigantic data sets parsimonious models → less parameters than covar.s estimation → use additional information Bayesian: Prior REML: Impose penalty P on likelihood → P = f(parameters) → designed to reduce S.V. K. M. / M. K. / D. G. | | AAABG 2011 2 / 13
  • 4. Penalized REML | Introduction Penalized REML Maximize: log LP = log L − 1 2 ψ P Tuning factor Penalty on parameters Standard, unpenalized REML log likelihood Objectives: Introduce new type of P → prior: Σ ∼ IW Compare efficacy of different P K. M. / M. K. / D. G. | | AAABG 2011 3 / 13
  • 5. Penalized REML | REML and penalties Penalties to improve estimates of ΣG ˆΣP estimated much more accurately than ˆΣG Idea: ‘Borrow strength’ from ˆΣP 1 Shrink canonical eigenvalues towards their mean → ‘bending’ (Hayes & Hill 1981) Pℓ λ ∝ i(log λi − ¯λ)2 λi : eig.values of ˆΣ −1 P ˆΣG K. M. / M. K. / D. G. | | AAABG 2011 4 / 13
  • 6. Penalized REML | REML and penalties Penalties to improve estimates of ΣG ˆΣP estimated much more accurately than ˆΣG Idea: ‘Borrow strength’ from ˆΣP 1 Shrink canonical eigenvalues towards their mean → ‘bending’ (Hayes & Hill 1981) Pℓ λ ∝ i(log λi − ¯λ)2 λi : eig.values of ˆΣ −1 P ˆΣG 2 a) Shrink ˆΣG towards ˆΣP → assume ΣG ∼ IW(Σ−1 P , ψ) → obtain penalty as minus log density of IW PΣ ∝ C log | ˆΣG| + tr ˆΣ−1 G ˆΣ0 P b) Shrink ˆRG towards ˆRP PR ∝ C log |ˆRG| + tr ˆR−1 G ˆR0 P K. M. / M. K. / D. G. | | AAABG 2011 4 / 13
  • 7. Penalized REML | Simulation study Simulation Paternal half-sib design: s = 100, n = 10 5 traits, h2 1 ≥ . . . ≥ h2 5 , MVN 60 sets of population values → vary mean & spread of λi 1000 replicates per case 3 penalties Pℓ λ regress log( ˆλi) towards their mean PΣ shrink ˆΣG towards ˆΣP PR shrink ˆRG towards ˆRP Obtain ˆΣψ G and ˆΣψ E for values of ψ, range 0 − 1000 K. M. / M. K. / D. G. | | AAABG 2011 5 / 13
  • 8. Penalized REML | Simulation study Simulation - cont. Estimate ψ using population values (V∞) → construct MSB & MSW → validation → ˆψ maximize log L in valid. data Evaluate effect of penalty → Loss: deviation of ˆΣX from ΣX L1(ΣX, ˆΣX) = tr(Σ−1 X ˆΣX) − log |Σ−1 X ˆΣX| − q → Percentage Reduction In Average Loss PRIAL = 100       1 − ¯L1(ΣX, ˆΣ ˆψ X ) ¯L1(ΣX, ˆΣ0 X )       K. M. / M. K. / D. G. | | AAABG 2011 6 / 13 penalized unpenalized
  • 9. Penalized REML | Results | PRIAL PRIAL in estimated covariance matrices Pλ l PΣ PR 40 60 80 100 q q q q q q q Genetic 71.4 70.6 72.3 Pλ l PΣ PR 0 20 40 60 80 Residual 43.4 13.3 37.1 Pλ l PΣ PR 0 2 4 6 8 q q q qq q qq q q q q q Phenotypic 1.2 1.2 2.2 K. M. / M. K. / D. G. | | AAABG 2011 7 / 13
  • 10. Penalized REML | Results | PRIAL PRIAL for ˆΣG – 60 individual cases in order of PRIAL for Pℓ λ 5K 5L 5J 5D 2K 5H 5F 2F 5E 2L 4F 4K 2H 4L 5I 2D 4H 3K 3F 2E 5C 1D 4D 3L 1C 3H 3D 2J 5G 4J 4E 5B 3E 5A 4I 1G 3C 3A 3B 1E 3G 3J 1F 4A 2I 4C 4B 1K 3I 4G 2C 1B 2G 1H 2B 2A 1L 1J 1A 1I 40 60 80 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Penalty Pλ l PΣ PR K. M. / M. K. / D. G. | | AAABG 2011 8 / 13
  • 11. Penalized REML | Results | PRIAL Extension: PRIAL “double” penalties P so far: improve ˆΣG “Double” → PΣ on ˆΣG & ˆΣE PR on ˆRG & ˆRE Pℓ λ on log λi & log(1−λi) → 1: joint ψ 2: separate ψ Pℓ λ PΣ PR ˆΣG G 71 71 72 G+E 1 73 70 73 2 74 73 74 ˆΣE G 43 13 37 G+E 1 62 54 60 2 65 65 63 Little impact on PRIAL for ˆΣG Can ↑↑ PRIAL for ˆΣE w/out ↓ PRIAL for ˆΣG Separate ˆψ: extra effort, limited add. improvement K. M. / M. K. / D. G. | | AAABG 2011 9 / 13
  • 12. Penalized REML | Results | PRIAL Summary: PRIAL Substantial improvements in ˆΣG & ˆΣE feasible → Results shown ‘optimistic’ → ˆψ using pop.values PRIAL heavily influenced by spread of can. eigenvalues Pℓ λ best if λi ≈ ¯λ; overshrink if not PΣ & PR worst if λi ≈ ¯λ; more robust than Pℓ λ if λi ≈ ¯λ similar canonical eigenvalues unlikely in practice? New type of P useful K. M. / M. K. / D. G. | | AAABG 2011 10 / 13
  • 13. Penalized REML | Results | Bias Mean estimates of can. eigenvalues Penalties act differently – illustrated for single case λi = 0.5, 0.2, 0.15, 0.1, 0.05; ¯λ = 0.2 0 0.2 0.4 λ1 λ3 λ5 q q q q q None q q q q q Pλ l λ1 λ3 λ5 q q q q q PΣ q q q q q q q q q q PR q Pop.value K. M. / M. K. / D. G. | | AAABG 2011 11 / 13
  • 14. Penalized REML | Results | Bias Mean relative bias (in %) 1 2 3 4 5 Can. eigenvalues None 9 27 17 -20 -79 Pℓ λ -4 16 29 58 101 PΣ 8 25 25 39 75 PR 1 16 21 37 57 Heritabilities None -1 4 5 7 12 Pℓ λ -7 5 12 23 45 PΣ 1 10 15 26 44 PR -2 2 5 9 17 K. M. / M. K. / D. G. | | AAABG 2011 12 / 13
  • 15. Penalized REML | Finale Conclusions Can obtain ‘better’ estimates of genetic cov. matrices → borrow strength from phenotypic covariance matrix → easy to implement by penalty on log L K. M. / M. K. / D. G. | | AAABG 2011 13 / 13
  • 16. Penalized REML | Finale Conclusions Can obtain ‘better’ estimates of genetic cov. matrices → borrow strength from phenotypic covariance matrix → easy to implement by penalty on log L Comparable improvement (PRIAL) from different P None best overall PR: Shrinking ˆRG towards ˆRP less variable less bias in estimates of genetic parameters easier to implement than Pℓ λ K. M. / M. K. / D. G. | | AAABG 2011 13 / 13
  • 17. Penalized REML | Finale Conclusions Can obtain ‘better’ estimates of genetic cov. matrices → borrow strength from phenotypic covariance matrix → easy to implement by penalty on log L Comparable improvement (PRIAL) from different P None best overall PR: Shrinking ˆRG towards ˆRP less variable less bias in estimates of genetic parameters easier to implement than Pℓ λ Penalized REML recommended! Small to moderate data sets, ≥ 4 traits K. M. / M. K. / D. G. | | AAABG 2011 13 / 13