Application of Bayesian and Sparse Network Models for Assessing Linkage Disequilibrium in Animals and Plants

Application of Bayesian and Sparse Network
Models for Assessing Linkage Disequilibrium in
Animals and Plants
C-36-6

Gota Morota
Department of Animal Sciences
University of Wisconsin-Madison

Aug 30, 2012

1 / 16
Systems Genetics

Figure 1: Multi-dimensional gene network

Purpose of this study
• take the view that loci associate and interact together as a
network
• evaluate LD reflecting the biological nature that loci interact as
a complex system
2 / 16
IAMB algorithm
Incremental Association Markov Blanket (Tsamardinos et al. 2003)
1. Compute Markov Blankets (MB)
2. Compute Graph Structure
3. Orient Edges

Figure 2: The Markov Blanket of a node xi

3 / 16
Identifying the MB of a node
• Growing phase
• heuristic function:
f (X ; T |CMB ) = MI(X ; T |CMB )

=
cmb ∈CMB





P (CMB ) 




P (X , T |CMB )



P (X , T |CMB ) log

P (X |CMB )P (T |CMB ) 


x ∈X t ∈T

• conditional independence tests (Pearson’s χ2 test):

H0 : P (X , T |CMB ) = P (X |CMB ) · P (T |CMB ) (do not add X )
HA : P (X , T |CMB )

P (X |CMB ) · P (T |CMB ) (add X to the CMB)

• Shrinking phase
• conditional independence tests (Pearson’s χ2 test):
H0 : P (X , T |CMB − X )

P (X |CMB − X ) · P (T |CMB − X ) (keep X )

HA : P (X , T |CMB − X ) = P (X |CMB − X ) · P (T |CMB − X ) (remove X
4 / 16
Network Structure
Algorithm
Suppose Y ∈ MB (T ). Then T and Y are connected if they are
conditionally dependent given all subsets of the smaller of
MB (T ) − (Y ) and MB (Y ) − (T ).
Example:
• MB (T ) = (A , B , Y ), MB (Y ) = (C , D , E , F , T )
• since MB (T ) < MB (Y ), independence tests are conditional
on all subsets of MB (T ) − (Y ) = (A , B ).
• if any of the
CI(T , Y |{}), CI(T , Y |{A }), CI(T , Y |{B }), andCI(T , Y |{A , B })
imply conditional independence,
↓
• T and Y are considered separate (spouses)
• repeat for T ∈ S and Y ∈ MB (T ),
5 / 16
Materials
1. Data
• 4,898 Holstein bulls (USDA-ARS AIPL)
• 37,217 SNP markers (MAF > 0.025)
• milk protein yield

2. Missing genotypes imputation
• fastPHASE (Scheet and Stephens, 2006)

3. Select 15 SNPs
• Bayesian LASSO

4. uncover associations among a set of marker loci found to
have the strongest effects on milk protein yield

6 / 16
Results – Top 15 SNPs
IAMB algorithm

Pairwise LD among SNPs (r2)

J

d

A

c
b
a
Z

L

Y
X

M

N

W
V
U

F

T
S

B

K

G

R
Q
P

H

O
N

O

E

M
L
K
J

I

I
H
G
F
E

C

D
C
B

R2 Color Key

A
0

Figure 3: r 2

0.2

0.4

0.6

0.8

1

D
Figure 4: IAMB
7 / 16
Conclusion and Possible Improvements

• LD relationships are of a multivariate nature
• r 2 gives an incomplete description of LD

⇓
• undirected networks
• sparsity

8 / 16
Pairwise Binary Markov Networks
We estimate the Markov network parameters Θp ×p by maximizing
a log-likelihood.






f (x1 , ..., xp ) =
exp 



Ψ(Θ)
1

p

θj ,j xj +
j =1

1≤j <k ≤p






θj ,k xj xk 




(1)

where
xj ∈ {0, 1}

Ψ(Θ) =
x ∈0 , 1

(2)






exp 




p

θj ,j xj +
j =1

1 ≤j <k ≤p






θj ,k xj xk 




(3)

• the first term is a main effect of binary marker xj (node
potential)
• the second term corresponds to an“interaction effect” between
binary markers xj and xk (link potential)
• Ψ(Θ) is the normalization constant (partition function)
9 / 16
Ravikumar et al. (2010)
The pseudo-likelihood based on the local conditional likelihood
associated with each binary marker can be represented as
n

p
x

φi ,ij,j (1 − φi ,j )1−xi,j

l (Θ) =

(4)

i =1 j =1

where φi ,j is the conditional probability of xi ,j = 1 given all other
variables. Using a logistic link function,

φi ,j = P(xi ,j = 1|xi ,k , k j ; θj ,k , 1 ≤ k ≤ p )
exp(θj ,j + k j θj ,k xi ,k )
=
1 + exp(θj ,j + k j θj ,k xi ,k )

(5)
(6)

10 / 16
Ravikumar et al. (2010) (cont.)
• L1 regularized logistic regressions problem
• regressing each marker on the rest of the markers
• the network structure is recovered from the sparsity pattern of
the regression coefficients


 0


 ˆ−2
 β

 1




ˆ  .
 .
Θ= .


 −(p −1)

ˆ
β

 1

 −p
 ˆ
β1

ˆ
β −1 ,
2
0

··· ,
··· ,

··· ,
0
ˆ−(p −1)
· · · , β p −2
ˆ p
· · · , β−−2
p
˜
Θ=

ˆ ˆ
Θ • ΘT

ˆ 1
β−−1
p
ˆ 2
β−−1
p

ˆp
β −1
ˆp
β −2
.
.
.
















··· ,




−(p −1) 

ˆp

0
β




−p
ˆ
β p −1
0

(7)

(8)
11 / 16
Materials
1. Data
• 599 inbred wheat lines (CIMMYT)
• 1447 Diversity Array Technology (DArT) binary markers
• mean grain yields

2. Select 30 SNPs
• Bayesian LASSO

3. Benchmark methods
• IAMB algorithm
• r2

12 / 16
lambda = CV

9

8

7

Bayesian Network

9

6

10

8

7

6

10

5

11

4
3

12

5

11

13

4
3

12
13

2

2

14

1

14

15

0

15

0

16

29

16

29

28

17
18

27
19

26
20

25
21

22

23

24

Figure 5: L1 regularization

1

28

17
18

27
19

26
20

25
21

22

23

24

Figure 6: IAMB
13 / 16
Summary
Interactions and associations among the cells and genes form a
complex biological system

⇓
• r 2 → association(m1, m2)|∅ (empty set)
• L1 regularized MN → association(m1, m2) | else

A final remark
• selecting tag SNPs unconditionally, as well as conditionally,
on other markers when the dimension of the data is high
• data generated from next generation sequence technologies

14 / 16
Acknowledgments

University of Wisconsin-Madison
• Daniel Gianola
• Guilherme Rosa

University College London
• Marco Scutari

• Kent Weigel
• Bruno Valente

15 / 16
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Application of Bayesian and Sparse Network Models for Assessing Linkage Disequilibrium in Animals and Plants

  • 1. Application of Bayesian and Sparse Network Models for Assessing Linkage Disequilibrium in Animals and Plants C-36-6 Gota Morota Department of Animal Sciences University of Wisconsin-Madison Aug 30, 2012 1 / 16
  • 2. Systems Genetics Figure 1: Multi-dimensional gene network Purpose of this study • take the view that loci associate and interact together as a network • evaluate LD reflecting the biological nature that loci interact as a complex system 2 / 16
  • 3. IAMB algorithm Incremental Association Markov Blanket (Tsamardinos et al. 2003) 1. Compute Markov Blankets (MB) 2. Compute Graph Structure 3. Orient Edges Figure 2: The Markov Blanket of a node xi 3 / 16
  • 4. Identifying the MB of a node • Growing phase • heuristic function: f (X ; T |CMB ) = MI(X ; T |CMB ) = cmb ∈CMB     P (CMB )     P (X , T |CMB )    P (X , T |CMB ) log  P (X |CMB )P (T |CMB )   x ∈X t ∈T • conditional independence tests (Pearson’s χ2 test): H0 : P (X , T |CMB ) = P (X |CMB ) · P (T |CMB ) (do not add X ) HA : P (X , T |CMB ) P (X |CMB ) · P (T |CMB ) (add X to the CMB) • Shrinking phase • conditional independence tests (Pearson’s χ2 test): H0 : P (X , T |CMB − X ) P (X |CMB − X ) · P (T |CMB − X ) (keep X ) HA : P (X , T |CMB − X ) = P (X |CMB − X ) · P (T |CMB − X ) (remove X 4 / 16
  • 5. Network Structure Algorithm Suppose Y ∈ MB (T ). Then T and Y are connected if they are conditionally dependent given all subsets of the smaller of MB (T ) − (Y ) and MB (Y ) − (T ). Example: • MB (T ) = (A , B , Y ), MB (Y ) = (C , D , E , F , T ) • since MB (T ) < MB (Y ), independence tests are conditional on all subsets of MB (T ) − (Y ) = (A , B ). • if any of the CI(T , Y |{}), CI(T , Y |{A }), CI(T , Y |{B }), andCI(T , Y |{A , B }) imply conditional independence, ↓ • T and Y are considered separate (spouses) • repeat for T ∈ S and Y ∈ MB (T ), 5 / 16
  • 6. Materials 1. Data • 4,898 Holstein bulls (USDA-ARS AIPL) • 37,217 SNP markers (MAF > 0.025) • milk protein yield 2. Missing genotypes imputation • fastPHASE (Scheet and Stephens, 2006) 3. Select 15 SNPs • Bayesian LASSO 4. uncover associations among a set of marker loci found to have the strongest effects on milk protein yield 6 / 16
  • 7. Results – Top 15 SNPs IAMB algorithm Pairwise LD among SNPs (r2) J d A c b a Z L Y X M N W V U F T S B K G R Q P H O N O E M L K J I I H G F E C D C B R2 Color Key A 0 Figure 3: r 2 0.2 0.4 0.6 0.8 1 D Figure 4: IAMB 7 / 16
  • 8. Conclusion and Possible Improvements • LD relationships are of a multivariate nature • r 2 gives an incomplete description of LD ⇓ • undirected networks • sparsity 8 / 16
  • 9. Pairwise Binary Markov Networks We estimate the Markov network parameters Θp ×p by maximizing a log-likelihood.      f (x1 , ..., xp ) = exp     Ψ(Θ) 1 p θj ,j xj + j =1 1≤j <k ≤p      θj ,k xj xk     (1) where xj ∈ {0, 1} Ψ(Θ) = x ∈0 , 1 (2)      exp     p θj ,j xj + j =1 1 ≤j <k ≤p      θj ,k xj xk     (3) • the first term is a main effect of binary marker xj (node potential) • the second term corresponds to an“interaction effect” between binary markers xj and xk (link potential) • Ψ(Θ) is the normalization constant (partition function) 9 / 16
  • 10. Ravikumar et al. (2010) The pseudo-likelihood based on the local conditional likelihood associated with each binary marker can be represented as n p x φi ,ij,j (1 − φi ,j )1−xi,j l (Θ) = (4) i =1 j =1 where φi ,j is the conditional probability of xi ,j = 1 given all other variables. Using a logistic link function, φi ,j = P(xi ,j = 1|xi ,k , k j ; θj ,k , 1 ≤ k ≤ p ) exp(θj ,j + k j θj ,k xi ,k ) = 1 + exp(θj ,j + k j θj ,k xi ,k ) (5) (6) 10 / 16
  • 11. Ravikumar et al. (2010) (cont.) • L1 regularized logistic regressions problem • regressing each marker on the rest of the markers • the network structure is recovered from the sparsity pattern of the regression coefficients   0    ˆ−2  β   1     ˆ  .  . Θ= .    −(p −1)  ˆ β   1   −p  ˆ β1 ˆ β −1 , 2 0 ··· , ··· , ··· , 0 ˆ−(p −1) · · · , β p −2 ˆ p · · · , β−−2 p ˜ Θ= ˆ ˆ Θ • ΘT ˆ 1 β−−1 p ˆ 2 β−−1 p ˆp β −1 ˆp β −2 . . .                ··· ,     −(p −1)   ˆp  0 β     −p ˆ β p −1 0 (7) (8) 11 / 16
  • 12. Materials 1. Data • 599 inbred wheat lines (CIMMYT) • 1447 Diversity Array Technology (DArT) binary markers • mean grain yields 2. Select 30 SNPs • Bayesian LASSO 3. Benchmark methods • IAMB algorithm • r2 12 / 16
  • 13. lambda = CV 9 8 7 Bayesian Network 9 6 10 8 7 6 10 5 11 4 3 12 5 11 13 4 3 12 13 2 2 14 1 14 15 0 15 0 16 29 16 29 28 17 18 27 19 26 20 25 21 22 23 24 Figure 5: L1 regularization 1 28 17 18 27 19 26 20 25 21 22 23 24 Figure 6: IAMB 13 / 16
  • 14. Summary Interactions and associations among the cells and genes form a complex biological system ⇓ • r 2 → association(m1, m2)|∅ (empty set) • L1 regularized MN → association(m1, m2) | else A final remark • selecting tag SNPs unconditionally, as well as conditionally, on other markers when the dimension of the data is high • data generated from next generation sequence technologies 14 / 16
  • 15. Acknowledgments University of Wisconsin-Madison • Daniel Gianola • Guilherme Rosa University College London • Marco Scutari • Kent Weigel • Bruno Valente 15 / 16