Presented at 26th International Biometric Conference. August 26-31, Kobe, Japan.

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
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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
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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
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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
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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 ),
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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
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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
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8. Conclusion and Possible Improvements
• LD relationships are of a multivariate nature
• r 2 gives an incomplete description of LD
⇓
• undirected networks
• sparsity
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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)
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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)
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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)
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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
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13. lambda = CV
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7
Bayesian Network
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3
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5
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4
3
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2
2
14
1
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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
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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
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15. Acknowledgments
University of Wisconsin-Madison
• Daniel Gianola
• Guilherme Rosa
University College London
• Marco Scutari
• Kent Weigel
• Bruno Valente
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