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Application of Bayesian and Sparse Network
Models for Assessing Linkage Disequilibrium in
Animals and Plants
Master’s Defense
Gota Morota

Dec 5, 2011

1 / 44
Outline

Overview of standard LD metrics

Bayesian Network

L1 regularized Markov Network

Exome sequence analysis

2 / 44
Outline

Overview of standard LD metrics

Bayesian Network

L1 regularized Markov Network

Exome sequence analysis

3 / 44
Outline

Overview of standard LD metrics

Bayesian Network

L1 regularized Markov Network

Exome sequence analysis

4 / 44
Outline

Overview of standard LD metrics

Bayesian Network

L1 regularized Markov Network

Exome sequence analysis

5 / 44
Linkage Disequilibrium (LD)

Definition
non-random association of alleles at different loci
• also known as gametic phase disequilibrium
• first used in 1960 (Lewontin and Kojima)
• has been used extensively in many area
1. genome enabled selection
2. genome-wide association study
3. understand past evolutionary and demographic events

6 / 44
Systems Genetics

Figure 1: A multi-dimensional gene feature/knowledge network

Purpose of the thesis
• 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
7 / 44
Graphical Models
• provide a way to visualize the structure of a model
• a graph is comprised of
1. nodes (vertices) → random variables
2. edges (links, arcs) → probabilistic relationship

8 / 44
Graphical Models
• provide a way to visualize the structure of a model
• a graph is comprised of
1. nodes (vertices) → random variables
2. edges (links, arcs) → probabilistic relationship

Figure 2: A soccer ball

8 / 44
Bayesian Networks (BN)
• directed acyclic graph (DAG) with graphical structure
G = (V , E )
• Application of BN to genomic
1. DNA microarrays (Friedman et al. 2000)
2. protein signaling networks (Sachs et al. 2005)
3. GWAS (Sebastiani et al. 2005)
4. genome-enabled prediction and classification in livestock
(Long et al. 2009)

Objective I
Apply BN to uncover associations among a set of marker loci found
to have the strongest effects on milk protein yield in Holstein cattle.

9 / 44
Factorization Properties

p (a , b , c ) = p (a )p (b |a )p (c |a , b )

(1)

Figure 3: The joint distribution of x1 , · · · , x3

10 / 44
Factorization Properties (cont.)

p

P (X1 , ..., Xp ) =

P (Xj |Pa (Xj ))

(2)

j =1

• Pa (Xj ) is a set of parent nodes of Xj
• a node is conditionally independent of its non-descendants
given its parents

One restriction
Must be no directed cycles (need to be DAG).

11 / 44
Structure Learning
• Local score-based algorithms
• Scoring metrics
•
•
•
•

Akaike Information Criterion (AIC)
Minimum Description Length (MDL)
K2
Bayesian Dirichlet Equivalent (BDe)

• Constraint-based algorithms (Causal inference algorithms)
• Koller-Sahami (KS)
• Grow Shrink (GS)
• Incremental Association Markov Blanket (IAMB)

12 / 44
IAMB algorithm
Incremental Association Markov Blanket (Tsamardinos et al. 2003)
1. Compute Markov Blankets (MB)
2. Compute Graph Structure
3. Orient Edges

Figure 4: The Markov Blanket of a node xi

13 / 44
Identifying the MB of a node
• Growing phase
• heuristic funtion:
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
14 / 44
Identifying the MB of a node (example)

Suppose we have target variable T , and a set of nodes
X = (A , B , C , D , E , F )
• Growing phase
1. MI(T , X |CMB = (∅)) → X = (C , E , A , D , B , F )
2. CI(T , C |CMB =(∅)) → CMB = (C )
3. MI(T , X |CMB = (C)) → X = (A , E , D , F , B )
4. CI(T , A |CMB =(C)) → CMB = (C , A )
• Shrinking phase
1. CI(T , C |CMB = (C , A ) − C )
2. CI(T , A |CMB = (C , A ) − A )

15 / 44
Network Structure
Algorithm
Suppose Y ∈ MB (T ). Then T and Y are connected if they are
conditionaly 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 ),
16 / 44
Data, Missing Genotype Imputation, and Subset
selection
1. Data
• 4,898 progeny tested Holstein bulls (USDA-ARS AIPL)
• 37,217 SNP markers (MAF > 0.025)
• Predicted Transmitting Ability (PTA) for milk protein yield

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

3. Select 15-30 SNPs
• Bayesian LASSO (BLR R pakcage, Perez et al, 2010)
´

4. SNPs ranking strategies
ˆ
• |βj |
ˆ
• |βj |/ Var (βj )
ˆ
• 2pj (1 − pj )β2
j
17 / 44
Results – Top 15 SNPs from Strategy 1
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 5: r 2

0.2

0.4

0.6

0.8

1

D
Figure 6: IAMB
18 / 44
Conclusion

The result confirms that LD relationships are of a multivariate
nature, and that r 2 gives an incomplete description and
understanding of LD.
• capture association among SNPs as a network
• no limitation with respect to the type of loci

19 / 44
Possible Improvements
• associations among loci are assumed bi-directional
• LD is expected to decline rapidly as the physical distance
between two loci increases, and that pairs of loci on different
chromosomes rarely show high LD
• conditional independence property

20 / 44
Possible Improvements
• associations among loci are assumed bi-directional
• LD is expected to decline rapidly as the physical distance
between two loci increases, and that pairs of loci on different
chromosomes rarely show high LD
• conditional independence property

⇓
therefore
• undirected networks
• sparisty
• conditional independence property

20 / 44
Undirected Graphical Models

Figure 7: An undirected graph

• Markov networks (Markov random fields)
• G = (V , E )
• express an affinity instead of a causal relationship

21 / 44
Pairwise Conditional Independence Property

pairwise conditional independence property
• an absence of edge between two nodes,
xj and xk , implies conditional
independence, given all other nodes

p (xj , xk |x−j ,−k ) = p (xj |x−j ,−k )p (xj |x−j ,−k )

(3)

In Figure (8), (a ⊥ d |b , c) and (b ⊥ d |a , c).

Figure 8: Example 1

22 / 44
Cliques
A clique is a subset of nodes in a graph such that every pair of
nodes are connected by edges
• (a) {X , Y }, {Y , Z }
• (b) {X , Y , W }, {Z }
• (c) {X , Y }, {Y , Z }, {Z , W }, {X , W }
• (d) {X , Y }, {Y , Z }, {Z , W }

Maximum cliques
Figure 9: Example 3

a maximum clique is defined as the
clique having the largest size
23 / 44
The Factorization of Markov Networks
the Hammersley-Clifford theorem
for any positive distributions, the distribution factorizes according to
the Markov network structure defined by cliques.
Consider X = xi , · · · , xn ,
p (X ) =

1
Z

φc (Xc )

(4)

C ∈G

where Z is a normalizing constant defined by

φc (Xc )

Z=

(5)

x C ∈G

and φ is called a potential function or a clique potential.

24 / 44
The Factorization of Markov Networks (cont.)

• the sets of two node cliques (a , b ), (a , d ), (b , d )
and (b , c ), (c , d ), (b , d )
• maximum cliques (a , b , d ) and (b , c , d )
respectively.
Figure 10: Example 5

1
φ1 (a , b , d ) · φ2 (b , c , d )
(6)
Z
1
(7)
P (a , b , c , d ) = φ1 (a , b ) · φ2 (a , d ) · φ3 (b , d ) · φ4 (b , c , d )
Z
1
P (a , b , c , d ) = φ1 (a , b ) · φ2 (a , d ) · φ3 (b , d ) · φ4 (a , b , d ) · φ5 (b , c , d )
Z
(8) 25 / 44

P (a , b , c , d ) =
Log-Linear Models



k







1




θq φq (Xq )
p (X ) = exp 






Z
q =1

(9)

where
• (X1 , ..., Xk ) are cliques in the MN
• (φ1 (X1 ), ..., φk (Xk )) are sets of clique potentials asoociated
with k th clique
• (θ1 , ..., θk ) are parameters of the log-linear models as weights

26 / 44
Pairwise Binary Markov Networks
We estimate the Markov network parameters Θp ×p by maximizing
a log-likelihood.

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

p

1

θj ,j xj +
j =1

1 ≤j <k ≤p






θj ,k xj xk 




(10)

where
xj ∈ {0, 1}

Ψ(Θ) =
x ∈0 , 1

(11)






exp 




p

θj ,j xj +
j =1

1 ≤j <k ≤p






θj ,k xj xk 




(12)

• 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)
27 / 44
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) =

(13)

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 )

(14)
(15)

28 / 44
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

(16)

(17)
29 / 44
L1 Regularization
n

log L =
i =1






[xi ,j 




p

k j






θk xi ,k  − log(1 + e




p
θ x
k j k i ,k

)] + λ(θ)

(18)

• Cyclic Coordinate Descent (CCD) algorithm (Friedman, 2010)
• the smallest λmax that shrinks every coefficient to zero
• λmin = λmax = 0.01λmax
• Tuning the LASSO:
• AIC
• BIC
• CV
• GCV
• Ten fold cross validation
• goodness of fit → deviance
30 / 44
Summary of Ravikumar et al. (2010)

• computation of the partition function is not needed
• p different regularization parameters
• leads to asymptotically consistent estimates of MN
parameters as well as to model selection.

⇓

Implementation
Implemented in R with glmnet and with igraph packages.

31 / 44
¨
Hofling and Tibshirani’s method (2009)
Aims to optimize jointly over Θ

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

p

θj ,j xj +
j =1

1 ≤j <k ≤p






θj ,k xj xk 




(19)

The log likelihood for all n observations is given by
n

l (Θ) =
i =1











p

θj ,j xij +
j =1

1 ≤j <k ≤p






θj ,k xij xik  − log(nΨ(Θ))




(20)

Now, adding the L1 penalty to equation (20) yields
n

log f (x1 , ..., xp ) − n||S • Θ||1

(21)

i =1

where S = 2R − diag (R ); R is a p × p lower triangular matrix of
containing the penalty parameter
32 / 44
¨
Hofling and Tibshirani’s method (2009)
Consider a local quadratic Taylor expansion of the log-likelihood
around Θ(m)
fΘ( m) (Θ) = C +
j ≥k

∂l
1 ∂2 l
(m )
(m)
(θjk − θjk ) +
(θ − θjk )2 − n||S • Θ||1
2 jk
∂θjk
2 (∂θjk )
(22)

the solution is soft thresholding because the Hessian is diagonal





sjk
ˆ
˜ ˜

θjk = sign(θjk ) |θjk | − 2


∂ l


(∂θjk )

∂ l
(∂θjk )2
2

(m )
˜
θjk = θjk −

−1











2

jk

∂
˜
|θjk | − sjk / (∂θ

l

2
jk )

∂
˜
if |θjk | > sjk / (∂θ

2

l

2
jk )

+

∂l
∂θjk

∂2
˜
The soft thresholding operator |θjk | − sjk / (∂θ l)2
2

(23)

(24)
returns
+

, and zero otherwise.
33 / 44
Reconstruction of the network

Since weak associations are shrunk toward zero,
• no need to conduct a series of multiple testings

Reconstruction of the LD network

ˆ
• if Θj ,k = 0, then (xj , ⊥ xk )|else
ˆ
• if Θj ,k

0, then (xj , not ⊥ xk )|else

The matrix entries can be considered as edge weights

34 / 44
Data, Subset selection and the reference models

• 599 inbred wheat lines with 1447 Diversity Array Technology
(DArT) binary markers (CIMMYT)
• grain yields
• Bayesian LASSO
• IAMB (Incremental Association Markov Blanket) algorithm for
learning BN
• r 2 metric

35 / 44
10th lambda

15th lambda

8 7 6
9 qqq
10
5
qq
q4
11
q
q
3
12
q
q
13
2
q
q
14
1
q
q
0
15
q
q
29
16
q
q
28
17
q
q
18
27
q
q
19
26
q 20
q
25
q q 22 23 q q
21
q q 24

8 7 6
9 qqq
10
5
qq
q4
11
q
q
3
12
q
q
13
2
q
q
14
1
q
q
0
15
q
q
29
16
q
q
28
17
q
q
18
27
q
q
19
26
q 20
q
25
q q 22 23 q q
21
q q 24

25th lambda

40th lambda

8 7 6
9 qqq
10
5
qq
q4
11
q
q
3
12
q
q
13
2
q
q
14
1
q
q
0
15
q
q
29
16
q
q
28
17
q
q
18
27
q
q
19
26
q 20
q
25
q q 22 23 q q
21
q q 24

8 7 6
9 qqq
10
5
qq
q4
11
q
q
3
12
q
q
13
2
q
q
14
1
q
q
0
15
q
q
29
16
q
q
28
17
q
q
18
27
q
q
19
26
q 20
q
25
q q 22 23 q q
21
q q 24

50th lambda

55th lambda

8 7 6
9 qqq
10
5
qq
q4
11
q
q
3
12
q
q
13
2
q
q
14
1
q
q
0
15
q
q
29
16
q
q
28
17
q
q
18
27
q
q
19
26
q 20
q
25
q q 22 23 q q
21
24
qq

8 7 6
9 qqq
10
5
qq
q4
11
q
q
3
12
q
q
13
2
q
q
14
1
q
q
0
15
q
q
29
16
q
q
28
17
q
q
18
27
q
q
19
26
q 20
q
25
q q 22 23 q q
21
24
qq

Figure 11: LD networks with 6 different λ values

36 / 44
lambda = CV

9

8

7

6

10

5

11

4
3

12
13

2

14

1

15

0

16

29
28

17
18

27
19

26
20

25
21

22

23

24

Figure 12: L1 regularized LD network learned by the method of
Ravikumar et al. with chosen by CV. Nodes denote 30 marker loci.
37 / 44
lambda = sqrt(log(p)/n)

9

8

7

6

10

5

11

4
3

12
13

2

14

1

15

0

16

29
28

17
18

27
19

26
20

25
21

22

23

24

¨
Figure 13: L1 regularized LD network learned by Hofling’s method with λ
chosen as log (p )/n = 0.075, where p = 30, n = 599. Each node
denotes a marker locus.

38 / 44
lambda = sqrt(log(p)/n)

lambda = CV

9

8

7

9
6

10

8

7

6

10
5

11

5

11

4

4
3

12
3

12

13
13

2

2
14

14

1

1
15

0

16

29

0

15

29

16

28

17
28

17

18
18

27

27
19
19

26
20

25
21

22

23

24

Figure 14: Ravikumar et al.

26
20

25
21

22

23

24

¨
Figure 15: Hofling and Tibshirani’s
method
39 / 44
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 16: Ravikumar et al.

1

28

17
18

27
19

26
20

25
21

22

23

24

Figure 17: IAMB
40 / 44
Summary
interactions and associations among the cells and genes form a
complex biological system

⇓
r 2 only capture superficial marginal correlations

⇓

explored the possibility of employing graphical models as an
alternative approach
• r 2 → association(m1, m2)|∅ (emtpyset)
• L1 regularized MN → association(m1, m2) | else

41 / 44
Summary (cont.)

• higher-order associations → Reproducing Kernel Hilbert
Spaces methods
• suitable for binary-valued variables only

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.

42 / 44
GAW17

GAW 17 = Genetic Analysis Workshop 17
• common disease common variant hypothesis vs. common
disease rare variant hypothesis
• exome sequence from the 1000 Genomes project
• 119/166 papers have been accepted for publication
• Bayesian hierarchical mixture model

GAW18
Scheduled for October 14-17, 2012.

43 / 44
Acknowledgments
University of Wisconsin-Madison
• Daniel Gianola
• Guilherme Rosa
• Kent Weigel
• Bruno Valente

University College London
• Marco Scutari

Unversity of Freiburg
• Holger Hofling
¨

• fellow graduate students in
the 4th and 6th floors

44 / 44

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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 Master’s Defense Gota Morota Dec 5, 2011 1 / 44
  • 2. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 2 / 44
  • 3. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 3 / 44
  • 4. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 4 / 44
  • 5. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 5 / 44
  • 6. Linkage Disequilibrium (LD) Definition non-random association of alleles at different loci • also known as gametic phase disequilibrium • first used in 1960 (Lewontin and Kojima) • has been used extensively in many area 1. genome enabled selection 2. genome-wide association study 3. understand past evolutionary and demographic events 6 / 44
  • 7. Systems Genetics Figure 1: A multi-dimensional gene feature/knowledge network Purpose of the thesis • 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 7 / 44
  • 8. Graphical Models • provide a way to visualize the structure of a model • a graph is comprised of 1. nodes (vertices) → random variables 2. edges (links, arcs) → probabilistic relationship 8 / 44
  • 9. Graphical Models • provide a way to visualize the structure of a model • a graph is comprised of 1. nodes (vertices) → random variables 2. edges (links, arcs) → probabilistic relationship Figure 2: A soccer ball 8 / 44
  • 10. Bayesian Networks (BN) • directed acyclic graph (DAG) with graphical structure G = (V , E ) • Application of BN to genomic 1. DNA microarrays (Friedman et al. 2000) 2. protein signaling networks (Sachs et al. 2005) 3. GWAS (Sebastiani et al. 2005) 4. genome-enabled prediction and classification in livestock (Long et al. 2009) Objective I Apply BN to uncover associations among a set of marker loci found to have the strongest effects on milk protein yield in Holstein cattle. 9 / 44
  • 11. Factorization Properties p (a , b , c ) = p (a )p (b |a )p (c |a , b ) (1) Figure 3: The joint distribution of x1 , · · · , x3 10 / 44
  • 12. Factorization Properties (cont.) p P (X1 , ..., Xp ) = P (Xj |Pa (Xj )) (2) j =1 • Pa (Xj ) is a set of parent nodes of Xj • a node is conditionally independent of its non-descendants given its parents One restriction Must be no directed cycles (need to be DAG). 11 / 44
  • 13. Structure Learning • Local score-based algorithms • Scoring metrics • • • • Akaike Information Criterion (AIC) Minimum Description Length (MDL) K2 Bayesian Dirichlet Equivalent (BDe) • Constraint-based algorithms (Causal inference algorithms) • Koller-Sahami (KS) • Grow Shrink (GS) • Incremental Association Markov Blanket (IAMB) 12 / 44
  • 14. IAMB algorithm Incremental Association Markov Blanket (Tsamardinos et al. 2003) 1. Compute Markov Blankets (MB) 2. Compute Graph Structure 3. Orient Edges Figure 4: The Markov Blanket of a node xi 13 / 44
  • 15. Identifying the MB of a node • Growing phase • heuristic funtion: 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 14 / 44
  • 16. Identifying the MB of a node (example) Suppose we have target variable T , and a set of nodes X = (A , B , C , D , E , F ) • Growing phase 1. MI(T , X |CMB = (∅)) → X = (C , E , A , D , B , F ) 2. CI(T , C |CMB =(∅)) → CMB = (C ) 3. MI(T , X |CMB = (C)) → X = (A , E , D , F , B ) 4. CI(T , A |CMB =(C)) → CMB = (C , A ) • Shrinking phase 1. CI(T , C |CMB = (C , A ) − C ) 2. CI(T , A |CMB = (C , A ) − A ) 15 / 44
  • 17. Network Structure Algorithm Suppose Y ∈ MB (T ). Then T and Y are connected if they are conditionaly 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 ), 16 / 44
  • 18. Data, Missing Genotype Imputation, and Subset selection 1. Data • 4,898 progeny tested Holstein bulls (USDA-ARS AIPL) • 37,217 SNP markers (MAF > 0.025) • Predicted Transmitting Ability (PTA) for milk protein yield 2. Missing genotypes imputation • fastPHASE (Scheet and Stephens, 2006) 3. Select 15-30 SNPs • Bayesian LASSO (BLR R pakcage, Perez et al, 2010) ´ 4. SNPs ranking strategies ˆ • |βj | ˆ • |βj |/ Var (βj ) ˆ • 2pj (1 − pj )β2 j 17 / 44
  • 19. Results – Top 15 SNPs from Strategy 1 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 5: r 2 0.2 0.4 0.6 0.8 1 D Figure 6: IAMB 18 / 44
  • 20. Conclusion The result confirms that LD relationships are of a multivariate nature, and that r 2 gives an incomplete description and understanding of LD. • capture association among SNPs as a network • no limitation with respect to the type of loci 19 / 44
  • 21. Possible Improvements • associations among loci are assumed bi-directional • LD is expected to decline rapidly as the physical distance between two loci increases, and that pairs of loci on different chromosomes rarely show high LD • conditional independence property 20 / 44
  • 22. Possible Improvements • associations among loci are assumed bi-directional • LD is expected to decline rapidly as the physical distance between two loci increases, and that pairs of loci on different chromosomes rarely show high LD • conditional independence property ⇓ therefore • undirected networks • sparisty • conditional independence property 20 / 44
  • 23. Undirected Graphical Models Figure 7: An undirected graph • Markov networks (Markov random fields) • G = (V , E ) • express an affinity instead of a causal relationship 21 / 44
  • 24. Pairwise Conditional Independence Property pairwise conditional independence property • an absence of edge between two nodes, xj and xk , implies conditional independence, given all other nodes p (xj , xk |x−j ,−k ) = p (xj |x−j ,−k )p (xj |x−j ,−k ) (3) In Figure (8), (a ⊥ d |b , c) and (b ⊥ d |a , c). Figure 8: Example 1 22 / 44
  • 25. Cliques A clique is a subset of nodes in a graph such that every pair of nodes are connected by edges • (a) {X , Y }, {Y , Z } • (b) {X , Y , W }, {Z } • (c) {X , Y }, {Y , Z }, {Z , W }, {X , W } • (d) {X , Y }, {Y , Z }, {Z , W } Maximum cliques Figure 9: Example 3 a maximum clique is defined as the clique having the largest size 23 / 44
  • 26. The Factorization of Markov Networks the Hammersley-Clifford theorem for any positive distributions, the distribution factorizes according to the Markov network structure defined by cliques. Consider X = xi , · · · , xn , p (X ) = 1 Z φc (Xc ) (4) C ∈G where Z is a normalizing constant defined by φc (Xc ) Z= (5) x C ∈G and φ is called a potential function or a clique potential. 24 / 44
  • 27. The Factorization of Markov Networks (cont.) • the sets of two node cliques (a , b ), (a , d ), (b , d ) and (b , c ), (c , d ), (b , d ) • maximum cliques (a , b , d ) and (b , c , d ) respectively. Figure 10: Example 5 1 φ1 (a , b , d ) · φ2 (b , c , d ) (6) Z 1 (7) P (a , b , c , d ) = φ1 (a , b ) · φ2 (a , d ) · φ3 (b , d ) · φ4 (b , c , d ) Z 1 P (a , b , c , d ) = φ1 (a , b ) · φ2 (a , d ) · φ3 (b , d ) · φ4 (a , b , d ) · φ5 (b , c , d ) Z (8) 25 / 44 P (a , b , c , d ) =
  • 28. Log-Linear Models  k      1     θq φq (Xq ) p (X ) = exp        Z q =1 (9) where • (X1 , ..., Xk ) are cliques in the MN • (φ1 (X1 ), ..., φk (Xk )) are sets of clique potentials asoociated with k th clique • (θ1 , ..., θk ) are parameters of the log-linear models as weights 26 / 44
  • 29. Pairwise Binary Markov Networks We estimate the Markov network parameters Θp ×p by maximizing a log-likelihood.      f (x1 , ..., xp ) = exp     Ψ(Θ) p 1 θj ,j xj + j =1 1 ≤j <k ≤p      θj ,k xj xk     (10) where xj ∈ {0, 1} Ψ(Θ) = x ∈0 , 1 (11)      exp     p θj ,j xj + j =1 1 ≤j <k ≤p      θj ,k xj xk     (12) • 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) 27 / 44
  • 30. 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) = (13) 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 ) (14) (15) 28 / 44
  • 31. 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 (16) (17) 29 / 44
  • 32. L1 Regularization n log L = i =1      [xi ,j     p k j      θk xi ,k  − log(1 + e    p θ x k j k i ,k )] + λ(θ) (18) • Cyclic Coordinate Descent (CCD) algorithm (Friedman, 2010) • the smallest λmax that shrinks every coefficient to zero • λmin = λmax = 0.01λmax • Tuning the LASSO: • AIC • BIC • CV • GCV • Ten fold cross validation • goodness of fit → deviance 30 / 44
  • 33. Summary of Ravikumar et al. (2010) • computation of the partition function is not needed • p different regularization parameters • leads to asymptotically consistent estimates of MN parameters as well as to model selection. ⇓ Implementation Implemented in R with glmnet and with igraph packages. 31 / 44
  • 34. ¨ Hofling and Tibshirani’s method (2009) Aims to optimize jointly over Θ      f (x1 , ..., xp ) = exp     Ψ(Θ) 1 p θj ,j xj + j =1 1 ≤j <k ≤p      θj ,k xj xk     (19) The log likelihood for all n observations is given by n l (Θ) = i =1          p θj ,j xij + j =1 1 ≤j <k ≤p      θj ,k xij xik  − log(nΨ(Θ))    (20) Now, adding the L1 penalty to equation (20) yields n log f (x1 , ..., xp ) − n||S • Θ||1 (21) i =1 where S = 2R − diag (R ); R is a p × p lower triangular matrix of containing the penalty parameter 32 / 44
  • 35. ¨ Hofling and Tibshirani’s method (2009) Consider a local quadratic Taylor expansion of the log-likelihood around Θ(m) fΘ( m) (Θ) = C + j ≥k ∂l 1 ∂2 l (m ) (m) (θjk − θjk ) + (θ − θjk )2 − n||S • Θ||1 2 jk ∂θjk 2 (∂θjk ) (22) the solution is soft thresholding because the Hessian is diagonal     sjk ˆ ˜ ˜  θjk = sign(θjk ) |θjk | − 2   ∂ l  (∂θjk ) ∂ l (∂θjk )2 2 (m ) ˜ θjk = θjk − −1           2 jk ∂ ˜ |θjk | − sjk / (∂θ l 2 jk ) ∂ ˜ if |θjk | > sjk / (∂θ 2 l 2 jk ) + ∂l ∂θjk ∂2 ˜ The soft thresholding operator |θjk | − sjk / (∂θ l)2 2 (23) (24) returns + , and zero otherwise. 33 / 44
  • 36. Reconstruction of the network Since weak associations are shrunk toward zero, • no need to conduct a series of multiple testings Reconstruction of the LD network ˆ • if Θj ,k = 0, then (xj , ⊥ xk )|else ˆ • if Θj ,k 0, then (xj , not ⊥ xk )|else The matrix entries can be considered as edge weights 34 / 44
  • 37. Data, Subset selection and the reference models • 599 inbred wheat lines with 1447 Diversity Array Technology (DArT) binary markers (CIMMYT) • grain yields • Bayesian LASSO • IAMB (Incremental Association Markov Blanket) algorithm for learning BN • r 2 metric 35 / 44
  • 38. 10th lambda 15th lambda 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 25th lambda 40th lambda 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 50th lambda 55th lambda 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 24 qq 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 24 qq Figure 11: LD networks with 6 different λ values 36 / 44
  • 39. lambda = CV 9 8 7 6 10 5 11 4 3 12 13 2 14 1 15 0 16 29 28 17 18 27 19 26 20 25 21 22 23 24 Figure 12: L1 regularized LD network learned by the method of Ravikumar et al. with chosen by CV. Nodes denote 30 marker loci. 37 / 44
  • 40. lambda = sqrt(log(p)/n) 9 8 7 6 10 5 11 4 3 12 13 2 14 1 15 0 16 29 28 17 18 27 19 26 20 25 21 22 23 24 ¨ Figure 13: L1 regularized LD network learned by Hofling’s method with λ chosen as log (p )/n = 0.075, where p = 30, n = 599. Each node denotes a marker locus. 38 / 44
  • 41. lambda = sqrt(log(p)/n) lambda = CV 9 8 7 9 6 10 8 7 6 10 5 11 5 11 4 4 3 12 3 12 13 13 2 2 14 14 1 1 15 0 16 29 0 15 29 16 28 17 28 17 18 18 27 27 19 19 26 20 25 21 22 23 24 Figure 14: Ravikumar et al. 26 20 25 21 22 23 24 ¨ Figure 15: Hofling and Tibshirani’s method 39 / 44
  • 42. 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 16: Ravikumar et al. 1 28 17 18 27 19 26 20 25 21 22 23 24 Figure 17: IAMB 40 / 44
  • 43. Summary interactions and associations among the cells and genes form a complex biological system ⇓ r 2 only capture superficial marginal correlations ⇓ explored the possibility of employing graphical models as an alternative approach • r 2 → association(m1, m2)|∅ (emtpyset) • L1 regularized MN → association(m1, m2) | else 41 / 44
  • 44. Summary (cont.) • higher-order associations → Reproducing Kernel Hilbert Spaces methods • suitable for binary-valued variables only 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. 42 / 44
  • 45. GAW17 GAW 17 = Genetic Analysis Workshop 17 • common disease common variant hypothesis vs. common disease rare variant hypothesis • exome sequence from the 1000 Genomes project • 119/166 papers have been accepted for publication • Bayesian hierarchical mixture model GAW18 Scheduled for October 14-17, 2012. 43 / 44
  • 46. Acknowledgments University of Wisconsin-Madison • Daniel Gianola • Guilherme Rosa • Kent Weigel • Bruno Valente University College London • Marco Scutari Unversity of Freiburg • Holger Hofling ¨ • fellow graduate students in the 4th and 6th floors 44 / 44