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