Analysis of dose-response microarray data using
Bayesian Variable Selection (BVS) methods:
Modeling and multiplicity adjustments
Ziv Shkedy, Martin Otava, Adetayo Kasim and Dan Lin
Center for Statistics (CenStat), Hasselt University, Belgium and Durham University, UK
7th meeting of the Eastern Mediterranean Region
of the International Biometric Society (EMR-IBS)
Tel – Aviv
22/04 – 25/04,2013
Research team
Hasselt University, Belgium
• Dan Lin.
• Ziv Shkedy.
• Martin Otava
Durham University, UK
•
Adetyo Kasim.
Imperial College, UK
•
Bernet Kato.
Johnson & Johnson
Pharmaceutical
•
•
•
•
Luc Bijnens.
Willem Talloen.
Hinrich Gohlmann.
Dhammika Amaratunga
Overview
• Introduction to dose-response modeling in microarray
experiments.
• Primary interest: selection of a subset of genes with significant
monotone dose-response relationship.
• Focus:
1. Estimation and inference under order
restrictions.
2. Multiplicity adjustment.
• Methodology: Bayesian Variable Selection models.
3
Dose-response microarray experiment
Example of four genes.
Different dose-response relationships.
4 dose levels.
16988 genes.
Primary Interest:
detection of genes
with monotone doseresponse relationship
4
Estimation and inference under order restrictions
•Primary interest: discovery of genes with monotone relationship
with respect to dose.
• Order restricted inference.
•Simple order (=monotone) alternatives.
H 0 : 0  1 ,...,  K
H1 : 0  1 ,...,  K
H01 , H02 ,...,H0 g ,...,H0m
16988 null hypotheses to test
5
Model formulation (1)
•Gene specific model
•One-way ANOVA with order restricted parameters.
•Simple order (monotone profiles).

Yij ~ N i ,
2

0  1 ,..., K
•The order constraints are build into the specification of the prior
distributions (Gelfand, Smith and Lee, 1992).
Model formulation (1)
•Likelihood:

Yij ~ N i , 2

•Specification of the prior :
i ~ N  , 2 I (i 1, i1 )
• i ~ N  , 2  unconstrained prior.

 N   ,  2
P |  ,    
0


i 1  i  i 1
otherwise
7
Model formulation (2)
•Re formulation of the mean
structure:

Yij ~ N i ,
2

i
i   0    
 1
dose
  0
  ~ N  ,  I (0, )
2
0 ~ N  , 2 
•For a dose-response
experiment with 4 dose levels
(control + 3 doses):
c
d1
d2
d3
mean
0
 0  1
 0  1   2
 0  1   2   3
 i  0  0  1 ,..., K
•We fitted two monotone
models:
g7 : 0  1  2  3
8.6
8.8
g7 : 0  1  2  3
8.4
g5 : 0  1  2  3
g5 : 0  1  2  3
8.2
gene expression
9.0
Example of one gene (13386)
1.0
1.5
2.0
2.5
dose
3.0
3.5
4.0
Equality constraints are replaced with a
single parameter.
Inference
g5 : 0  1  2  3
•Simple order alternative.
dose
H 0 : 0  1 ,...,  K
c
d1
d2
d3
H1 : 0  1 ,...,  K
i
i   0    
mean
0
0
0   2
0   2   3
1  0
 1
8.4
8.6
8.8
9.0
g5 : 0  1  2  3
8.2
gene expression
   0  i  i1
1.0
1.5
2.0
2.5
dose
3.0
3.5
4.0
10
All possible monotone dose-response models
•Simple order alternative.
g 0 :  0  1   2  3
g1 :  0  1   2  3
g 2 :  0  1   2  3
g 3 :  0  1   2  3
g 4 :  0  1   2  3
g 5 :  0  1   2  3
g 6 :  0  1   2  3
g 7 :  0  1   2  3
•The null model
H1 : 0  1  2  3
•We decompose the simple order
alternative to all sub alternative.
11
All possible monotone dose-response models
•4 dose levels:
g5 : 0  1  2  3
g0 : 0  1  2  3
1  0,  2  0,  3  0
1  0,  2  0,  3  0
12
Bayesian variable selection: model formulation for
order restricted model
•The mean structure:
i
i   0    
 1
•Bayesian Variable Selection: a procedure of deciding which of
the model parameters is equal to zero.
•Define an indicator variable:
1  i
zi  
0  i
included in the model
not Included in the model
13
Bayesian variable selection: model formulation for
order restricted model
•The mean structure for a candidate model:

gr   S
K 1

:   0 , 1,...,K 
K
 i   0   zi   i
i 1
0 ~ N  , 2 
 i ~ N  , 2 I (0, )
zi ~ B( i )
 i ~ U (0,1)
Order restrictions
Variable selection
ESTIMATION
INFERENCE and MODEL
SELECTION
14
The posterior probability of the null model
•The posterior probability that the triplet equal to zero: z  (0,0,0)
p( z  ( z1  0, z2  0, z3  0) | data, R)
K
i  0   zi   i  0  1  2  3
i 1
p( z  (0,0,0) | data, R)  p( g0 | data, R)
15
Example: gene 3413
0.5
5.8
•The highest posterior probability is obtained for the null model
(0.514).
•Shrinkage through the overall mean.
0.4
p( g0 | data, R)  0.514
0.3
5.4
0.2
5.2
0.1
5.0
BVS
0.0
4.8
gene expression
5.6
g_7
BVS
null
1.0
1.5
2.0
2.5
dose
3.0
3.5
4.0
g0
g3
g2
g6
g1
g4
g5
g7
16
Example: gene 13386
0  1  2  3 p( g0 | data, R)  0.001
0  1  2  3
p( g1 | data, R)  0.4059
0  1  2  3
p( g5 | data, R)  0.4186
0.4
•The highest posterior
probability is obtained for
model g5.
•Data do not support the
null model.
0.3
0.2
8.8
8.6
0.1
8.4
0.0
8.2
gene expression
9.0
g_7
g_5
BVS
1.0
1.5
2.0
2.5
dose
3.0
3.5
4.0
g0
g3
g2
g6
g1
g4
g5
g7
Multiplicity adjustment
•Primary interest: discovery of subset of genes with monotone
relationship with respect to dose.
N ( ) The number of genes in the discovery list.
1
Ig  
0
pg ( g 0 | data, R)  
gene g is included in the discovery
list
pg ( g 0 | data, R)  
gene g is not included in the
discovery list
m
cFD( )
cFDR( ) 

N ( )
 P g
g 1
g
0
| data, R   I g
N ( )
Multiplicity adjustment
τ
 p g
g
cFDR(0.102) 
0
| z , data, R 
3295
 5%
The expected error rate for the
list with all genes for which the
posterior probability of the null
model < 0.102 are included.
19
Discussion & To Do list
• BVS methods: estimation and inference.
• Multiplicity adjustment is based on the posterior probability
of the null model.
• Connection between BVS and MCT.
• Connection between BVS and Bayesian model averaging.
• BVS for order restricted but non monotone alternatives
(umbrella alternatives/partial order alternatives).
• Posterior probabilities for the number of levels and the level
probabilities for isotonic regressions.
Thank you!
21