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Analysis of dose-response microarray data using Bayesian Variable Selection (BVS) methods:
Modeling and multiplicity adjustments
7th meeting of the Eastern Mediterranean Region of the International Biometric Society (EMR-IBS)
Tel – Aviv22/04 – 25/04,2013
Ziv Shkedy, Martin Otava, Adetayo Kasim and Dan Lin
Center for Statistics (CenStat), Hasselt University, Belgium and Durham University, UK
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Research team
• Dan Lin.• Ziv Shkedy.• Martin Otava
• Luc Bijnens.• Willem Talloen.• Hinrich Gohlmann.• Dhammika Amaratunga
Hasselt University, Belgium Johnson & Johnson Pharmaceutical
Durham University, UK• Adetyo Kasim.
Imperial College, UK
• Bernet Kato.
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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
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Dose-response microarray experiment
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Example of four genes.Different dose-response relationships.
Primary Interest: detection of genes with monotone dose-response relationship
4 dose levels.16988 genes.
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Estimation and inference under order restrictions
5
K
K
H
H
,...,:
,...,:
101
100
•Primary interest: discovery of genes with monotone relationship with respect to dose.• Order restricted inference.•Simple order (=monotone) alternatives.
mg HHHH 000201 ,...,,...,,
16988 null hypotheses to test
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Model formulation (1)
2,~ iij NY
K ,...,10
•Gene specific model •One-way ANOVA with order restricted parameters.•Simple order (monotone profiles).
•The order constraints are build into the specification of the prior distributions (Gelfand, Smith and Lee, 1992).
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Model formulation (1)
7
2,~ iij NY
),(,~ 112
iii IN
otherwise
NP iii
0
,,| 11
2
•Specification of the prior :
• unconstrained prior.
•Likelihood:
2,~ Ni
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Model formulation (2)
2,~ iij NY
01
0
i
i
),0(,~ 2 IN
32103
2102
101
0
d
d
d
cdose mean
20 ,~ N
•Re formulation of the mean structure:
•For a dose-response experiment with 4 dose levels (control + 3 doses):
Ki ,...,0 10
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Example of one gene (13386)
32105 : g
32107 : g
1.0 1.5 2.0 2.5 3.0 3.5 4.0
8.2
8.4
8.6
8.8
9.0
dose
gene
exp
ress
ion
32107 : g
32105 : g
•We fitted two monotone models:
Equality constraints are replaced with a single parameter.
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Inference
10
K
K
H
H
,...,:
,...,:
101
100
i
i1
0
10 ii
3203
202
01
0
d
d
d
cdose mean
01
1.0 1.5 2.0 2.5 3.0 3.5 4.0
8.2
8.4
8.6
8.8
9.0
dose
ge
ne
exp
ressio
n 32105 : g
32105 : g•Simple order alternative.
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All possible monotone dose-response models
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32107
32106
32105
32104
32103
32102
32101
32100
::
:
:
:
:
:
:
gg
g
g
g
g
g
g
32101 : H
•We decompose the simple order alternative to all sub alternative.
•The null model
•Simple order alternative.
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All possible monotone dose-response models
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32105 : g
0,0,0 321
32100 : g
0,0,0 321
•4 dose levels:
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Bayesian variable selection: model formulation for order restricted model
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i
iiz
0
1
•The mean structure:
included in the model
not Included in the model
i
i1
0
•Bayesian Variable Selection: a procedure of deciding which of the model parameters is equal to zero.
•Define an indicator variable:
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14
K
iiii z
10
KK
r Sg ,...,,: 101
),0(,~ 2 INi )(~ ii Bz )1,0(~Ui
Bayesian variable selection: model formulation for order restricted model
20 ,~ N
•The mean structure for a candidate model:
Order restrictions Variable selection
ESTIMATION INFERENCE and MODEL SELECTION
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The posterior probability of the null model
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),|(),|)0,0,0(( 0 RdatagpRdatazp
32101
0
K
iiii z
),|)0,0,0(( 321 Rdatazzzzp
•The posterior probability that the triplet equal to zero: )0,0,0(z
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Example: gene 3413
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514.0),|( 0 Rdatagp
1.0 1.5 2.0 2.5 3.0 3.5 4.0
4.8
5.0
5.2
5.4
5.6
5.8
dose
gene
exp
ress
ion
g_7BVSnull
g0 g3 g2 g6 g1 g4 g5 g7
0.0
0.1
0.2
0.3
0.4
0.5
•The highest posterior probability is obtained for the null model (0.514).•Shrinkage through the overall mean.
BVS
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Example: gene 13386
4186.0),|( 5 Rdatagp
001.0),|( 0 Rdatagp
1.0 1.5 2.0 2.5 3.0 3.5 4.0
8.2
8.4
8.6
8.8
9.0
dose
ge
ne
exp
ress
ion
g_7g_5BVS
g0 g3 g2 g6 g1 g4 g5 g7
0.0
0.1
0.2
0.3
0.4
4059.0),|( 1 Rdatagp
3210
3210
3210
•The highest posterior probability is obtained for model g5. •Data do not support the null model.
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Multiplicity adjustment
),|(0
),|(1
0
0
Rdatagp
RdatagpI
g
gg
)(N
gene g is included in the discovery list
gene g is not included in the discovery list
The number of genes in the discovery list.
•Primary interest: discovery of subset of genes with monotone relationship with respect to dose.
)(
,|
)(
)()( 1
0
N
IRdatagP
N
cFDcFDR
m
ggg
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Multiplicity adjustment
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%5
3295
,,|
)102.0(0
Rdatazgp
cFDRg
The expected error rate for the list with all genes for which the posterior probability of the null model < 0.102 are included.
τ
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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.
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Thank you!
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