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

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.

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

4

Example of four genes.Different dose-response relationships.

Primary Interest: detection of genes with monotone dose-response relationship

4 dose levels.16988 genes.

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

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).

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

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

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.

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.

All possible monotone dose-response models

11

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.

All possible monotone dose-response models

12

32105 : g

0,0,0 321

32100 : g

0,0,0 321

•4 dose levels:

Bayesian variable selection: model formulation for order restricted model

13

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:

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

The posterior probability of the null model

15

),|(),|)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

Example: gene 3413

16

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

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.

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

Multiplicity adjustment

19

%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

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!

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