Statistical Methods for Microarray Experiments: …...Statistical Methods for Microarray Experiments: Analysis of Dose-response Studies and Software Development in R Setia Pramana

Statistical Methods for Microarray Experiments:Analysis of Dose-response Studies and

Software Development in R

Setia Pramana

Promotors:Prof. dr. Ziv Shkedy

Prof. dr. Dan Lin

Setia Pramana () Dose-response in Microarray Experiments Diepenbeek, 09/09/2011 1 / 50

Outline

Introduction to dose-response microarray analysis

Content of the dissertation.

Focus: dose-response modeling and model averaging of targetdoses in microarray experiments

Discussion, conclusion, and future research


1. Introduction


Dose-response Study

The fundamental study in drug development.

Main goal: obtain safe and efficacious dose.

Investigate the dependence of the phenotypic response on doses.

Interest to discover monotone dose-response relationship

Monotonicity − > order restricted profile.


Dose-response Study

The fundamental study in drug development.

Main goal: obtain safe and efficacious dose.

Investigate the dependence of the phenotypic response on doses.

Interest to discover monotone dose-response relationship

Monotonicity − > order restricted profile.

Dose-response Microarray Studies:

Monitoring of gene expression with respect to increasing doses.

To identify a subset of genes with a monotonic trend.


Dose-response Microarray studies

The StudyCompound

Dose 0/Control Dose 1 Dose 2 … Dose K

Data Structure

mKnmKmnmmnm

KnKnn

KnKnn

K

K

K

YYYYYY

YYYYYY

YYYYYY

Y

...........

.

.

.

.....

.....

.....

.

.

.

.

.

.

...........

...........

1111001

2122111220012

1111111110011

10

10

10Gene 1

Gene 2

.

.

.

Gene m

d0 d1 ….. dK

Dose levels


Dose-response Microarray Studies

Case study: a study focuses on antipsychotic compounds.

6 dose levels with 4-5 samples at each dose level.

Each array consists of 11,565 genes.

One gene profile:

Doses

Gen

e E

xpre

ssio

n

0 0.16 0.63 2.5 10 40

67

89

+

+

+

+

+

+

* *

*

*

*

*

Gene:


2. Contents of the Dissertation


Contents of the Dissertation








3. Dose-response Modeling and ModelAveraging in Microarray Experiments


The Framework

Dose-response relationship for each gene:

Yij = f (di , θ) + εij , i = 0, 2, . . . , K , j = 1, 2, . . . , ni ,

A secondary analysis to investigate dose-response relationship.

f (di , θ): linear or non-linear models (e.g., Emax , logistic).

More than one model fit the data.

Focus only on monotone relationship.


The Framework

Feature Selection Significant test: Genes with a

monotonic trend

Parametric Modeling Target dose estimation

Model Uncertainty

with Model Averaging


Feature Selection: Case Study

11,565 genes.

Test for monotone trend : Likelihood Ratio Test (Bartholomew,1959).

BH-FDR for multiplicity adjustment (Benjamini and Hochberg,1995.

Number of significant genes (FDR=0.05)

JNJa JNJb JNJc CompB CompA CompC211 251 164 332 72 242


Next Step

Focus on the CompA: We have 72 significant genes.


Next Step


Rank the genes.


Next Step


Rank the genes.

Compare the compounds.


Next Step


Rank the genes.

Compare the compounds.

Based on what criterion?


Rank Genes Based on the Fold-change? or t-typeStatistics?

Fold-changes: after log2 transformed, µ̂∗

K − µ̂∗

0




K − µ̂∗

0

Modified t-type statistics:µ̂∗

K −µ̂∗

0se − > Marcus (1976)




K − µ̂∗

0

Modified t-type statistics:µ̂∗

K −µ̂∗

0se − > Marcus (1976)

Two genes with similar fold-changes and t-type tests:

0 1 2 3 4

01

23

45

Dose

Gen

e ex

pres

sion



More suitable parameter: the ED50

0 1 2 3 4

01

23

45

Dose

Gen

e ex

pres

sion

genes 1, ED50=1mg/mlgenes 2, ED50=3mg/ml

Two genes with similar fold-changes, t-type tests, and differentED50 estimates.


The Target Dose: ED50

The ED50: dose which induces a response halfway between thebaseline and maximum.


The Target Dose: ED50

The ED50: dose which induces a response halfway between thebaseline and maximum.

0 1 2 3 4

01

23

45

Dose

Gen

e ex

pres

sion

compound 1, ED50=1mg/mlcompound 2, ED50=2mg/ml

ED50 reflects the potency of the tested drug or compound.

Rank genes or compare compounds based on the ED50.


ED50 Within Dose Range

To avoid problems, the ED50 is restricted to lie within the interval(d0, dK ].

ED50, 1 ED50, 2


ED50 Within Dose Range

To avoid problems, the ED50 is restricted to lie within the interval(d0, dK ].

ED50, 1 ED50, 2

For gene expression data, the ED50 within dose-range = dose ofhalf way to the fold-change.


Dose-response Model

Dose

Res

pons

e

8

9

10

11

12

13

emax

0 10 20 30 40

exponential linear

0 10 20 30 40

linlog logistic

0 10 20 30 40

8

9

10

11

12

13

sigEmax

Responses Model Predictions 0.95 Pointwise CI Estim. Dose


Dose-response Modeling: Pros and Cons

Assume a functional relationship between the response and thedose according to a pre-specified parametric model.

The dose is taken as a quantitative factor.

Provides flexibility in investigating the effect of doses not used inthe actual study.


Dose-response Modeling: Pros and Cons

Assume a functional relationship between the response and thedose according to a pre-specified parametric model.

The dose is taken as a quantitative factor.

Provides flexibility in investigating the effect of doses not used inthe actual study.

Its result validity depends on the correct choice of thedose-response model, which is a priori unknown.

Multiple models describe the data equivalently, but the target doseestimates are different.


The Framework: Model Averaging


monotonic trend

Parametric Modeling Target dose estimation

Model Uncertainty

with Model Averaging


Model Averaging

Account for model uncertainty.

All fits are taken into consideration.

Combines results from different models.

Poor fits receive small weights.


Model Averaging

Fit R candidate models:

f1, f2, f3, . . . , fR


Model Averaging


f1, f2, f3, . . . , fR

Calculate the Information Criterion (IC) of each model.

IC1, IC2, IC3, . . . , ICR


Model Averaging


f1, f2, f3, . . . , fR


IC1, IC2, IC3, . . . , ICR

IC: measure relative goodness of fit and account for modelcomplexity e.g, AIC, BIC.AIC = −2logL(θ|data) + 2M.


Model Averaging


f1, f2, f3, . . . , fR


IC1, IC2, IC3, . . . , ICR

IC: measure relative goodness of fit and account for modelcomplexity e.g, AIC, BIC.AIC = −2logL(θ|data) + 2M.

Model selection: Choose the model with minimum AIC, ignoringthe other results.


Model Averaging

Model Averaging: Each model is given a weight:

ω1, ω2, ω3, . . . , ωR


Model Averaging

Model Averaging: Each model is given a weight:

ω1, ω2, ω3, . . . , ωR

ωr =exp(−∆ICr/2)∑

exp(−∆ICr/2)

∆ICr = ICr − ICmin

0 ≤ ωr ≤ 1,

R∑

r

ωr = 1


Model Averaging

θ is the parameter of interest: ED50

θ̂MA =

R∑

r

ωr × θ̂r ,

θ̂r is the estimate of θ from model r

v̂ar(θ̂MA) is obtained using bootstrap.


Model Averaging

θ is the parameter of interest: ED50

θ̂MA =

R∑

r

ωr × θ̂r ,

θ̂r is the estimate of θ from model r

v̂ar(θ̂MA) is obtained using bootstrap.

Given the fits of R models, we can estimate the MAdose-response curve as:

f̂MA(d) =

R∑

r

ωr × f̂ (θ, d)r .


Dose-response Models for One Gene (FOS)

Data and fitted value for Gene FOS

0 10 20 30 40

910

1112

Dose

Gen

e E

xpre

ssio

n

LinearLinlogExponentialEmaxsigEmaxLogisticModel Average

0 10 20 30 409

1011

12Dose

Gen

e E

xpre

ssio

n

EmaxModel Average


Model Averaging for One Gene (FOS)

ED50, AIC, and AIC weightsfor the gene FOS

Model ED50 AIC WeightsLinear 20.000 86.37 <0.001Linear log-dose 5.405 69.51 0.029Exponential 22.502 89.78 <0.0014P Logistic 2.042 67.53 0.077Hyperbolic Emax 1.241 63.30 0.640Sigmoidal Emax 1.241 65.15 0.254Model Average ED50 1.423

MA ED50 and its confidenceinterval for each model

ED50M

odel

s

Lin

LinL

ogE

xpon

entia

lH

Em

axS

Em

axLo

gis

MA

0 5 10 15 20


What Is Next?


monotonic trend

Parametric Modeling Target dose estimation (ED50)

Model Averaging

MA ED50

What is the next

step?


What Is Next?

Model Averaging

MA ED50

Gene Ranking in a

Single Compound

Comparison between

Compounds


Gene Ranking in a Single Compound

0 10 20 30 40

010

2030

index

ED

50

42 genes have a significantupward trend (FDR=0.05).

Genes with a smaller ED50

react faster to the compound(genes need smaller dose tobe expressed).

Genes with high ED50 are lessinteresting.



Genes profile with the smallest and highest MA ED50.

0 10 20 30 40

7.5

8.0

8.5

9.0

9.5

MA ED50 = 0.04

doses

gene

exp

ress

ion

0 10 20 30 40

2.6

2.8

3.0

3.2

MA ED50 = 30.78520

doses

gene

exp

ress

ion



3D scatter plot: Fold Change, ED50, µ̂∗

0

0 5 10 15 20 25 30 35

0.0

0.5

1.0

1.5

2.0

2.5

3.0

2

4

6

8

10

12

14

ED50

Mu0

Fol

d ch

ange



3D scatter plot:Fold Change, ED50, µ̂∗

0

0 5 10 15 20 25 30 35

0.0

0.5

1.0

1.5

2.0

2.5

3.0

2

4

6

8

10

12

14

ED50

Mu0

Fol

d ch

ange

0 10 20 30 40

910

1112

MA ED50=1.42, FC =2.56, Mu0= 9.20

doses

gene

exp

ress

ion

0 10 20 30 40

45

67

8

MA ED50=4.82, FC =2.90, Mu0= 4.32

doses

gene

exp

ress

ion



Genes with similar MA ED50

different E0

0 10 20 30 40

45

67

8

doses

gene

exp

ress

ion

Genes with similar Fold change,different E0 and MA ED50

0 10 20 30 40

46

810

12

doses

gene

exp

ress

ion


Compounds Comparison

Model Averaging

MA ED50

Gene Ranking in a

Single Compound

Comparison between

Compounds


Compounds Comparison Based on the MA ED50

Plot of MA ED50

compound JnJa vs. CompA

5 10 15 20

510

1520

CompA

JnJa



Plot of MA ED50


5 10 15 20

510

1520

CompA

JnJa

0 10 20 30 40

3.5

4.0

4.5

5.0

CompA, MA ED50 = 16.89

doses

gene

exp

ress

ion

0 10 20 30 40

3.4

3.6

3.8

4.0

4.2

JnJa, MA ED50 = 3.41

doses

gene

exp

ress

ion



Plot of MA ED50


5 10 15 20

510

1520

CompA

JnJa

0 10 20 30 40

7.0

7.5

8.0

8.5

9.0

CompA, MA ED50 = 2.47

doses

gene

exp

ress

ion

0 10 20 30 40

8.0

8.5

9.0

JnJa, MA ED50 = 12.46

doses

gene

exp

ress

ion


Testing for Compound Effect

Investigate the activity of different compounds on the expressionlevel of a single gene.



Investigate the activity of different compounds on the expressionlevel of a single gene.

Based on a specific dose-response model (the Emax model).

Emax model: f (d) = E0 + d×(EMax )d+ED50

E0 + Emax

E0

Emax

ED50



The parameters are formulated as a linear function of thecompounds:

E0k

ED50k

Emaxk

=

θ1 + γk

θ2 + δk

θ3 + ηk

, k = 2, ..., K ,

For the ED50 parameter, we test the null hypothesis of nocompound effect:

H0 : δk = 0,H1 : δk 6= 0.



Model with common E0 for allcompounds.

doses

Gen

e ex

pres

sion

0 0.16 0.63 2.5 10 40

89

1011

1213

CompAJnJaCompBCompCJnJbJnJc

doses

Gen

e ex

pres

sion

0 0.16 0.63 2.5 10 40

89

1011

1213




Model with common E0 for allcompounds.

doses

Gen

e ex

pres

sion

0 0.16 0.63 2.5 10 40

89

1011

1213


doses

Gen

e ex

pres

sion

0 0.16 0.63 2.5 10 40

89

1011

1213


Model with common Emax and E0

for all compounds.

doses

Gen

e ex

pres

sion

0 0.16 0.63 2.5 10 40

89

1011

1213


doses

Gen

e ex

pres

sion

0 0.16 0.63 2.5 10 40

89

1011

1213



Application to Another Target Dose: MED

The previous approach can also be implemented to other targetdoses such as: Minimum Effective Dose (MED).

MED is defined as ”the lowest dose producing a clinicallyimportant response that can be declared statistically, significantlydifferent from the placebo response”

(Pinheiro, Bretz, and Branson, 2006)


The Minimum Effective Dose

MED is formulated as:

MED = argmind∈(d0,dk ](f (d , θ) > f (d0, θ) + ∆). (1)

The clinically relevant difference (∆): is the smallest relevantdifference, by which we expect a dose to be better than theplacebo.

∆ = τ × µ̂∗

0, where τ is the percentage change from the isotonicmean response in the placebo (µ̂∗

0)

(Bretz et.al, 2010)


The Minimum Effective Dose

MED graphical representation


4. Discussion, Conclusion, and FutureResearch


Discussion

Parametric modeling is aimed to investigate in more details theresponse of genes to the dose.


Discussion


There is no single model which fits all genes.


Discussion



Three steps approach:


Discussion



Three steps approach:Select the genes with a monotone trend.


Discussion




Fit the selected genes with the candidate models to get a targetdose.


Discussion





Average the target dose from the candidate models.


Discussion





Average the target dose from the candidate models.

These procedures combine the advantage of testing for amonotone trend, model-based, and model averaging.


Conclusion

The three steps dose-response modeling approach:Overcome the weakness of model selection approach.

Provide a realistic target dose.

Can be implemented in other target doses.

The model-average ED50 can be used to rank the genes andcompare the tested compounds.


Future Research

Investigate two possible approaches: Average the ED50 orAverage Model?


Future Research

Investigate two possible approaches: Average the ED50 orAverage Model?

Probe level dose-response modeling.


Thank you for your attention...

” All things are poison and nothing is without poison;only the dose makes that a thing is no poison.”

(Paracelsus)


Documents

Statistical Methods for Microarray Experiments: …...Statistical Methods for Microarray Experiments: Analysis of Dose-response Studies and Software Development in R Setia Pramana