dose-response modeling of gene expression data in pre-clinical microarray experiments
TRANSCRIPT
Introduction Testing and Modeling Testing for Trend Model Based Model Averaging Application Discussion References
Dose-Response Modeling of Gene ExpressionData in pre-clinical Microarray Experiments
Setia Pramana
Workshop on Multiplicity and Microarray Analysis
Diepenbeek, 19 February 2010
Introduction Testing and Modeling Testing for Trend Model Based Model Averaging Application Discussion References
Outline
IntroductionDose-response StudiesDose-response in Microarray Experiments
Modeling Dose-response study in Microarray settingDose-response ModelingTesting for TrendModel BasedModel Averaging
ApplicationAntipsychotic StudyResults
Discussion
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Dose-response studies
The fundamental study in drug developments.
Too high dose can result in an unacceptable toxicity profile.
Too low dose decreases the chance of it showingeffectiveness.Main aim: find dose or range of dose that is:
efficacious (for improving or curing the intended diseasecondition)safe (with acceptable risk of adverse effects)
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Dose-response studies
Dose-response study investigates the dependence of theresponse on doses.
Is there any dose-response relationship?
What doses exhibit a response different from the control?
What is the shape of the relationship?
Estimates the target dose: minimum effective dose (MED),maximally tolerated dose (MTD) or half maximal effectiveconcentration/dose (EC50), Ruberg,1995.
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Dose-response in Microarray Experiments
This research focuses on a dose-response study within amicroarray setting.Monitoring of gene expression with respect to increasingdose of a compound.
dose
gene
exp
ress
ion
0 0.01 0.04 0.16 0.63 2.5
67
89
10
No prior info about the dose-response shape.Genes have different shapes.Many ”noisy” genes hence need for initial filtering.
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Steps
Remove genes that do not show a monotone trend, usingthe monotonic trend test statistics, i.e., Likelihood RatioTest (E2
01).
Fit several dose-response models in genes with amonotone trend.
Estimate the target dose, e.g., EC50.
Apply model averaging.
Introduction Testing and Modeling Testing for Trend Model Based Model Averaging Application Discussion References
Testing for Trend
Dose-response relationship
01
23
45
dose
gene
exp
ress
ion
+
++
+
*
**
*
Gene a: increasing monotonic trend
0 1 10 100
−1
01
23
4
dose
gene
exp
ress
ion
+
+
+
+
*
* *
*
Gene b: decreasing monotonic trend
0 1 10 100
01
23
dose
gene
exp
ress
ion
++
+
+
Gene c: non−monotonic trend
0 1 10 100
−0.
50.
51.
5
dose
gene
exp
ress
ion
++
+
+
Gene d: no dose−response relationship
0 1 10 100
Introduction Testing and Modeling Testing for Trend Model Based Model Averaging Application Discussion References
Testing for Monotonic Trend
For gene i (i = 1, · · · , m) with K doses (j = 0, · · · , K )
H0 : µ(d0) = µ(d1) = · · · = µ(dK )
HUp1 : µ(d0) ≤ µ(d1) ≤ · · · ≤ µ(dK )
or
HDown1 : µ(d0) ≥ µ(d1) ≥ · · · ≥ µ(dK )
with at least one inequality.
Introduction Testing and Modeling Testing for Trend Model Based Model Averaging Application Discussion References
Test Statistics for Trend Test
Test statistic Formula
Likelihood Ratio E201 =
∑ij (yij−µ)2
−
∑ij (yij−µ⋆
i )2
∑ij (yij−µ)2
Test (LRT)
Williams t = (µ⋆K − y0)/
√2 ×
∑Ki=0
∑nij=1(yij − µi )2/(ni (n − K ))
Marcus t = (µ⋆K − µ⋆
0 )/√
2 ×
∑Ki=0
∑nij=1(yij − µi )2/(ni (n − K ))
M M = (µ⋆K − µ⋆
0 )/√∑K
i=0∑ni
j=1(yij − µ⋆i )2/(n − K )
Modified M (M’) M′ = (µ⋆K − µ⋆
0 )√∑K
i=0∑ni
j=1(yij − µ⋆i )2/(n − I)
More detail see Lin et.al 2007.
In this case the LRT E201 is used.
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Model Based
Assumes a functional relationship between the responseand the dose, taken as a quantitative factor, according to apre-specified parametric model.
Provides flexibility in investigating the effect of doses notused in the actual study
Its result validity depends on the correct choice of the doseresponse model, which is a priori unknown.
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Dose-response Models
Model Name FunctionLinear f (x) = E0 + δxLinear Log-dose f (x) = E0 + βlog(x + c)
Exponential f (x) = E0 + E1(ex/ϑ− 1)
Four parameter logistic f (x) = E0 +Emax−E0
1+exp[(EC50−x)/φ]
Five parameter logistic f (x) = E0 +Emax−E0
(1+exp[(EC50−x)/φ])γ
Hyperbolic Emax f (x) = E0 +x×(EMax−E0)
x+EC50
Sigmoidal Emax f (x) = E0 +xN
×(EMax−E0)
xN+ECN50
Gompertz f (x) = E0 + (EMax − E0)e−exp(ϕ(EC50−x))
Weibull 1 f (x) = E0 + (Emax − E0)e−exp(b(log(x)−log(EC50)))
Weibull 2 f (x) = E0 + (Emax − E0)(1 − e−exp(b(log(x)−log(EC50))))
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Dose-response Profiles
Dose-response profiles for each model
0.0 1.5 3.0
02
46
810
Linear
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
Linear Log−dose
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
3P Exponential
dosege
ne e
xpre
ssio
n
0.0 1.5 3.0
02
46
810
4P Logistic
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
5P Logistic
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
Hyperbolic E−max
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
Sigmoid E−max
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
4P Gompertz
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
Weibull 1
dose
gene
exp
ress
ion
0.0 1.5 3.0
02
46
810
Weibull 2
dosege
ne e
xpre
ssio
n
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Target Dose: EC50
The EC50: dose/concentration which induces a responsehalfway between the baseline and maximum.
YEC50= E0 +
E0 + Emax
2(1)
max0EE
0E
Slope (N)
maxE
Dose
50IC
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Model Averaging
Combines results from different models.
Account for model uncertainty.
All fits are taken into consideration.
Poor fits receive a small weights.
Let θ be a quantity in which we are interested in and wecan estimate θ from R models, the model averaged θ isdefined as:
θ =R∑
i
ωiθi ,
where θi is the value of θ from model i and ωi thedata-driven weights that sum to one assigned to model i .
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Model Averaging
Model averaging uses all (or most) of the candidatemodels whereas model selection selects the best model(the model with the highest value of ωi ).
Akaike’s weights:
ωi(AIC) =exp(−1
2∆AICi)∑Ri=1 exp(−1
2∆AICi)(2)
where ∆AICi = AICi − AICmin
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Model Averaged EC50
The model-averaged EC50 is defined as:
EC50 =R∑
i=1
ωi(AIC)EC50,i , (3)
where ωi : the Akaike’s weight and EC50,i : the EC50 of model i .The estimator for variance of EC50 is defined as:
var(EC50) =
[R∑
i=1
ωi(AIC)
√var(EC50,i |Mi) + (EC50,i − EC50)2
]2
,
(4)where var(EC50,i |Mi) is the variance of EC50 in model i .
Introduction Testing and Modeling Testing for Trend Model Based Model Averaging Application Discussion References
Antipsychotic Study
Case study: a study focuses on an antipsychoticcompound.6 dose levels with 4-5 samples at each dose level.Each array consists of 11,565 genes.
dose
gene
exp
ress
ion
0 0.01 0.04 0.16 0.63 2.5
67
89
10
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Results
250 genes have a significant monotonic trend (FDR=0.05)using E2 test statistic with 1000 permutations.
Data and isotonic trend of four best genes
0.0 0.5 1.0 1.5 2.0 2.5
67
89
10
dose
gene
exp
ress
ion
+
+
+
+
+ +
*
*
*
*
* *
1
0.0 0.5 1.0 1.5 2.0 2.5
89
1011
dose
gene
exp
ress
ion
+
+
+
+
+ +
*
*
*
*
* *
2
0.0 0.5 1.0 1.5 2.0 2.5
6.0
7.0
8.0
9.0
dose
gene
exp
ress
ion
+
+
+
++ +
*
*
*
** *
3
0.0 0.5 1.0 1.5 2.0 2.5
910
1112
dose
gene
exp
ress
ion
+
+
+
+ + +
*
*
*
* * *
4
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Results
Data and isotonic trend of four other genes
0.0 0.5 1.0 1.5 2.0 2.5
5.0
5.5
6.0
6.5
7.0
7.5
dose
gene
exp
ress
ion
+
++
+
+
+
***
*
*
*
0.0 0.5 1.0 1.5 2.0 2.5
6.1
6.3
6.5
6.7
dose
gene
exp
ress
ion
+++
+
+
+
*** *
*
*
0.0 0.5 1.0 1.5 2.0 2.5
12.9
013
.00
13.1
0
dose
gene
exp
ress
ion
+
++
+
+
+
*** *
*
*
0.0 0.5 1.0 1.5 2.0 2.5
13.1
513
.25
13.3
5
dose
gene
exp
ress
ion
+++
+
+
+
***
*
*
*
Introduction Testing and Modeling Testing for Trend Model Based Model Averaging Application Discussion References
Results
Number of models that converged and number of times asthe best model:
Model Number of models Number selected asconverge the best model
Linear 250 18Linear log-dose 250 31Three parm exponential 45 15Four parm logistic 199 4Five parm logistic 135 0Sigmoidal Emax 25 1Hyperbolic Emax 250 153Four parm Gompertz 8 0Weibull 1 213 22Weibull 2 213 6
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Results
Data and fitted value for the best gene
0.0 0.5 1.0 1.5 2.0 2.5
67
89
10
dose
gene
exp
ress
ion
linear Linear LogdoseHyperbolic EmaxWeibull 1Weibull 2Averaged Model
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Results
AIC, EC50, and Akaike’s weight for the best geneModel AIC EC50 WeightLinear 93.062 1.250 1.027e-14Linear log-dose 93.062 0.871 1.212e-13Hyperbolic Emax 29.656 0.039 0.603Weibull 1 31.709 1.023 0.216Weibull 2 32.064 1.061 0.181Model Average - 0.436 -
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Results
Plot of Confidence Interval of EC50
EC50
Mod
els
Lin
LinL
oghi
pEm
axw
eibu
ll1w
eibu
ll2M
AIC
50
−0.5 0.0 0.5 1.0 1.5 2.0 2.5
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Results
Data and fitted value for Gene 2
0.0 0.5 1.0 1.5 2.0 2.5
11.4
11.5
11.6
11.7
11.8
11.9
dose
gene
exp
ress
ion
Linear Linear Logdose4 parm Logistic5 parm LogisticHyperbolic EmaxWeibull 1Weibull 2Averaged Model
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Results
AIC, EC50, and Akaike’s weight for Gene 2Model AIC EC50 WeightLinear -35.03 1.25 0.021Linear log-dose -37.06 0.871 0.056Four parameter logistic -39.61 0.073 0.204Five parameter logistic -37.81 0.373 0.083Hyperbolic Emax -39.67 0.095 0.211Weibull 1 -39.91 1.240 0.236Weibull 2 -39.44 1.141 0.187Model Average - 0.706 -
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Discussion
In dose-response modeling in a microarray setting, fittingdirectly the proposed models to all genes (which can betens thousands) can create problems, such as complexityand time consumption.We propose a three steps approach:
Select the genes with a monotone trend using the LRT (E2)test statistic.Fit the selected genes with the candidate models toestimate the target dose.Average the target dose from the candidate models.
Testing for trendAssumes no specific dose-response relationship shape.Filters genes with a non monotonic trend
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Discussion
Model-based approach:Assumes a functional relationship.Provides flexibility.
Model averaging:Accounts for uncertainty.Takes in to account all the proposed models.
Genes then can be ranked based on the Model AveragedEC50
Software: IsoGene and IsoGeneGUI packages fortesting for trend.
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Selected References
Barlow, R.E., Bartholomew, D.J., Bremner, M.J. and Brunk, H.D.(1972) Statistical Inference Under Order Restriction, New York:Wiley.
Benjamini, Y. and Hochberg, Y. (1995) Controlling the falsediscovery rate: a practical and powerful approach to multipletesting, J. R. Statist. Soc. B, 57, 289-300.
Lin, Dan, Shkedy, Ziv, Yekutieli, Dani, Burzykowski, Tomasz,Gohlmann, Hinrich, De Bondt, An, Perera, Tim, Geerts, Tamaraand Bijnens, Luc.(2007) Testing for Trends in Dose-ResponseMicroarray Experiments: A Comparison of Several TestingProcedures, Multiplicity and Resampling-Based Inference,Statistical Applications in Genetics and Molecular Biology: Vol. 6: Iss. 1, Article 26.
Pinheiro, J.C. and Bates, D.M. (2000) Mixed Effects Models in Sand S-Plus. Springer-Verlag, New York.
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