Linear Models and Empirical Bayes Methods for · PDF fileLinear Models for Microarray Data...

IntroductionLinear Models for Microarray Data

Linear Models and Empirical Bayes for MicroarraysImplementation and Examples

SummaryReferences

Linear Models and Empirical Bayes Methodsfor Microarray Data Analysis

Alex Sánchez

Department d’EstadísticaUniversitat de Barcelona

April 13, 2009

Linear Models and Empirical Bayes



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Outline1 Introduction2 Linear Models for Microarray Data

Linear modelsExample 1: Swirl–Zebrafish experimentExample 2: Three-groups comparisonExample 3: A factorial experimentEstimation and inference with the linear modelStrength and weakness

3 Linear Models and Empirical Bayes for MicroarraysOverview of G.Smyth’s approachLönnsted and Speed’s B–statisticThe moderated-t

4 Implementation and Examples




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Overview of the presentation

This presentation treats two complementary aspectsThe use of a general linear model approach to analyzemicroarray data, specifically to select differentiallyexpressed genes in statistically designed microarrayexperiments.The enhancement suggested by Smyth (2004) to solvesome weaknesses of this approach when applied tomicroarray data.

Along the presentation some examples are introduced.The experimental design is presented but the analysis isreferred to the limma user’s guide.




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What is a Linear Model?

The linear model (Faraway, 2004) is a general frame formodelling and data analysis in statistics.Consists of defining linear relation between observedvalues and experimental conditions.If some assumptions on the data are true one can...

Obtain good estimators for the model parameters and theirstandard errors.(With some extra conditions) make inference about theexperiment.

Regression and Analysis of the Variance can be bothformulated as special cases of the linear models.




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How to use Linear Models to analyze microarray data

The application of linear models can be seen as amulti–step sequential process.

1 Start by specifying the design of the experiment: whichsamples are allocated to which conditions.

2 (Re–)Write a linear model for this design in the form ofY = Xβ + ε, where X is the design matrix.

3 If needed re–state the questions to answer as linearcontrasts on the parameters of the model.

4 Once the model is specified apply the general theory toestimate the parameters and the contrasts and,

5 If the appropriate validity conditions hold, perform inferenceon the model parameters based on the estimates.

This process will be illustrated in the examples that follow.




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Example 1: Swirl–Zebrafish experimentDescription

Swirl is a point mutation causing defects in the organizationof the developing embryo along its ventral-dorsal axis.As a result some cell types are reduced and others areexpanded.A goal of this experiment was to identify genes with alteredexpression in the swirl mutant compared to the wildzebrafish




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Example 1: Experimental design

Slide Cy3 Cy51 W M2 M W3 W M4 M W

Each microarray contained 8848cDNA probes (either genes orEST sequences).4 replicate slides: 2 sets ofdye-swap pairs.For each pair, target cDNA of theswirl mutant (S) was labeledusing one of Cy5 or Cy3 and thetarget cDNA of the wild type (W )mutant was labeled using theother dye.




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Example 1: Linear model

Parameter of interest is: α = E(

log SW

).

Samples 1 and 3 are labelled as : S (Green) and W (Red).Samples 2 and 4 are dye-swapped.The model, y = Xα+ ε, is:

y1 = α + ε1y2 = −α + ε2y3 = α + ε3y4 = −α + ε4

=⇒

y1y2y3y4

=

1−1

1−1

︸︷︷︸

Design Matrix,X

α+

ε1ε2ε3ε4




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Example 2: Comparison of 3 groupsDescription

IncHI plasmids encode multiple–antibiotic resistance in S.enterica.Plasmid R27, the “wild type” is thermosensitive for transfer.Some mutant phenotypes associated to both chromosomalhha and hns participate in different metabolic processes ofinterest in termoregulated conjugation.The goal of the experiment is to find genes which aredifferentially expressed in three different mutant types, sayM1, M2 and M3.




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Example 2: Possible design strategies

This experiment might be implemented differentlydepending on the type of chips used (two or one colour)and on which comparisons are of higher interest.

Using two colour–slidesA reference design: Hybridize each Mutant (Mi ) vs. Wildtype (W ).A loop design: Hybridize each mutant to each other in adouble loop that includes dye-swapping.

Using one colour slides: hybridize mutants and wild typesseparately.




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Design representation (1)

Allows for directcomparison of Mutant vsWild.Number of parameters toestimate=3, intuitiverelation between number ofparameters and mutants.Mutant vs Mutantcomparisons are lessefficient.




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Allows for directcomparison of Mutant vsMutant.Number of parameters toestimate=2. Less intuitive.Mutant vs Mutantcomparisons are moreefficient.




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Allows for directcomparison of

Mutant vs Wild andMutant vs Mutant.

Number of parameters toestimate=4.All comparisons can bemade efficiently.




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Linear model for reference design IModel, y = Xα+ ε, and contrasts C′β

Model parameters:

α1 = E(

logM1

W

), α2 = E

(log

M2

W

), α3 = E

(log

M3

W

).

Contrasts: Interesting comparisons.

β1 = α1 − α2,

β2 = α1 − α3,

β3 = α2 − α3.




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Linear model for reference design IIModel, y = Xα+ ε, and contrasts C′β

y1y2y3y4y5y6

=

1 0 00 1 00 0 1−1 0 0

0 −1 00 0 −1

︸︷︷︸

Design Matrix,X

α1α2α3

+

ε1ε2ε3ε4ε5ε6

β1

β2β3

=

1 −1 01 0 −10 1 −1

︸︷︷︸

Contrast Matrix,C

α1α2α3

.




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Linear model for loop design IModel, y = Xα+ ε, and contrasts C′β

Model parameters:

α1 = E(

logM1

M2

), α2 = E

(log

M2

M3

).

α3 is not necessary: log(

M1M3

)= log

(M1M2

)− log

(M2M3

).

Contrasts: Some of the desired comparisons are preciselythe parameters.

β1 = α1,

β2 = α2,

β3 = α1 + α2.




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Linear model for loop design IIModel, y = Xα+ ε, and contrasts C′β

y1y2y3y4y5y6

=

1 00 11 −1−1 0

0 −1−1 1

︸︷︷︸

Design Matrix,X

(α1α2

)+

ε1ε2ε3ε4ε5ε6

β1

β2β3

=

1 00 11 +1

︸︷︷︸Contrast Matrix,C

(α1α2

).




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Linear model for one colour arrays design IModel, y = Xα+ ε, and contrasts C1′

β, C2′β

Model parameters:

α1 = E(logM1), α2 = E(logM2), α3 = E(logM3), α4 = E(logW ).

Contrasts: Two possible sets of interesting comparisons.1 Comparison between mutant types (C1′

β)

β11 = α1 − α2,

β12 = α3 − α2,

β13 = α2 − α3.




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Linear model for one colour arrays design IIModel, y = Xα+ ε, and contrasts C1′

β, C2′β

2 Comparison between each mutant and the wild type (C2′β)

β21 = α4 − α1,

β22 = α3 − α1,

β23 = α2 − α1.




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Linear model for one colour arrays design IIIModel, y = Xα+ ε, and contrasts C1′

β, C2′β

y1y2y3y4y5y6y7y8

=

1 0 0 01 0 0 00 1 0 00 1 0 00 0 1 00 0 1 00 0 0 10 0 0 1

︸︷︷︸

Design Matrix,X

α1α2α3α4

+

ε1ε2ε3ε4ε5ε6ε7ε8




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Linear model for one colour arrays design IVModel, y = Xα+ ε, and contrasts C1′

β, C2′β

β11β1

2β1

3

=

1 −1 0 01 0 −1 00 1 −1 0

︸︷︷︸

Contrast Matrix,C1

α1α2α3α4

.

β21β2

2β2

3

=

1 0 0 −10 1 0 −10 0 1 −1

︸︷︷︸

Contrast Matrix,C2

α1α2α3α4

.




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Example 3: Estrogen experimentDescription

Goal: to study the effect of estrogen on the genes in ER+breast cancer cells over time.After serum starvation of eight samples, four samplesexposed to estrogen, and mRNA abundance measuredafter 10 hours (2 samples) and 48 hours (other two).Remaining four samples left untreated, and mRNAabundance measured similarly (10 hours for two samples,48 hours for the other two).Experiment with 2x2 factorial design: two factors (estrogenand time), each at two levels (present or absent,10 hoursor 48 hours).




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Example 3: Experimental design

Slide Estrogen Time1 Absent 102 Absent 103 Present 104 Present 105 Absent 486 Absent 487 Present 488 Present 48

One channel microarrays(Affymetrix) used.Each condition replicated twice.Specific questions to answer:

Estrogen effect after 10 hours.Estrogen effect after 48 hours.Time effect when no estrogenapplied.




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Example 3: Linear model

This experiment admits different parametrizationsSeparate factors with 2 levels each for estrogen (Abs/Pres),time (10h/48h) and interaction:

Yijk = αi︸︷︷︸Estrogen

+ βj︸︷︷︸Time

+ γij︸︷︷︸interaction

+εijk , i = 1,2, j = 1,2, k = 1,2

This first parametrization seems more natural but it is morecomplicated to rely on it to answer the questions posed.One combinate factor with 4 levels(Abs.10h, Abs.48h, Pres.10h, Pres.48h)

Yij = αi + εij , i = 1, ...,4, j = 1,2.

This parametrization seems more rigid but it is betteradapted to answer the questions posed.

The second parametrization is adopted here.Linear Models and Empirical Bayes



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Linear model for factorial design (1)

Model parameters:

α1 = E(logAbs.10h), α2 = E(logAbs.48h),

α3 = E(logPres.10h), α4 = E(logPres.48h).

Contrasts: Interesting questions are straightforward.

β11 = α3 − α1, Estrogen effect after 10 hoursβ1

2 = α4 − α2, Estrogen effect after 48 hoursβ1

3 = α2 − α1, Time effect in absence of estrogen




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Linear models for one colour arrays design (2) IModel, y = Xα+ ε, and contrasts C′β

y1y2y3y4y5y6y7y8

=

1 0 0 01 0 0 00 1 0 00 1 0 00 0 1 00 0 1 00 0 0 10 0 0 1

︸︷︷︸

Design Matrix,X

α1α2α3α4

+

ε1ε2ε3ε4ε5ε6ε7ε8




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Linear models for one colour arrays design (2) IIModel, y = Xα+ ε, and contrasts C′β

β11β1

2β1

3

=

−1 0 1 00 −1 0 1−1 1 0 0

︸︷︷︸

Contrast Matrix,C

α1α2α3α4

.




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Estimation and inference I

Having expressed the experiment as a linear model:

E(yg) = Xαg, var(yg) = Wgσg ,

allows to use standard linear model theory to obtain ....Parameter estimates: α̂g(≈ α).Standard deviation estimates: σ̂g = sg(≈ σ).Standard error estimates:v̂arα̂g = Vg s2

g .




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Estimation and inference II

These estimates are the basis to perform inference aboutα i.e. test H0 : α = 0?, based on the fact that:

tgj =αgj

sg√vgj

∼ Student distribution.

Similar result holds for α1 − α2.

The estimation and inferential procedures do not dependon which parametrization has been adopted, althoughdifferent numerical values may be, of course, obtained.




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Strength and Weakness of Linear Models

Linear model approach is flexible and powerfulCan be adapted to many different and complex situations.Always yields good (BLUE) estimates.If assumptions are true it provides a basis for inference.

However...If assumptions don’t hold conclusions are not to be trusted.Even if they hold they may be affected by small samplesizes, so that high variances estimates may yield nonsignificant t-values.

The methodology developed by Smyth (2004) based uponresults of Lönsted & Speed (2002) addresses how to dealwith these weaknesses.




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Overview of G.Smyth’s approachLönnsted and Speed’s B–statisticThe moderated-t








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General overview I

Smyth (2004) considers the problem of identifying geneswhich are differentially expressed across specifiedconditions in designed microarray experiments.He addresses the fact that

the variability of the expression values differs betweengenes, butthe parallel nature of the inference in microarrays allowssome possibilities for borrowing information from theensemble of genes which can assist in inference abouteach gene individually.




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General overview II

Smyth (2004) develops in 3 steps the hierarchical model ofLönnstedt and Speed (2002) into a practical approach.

The first step is to re–state it in the context of general linearmodels.

The second step is to derive consistent, closed formestimators for the hyperparameters. These estimatorshave robust behavior even for small numbers of arrays.

The third step is to re–formulate the posterior odds statisticin terms of a moderated t-statistic.




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The B–statistic

L-S (2002) addressed the problem of improving usualmeasures of differential expression such as M = log(R/G)or t = M

/(s/√

n) .Other attempts: Tibshirani et al. SAM’s Statistic.

They rely on the (log) ratio of two probabilities: theprobability of the gene being expressed vs. the probabilityof not being expressed.

This is a common approach in clinical studies or geneticsand is called a (log) odds-ratio or LODS.

B = logP[Affected|Mij ]

P[Not Affected|Mij ],

gene=i (i = 1...N), replication=j (j = 1, ...,n).




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Empirical Bayes approach

Assume that the mean and variance of log-ratios for eachgene follow a priori fixed distributions.Combine the information from all the genes to estimatetheir parameters.Use the Bayesian method to derive an expression of Bwhich combines both the information of each gene and theinformation obtained from all the genes in a posteriorlog–odds–ratio.

Bg = const + log

2an + s2 + M2

g

2an + s2 +

M2g

1+nc




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Pro’s and Con’s of B.

Bg = const+

log

2an + s2 + M2

g

2an + s2 +

M2g

1+nc

Useful to rank genes ...Bg increases with Mg ,If Mg is small, a ensures that the ratiowill not be expanded by a very smallvariance.B ≈ Mg/sg for large n.

However...Still no p − values.Depends on many parameters.




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Hierarchical model

A hierarchichal model is introduced to describe how theunknown coefficients βgj and unknown variances σgvaryacross genes.This is done adopting a Bayesian approach that puts priordistributions for these sets of parameters.

Normal Model Priors

β̂gj ∼ N(βgj , vgjσ2g)

P(βgj 6= 0) = pβgj |σ2

g , βgj 6= 0 ∼ N(0, v0jσ2g)

s2g ∼ σ2

gχ2dg

σ2g ∼ s2

0

(χ2

d0/d0

)−1




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Posterior statistics

Posterior variance estimators

s̃2g = E(σ2

g |s2g) =

dos2o + dgs2

g

d0 + dg

The posterior values shrink the observed variances towards the prior values with the degree of shrinkage

depending on the relative sizes of the observed and prior degrees of freedom

The moderated t−statistic is:

t̃gj =β̂gj

s̃g√vgj

This distributional result assumes d0 and s0 to be givenvalues. In practice they need to be estimated from the data




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Implementation and Examples

This approach has become very popular betweenmicroarray users mainly due to the fact that it isimplemented in an excellently well documentedBioconductor package: limma.The limma user guide (available after installation) containsthe analysis of the Swirl and the estrogen data as well asmany other examples.




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Summary

Linear models provide a flexible and powerful approach tomodelling and analyzing microarray experiments.The hierarchical model presented gives moderatedstatistics that help to borrow the information across genesto compensate for the usually small number of replicates.The programs Limma, LimmaGUI and LimmaAffyGUIallow a direct application of these approaches.




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References

Faraway, J. (2004) Linear models with R Chapman andHall (CRC).Lönnstedt, I. and Speed, T. (2002) Replicated MicroarrayData Statistica Sinica 12(2002), 31–46.Smyth G.K. (2004) Linear models and empirical bayesmethods for assessing diferential expression in microarrayexperiments. Statistical Applications in Genetics andMolecular Biology, 3:Article 3, 2004.


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