using limma for differential expressionmaster.bioconductor.org/help/course-materials/2009/... ·...
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![Page 1: Using limma for Differential Expressionmaster.bioconductor.org/help/course-materials/2009/... · Bioconductor Introduction Design Matrices Contrast Matrices Examples Two-color Data](https://reader034.vdocuments.us/reader034/viewer/2022050302/5f779cbd8ec62a0f0617df13/html5/thumbnails/1.jpg)
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Using limma for Differential Expression
James W. [email protected]
BioC 2009July 27, 2009
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Overview
Typical analysis using limma:
Read in data
Preprocess two-color data
Create design matrix
Create contrast matrix
Fit model
Make comparisons
Output interesting results
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Goals
Goals for this workshop:
Statistics
Linear models (esp ANOVA)Design matricesContrast matrices
Other considerations
Intra-group correlationArray weights
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Introduction
DesignMatrices
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Examples
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Intra-groupCorrelation
Array Weights
Linear model
0 2 4 6 8 10
02
46
810
X
Y
m
y = mx + b
b
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Introduction
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ANOVA
Predictors are factors (unordered)
tumor/normalexperimental/controlmutant/wild type
Simplest form of ANOVA is t-test
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
ANOVA
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
ANOVA
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
ANOVA
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
ANOVA
Basic form of the t-test:
Differencebetweengroupmeans
Variabilitywithingroups(1)
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ANOVA Model
yij = µj + εij ,
– OR –tumor sample = mean tumor value + errornormal sample = mean normal value + error
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Introduction
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Array Weights
ANOVA (Graphically)
ANOVA is a ’mean decomposition’ of observed data
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Mean tumor = 5.5
This sample = 5.3Error = −0.2
Sample value = 5.5 + (−0.2) = 5.3
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Linear Algebra - Background
Basic algebra – we need N equations to solve for N unknowns4 = y - 3x1 = -2y + 4xSolve one equation for y, insert answer in other equationy = 3x - 41 = -2(3x + 4) + 4x2x = -9x = -9/2y = -19/2
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Linear Algebra
We can do the same thing using matrix multiplication:[41
]=
[1 −3−2 4
] [yx
]
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Linear Algebra
Solve for x and y :[yx
]=
[1 −3−2 4
]−1 [41
]In R:
> mat <- matrix(c(1,-2,-3,4), ncol = 2)
> solve(mat) %*% c(4,1)
[,1][1,] -9.5[2,] -4.5
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ANOVA Model (again)
Our current model:tumor sample = mean tumor value + errornormal sample = mean normal value + error
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ANOVA Model (again)
Substitute x and y :tumor sample = x + errornormal sample = y + error
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ANOVA Model (again)
Expand a bit:tumor sample = 1x + 0y + errornormal sample = 0x + 1y + error
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ANOVA Model (again)
Plug in data:5.3 = 1x + 0y + error5.5 = 1x + 0y + error5.7 = 1x + 0y + error3.5 = 0x + 1y + error3.1 = 0x + 1y + error3.4 = 0x + 1y + error
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Introduction
DesignMatrices
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Intra-groupCorrelation
Array Weights
ANOVA Model (again)
In linear algebra terms:
5.35.55.73.53.13.4
=
1 01 01 00 10 10 1
[
xy
]+
errorerrorerrorerrorerrorerror
First row is:5.3 = 1x + 0y + error
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ANOVA Model (again)
To check if our design matrix really does what we think...Say we call our design matrix ’D’And our vector of observations ’O’To solve for (x,y) we use[
xy
]=[
D ′D]−1
D ′O
> mat <- matrix(rep(c(1,0,0,1), each = 3), ncol = 2)
> solve(crossprod(mat)) %*% t(mat)
[,1] [,2] [,3] [,4] [,5] [,6][1,] 0.33 0.33 0.33 0.00 0.00 0.00[2,] 0.00 0.00 0.00 0.33 0.33 0.33
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Parameterization
Parameter = thing estimated by modelCurrent model:
Mean of tumor samplesMean of normal samples
Different model:
Mean of normal samplesMean of tumor samples - mean of normal samples
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Setup:x = mean of tumor samplesy = mean of normal samplesz = mean of tumor - mean of normalnormal sample = 1y + 0z + errortumor sample = 1y + 1z + errorsotumor sample = 1y + (1x - 1y) + error
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Alternate Parameterization
In linear algebra terms:
5.35.55.73.53.13.4
=
1 11 11 11 01 01 0
[
yz
]+
errorerrorerrorerrorerrorerror
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Creating Design Matrices
By hand:
> mat <- cbind(c(1,1,1,0,0,0), c(0,0,0,1,1,1))
> dimnames(mat) <- list(paste("Sample", 1:6),
+ c("Tumor","Normal"))
> mat
Tumor NormalSample 1 1 0Sample 2 1 0Sample 3 1 0Sample 4 0 1Sample 5 0 1Sample 6 0 1
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Creating Design Matrices
Using model.matrix():
> samps <- factor(rep(c("Tumor","Normal"), each = 3))
> model.matrix(~samps)
(Intercept) sampsTumor1 1 12 1 13 1 14 1 05 1 06 1 0attr(,"assign")[1] 0 1attr(,"contrasts")attr(,"contrasts")$samps[1] "contr.treatment"
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Creating Design Matrices
Using model.matrix():
> model.matrix(~0 + samps)
sampsNormal sampsTumor1 0 12 0 13 0 14 1 05 1 06 1 0attr(,"assign")[1] 1 1attr(,"contrasts")attr(,"contrasts")$samps[1] "contr.treatment"
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Creating Design Matrices
Decide on model, then create design matrixDo not create design matrix and then figure outwhat the model is...
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Contrast Matrices
What is a contrast?
Difference between parameter estimates
Coefficients must equal zero
Trivial example using our data:1x - 1y
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Linear algebra again...5.34 2.916.54 8.562.35 2.565.55 6.458.78 6.98
[
1−1
]=
2.43−2.02−0.21−0.9
1.8
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Rules:
Coefficients must equal zero
Rows of matrix must equal number of parameters in model
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Contrast Matrices
Example: Three sample types (cont, trt1, trt2). Compare eachtrt to cont.
> contrast <- matrix(c(-1,1,0,-1,0,1), ncol = 2)
> dimnames(contrast) <- list(c("cont","trt1","trt2"),
+ c("trt1 - cont",
+ "trt2 - cont"))
> contrast
trt1 - cont trt2 - contcont -1 -1trt1 1 0trt2 0 1
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Contrast Matrices
Example: Three sample types (cont, trt1, trt2). Compare themean of trt samples to cont.
> contrast <- matrix(c(-1,0.5,0.5), ncol = 1)
> dimnames(contrast) <- list(c("cont","trt1","trt2"),
+ "mean trt - cont")
> contrast
mean trt - contcont -1.0trt1 0.5trt2 0.5
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Two Factor ANOVA
Setup:
Two sample types (say, tumor and normal)
Two treatments (say, chemo drug and ’carrier solution’)
Three replicates (12 samples total)
Comparisons:
untreated tumor vs untreated normaltreated tumor vs untreated tumordifference in effect of treatment
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Two Factor ANOVA
Parameters we need:
mean of treated normal samples
mean of untreated normal samples
mean of treated tumor samples
mean of untreated tumor samples
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Design matrix:
1 0 0 01 0 0 01 0 0 00 1 0 00 1 0 00 1 0 00 0 1 00 0 1 00 0 1 00 0 0 10 0 0 10 0 0 1
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Two Factor ANOVA
In R:
> mat <- cbind(c(1,1,1,rep(0,9)),
+ c(0,0,0,1,1,1,rep(0,6)),
+ c(rep(0,6),1,1,1,0,0,0),
+ c(rep(0,9),1,1,1))
> colnames(mat) <- c("nu","nt","tu","tt")
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Two Factor ANOVA
> mat
nu nt tu tt[1,] 1 0 0 0[2,] 1 0 0 0[3,] 1 0 0 0[4,] 0 1 0 0[5,] 0 1 0 0[6,] 0 1 0 0[7,] 0 0 1 0[8,] 0 0 1 0[9,] 0 0 1 0[10,] 0 0 0 1[11,] 0 0 0 1[12,] 0 0 0 1
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Contrast matrix:
> makeContrasts(tu - nu, tt - tu,
+ tt - tu - nt + nu,
+ levels = mat)
ContrastsLevels tu - nu tt - tu tt - tu - nt + nu
nu -1 0 1nt 0 0 -1tu 1 -1 -1tt 0 1 1
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Intra-groupCorrelation
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Two Factor ANOVA
Using model.matrix()
> typ <- factor(rep(c("Norm","Tum"), each = 6))
> trt <- factor(rep(c("Untrt", "Trt"),each = 3, times = 2))
> typ
[1] Norm Norm Norm Norm Norm Norm Tum Tum[9] Tum Tum Tum TumLevels: Norm Tum
> trt
[1] Untrt Untrt Untrt Trt Trt Trt[7] Untrt Untrt Untrt Trt Trt TrtLevels: Trt Untrt
> mat <- model.matrix(~0+trt*typ)
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Introduction
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> mat
trtTrt trtUntrt typTum trtUntrt:typTum1 0 1 0 02 0 1 0 03 0 1 0 04 1 0 0 05 1 0 0 06 1 0 0 07 0 1 1 18 0 1 1 19 0 1 1 110 1 0 1 011 1 0 1 012 1 0 1 0attr(,"assign")[1] 1 1 2 3attr(,"contrasts")attr(,"contrasts")$trt[1] "contr.treatment"
attr(,"contrasts")$typ[1] "contr.treatment"
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Two Factor ANOVA
Questions:
What parameters are being estimated?
What contrasts matrix do we need?
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Batch Effect
Setup:
Two sample types (say, treated and control)
Two experiments, one week apart
Three replicates (12 samples total)
Compare treated vs normal
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Batch Effects
> typ <- factor(rep(c("trt","cont"),
+ each = 3, times = 2))
> bat <- factor(rep(1:2, each = 6))
> mat <- model.matrix(~ 0 + typ + bat)
![Page 44: Using limma for Differential Expressionmaster.bioconductor.org/help/course-materials/2009/... · Bioconductor Introduction Design Matrices Contrast Matrices Examples Two-color Data](https://reader034.vdocuments.us/reader034/viewer/2022050302/5f779cbd8ec62a0f0617df13/html5/thumbnails/44.jpg)
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Introduction
DesignMatrices
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Two-colorData
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Batch Effects
> mat
typcont typtrt bat21 0 1 02 0 1 03 0 1 04 1 0 05 1 0 06 1 0 07 0 1 18 0 1 19 0 1 110 1 0 111 1 0 112 1 0 1attr(,"assign")[1] 1 1 2attr(,"contrasts")attr(,"contrasts")$typ[1] "contr.treatment"
attr(,"contrasts")$bat[1] "contr.treatment"
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Introduction
DesignMatrices
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Two-colorData
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Intra-groupCorrelation
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Two-color Data
Design matrices are a bit more tricky:
We are working with ratios
Max number of parameters = sample types - 1
Dye swaps
Experimental design considerations
modelMatrix()
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Introduction
DesignMatrices
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Examples
Two-colorData
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Intra-groupCorrelation
Array Weights
Two Samples
Setup:
Two sample types (say, wild type and mutant)
All mutant labeled with Cy5
All wild type labeled with Cy3
Four of each sample (four arrays)
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Two Samples
> targets <- data.frame(Cy5 = rep("mut", 4),
+ Cy3 = rep("wt", 4))
> modelMatrix(targets, ref = "wt")
Found unique target names:mut wt
mut[1,] 1[2,] 1[3,] 1[4,] 1
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Dye Swaps
Same setup, but every other array we swap dyes
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Dye Swaps
> targets <- data.frame(Cy5 = rep(c("mut","wt"), 2),
+ Cy3 = rep(c("wt","mut"),2))
> modelMatrix(targets, ref = "wt")
Found unique target names:mut wt
mut[1,] 1[2,] -1[3,] 1[4,] -1
![Page 50: Using limma for Differential Expressionmaster.bioconductor.org/help/course-materials/2009/... · Bioconductor Introduction Design Matrices Contrast Matrices Examples Two-color Data](https://reader034.vdocuments.us/reader034/viewer/2022050302/5f779cbd8ec62a0f0617df13/html5/thumbnails/50.jpg)
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Dye Effect
Same setup, but we want to test for dye effect
> cbind(dye = 1, mut = rep(c(1,-1), 2))
dye mut[1,] 1 1[2,] 1 -1[3,] 1 1[4,] 1 -1
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
’Connected’ Experiment
![Page 52: Using limma for Differential Expressionmaster.bioconductor.org/help/course-materials/2009/... · Bioconductor Introduction Design Matrices Contrast Matrices Examples Two-color Data](https://reader034.vdocuments.us/reader034/viewer/2022050302/5f779cbd8ec62a0f0617df13/html5/thumbnails/52.jpg)
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Introduction
DesignMatrices
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Examples
Two-colorData
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Array Weights
’Connected’ Experiment
Targets file:
Cy3 Cy51 cont trt12 trt1 cont3 trt1 trt24 trt2 trt15 trt2 cont6 cont trt2
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Introduction
DesignMatrices
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Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
’Connected’ Experiment
Design matrix:
> modelMatrix(tgts, ref = "cont")
Found unique target names:cont trt1 trt2
trt1 trt2[1,] 1 0[2,] -1 0[3,] -1 1[4,] 1 -1[5,] 0 -1[6,] 0 1
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Introduction
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’Connected’ Experiment
By Hand:
Control Trt1 Trt20 1 01 0 00 0 10 1 01 0 00 0 1
−
Control Trt1 Trt21 0 00 1 00 1 00 0 10 0 11 0 0
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Bioconductor
Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
’Unconnected’ Experiment
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Introduction
DesignMatrices
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Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
’Unconnected’ Experiment
Targets file:
Cy5 Cy31 cont1 trt12 trt1 cont13 cont2 trt24 trt2 cont2
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Introduction
DesignMatrices
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Two-colorData
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Array Weights
’Unconnected’ Experiment
Incorrect design matrix:
> modelMatrix(tgts, ref = "cont1")
Found unique target names:cont1 cont2 trt1 trt2
cont2 trt1 trt2[1,] 0 -1 0[2,] 0 1 0[3,] 1 0 -1[4,] -1 0 1
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Introduction
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’Unconnected’ Experiment
> dat <- matrix(rnorm(40), ncol=4)
> fit <- lmFit(dat, modelMatrix(tgts, ref = "cont1"))
Found unique target names:cont1 cont2 trt1 trt2Coefficients not estimable: trt2
> head(fit$coef)
cont2 trt1 trt2[1,] -1.91 0.27 NA[2,] -0.27 -1.46 NA[3,] -0.48 0.71 NA[4,] 0.86 -0.22 NA[5,] -1.11 -0.75 NA[6,] 0.15 0.17 NA
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Introduction
DesignMatrices
ContrastMatrices
Examples
Two-colorData
Otherconsiderations
Intra-groupCorrelation
Array Weights
Intra-group correlation
Batch effect
Technical replication
Correlated samples
Multiple littersLongitudinal data
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Introduction
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Two-colorData
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Intra-group correlation
Fixed batch effect
Mixed model
Estimate correlation (duplicateCorrelation)Fit model (’correlation’ argument)
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Array weights
With arrays of lesser quality you can
Discard array dataDown-weight arrays
Weights based on linear fit
arrayWeights/arrayWeightsSimple
’weight’ argument to lmFit()