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Multi-way Anova. Identifying and quantifying sources of variation Ability to "factor out" certain sources - ("adjusting") For the beginning, we will reproduce paired t-test results by assuming that arrays are one of the factors in a Two-way ANOVA

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Multi-way Anova


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multi way anova
Multi-way Anova
  • Identifying and quantifying sources of variation
  • Ability to "factor out" certain sources - ("adjusting")
  • For the beginning, we will reproduce paired t-test results by assuming that arrays are one of the factors in a Two-way ANOVA
  • Second, adjusting for the dye effects in a Three-way ANOVA
  • Third, four and more - way ANOVA when having multiple factors of interest
statistical inference in two way anova

Statistical Model

Parameter Estimates

Null Hypotheses

Null Distributions

Statistical Inference in Two-way ANOVA
alternative formulations of the two way anova

No-intercept Model

Null Hypotheses

Null Distributions

Alternative Formulations of the Two-way ANOVA
  • Parameter Estimates
  • Gets complicated
  • Regardless of how the model is parametrized certain parametersremain unchanged (Trt2-Trt1)
  • In this sense all formulations are equivalent
re organizing data for anova
Re-organizing data for ANOVA

> DAnova<-NNBLimmadataC

> DAnova$weights<-cbind(DAnova$weights,DAnova$weights)

> DAnova$M<-cbind((DAnova$A+DAnova$M/2),DAnova$A-(DAnova$M/2))

> DAnova$A<-cbind(DAnova$A,DAnova$A)

> attributes(DAnova)

$names

[1] "weights" "targets" "genes" "printer" "M" "A"

$class

[1] "MAList"

attr(,"package")

[1] "limma"

> NNBLimmadataC$M[1,]

51-C1-3-vs-W1-5 60-W2-3-vs-C2-5 72-C3-3-vs-W3-5 79-W4-3-vs-C4-5 82-C5-3-vs-W5-5 97-W6-3-vs-C6-5

-0.05280598 -0.15767422 0.40130216 -0.35292771 -0.22061576 -0.21653047

> NNBLimmadataC$A[1,]

51-C1-3-vs-W1-5 60-W2-3-vs-C2-5 72-C3-3-vs-W3-5 79-W4-3-vs-C4-5 82-C5-3-vs-W5-5 97-W6-3-vs-C6-5

6.289658 6.129577 6.483613 6.452317 6.510143 7.106134

> DAnova$M[1,]

51-C1-3-vs-W1-5 60-W2-3-vs-C2-5 72-C3-3-vs-W3-5 79-W4-3-vs-C4-5 82-C5-3-vs-W5-5 97-W6-3-vs-C6-5

6.263255 6.050740 6.684264 6.275853 6.399835 6.997869

51-C1-3-vs-W1-5 60-W2-3-vs-C2-5 72-C3-3-vs-W3-5 79-W4-3-vs-C4-5 82-C5-3-vs-W5-5 97-W6-3-vs-C6-5

6.316061 6.208414 6.282962 6.628781 6.620451 7.214399

setting up the design matrix for limma
Setting-up the design matrix for limma

> targets

SlideNumber FileName Cy3 Cy5 Date

51-C1-3-vs-W1-5 51 51-C1-3-vs-W1-5.gpr C W 11/8/2004

60-W2-3-vs-C2-5 60 60-W2-3-vs-C2-5.gpr W C 11/8/2004

72-C3-3-vs-W3-5 72 72-C3-3-vs-W3-5.gpr C W 11/8/2004

79-W4-3-vs-C4-5 79 79-W4-3-vs-C4-5.gpr W C 11/8/2004

82-C5-3-vs-W5-5 82 82-C5-3-vs-W5-5.gpr C W 11/8/2004

97-W6-3-vs-C6-5 97 97-W6-3-vs-C6-5.gpr W C 11/8/2004

> trt<-c(targets$Cy5,targets$Cy3)

> trt

[1] "W" "C" "W" "C" "W" "C" "C" "W" "C" "W" "C" "W"

> array<-c(1:6,1:6)

> array

[1] 1 2 3 4 5 6 1 2 3 4 5 6

setting up the design matrix for limma7
Setting-up the design matrix for limma

> designa<-model.matrix(~-1+factor(array)+factor(trt))

> designa

factor(array)1 factor(array)2 factor(array)3 factor(array)4 factor(array)5 factor(array)6 factor(trt)W

1 1 0 0 0 0 0 1

2 0 1 0 0 0 0 0

3 0 0 1 0 0 0 1

4 0 0 0 1 0 0 0

5 0 0 0 0 1 0 1

6 0 0 0 0 0 1 0

7 1 0 0 0 0 0 0

8 0 1 0 0 0 0 1

9 0 0 1 0 0 0 0

10 0 0 0 1 0 0 1

11 0 0 0 0 1 0 0

12 0 0 0 0 0 1 1

attr(,"assign")

[1] 1 1 1 1 1 1 2

attr(,"contrasts")

attr(,"contrasts")$"factor(array)"

[1] "contr.treatment"

attr(,"contrasts")$"factor(trt)"

[1] "contr.treatment"

setting up the design matrix for limma8
Setting-up the design matrix for limma

> colnames(designa)<-c(paste("A",1:6,sep=""),"W")

> designa

A1 A2 A3 A4 A5 A6 W

1 1 0 0 0 0 0 1

2 0 1 0 0 0 0 0

3 0 0 1 0 0 0 1

4 0 0 0 1 0 0 0

5 0 0 0 0 1 0 1

6 0 0 0 0 0 1 0

7 1 0 0 0 0 0 0

8 0 1 0 0 0 0 1

9 0 0 1 0 0 0 0

10 0 0 0 1 0 0 1

11 0 0 0 0 1 0 0

12 0 0 0 0 0 1 1

attr(,"assign")

[1] 1 1 1 1 1 1 2

attr(,"contrasts")

attr(,"contrasts")$"factor(array)"

[1] "contr.treatment"

attr(,"contrasts")$"factor(trt)"

[1] "contr.treatment"

comparing to paired t test
Comparing to paired t-test

> Anova<-lmFit(DAnova,designa)

>

> Anova$coefficients[2,]

A1 A2 A3 A4 A5 A6 W

8.36197475 9.90627295 11.34002704 10.77586480 9.35096212 9.92299117 -0.03068036

> LimmaPTT$coefficients[2]

[1] -0.03068036

> Anova$coefficients[1,]

A1 A2 A3 A4 A5 A6 W

NA NA 6.3898451 6.3585492 6.4163747 7.0123661 0.1875361

> LimmaPTT$coefficients[1]

[1] 0.1875361

>

comparing to paired t test10
Comparing to paired t-test

> plot(Anova$coefficients[,"W"],LimmaPTT$coefficients)

> plot(Anova$sigma,LimmaPTT$sigma)

> plot(Anova$stdev.unscaled[,"W"],LimmaPTT$stdev.unscaled)

> plot(Anova$sigma*Anova$stdev.unscaled[,"W"],LimmaPTT$sigma*LimmaPTT$stdev.unscaled)

> plot(Anova$df.residual,LimmaPTT$df.residual)

adjusting for dye three way anova
Adjusting for Dye - Three-way ANOVA

> dye<-c(rep("Cy5",6),rep("Cy3",6))

>

> designad<-model.matrix(~-1+factor(array)+factor(dye)+factor(trt))

> #designad

> colnames(designad)<-c(paste("A",1:6,sep=""),"Cy5","W")

> designad

A1 A2 A3 A4 A5 A6 Cy5 W

1 1 0 0 0 0 0 1 1

2 0 1 0 0 0 0 1 0

3 0 0 1 0 0 0 1 1

4 0 0 0 1 0 0 1 0

5 0 0 0 0 1 0 1 1

6 0 0 0 0 0 1 1 0

7 1 0 0 0 0 0 0 0

8 0 1 0 0 0 0 0 1

9 0 0 1 0 0 0 0 0

10 0 0 0 1 0 0 0 1

11 0 0 0 0 1 0 0 0

12 0 0 0 0 0 1 0 1

attr(,"assign")

[1] 1 1 1 1 1 1 2 3

attr(,"contrasts")

attr(,"contrasts")$"factor(array)"

[1] "contr.treatment"

attr(,"contrasts")$"factor(dye)"

[1] "contr.treatment"

attr(,"contrasts")$"factor(trt)"

[1] "contr.treatment"

adjusting for dye three way anova12
Adjusting for Dye - Three-way ANOVA

>Anovad<-lmFit(DAnova,designad)

>

> Anova$coefficients[2,]

A1 A2 A3 A4 A5 A6 W

8.36197475 9.90627295 11.34002704 10.77586480 9.35096212 9.92299117 -0.03068036

> LimmaPTT$coefficients[2]

[1] -0.03068036

> Anovad$coefficients[2,]

A1 A2 A3 A4 A5 A6 Cy5 W

8.38440701 9.92870520 11.36245930 10.79829705 9.37339438 9.94542342 -0.04486451 -0.03068036

> Anova$coefficients[1,]

A1 A2 A3 A4 A5 A6 W

NA NA 6.3898451 6.3585492 6.4163747 7.0123661 0.1875361

> LimmaPTT$coefficients[1]

[1] 0.1875361

> Anovad$coefficients[1,]

A1 A2 A3 A4 A5 A6 Cy5 W

NA NA 6.43844153 6.40714571 6.46497115 7.06096259 -0.09719295 0.18753615

>

comparing to paired t test13
Comparing to paired t-test

> plot(Anovad$coefficients[,"W"],LimmaPTT$coefficients)

> plot(Anovad$sigma,LimmaPTT$sigma)

> plot(Anovad$stdev.unscaled[,"W"],LimmaPTT$stdev.unscaled)

> plot(Anovad$sigma*Anova$stdev.unscaled[,"W"],LimmaPTT$sigma*LimmaPTT$stdev.unscaled)

> plot(Anovad$df.residual,LimmaPTT$df.residual)

comparing to paired t test14
Comparing to paired t-test

> AnovaB<-eBayes(Anova)

> AnovadB<-eBayes(Anovad)

> volcanoplot(AnovaB,coef=7)

> volcanoplot(AnovadB,coef=8)