Extending the Loop Design for Microarray Experiments

Extending the Loop Extending the Loop Design for Design for Microarray Microarray

Experiments Experiments Naomi S. Altman, Naomi S. Altman,

Pennsylvania State University), Pennsylvania State University), [email protected]@stat.psu.edu

Interface Meetings May 04Interface Meetings May 04

Expt Design and Expt Design and MicroarraysMicroarrays

Microarrays are Microarrays are ExpensiveExpensiveNoisyNoisy

A perfect situation for A perfect situation for optimal designoptimal design

OutlineOutline

Reference DesignReference Design Loop DesignsLoop Designs ReplicationReplication Optimal Design/AnalysisOptimal Design/Analysis Incorporating Multiple Factors Incorporating Multiple Factors

and Blocksand Blocks

Arrow NotationArrow Notation

Introduced by Kerr and Churchill Introduced by Kerr and Churchill (2001)(2001)

Each array is represented by an arrow.Each array is represented by an arrow.

Red Green

Reference DesignReference Design

Reference

A

B

C

D4 arrays

1 sample/treatment

4 reference samples

Loop DesignLoop Design(Kerr and Churchill 2001)(Kerr and Churchill 2001)

A

C

B

D

4 arrays

2 samples/treatment

ReplicationReplicationOften there is confusion among:Often there is confusion among:

Biological replicatesBiological replicates

Technical replicatesTechnical replicatesrepeated samplesrepeated samplessplit sample and relabelsplit sample and relabelspot replicationspot replication

In this presentation: We consider only In this presentation: We consider only one spot/gene/arrayone spot/gene/arrayany technical replicates are averagedany technical replicates are averagedeach sample is an each sample is an independent biological independent biological replicatereplicate

Linear Mixed Model for Linear Mixed Model for Microarray DataMicroarray Data

is the response of the gene in one channelis the response of the gene in one channel

is the mean response of the gene over all is the mean response of the gene over all treatments, channels, arraystreatments, channels, arrays

is the effect of treatment iis the effect of treatment i

the effect of dye jthe effect of dye j

is the effect of the array k (or spot on the array)is the effect of the array k (or spot on the array)

is the random deviation from the other effects is the random deviation from the other effects and includes biological variation, technical and includes biological variation, technical variation and random errorvariation and random error

ijkkjiijkY

ji

ijkY

ijkk

Linear Mixed Model for Linear Mixed Model for Microarray DataMicroarray Data

The 2 channels on a single spot are correlatedThe 2 channels on a single spot are correlated

→ → array should be treated as a random effectarray should be treated as a random effect

ijkkjiijkY

Differencing Channels on Differencing Channels on an Arrayan Array

Often the difference between samples Often the difference between samples on a single array is the unit of on a single array is the unit of analysis:analysis:

rGkiRkktir YY )).((

Normalization is almost always done on this quantity.

In a reference design, the difference between treatments A and B can be estimated from 2 arrays by

)).(()).((ˆˆ

luBrktArBA

But there can be a large loss of information.

Var()=0.126 Var(M)=0.453

)).(( ktAr

Drosophila arrays courtesy of

Bryce MacIver, PSU

Reference DesignReference Design

The reference sample is the same biological The reference sample is the same biological material on every arraymaterial on every array

T treatments, T treatments, k replicates,k replicates, kT arrayskT arrays

If there are technical dye-swaps, these are If there are technical dye-swaps, these are averaged to form 1 replicate.averaged to form 1 replicate.

If all comparisons are between treatments, If all comparisons are between treatments, there is no need to dye-swap. If there are there is no need to dye-swap. If there are dye-swaps, these should be balanced by dye-swaps, these should be balanced by treatment.treatment.

Reference Design – Usual Reference Design – Usual AnalysisAnalysis

Usually the analysis is done on Usually the analysis is done on E.g.E.g.

).()().()(ˆˆ

BrArBA

24

and with k replicates, the variance of the estimated difference is k/4 2

Using the linear mixed model, we see that the variance of one pair is

The optimal w is

The resulting variance for a single replicate is

and with k replicates, the variance of the estimated difference is

Reference Design – Optimal Reference Design – Optimal WeightsWeights

Consider using Consider using

ThenThen )).(()).((ˆˆluBr

wktAr

wBA

rGkiRkktirw wYY )).((

)/( 222

)/(24 2242 )/(2 224

)(/2/4 2242 kk )(/2 224 k

)/(24 2242min Var 22222 /22

)/( 222 optw

Reference Design – Optimal Reference Design – Optimal WeightsWeights

We do not know the optimal weights but

if we use mixed model ANOVA such as those available in SAS, Splus or R, the weights are approximated from the data – leading to more efficient computations.

Loop DesignsLoop Designs

A

C

B

D

A loop is balanced for dye effects and has two replicates at each node.

T treatments, 2k replicates, Tk arrays

Recall: for a reference design we get only k replicates on Tk arrays

Using optimal weighting

Var(A-B)=Var(A-D) =

Var(A-C)=

Both are smaller than the variance of the reference design with 4 arrays

Loop Designs T=4, 4 Loop Designs T=4, 4 arraysarrays

22222 2/ A

C

B

D

22222 /

22222 /22

Loop Designs T=4Loop Designs T=4

A

C

B

D

A

B

C

D

A

D

B

C

Design L4C Design L4B Design L4D

Loop Design – 3 loops = 6 replicates/treatments

3* L4C Var(A-B)=

Var(A-C)=

L4B+L4C+L4D

Var(difference) =

T=4, 12 arraysT=4, 12 arrays

22222 6/3/

Reference Design – 3 replicates/treatment

Var(difference) =

)(3/23/2 22222

22222 3/3/

22222 343/23/

Loop Design – 3 loops = 6 replicates/treatments

3* L4C Var(A-B)= 0.46

Var(A-C)= 0.58

L4B+L4C+L4D

Var(difference) = 0.47

T=4, 12 arraysT=4, 12 arraysAssuming Assuming

Reference Design – 3 replicates/treatment

Var(difference) = 0.83

3/ 22

2

22

2

Incorporating 2x2 FactorialIncorporating 2x2 Factorialin a Loop in a Loop

The design is 2 genotypes G,g and 2 tissuesT,tOnly within genotype and within tissue comparisons are of interest

GT

gt

gT

Gt

An 8 Treatment ExampleAn 8 Treatment ExampleA

C

B

DG

F E

H

An 8 Treatment ExampleAn 8 Treatment ExampleA

C

B

DG

F E

H

2 Complete Blocks

An 8 Treatment ExampleAn 8 Treatment ExampleA

C

B

DG

F E

H

Replication:

Yellow loop?

Red “loop”?

And now for the rest of And now for the rest of the storythe story

Missing arrays – Missing arrays – not fatal but not fatal but reduce reduce efficiencyefficiency

Added Added treatmentstreatments

A

C

B

D

A

C

B

D

E

And now for the rest of And now for the rest of the storythe story

Missing arrays – Missing arrays – not fatal but not fatal but reduce reduce efficiencyefficiency

Added Added treatmentstreatments

A

C

B

D

A

C

B

D

E

The Moral of the StoryThe Moral of the Story Loop designs are very efficientLoop designs are very efficient

Can incorporate factorial arrangementsCan incorporate factorial arrangements Can incorporate blocksCan incorporate blocks Can be replicated in various ways to Can be replicated in various ways to

improve efficiencyimprove efficiency Optimal design can help determine Optimal design can help determine

which (generalized) loop design to which (generalized) loop design to useuse

ANOVA-type analyses on the ANOVA-type analyses on the individual channels – not differencing individual channels – not differencing – should be used for analysis.– should be used for analysis.

C2

B2

A1

C1

B1

A2