Hybridization Design for 2-Channel Microarray Experiments Naomi S. Altman, Pennsylvania State University), NSF_RCN Meetings 04

Expt Design and Microarrays Microarrays are Microarrays are Expensive Expensive Noisy Noisy A perfect situation for optimal design A perfect situation for optimal design

Outline Designing a Microarray Study Designing a Microarray Study Reference Design Reference Design Loop Designs Loop Designs Replication Replication Optimal Design/Analysis Optimal Design/Analysis Incorporating Multiple Factors and Blocks Incorporating Multiple Factors and Blocks

Designing a Microarray Experiment Define objectives Define objectives Determine factors and treatments Determine factors and treatments Determine appropriate analysis method Determine appropriate analysis method Determine sample design (biological and technical replication) Determine sample design (biological and technical replication) Determine platform Determine platform Design spots for custom arrays Design spots for custom arrays Determine hybridization pairs Determine hybridization pairs Perform experiment Perform experiment

Designing a Microarray Experiment Define objectives Define objectives Determine factors and treatments Determine factors and treatments Determine appropriate analysis method Determine appropriate analysis method Determine sample design (biological and technical replication) Determine sample design (biological and technical replication) Determine platform Determine platform Design spots for custom arrays Design spots for custom arrays Determine hybridization pairs ← Determine hybridization pairs ← Perform experiment Perform experiment

Arrow Notation Introduced by Kerr and Churchill (2001) Each array is represented by an arrow. RedGreen

Reference Design Reference A B C D 4 arrays 1 sample/treatment 4 reference samples

Loop Design (Kerr and Churchill 2001) A C B D 4 arrays 2 samples/treatment

Replication Often there is confusion among: Biological replicates Technical replicates repeated samples split sample and relabel spot replication In this presentation: We consider only one spot/gene/array any technical replicates are averaged each sample is an independent biological replicate

Linear Mixed Model for Microarray Data is the response of the gene in one channel is the response of the gene in one channel is the mean response of the gene over all treatments, channels, arrays is the mean response of the gene over all treatments, channels, arrays is the effect of treatment i is the effect of treatment i the effect of dye j the 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 and includes biological variation, technical variation and random error is the random deviation from the other effects and includes biological variation, technical variation and random error

Linear Mixed Model for Microarray Data The 2 channels on a single spot are correlated → array should be treated as a random effect

Differencing Channels on an Array Often the difference between samples on a single array is the unit of analysis: 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 But there can be a large loss of information.

Var(  )=0.126 Var(M)=0.453 Drosophila arrays courtesy of Bryce MacIver, PSU

Reference Design The reference sample is the same biological material on every array T treatments, k replicates,kT arrays If there are technical dye-swaps, these are averaged to form 1 replicate. If all comparisons are between treatments, there is no need to dye-swap. If there are dye-swaps, these should be balanced by treatment.

Reference Design – Usual Analysis Usually the analysis is done on  E.g. and with k replicates, the variance of the estimated difference is 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 Weights Consider using  Then

Reference Design – Optimal Weights 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 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 arrays A C B D

Loop 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 arrays T=4, 12 arrays Reference Design – 3 replicates/treatment Var(difference) =

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 arrays Assuming T=4, 12 arrays Assuming Reference Design – 3 replicates/treatment Var(difference) = 0.83

An 8 Treatment Example A C B D G FE H

A C B D G FE H 2 Complete Blocks

An 8 Treatment Example A C B D G FE H Replication: Yellow loop? Red “loop”?

Incorporating 2x2 Factorial in a Loop GT gt gT Gt GT gT gt Gt Which Arrangement is Better?

Incorporating 2x2 Factorial in a Loop The contrasts of interest can be written (in terms of the means – not the observations) ½(A+B)- ½ (C+D) ½(A+D)-½ (B+C) ½(A+C)-½ (B+D) A C B D

Incorporating 2x2 Factorial in a Loop The optimal variances are: ½(A+B)-½ (C+D) ½(A+D)-½ (B+C) ½(A+C)-½ (B+D) A C B D

Incorporating 2x2 Factorial in a Loop GT gt gT Gt GT gT gt Gt Best arrangement for estimating interaction Best arrangement for estimating time main effect

And now for the rest of the story Missing arrays – not fatal but reduce efficiency Added treatments A C B D A C B D E

And now for the rest of the story Missing arrays – not fatal but reduce efficiency Added treatments A C B D A C B D E

Optimal Design? The loop design has not been shown to be optimal The loop design has not been shown to be optimal There are lots of other BIBDs for 2 samples/block There are lots of other BIBDs for 2 samples/block General BIBDs can be adapted as more channels become available General BIBDs can be adapted as more channels become available Loop designs are particularly appealing due to the dye balance and graphical representation Loop designs are particularly appealing due to the dye balance and graphical representation

The Moral of the Story Loop designs are very efficient Loop designs are very efficient Can incorporate factorial arrangements Can incorporate factorial arrangements Can incorporate blocks Can incorporate blocks Can be replicated in various ways to improve efficiency Can be replicated in various ways to improve efficiency Optimal design ideas can help determine which BIBD to use Optimal design ideas can help determine which BIBD to use ANOVA-type analyses on the individual channels – not differencing – should be used for analysis. ANOVA-type analyses on the individual channels – not differencing – should be used for analysis.

References Kerr and Churchill (2001), Experimental design for gene expression microarrays, Biostatistics, 2: Kerr and Churchill (2001), Experimental design for gene expression microarrays, Biostatistics, 2: Kerr (2003) Design Considerations for efficient and effective microarray studies, Biometrics, 59: Kerr (2003) Design Considerations for efficient and effective microarray studies, Biometrics, 59: Yang and Speed (2002) Design Issues for cDNA Microarray Experiments Nature Reviews Genetics 3, Yang and Speed (2002) Design Issues for cDNA Microarray Experiments Nature Reviews Genetics 3,

C2 B2 A1 C1 B1 A2