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How many mice and how many arrays? Replication in cDNA microarray - - PowerPoint PPT Presentation
How many mice and how many arrays? Replication in cDNA microarray - - PowerPoint PPT Presentation
How many mice and how many arrays? Replication in cDNA microarray experiment Xiangqin Cui The Jackson Laboratory Data preprocess: Download data Trim to 5507 clones in all organs Extract foreground only log 2 transformation Intensity-Lowess
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Mixed linear model for each gene:
2 m
σ
ijk k j i ijk
R A D M u y ε + + + + + =
Log2(signal) Gene mean (fixed) Mouse effect (random ) Dye effect (fixed) Array effect (random ) Reference mean (fixed)
- Meas. error
(random )
2 a
σ
2 e
σ
) , ( ~
2 m i
N M σ
) , ( ~
2 e ijk
N σ ε
) , ( ~
2 a k
N A σ
Assumptions:
lijk k j i l lijk
R A D M O u y ε + + + + + + =
Organ (fixed effect) Combined data: Individual organ:
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Estimated variance components from kidney
Significant gene
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Estimated variance components from all organs
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Gallstone Brain cortex
Variance components from two other data sets:
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MSE for treatment effects:
mnr mnr mn m MSE
e s a m 2 2 2 2
σ σ σ σ + + + =
m : number of mice per treatment n : number of arrays per mouse r: number of spots for each clone on array
Reference Design
Trt A Trt B m1 m2 m3 m4 ( m = 2 ) R R R R ( n = 2 )
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4 mice / trt, 2 arrays / mouse 4 mice / trt, 4 arrays / mouse 2 mice / trt, 2 arrays / mouse 2 mice / trt, 4 arrays / mouse 2 mice / trt, 6 arrays / mouse 4 mice / trt, 6 arrays / mouse
Power at different fold changes
1.5 fold change kidney
Log2(fold change)
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Power to detect 1.5 fold change in kidney
2 mice / trt 4 mice / trt 6 mice / trt 8 mice / trt 8 12 16 24 32
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Power to detect 1.5 fold change
2 mice / trt 4 mice / trt 6 mice / trt 8 mice / trt
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Resource allocation:
a m
nC m mC Cost ⋅ + =
m mice / trt n arrays / mouse r spot replicates on array Cm cost / mouse Ca cost / array
The optimum number of arrays per mouse:
a m m e s a
C C r r n ⋅ + + =
2 2 2 2
σ σ σ σ
spot variation
2 s
σ
mnr mnr mn m MSE
e s a m 2 2 2 2
σ σ σ σ + + + =
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Examples for resource allocation
Mouse price Array price # of arrays per mouse $15 $300 2 $300 $300 7 $1500 $300 16
- Based on the variance components estimated from Project Normal data.
- r = 1 ( no replicated spots on array )
- Reference design
More efficient array level designs, such as direct comparisons and loop designs, can reduce the optimum number of arrays per mouse.
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Pooling mice
2 2
1
m pool
k σ σ
α
=
k : pool size α : constant for the effect of pooling. 0 < α < 1 α = 0, pooling has no effect. α = 1, pooling has maximum effect. Pooling can reduce the mouse variance but not the technical variances
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Power increase to detect 1.5 fold change by pooling
2 mice / trt 4 mice / trt 6 mice / trt 8 mice / trt 2 pools / trt 4 pools / trt 6 pools / trt 8 pools / trt
( Pool size k = 3, α = 1 )
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2 mice / trt 4 mice / trt 6 mice / trt 8 mice / trt 2 pools / trt 4 pools / trt 6 pools / trt 8 pools / trt
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Conclusions
* Technical variation is larger than biological variation for
most genes.
* Increase of technical replication can improve the power of
the experiment effectively.
* Biological replication is essential for making broad-sense
- inferences. Increase of it is more effective in improving the
power of the experiment.
* Pooling can reduce biological variation. However, the
effect can be small due to relative small biological variation.
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