ANOVA and the Bootstrap - Computational Diagnostics Group

27 Pages·2003·0.05 MB·English

Practical microarray analysis – resampling and the bootstrap
ANOVA and the Bootstrap
Ulrich Mansmann
[email protected]
Practical microarray analysis
March 2003
Heidelberg
Heidelberg, March 2003 1
Practical microarray analysis – resampling and the bootstrap
Probe conservation and gene expression
Question of interest:
Does time between probe harvesting and probe hybridisation have a relevant
influence on gene expression?
Are there degradation effects or time pattern?
Data:
Probes of three patients were hybridised at day 0, 1, and 2
Nine Affimetrix – chips with 12625 genes each
Methodological approach:
Quantify time effects: general or gene-specific
Is there evidence for time effects?
How to quantify variability between patients?
How to quantify variability over time (within patients)?
How to operationalise the idea of relevant influence on gene expression)
Heidelberg, March 2003 2
Practical microarray analysis – resampling and the bootstrap
The one gene scenario - Table
time T T T
1 2 3
probe/patient
P X X X
1 11 12 13
P X X X
2 21 22 23
P X X X
3 31 32 33
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Practical microarray analysis – resampling and the bootstrap
The one gene scenario - Graph
Affi-ID 36518_at
0
5.
1
8
4.
1
6
al 4.
n 1
g
si
sf.
n 4
N tra 14.
S
V
2
4.
1
0
4.
1
8
3.
1
1.0 1.5 2.0 2.5 3.0
Time
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Practical microarray analysis – resampling and the bootstrap
First ideas on variability
3 3 3 3
= (cid:229) (cid:229) - 2 1 (cid:229) (cid:229)
Quantifying variability: SS (x x) with x = x , the global mean
ij ij
9
= = = =
i 1j 1 i 1j 1
Separate variability between patients from the variability of individual time
courses (within patients)
3 3
= (cid:229) (cid:229) - 2
SS (x x)
ij
= =
i 1j 1
3 3 3
(cid:229) (cid:229) - + - 2 1 (cid:229)
= (x x x x) with x = x mean patient value
ij i i i ij
3
= = =
i 1j 1 j 1
3 3 3
(cid:229) (cid:229) - 2 (cid:229) - 2
= (x x ) + 3 (x x)
ij i i
= = =
i 1j 1 i 1
within patient variability between patient variability
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Practical microarray analysis – resampling and the bootstrap
ANOVA – Analysis of variance (1)
Analysis of variance(ANOVA) studies differences between then mean values
of normal distributed data from groups.
We are interested in an analysis of y = x - x , time course measurements
ij ij i
adjusted for individual probe levels
The biological variability between individual probes is not of interest for the next
step and is therefore separated.
We only look at individually adjusted time courses.
3
(cid:229)
Important boundary condition: y = 0 for all i (i = 1, 2, 3)
ij
=
j 1
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Practical microarray analysis – resampling and the bootstrap
Affi-ID 36518_at
8
0.
6
0.
4
d) 0.
e
st
u
dj
a 2
al ( 0.
n
g
si
ansf. 0.0
N tr
S
V
2
0.
-
4
0.
-
1.0 1.5 2.0 2.5 3.0
Time
Is there evidence for a systematic pattern?
What is a systematic pattern?
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Practical microarray analysis – resampling and the bootstrap
ANOVA – Analysis of variance (2)
A formal view on the data of patient i (probe i)
(cid:230) 1 (cid:246)
(cid:230) 1 (cid:246) (cid:231) (cid:247)
(cid:230) y (cid:246) (cid:231) (cid:247) (cid:231) 6 (cid:247) (cid:230) e (cid:246)
(cid:231) i1 (cid:247) (cid:231) 2 (cid:247) (cid:231) i1 (cid:247)
(cid:231) 2 (cid:247)
= (cid:215) + (cid:215) - + e
(cid:231) y (cid:247) a (cid:231) 0 (cid:247) b (cid:231) (cid:247)
i2 (cid:231) (cid:247) i2
(cid:231) (cid:247) (cid:231) 1 (cid:247) 6 (cid:231) (cid:247)
Ł y ł (- 1)(cid:215) (cid:231) (cid:247) Ł e ł
(cid:231) (cid:247) 1
i3 i3
Ł 2 ł (cid:231) (cid:247)
Ł 6 ł
A sequence of three measurements with mean 0 can be decomposed
into a linear term, a quadratic term, and residuals.
How to calculate the coefficients a and b?
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Practical microarray analysis – resampling and the bootstrap
ANOVA – Analysis of variance (3)
y = a 2-1/2 + b 6-1/2 + e
11 11
y = b (-2)(cid:215) 6-1/2 + e
12 12
y = a (-2-1/2) + b 6-1/2 + e
13 13
y = a 2-1/2 + b 6-1/2 + e
21 21
y = b (-2)(cid:215) 6-1/2 + e
22 22
y = a (-2-1/2) + b 6-1/2 + e
23 23
y = a 2-1/2 + b 6-1/2 + e
31 31
y = b (-2)(cid:215) 6-1/2 + e
32 32
y = a (-2-1/2) + b 6-1/2 + e
33 33
Least square estimates for coefficients:
3 3
2 6
(cid:229) - (cid:229) + -
a = (y y ) and b = (y y y )
(cid:215) i1 i3 (cid:215) (cid:215) i1 i3 i2
3 2 3 4
= =
i 1 i 1
e
Residuals are calculated by taking the difference
between observed and model based values.
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Practical microarray analysis – resampling and the bootstrap
ANOVA – Analysis of variance (4)
The final decomposition of the variance
3 3 3 3 3
= (cid:229) (cid:229) - 2 (cid:229) (cid:229) - 2 (cid:229) - 2
SS (x x) = (x x ) + 3 (x x) =
ij ij i i
= = = = =
i 1j 1 i 1j 1 i 1
3 3 3
(cid:229) (cid:229) e 2 + 2 2 (cid:229) - 2
( ) 3a + 3b + 3 (x x)
ij i
= = =
i 1j 1 i 1
unexplained model based patient level
variability variability variability
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