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Tom Kepler
Santa Fe Institute
Normalization and Analysis
of
DNA Microarray Data
by
Self-Consistency
and
Local Regression
[email protected]
Rat mesothelioma cells
control
Rat mesothelioma cells
treated with KBrO2
Normalization
Method to be improved:
1. Assume that some genes will not
change under the treatment under
investigation.
2. Identify these core genes in advance
of the experiment.
3. Normalize all genes against these
genes assuming they do not change
Normalization
New Method:
1. Assume that some genes will not
change under the treatment under
investigation.
2. Choose these core genes arbitrarily.
3. Normalize (provisionally) all genes
against these genes assuming they do
not change.
4. Determine which genes do not
change under this normalization.
5. Make this set the new core. If this
core differs from the previous core, go
to 3. Else, done.
Error Model
I  c[mRNA]
I = spot intensity
[mRNA] = concentration of specific mRNA
c = normalization constant
Error Model
I  c[mRNA]
I = spot intensity
[mRNA] = concentration of specific mRNA
c = normalization constant
 = lognormal multiplicative error
Error Model
Iijk  cij [mRNA]ik ijk
I = spot intensity
[mRNA] = concentration of specific mRNA
c = normalization constant
 = lognormal multiplicative error
index 1, i: treatment group
index 2, j: replicate within treatment
index 3, k: spot (gene)
Yijk  log( Iijk )  log(cij )  log([mRNA]ik )  log(ijk )
Yijk  ij  (k  ik )  ijk
Y = log spot intensity
 = mean log concentration of specific mRNA
 = treatment effect (conc. specific mRNA)
 = normalization constant
 = normal additive error
index 1, i: treatment group
index 2, j: replicate within treatment
index 3, k: spot (gene)
Yijk  ij  (k  ik )  ijk
Model:
Identifiability constraints:
 k  0
k
 niik  0
i
Estimate by ordinary least squares:
xk  Y k  Y
aij  i   Yij 
dik  i   Yi  k  Y k  Yi   Y
Yijk  ij  (k  ik )  ijk
Model:
Identifiability constraints:
 k  0
k
 niik  0
i
But note: cannot identify between  and 
Self-consistency:
 wk ( )ik  0
k
The weight wk() is small if the kth
gene is judged to be changed; close to
one if it is judged to be unchanged.
Procedure is iterative.
log intensity, array 2
6
4
2
0
-2
-2
0
2
4
log intensity, array 1
6
log intensity, array 2
6
4
2
0
-2
-2
0
2
4
log intensity, array 1
6
Failure of Model
Generalized Model
Yijk  ij (k )  (k  ik )   ij (k )ijk
The normalization ij(k) and the heteroscedasticity
function ij(k) are slowly varying functions
of the intensity, .
Estimate by Local Regression
Local Regression
data
Predict value at x=50: weight, linear regression
Predict whole function similarly
Compare to known true function
Simulation-based Validation
1. Reproduce observed bias.
Simulation-based Validation
2. Reproduce observed heteroscedasticity.
Test based on z statistic:
d 2 k  d1k
zk 
1 1
sk

n1 n2
Choice of significance level:
expected number of false positives:
E(false positives) =  N
But minimum detectable difference increases as
 gets smaller

E(fp) min diff
min ratio
0.05
0.01
0.001
0.0001
250
50
5
0.5
2.5
3
3.6
5
0.916
1.09
1.29
1.61
bias
“-fold change”
Proportion changed spots
Validation of method against simulated data
3. Hypothesis testing: Simulated from stated model
“rate false pos.” = mean observed / expected
Simulated data:
mis-specified model — multiplicative + additive noise
bias
“-fold change”
Proportion changed spots
Validation of method against simulated data
4. Hypothesis testing: Simulated from “wrong” model:
additive + multiplicative noise.
Acknowledgments
Lynn Crosby
North Carolina State University
Kevin Morgan
Strategic Toxicological Sciences
GlaxoWellcome
Santa Fe Institute
www.santafe.edu
postdoctoral fellowships available
(apply before the end of the year)
[email protected]