Corrections and Normalization
in microarrays
data analysis
Mauro Delorenzi
Acknowledgments
Uni. Cal. Statistics Berkeley /
WEHI Bioinformatics
Terry Speed (Berkeley / WEHI)
Yee Hwa Yang (Berkeley)
Sandrine Dudoit (Stanford)
Ingrid Lönnstedt (Uppsala)
Yongchao Ge (Berkeley)
Natalie Thorne (WEHI)
Mauro Delorenzi (WEHI)
Collaborations with:
Peter Mac CI, Melb.
Brown-Botstein lab, Stanford
Matt Callow (LBNL)
CSIRO Image Analysis Group
Most slides
were taken
from our
collection
Biological question
Gene regulation
Class prediction
Experimental design
Microarray experiment
16-bit TIFF files
Image analysis
(Rfg, Rbg), (Gfg, Gbg)
Normalization
R, G
Estimation
Testing
Clustering
Biological verification
and interpretation
Discrimination
excitation
cDNA clones
(probes)
laser 2
PCR product amplification
purification
printing
scanning
laser 1
emission
mRNA target)
overlay images and normalise
0.1nl/spot
microarray
Hybridise
target to
microarray
analysis
Scanner's Spots
Part of the image of one channel false-coloured on a white (v. high) red (high)
through yellow and green (medium) to blue (low) and black scale.
Gene Expression Data
Gene expression data on p genes for n samples
Slides
Genes
1
2
3
4
5
slide 1
slide 2
slide 3
slide 4
slide 5
…
0.46
-0.10
0.15
-0.45
-0.06
0.30
0.49
0.74
-1.03
1.06
0.80
0.24
0.04
-0.79
1.35
1.51
0.06
0.10
-0.56
1.09
0.90
0.46
0.20
-0.32
-1.09
...
...
...
...
...
Gene expression level of gene 5 in slide 4 j
= Log2( Red intensity / Green intensity)
These values are conventionally displayed on a red (>0) yellow (0) green (<0) scale.
Some statistical questions
Image analysis: addressing, segmenting, quantifying
Normalisation: within and between slides
Quality: of images, of spots, of (log) ratios
Which genes are (relatively) up/down regulated?
Assigning p-values to tests / confidence to results
Planning of experiments: design, sample size
Discrimination and allocation of samples
Clustering, classification: of samples, of genes
Selection of genes relevant to any given analysis
Analysis of time course, factorial and other special experiments
……………………& more
I. The simplest problem is identifying differentially
expressed genes using one slide
• This is a common enough hope
• Efforts are frequently successful
• It is not hard to do by eye
• The problem is probably beyond formal statistical inference
(valid p-values, etc) for the foreseeable future.
Objectives
Important aspects of a statistical analysis include:
• Tentatively separating systematic sources of variation
("artefacts"), that bias the results, from random sources of
variation ("noise"), that hide the truth.
• Removing the former and quantifying the latter
• Identifying and dealing with the most relevant source of
variation in subsequent analyses
Only if this is done can we hope to make more or less valid
probability statements about the confidence in the results
Every Correction is a new source of variability. There is a trade-off
between gains and losses. The best method depends on the
characteristic of the data and this can vary.
Typical Statistical Approach
Measured value
= real value + systematic errors + noise
Corrected value
= real value
+ noise
• Analysis of Corrected value =>
(unbiased) CONCLUSIONS
• Estimation of Noise =>
quality of CONCLUSIONS, statistical significance
(level of confidence) of the conclusions
Step 1: Background Correction
Image Analysis => Rfg ; Rbg ; Gfg ; Gbg (fg = foreground, bg =
background.) For each spot on the slide we calculate
Red intensity = R = Rfg - Rbg
Green intensity = G = Gfg - Gbg
M = Log2( Red intensity / Green intensity)
Subtraction of background values (additive background model
assuming to be locally constant …)
Sources of background: probe unspecifically sticking on slide,
irregular / dirty slide surface, dust, noise in the scanner
measurement
Not included: real cross-hybridisation and unspecific
hybridisation to the probe
The intensity pairs (R, G) are highly processed data and the methods of
image processing and background correction of the laser scan images
can have a large impact. Before applying normalisation, inference,
cluster analysis and the like, it is important to identify and remove
systematic sources of variation such as due to different labeling
efficiencies and scanning properties of the two dyes or spatial
inhomogeneities.
With many different users and protocols, the portion of the variation due to
systematic effects can vary substantially.
There are many sources of systematic variation which affect the measured
gene expression levels. Normalisation is the term used to describe the
process of re moving such variation.
Until the variation is properly accounted for or modelled, there is no
question of the system being in statistical control and hence no basis for
a statistical model to describe chance variation.
Step 2: An M vs A (MVA) Plot
M = log R/G = logR - logG
Lowess
curve
blanks
Positive controls
Negative
controls
(spotted in varying concentrations)
A = ( logR + logG ) /2
A reminder on logarithms
A numerical example
Why use an M vs A plot ?
1. Logs stretch out region we are most interested in.
2. Can more clearly see features of the data such as intensity
dependent variation, and dye-bias.
3. Differentially expressed genes more easily identified.
4. Intuitive interpretation
MVA plot: looking at data 1
Spot identifier
Lowess curve
S1.n. Control Slide: Dye Effect, Spread.
MVA plot: looking at data 2
S1.p . Normalised data. Spread.
MVA plot: looking at data 3
S4. A-dependent variability.
MVA plot: analysing data 4
S17. Saturation
MVA plot: looking at data 5: Unique
effects of different scanners
Step 3: Normalisation - median
• Assumption: Changes roughly symmetric
• First panel: smooth density of log2G and log2R.
• Second panel: M vs A plot with median put to zero
Step 4: Normalisation - lowess
• Assumption: changes roughly symmetric at all intensities.
A hypothetical quantitative model
a. linear response
A realistic hypothetical quantitative model
b. power functionresponse
Median
Effect
Scale
Effect
Dye-Intensity
Effect
Step 5: Normalisation - between groups
Print-tip groups
• After within slide global lowess normalization.
• Likely to be a spatial effect.
Normalization between groups (ctd)
Print-tip groups
• After print-tip location- and scale- normalization.
Effects of
Location
Normalisati
on
(example)
Before
After
Step 6: Rescaling (Spread-Normalisation)
Assumption:
All (print-tip-)groups should have the same spread in M
True ratio is ij where i represents different (print-tip)-groups and j represents
different spots. Observed is Mij, where Mij = ai * log( ij)
Robust estimate of ai is
Corrected values are calculated as:
Illustration: print-tip-group - Normalisation
Assumption: For every print group:
changes roughly symmetric
at all intensities.
Glass Slide
Array of bound cDNA probes
4x4 blocks = 16 pin groups
Step 7: Assessing Significance
MVA-plot and critical curves
Newton’s, Sapir & Churchill’s and Chen’s single slide method
Other Approaches
These normalisation procedures are based on the assumption that spots are as likely
to be higher in the first or the second dye. They work well with a high number
of independent spots.
If (a few) genes were selected another approach might be needed.
For the correction of dye-effects we recommend to use either:
1.
Paired dye-swapped slides and/or
2.
Internal Controls as spikes or a dilution series
In the second case, instead of all genes only the control spots are used to compute
the corrections.
In the first case, the data from the two slides can be combined. Assuming identical
dye-intensity interactions in the two slides, the effect is corrected by taking:
A = 0,5 (A1 + A2)
M= 0,5 (M1 – M2)
This procedure is called self-normalisation, as it is done spot-by-spot. A number of
controls give indication if it is working well. It also deals with some artifacts that
cause some genes to be always higher in one dye than in the other.
II. The second simplest problem is identifying differentially
expressed genes using replicated slides
There are a number of different aspects:
• First, between-slide normalization; then
• What should we look at: averages, SDs t-statistics,
other summaries?
• How should we look at them?
• Can we make valid probability statements?
Selecting genes up/down regulated 1
•M
•t
•t M
Results from the Apo AI ko experiment
Selecting genes up/down regulated
Two samples.
e.g. KO vs. WT or
mutant vs. WT
n
T
C
n
For each gene form the t statistic:
average of n trt Ms
sqrt(1/n (SD of n trt Ms)2)
Two samples with a
reference (e.g.
pooled control)
T
C
n
n
C*
C*
• For each gene form the t statistic:
average of n trt Ms - average of n ctl Ms
sqrt(1/n (SD of n trt Ms)2 + (SD of n ctl Ms)2)
Which genes have changed?
When permutation testing is possible
1. For each gene and each hybridisation (8 ko + 8 ctl), use
M=log2(R/G).
2. For each gene form the t statistic:
average of 8 ko Ms - average of 8 ctl Ms
sqrt(1/8 (SD of 8 ko Ms)2 + (SD of 8 ctl Ms)2)
3. Form a histogram of 6,000 t values.
4. Do a normal Q-Q plot; look for values “off the line”.
5. Permutation testing.
6. Adjust for multiple testing.
Histogram & qq plot
ApoA1
Adjusted and Unadjusted p-values for the 50 genes
with the largest absolute t-statistics.
Which genes have changed?
When Permutation testing is not possible
Our current approach is to use M-averages, SDs, tstatistics and a new statistic we call B, inspired by
empirical Bayes.
We hope in due course to calibrate B and use that as our
main tool.
Empirical Bayes log posterior odds ratio
 2a
2
2 
 s  M
 n

B  const  log
2
2a
M
  s2 


 n
1  nc 
•T
•B
•t  M
B
• t B
Remarks for multiarrays experiments
• Microarray experiments typically have thousands of genes,
but only few (1-10) replicates for each gene.
• Averages can be driven by outliers.
• Ts can be driven by tiny variances.
• B = LOR will, we hope
– use information from all the genes
– combine the best of M. and T
– avoid the problems of M. and T
Some web sites:
Technical reports, talks, software etc.
http://www.stat.berkeley.edu/users/terry/zarray/Html/
Especially:
Dudoit et al: “Statistical methods for …”
Yee Hwa Yang et al. “Normalization for cDNA Microarray Data”
Statistical software R “GNU’s S” http://lib.stat.cmu.edu/R/CRAN/
Packages within R environment:
-- Spot http://www.cmis.csiro.au/iap/spot.htm
-- SMA (statistics for microarray analysis)
http://www.stat.berkeley.edu/users/terry/zarray/Software
/smacode.html