1. Corrections and Normalization in microarrays data analysis
Mauro Delorenzi

2. Acknowledgments Terry Speed (Berkeley / WEHI) Yee Hwa Yang (Berkeley)
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)
Most slides were taken from our collection
Collaborations with:
Peter Mac CI, Melb.
Brown-Botstein lab, Stanford
Matt Callow (LBNL)
CSIRO Image Analysis Group

3. 16-bit TIFF files (Rfg, Rbg), (Gfg, Gbg) R, G 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
Discrimination
Biological verification
and interpretation

4. overlay images and normalise
excitation
scanning
cDNA clones
(probes)
laser 2
laser 1
emission
PCR product amplification
purification
printing
mRNA target)
overlay images and normalise
0.1nl/spot
Hybridise target to microarray
microarray
analysis

5. 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.

6. Gene Expression Data slide 1 slide 2 slide 3 slide 4 slide 5 …
Gene expression data on p genes for n samples
Slides
slide 1 slide 2 slide 3 slide 4 slide 5 …
Genes 3
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.

7. 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 4

8. 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. 4

9. 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. 4

10. 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 4

11. 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 4

12. 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. 4

13. Step 2: An M vs A (MVA) Plot
M = log R/G = logR - logG
Lowess curve
blanks
Positive controls
(spotted in varying concentrations)
Negative controls
A = ( logR + logG ) /2

14. A reminder on logarithms

15. A numerical example

16. Why use an M vs A plot ?
Logs stretch out region we are most interested in.
Can more clearly see features of the data such as intensity dependent variation, and dye-bias.
Differentially expressed genes more easily identified.
Intuitive interpretation

17. MVA plot: looking at data 1
Spot identifier
Lowess curve
S1.n. Control Slide: Dye Effect, Spread.

18. MVA plot: looking at data 2
S1.p . Normalised data. Spread.

19. MVA plot: looking at data 3
S4. A-dependent variability.

20. MVA plot: analysing data 4
S17. Saturation

21. MVA plot: looking at data 5: Unique effects of different scanners

22. Normalisation - Median
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

23. Step 4: Normalisation - lowess
Assumption: changes roughly symmetric at all intensities.

24. A hypothetical quantitative model
a. linear response

25. A realistic hypothetical quantitative model
b. power function-response
Median Effect
Scale Effect
Dye-Intensity Effect

26. Step 5: Normalisation - between groups
Log-ratios
Print-tip groups
After within slide global lowess normalization.
Likely to be a spatial effect.

27. Normalization between groups (ctd)
Log-ratios
Print-tip groups
After print-tip location- and scale- normalization.

28. Effects of Location Normalisation (example)
Before
After

29. Taking varying scale into account
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:

30. 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

31. Step 7: Assessing Significance
MVA-plot and critical curves Newton’s, Sapir & Churchill’s and Chen’s single slide method

32. 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:
Paired dye-swapped slides and/or
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. 4

33. 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? 4

34. Selecting genes up/down regulated 1M t
t M
Results from the Apo AI ko experiment

35. Which genes are (relatively) up/down regulated?
Selecting genes up/down regulated
Two samples.
e.g. KO vs. WT or mutant vs. WT
Two samples with a reference (e.g. pooled control) T C
 n T C*
 n C
For each gene form the t statistic:
average of n trt Ms
sqrt(1/n (SD of n trt Ms)2)
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)

36. 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. 9

37. Histogram & qq plot
ApoA1

38. Adjusted and Unadjusted p-values for the 50 genes with the largest absolute t-statistics.

39. Which genes have changed? When Permutation testing is not possible
Our current approach is to use M-averages, SDs, t-statistics 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 9

40. T B
t  M B
t B

41. 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

42. Some web sites:
Technical reports, talks, software etc.
Especially:
Dudoit et al: “Statistical methods for …”
Yee Hwa Yang et al. “Normalization for cDNA Microarray Data”
Statistical software R “GNU’s S”
Packages within R environment:
-- Spot
-- SMA (statistics for microarray analysis) /smacode.html