1. Lecture 9
Microarray experiments
MA plots
Normalization of microarray data
Tests for differential expression of genes
Multiple testing and FDR

2. DNA Microarray
Typical microarray chip
Though most cells in an organism contain the same genes, not all of the genes are used in each cell.
Some genes are turned on, or "expressed" when needed in particular types of cells.
Microarray technology allows us to look at many genes at once and determine which are expressed in a particular cell type.

3. DNA Microarray
Typical microarray chip
DNA molecules representing many genes are placed in discrete spots on a microscope slide which are called probes.
Messenger RNA--the working copies of genes within cells is purified from cells of a particular type.
The RNA molecules are then "labeled" by attaching a fluorescent dye that allows us to see them under a microscope, and added to the DNA dots on the microarray.
Due to a phenomenon termed base-pairing, RNA will stick to the probe corresponding to the gene it came from

4. The probes are typically spaced widely along the sequence
DNA Microarray
Usually a gene is interrogated by 11 to 20 probes and usually each probe is a 25-mer sequence
The probes are typically spaced widely along the sequence
Sometimes probes are choosen closer to the 3’ end of the sequence
A probe that is exactly complementary to the sequence is called perfect match (PM)
A mismatch probe (MM) is not complementary only at the central position
In theory MM probes can be used to quantify and remove non specific hybridization
Source: PhD thesis by Benjamin Milo Bolstad, 2004, University of California, Barkeley

5. Sample preparation and hybridization
Source: PhD thesis by Benjamin Milo Bolstad, 2004, University of California, Barkeley

6. This may fall prey to spatial defects
Sample preparation and hybridization
During the hybridization process cRNA binds to the array
Earlier probes had all the probes of a probset located continuously on the array
This may fall prey to spatial defects
Newer chips have all the probes spread out across the array
A PM and MM probe pair are always adjacent on the array
Source: PhD thesis by Benjamin Milo Bolstad, 2004, University of California, Barkeley

7. Samples can be taken at different stages of the growth curve
Growth curve of bacteria
Samples can be taken at different stages of the growth curve
One of them is considered as control and others are considered as targets
Samples can be taken before and after application of drugs
Sample can be taken under different experimental conditions e.g. starvation of some metabolite or so
What types of samples should be used depends on the target of the experiment at hand.

8. DNA Microarray
Typical microarray chip
After washing away all of the unstuck RNA, the microarray can be observed under a microscope and it can be determined which RNA remains stuck to the DNA spots
Microarray technology can be used to learn which genes are expressed differently in a target sample compared to a control sample (e.g diseased versus healthy tissues)
However background correction and normalization are necessary before making useful decisions or conclusions

9. The M stands for minus and A stands for add
MA plots
MA plots are typically used to compare two color channels, two arrays or two groups of arrays
The vertical axis is the difference between the logarithm of the signals(the log ratio) and the horizontal axis is the average of the logarithms of the signals
The M stands for minus and A stands for add
MA is also mnemonic for microarray
Mi= log(Xij) - log(Xik) = Log(Xij/Xik) (Log ratio)
Ai=[log(Xij) + log(Xik)]/2 (Average log intensity)

10. A typical MA plot
From the first plot we can see differences between two arrays but the non linear trend is not apparent
This is because there are many points at low intensities compared to at high intensities
MA plot allows us to assess the behavior across all intensities

11. Normalization of microarray data
Normalization is the process of removing unwanted non-biological variation that might exist between chips in microarray experiments
By normalization we want to remove the non-biological variation and thus make the biological variations more apparent.

12. Typical microarray data
・・・
Array j
Array m
Gene 1
X11
X12
X1j
X1m
Gene 2
X21
X22
X2j
X2m
Gene i
Xi1
Xi2
Xij
Xim
Gene n
Xn1
Xn2
Xnj
Xnm
Mean X1 X2 Xj Xm SD σ1 σ2 σj σm

13. Sij = Xij － Xj Cij = ( Xij － Xj ) / σj
Normalization within individual arrays
Array 1
Array 2
・・・
Array j
Array m
Gene 1
X11
X12
X1j
X1m
Gene 2
X21
X22
X2j
X2m
Gene i
Xi1
Xi2
Xij
Xim
Gene n
Xn1
Xn2
Xnj
Xnm
Mean X1 X2 Xj Xm SD σ1 σ2 σj σm
Sij = Xij － Xj
Scaling:
Cij = ( Xij － Xj ) / σj
Centering:

14. Effect of Scaling and centering normalization
Original Data
Centering
Scaling
Mean = 0
Standard deviation = 1
Mean = 0

15. Normalization between a pair of arrays: Loess(Lowess) Normalization
Lowess normalization is separately applied to each experiment with two dyes
This method can be used to normalize Cy5 and Cy3 channel intensities (usually one of them is control and the other is the target) using MA plots

16. Normalization between a pair of arrays: Loess(Lowess) Normalization
Genei-1
Ci-1
Ti-1
Genei Ci Ti
Genei+1
Ci+1
Ti+1
2 channel data
Mi=Log(Ti/Ci) (Log ratio)
Ai=[log(Ti) + log(Ci)]/2 (Average log intensity)
Mi=Log(Ti/Ci)
Each point corresponds to a single gene
Ai=[log(Ti) + log(Ci)]/2

17. Normalization between a pair of arrays: Loess(Lowess) Normalization
Mi=Log(Ti/Ci) (Log ratio)
Ai=[log(Ti) + log(Ci)]/2 (Average log intensity)
Mi=Log(Ti/Ci)
Each point corresponds to a single gene
Typical regression line
The MA plot shows some bias
Ai=[log(Ti) + log(Ci)]/2

18. Normalization between a pair of arrays: Loess(Lowess) Normalization
Mi=Log(Ti/Ci) (Log ratio)
Ai=[log(Ti) + log(Ci)]/2 (Average log intensity)
Mi=Log(Ti/Ci)
Each point corresponds to a single gene
The MA plot shows some bias
Usually several regression lines/polynomials are considered for different sections
Ai=[log(Ti) + log(Ci)]/2
The final result is a smooth curve providing a model for the data. This model is then used to remove the bias of the data points

19. Normalization between a pair of arrays: Loess(Lowess) Normalization
Bias reduction by lowess normalization

20. Normalization between a pair of arrays: Loess(Lowess) Normalization
Unnormalized fold changes
fold changes after Loess normalization

21. Normalization across arrays
Here we are discussing the following two normalization procedure applicable to a number of arrays
Quantile normalization
Baseline scaling normalization

22. Quantile normalization
Normalization across arrays
The goal of quantile normalization is to give the same empirical distribution to the intensities of each array
If two data sets have the same distribution then their quantile- quantile plot will have straight diagonal line with slope 1 and intercept 0.
Or projecting the data points of the quantile- quantile plot to 45-degree line gives the transformation to have the same distribution.
Quantile normalization
quantile- quantile plot motivates the quantile normalization algorithm

23. Quantile normalization Algorithm
Normalization across arrays
Quantile normalization Algorithm
Source: PhD thesis by Benjamin Milo Bolstad, 2004, University of California, Barkeley

24. Quantile Normalization:
Normalization across arrays
Quantile Normalization:
Original data
No.
Exp.1
Exp.2
Mean 5
0.4
0.8
0.6 = ( )/2 2
0.6 1
1.2
0.9 4 3
1.8
1.3
1.6
2.8
2.2
3.8
No.
Exp.1
Exp.2 1
1.6
1.2 2
0.6
2.8 3
1.8 4
0.8
3.8 5
0.4
Sort
1. Sort each column of X (values)
2. Take the means across rows of X sort
No.
Exp.1
Exp.2 5
0.6 2
0.9 1 4
1.3 3
2.2
2.8
No.
Exp.1
Exp.2 1
2.2
0.9 2 3
2.8
1.3 4 5
0.6
Sort
3. Assign this mean to each element
in the row to get X' sort
4. Get X normalized by rearranging each column of X'
sort to have the same ordering as original X

25. Normalization across arrays
Raw data
After quantile normalization

26. Baseline scaling method
Normalization across arrays
Baseline scaling method
In this method a baseline array is chosen and all the arrays are scaled to have the same mean intensity as this chosen array
This is equivalent to selecting a baseline array and then fitting a linear regression line without intercept between the chosen array and every other array

27. Baseline scaling method
Normalization across arrays
Baseline scaling method

28. Normalization across arrays
Raw data
After Baseline scaling normalization

29. Tests for differential expression of genes
Let x1…..xn and y1…yn be the independent measurements of the same probe/gene across two conditions.
Whether the gene is differentially expressed between two conditions can be determined using statistical tests.

30. Tests for differential expression of genes
Important issues of a test procedure are
Whether the distributional assumptions are valid
Whether the replicates are independent of each other
Whether the number of replicates are sufficient
Whether outliers are removed from the sample
Replicates from different experiments should not be mixed since they have different characteristics and cannot be treated as independent replicates

31. Tests for differential expression of genes
Most commonly used statistical tests are as follows:
(a) Student’s t-test
(b) Welch’s test
(c) Wilcoxon’s rank sum test
(d) Permutation tests
The first two test assumes that the samples are taken from Gaussian distributed data and the p-values are calculated by a probability distribution function
The later two are nonparametric and the p values are calculated using combinatorial arguments.

32. Student’s t-test
Assumptions: Both samples are taken from Gaussian distribution that have equal variances
Degree of freedom: m+n-2
Welch’s test is a variant of t-test where t is calculated as follows
Welch’s test does not assume equal population variances

33. The value of t is supposed to follow a t-distribution.
Student’s t-test
The value of t is supposed to follow a t-distribution.
After calculating the value of t we can determine the p-value from the t distribution of the corresponding degree of freedom

34. Wilcoxon’s rank sum test
Let x1…..xn and y1…ym be the independent measurements of the same probe/gene across two conditions.
Consider the combined set x1…..xn ,y1…ym
The test statistic of Wilcoxon test is
Where is the rank of xi in the combined series
Possible Minimum value of T is
Possible Maximum value of T is
Minimum and maximum values of T occur if all X data are greater or smaller than the Y data respectively i.e. if they are sampled from quite different distributions

35. Expected value and variance of T under null hypothesis are as follow:
Now unusually low or high values of T compared to the expected value indicate that the null hypothesis should be rejected i.e. the samples are not from the same population
For larger samples i.e. m+n >25 we have the following approximation

36. Wilcoxon’s rank sum test (Example)
Data x1 7 x2 8 x3 5 x4 9 x5 Y
Data y1 5 y2 6 y3 8 y4 4
X&Y
Data
Rank x4 9 1 x2 8 2 y3 3 x5 7 4 x1 5 y2 6 y1 x3 y4
n=5. m=4
T=R(x1)+R(x2)+R(x3)+R(x4)+R(x5)
= = 20
EH0(T)=n(m+n+1)/2= 5(4+5+1)/2=25
VarH0(T)=mn(m+n+1)/12= 5*4(4+5+1)/12=50/3=16.66
P-value = (From chart)

37. Example

38. Multiple testing and FDR
The single gene analysis using statistical tests has a drawback.
This arises from the fact that while analyzing microarray data we conduct thousands of tests in parallel.
Let we select genes with a significant level α=0.05 i.e a false positive rate of 5%
This means we expect that 500 individual tests are false which is not at logical
Therefore corrections for multiple testing are applied while analyzing microarray data

39. Multiple testing and FDR
Let αg be the global significance level and αs is the significance level at single gene level
In case of a single gene the probability of making a correct decision is
Therefore the probability of making correct decision for all n genes (i.e. at global level)
Now the probability of drawing the wrong conclusion in either of n tests is
For example if we have 100 different genes and αs=0.05
the probability that we make at least 1 error is this is very high and this is called family-wise error rate (FWER)

40. Multiple testing and FDR
Using binomial expansion we can write
Thus
Therefore the Bonferroni correction of the single gene level is the global level divided by the number of tests
Therefore for FWER of for n= genes the P-value at single gene level should be 10-6
Usually very few genes can meet this requirement
Therefore we need to adjust the threshold p-value for the single gene case.

41. Multiple testing and FDR
A method for adjusting p-value is given in the following paper
Westfall P. H. and Young S. S. Resampling based multiple testing : examples and methods for p-value adjustment(1993), Wiley, New York

42. Multiple testing and FDR
An alternative to controlling FWER is the computation of false discovery rate(FDR)
The following papers discuss about FDR
Storey J. D. and Tibshirani R. Statistical significance for genome wise studies(2003), PNAS 100,
Benjamini Y and Hochberg Y Controlling the false discovery rate : a practical and powerful approach to multiple testing(1995) J Royal Statist Soc B 57,
Still the practical use of multiple testing is not entirely clear.
However it is clear that we need to adjust the p-value at single gene level while testing many genes together.