Lecture 8 Microarray experiments MA plots Normalization of microarray data

Lecture 8

Microarray experiments

MA plots

Normalization of microarray data

Tests for differential expression of genes

Multiple testing and FDR

•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.

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

DNA Microarray


Source: PhD thesis by Benjamin Milo Bolstad, 2004, University of California, Barkeley

Usually a gene is interrogated by 11 to 20 probes and usually each probe is a 25-mer sequenceThe probes are typically spaced widely along the sequenceSometimes probes are choosen closer to the 3’ end of the sequenceA probe that is exactly complementary to the sequence is called perfect match (PM)A mismatch probe (MM) is not complementary only at the cemtral positionIn theory MM probes can be used to quantify and remove non specific hybridization

DNA Microarray

Sample preparation and hybridization



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 arrayThis may fall prey to spatial defectsNewer chips have all the probes spread out across the arrayA PM and MM probe pair are always adjacent on the array

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.

•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

DNA Microarray


MA plots

MA plots are typically used to compare two color channels, two arrays or two groups of arraysThe 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 signalsThe M stands for minus and A stands for addMA is also mnemonic for microarray

Mi= log(Xij) - log(Xik) = Log(Xij/Xik) (Log ratio)

Ai=[log(Xij) + log(Xik)]/2 (Average log intensity)

A typical MA plot

From the first plot we can see differences between two arrays but the non linear trend is not apparentThis is because there are many points at low intensities compared to at high intensitiesMA plot allows us to assess the behavior across all intensities

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.

Array 1 Array 2 ・・・

Array j ・・・

Array m

Gene 1 X11 X12 X1j X1m


・・・Gene i Xi1 Xi2 Xij Xim

・・・Gene n Xn1 Xn2 Xnj Xnm

Mean X1 X2 Xj Xm

SD σ1 σ2 σj σm

Typical microarray data

Array 1 Array 2 ・・・

Array j ・・・

Array m



・・・Gene i Xi1 Xi2 Xij Xim

・・・Gene n Xn1 Xn2 Xnj Xnm

Mean X1 X2 Xj Xm

SD σ1 σ2 σj σm

Normalization within individual arrays

Scaling: Sij = Xij － Xj

Centering: Cij = ( Xij － Xj ) / σj

Original Data

Scaling Centering

Effect of Scaling and centering normalization

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

Genei-1 Ci-1 Ti-1

Genei Ci Ti

Genei+1 Ci+1 Ti+1

Mi=Log(Ti/Ci) (Log ratio)

Ai=[log(Ti) + log(Ci)]/2 (Average log intensity)

Mi=

Log(

Ti/C

i)

Ai=[log(Ti) + log(Ci)]/2

Each point corresponds to a single gene

2 channel data




Mi=

Log(

Ti/C

i)



The MA plot shows some bias

Typical regression line




Mi=

Log(

Ti/C

i)



The MA plot shows some bias

Usually several regression lines/polynomials are considered for different sections

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


Bias reduction by lowess normalization


Unnormalized fold changes

fold changes after Loess normalization


Normalization across arrays

Here we are discussing the following two normalization procedure applicable to a number of arrays

1.Quantile normalization2.Baseline scaling normalization

Quantile normalization The goal of quantile normalization is to give the same empirical distribution to the intensities of each arrayIf 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- quantile plot motivates the quantile normalization algorithm


Quantile normalization Algorithm



No. Exp.1 No. Exp.2

1 1.6 1 1.2

2 0.6 2 2.8

3 1.8 3 1.8

4 0.8 4 3.8

5 0.4 5 0.8

No. Exp.1 No. Exp.2 Mean

5 0.4 5 0.8 0.6 = (0.4+0.8)/2

2 0.6 1 1.2 0.9

4 0.8 3 1.8 1.3

1 1.6 2 2.8 2.2

3 1.8 4 3.8 2.8

No. Exp.1 No. Exp.2

5 0.6 5 0.6

2 0.9 1 0.9

4 1.3 3 1.3

1 2.2 2 2.2

3 2.8 4 2.8

No. Exp.1 No. Exp.2

1 2.2 1 0.9

2 0.9 2 2.2

3 2.8 3 1.3

4 1.3 4 2.8

5 0.6 5 0.6

Original data

4. Get X normalized by rearranging each column of X' sort to have the same ordering as original X

1. Sort each column of X (values)2. Take the means across rows of X sort

3. Assign this mean to each elementin the row to get X' sort

Sort

Sort

Quantile Normalization:Normalization across arrays

Raw data

After quantile normalization


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


Baseline scaling methodNormalization across arrays

After Baseline scaling normalization

Raw data



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.

Important issues of a test procedure are(a)Whether the distributional assumptions are valid(b)Whether the replicates are independent of each other(c)Whether the number of replicates are sufficient(d)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


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 functionThe later two are nonparametric and the p values are calculated using combinatorial arguments.


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

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

Wilcoxon’s rank sum testLet 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

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

X Data

x1 7

x2 8

x3 5

x4 9

x5 7

Y Data

y1 5

y2 6

y3 8

y4 4

X&Y Data Rank

x4 9 1

x2 8 2

y3 8 3

x5 7 4

x1 7 5

y2 6 6

y1 5 7

x3 5 8

y4 4 9

Wilcoxon’s rank sum test (Example)

n=5. m=4

T=R(x1)+R(x2)+R(x3)+R(x4)+R(x5)=5+2+8+1+4= 20EH0(T)=n(m+n+1)/2= 5(4+5+1)/2=25VarH0(T)=mn(m+n+1)/12= 5*4(4+5+1)/12=50/3=16.66

P-value = .1112 (From chart)

Example


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 10000 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

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=0.05the probability that we make at least 1 error is 0.994 ---this is very high and this is called family-wise error rate (FWER)


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 0.01 for n= 10000 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.


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


An alternative to controlling FWER is the computation of false discovery rate(FDR)

The following papers discuss about FDRStorey J. D. and Tibshirani R. Statistical significance for genome wise studies(2003), PNAS 100, 9440-9445

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, 289-300

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.


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Lecture 8 Microarray experiments MA plots Normalization of microarray data