Transcript Slide 1

Lecture 8
Microarray experiments
MA plots
Normalization of microarray data
Tests for differential expression of genes
Multiple testing and FDR
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.
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
Usually a gene is interrogated by 11 to 20 probes and usually each probe is a 25mer 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 cemtral 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
Sample preparation and hybridization
Source: PhD thesis by Benjamin Milo Bolstad, 2004, University of California, Barkeley
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
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.
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
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)
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
Normalization of microarray data
Normalization is the process of removing unwanted nonbiological 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.
Typical microarray data
・・・
Array j
・・・
Array 1
Array 2
Array m
Gene 1
X11
X12
X1j
X1m
Gene 2
X21
X22
X2j
X2m
Xi1
Xi2
Xij
Xim
Gene n
Xn1
Xn2
Xnj
Xnm
Mean
X1
X2
Xj
Xm
SD
σ1
σ2
σj
σm
・・・
Gene i
・・・
Normalization within individual arrays
Array 1 Array 2 ・・・ Array j
・・・
Array m
Gene 1
X11
X12
X1j
X1m
Gene 2
X21
X22
X2j
X2m
Xi1
Xi2
Xij
Xim
Gene n
Xn1
Xn2
Xnj
Xnm
Mean
X1
X2
Xj
Xm
SD
σ1
σ2
σj
σm
・・・
Gene i
・・・
Scaling:
Centering:
Sij = Xij － Xj
Cij = ( Xij － Xj ) / σj
Effect of Scaling and centering normalization
Original Data
Scaling
Centering
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
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)
Mi=Log(Ti/Ci)
Ai=[log(Ti) + log(Ci)]/2 (Average log intensity)
Each point corresponds to a
single gene
Ai=[log(Ti) + log(Ci)]/2
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
Typical regression line
Ai=[log(Ti) + log(Ci)]/2
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
Ai=[log(Ti) + log(Ci)]/2
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
Normalization between a pair of arrays: Loess(Lowess)
Normalization
Bias reduction by lowess normalization
Normalization between a pair of arrays: Loess(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 normalization
2. Baseline scaling normalization
Normalization across arrays
Quantile normalization
quantile- quantile plot
motivates the quantile
normalization algorithm
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 quantilequantile plot to 45-degree
line gives the
transformation to have the
same distribution.
Normalization across arrays
Quantile normalization Algorithm
Source: PhD thesis by Benjamin Milo Bolstad, 2004, University of California, Barkeley
Normalization across arrays
Quantile Normalization:
Original data
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
Sort
1. Sort each column of X (values)
2. Take the means across rows of X sort
No.
Exp.1
No.
Exp.2
No.
Exp.1
No.
Exp.2
5
0.6
5
0.6
1
2.2
1
0.9
2
0.9
1
0.9
2
0.9
2
2.2
4
1.3
3
1.3
3
2.8
3
1.3
1
2.2
2
2.2
4
1.3
4
2.8
3
2.8
4
2.8
5
0.6
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
Normalization across arrays
Raw data
After quantile normalization
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
Normalization across arrays
Baseline scaling method
Normalization across arrays
Raw data
After Baseline scaling
normalization
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.
Tests for differential expression of genes
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
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 pvalues are calculated by a probability distribution
function
The 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 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
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
Wilcoxon’s rank sum test (Example)
X
Data
Y
Data
X&Y
Data
Rank
x1
7
y1
5
x4
9
1
x2
8
y2
6
x2
8
2
x3
5
y3
8
y3
8
3
x4
9
y4
4
x5
7
4
x5
7
x1
7
5
y2
6
6
y1
5
7
x3
5
8
y4
4
9
n=5. m=4
T=R(x1)+R(x2)+R(x3)+R(x4)+R(x5)
=5+2+8+1+4= 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 = .1112 (From chart)
Example
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 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
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 0.994 ---this is
very high and this is called family-wise error rate (FWER)
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 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.
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
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, 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.