Transcript Statistical Analysis of Microarray Data

Statistical Analysis of Microarray Data
Ka-Lok Ng
Statistical Analysis of Microarray Data
Ratios and reference samples
• Compute the ratio of fluorescence intensities for two samples that are
competitively hybridized to the same microarray. One sample acts as a
control , or “reference” sample, and is labeled with a dye (Cy3) that has a
different fluorescent spectrum from the dye (Cy5) used to label the
experimental sample.
• A convention emerged that two-fold induction or repression of an experimental
sample, relative to the reference sample, were indicative of a meaningful
change in gene expression.
• This convection does not reflect standard statistical definition of significance
• This often has the effect of selecting the top 5% or so of the clones present on
the microarray
| Log 2 (Ti ) | 1
Statistical Analysis Microarray Data
Reasons for adopting ratios as the standard for comparison
of gene expression
(1) Microarrays do not provide data on absolute expression
levels. Formulation of a ratio captures the central idea that
it is a change in relative level of expression that is
biological interesting.
(2) removes variation among arrays from the analysis.
Differences between microarray – such as (1) the absolute
amount of DNA spotted on the arrays, (2) local variation
introduced either during the sliding preparation and
washing, or during image capture.
Statistical Analysis of Microarray Data
All microarray experiments must be normalized to ensure that biases inherent in
each hybridization are removed.
True whether use ratios or raw fluorescent intensities are adopted as the
measure of transcript abundance.
Simple normalization of microarray data. The difference between the raw fluorescence is a meaningless number. Computing ratios allows immediate
visualization of which genes are higher in the red channel than the green channel, but logarithmic transformation of this measure on the base 2 scale
results in symmetric distribution of values. Finally, normalization by subtraction of the mean log ratio adjusts for the fact that the red channel was generally
more intense than the green channel, and centers the data around zero.
Statistical Analysis of Microarray Data
Statistical Analysis of Microarray Data
Calculate which genes are
differentially expressed
The fluorescence intensity for the
Cy3 or Cy5 channel after
background subtraction. Calculate
which genes are at least twofold
different in their abundance on
this array using two different
approaches: (a) by formulating the
Cy3:Cy5 ratio, and (b) by
calculating the difference in the log
base 2 transformed values. In both
cases, make sure that you adjust
for any overall difference in intensity
for the two dyes and comment on
whether this adjustment affects your
conclusions.
Calculate which genes are differentially expressed.
Statistical Analysis of Microarray Data
Divide by 0.954
Statistical Analysis of Microarray Data
Using the ratio method, without adjustment for overall dye effects, genes 2 and 9
appear to have Cy3/Cy5 < 0.5, suggesting that they are differentially regulated.
No genes have Cy3/Cy5 > 2. However, the average ratio is 0.95, indicating that
overall fluorescence is generally 5% greater in the Cy5 (RED) channel. One way
to adjust for this is to divide the individual ratios by the average ratio, which
results in the adjusted ratio column. This confirm that gene 2 is
underexpressed in Cy3, but not gene 9, whereas gene 5 may be
overexpressed.
Statistical Analysis of Microarray Data
Using the log transformation method, you get very similar results (-1
and +1).
The adjusted columns indicate the difference between the log2
fluorescenec intensity and the mean log2 intensity for the respective
dye, and hence express the relative fluorescence intensity, relative to
the sample mean. The difference between these values gives the final
column, indicating that genes 2 and 5 may differentially expressed by
twofold or more.
Statistical Analysis of Microarray Data
If you just subtract the raw log2 values, you will see that gene 9 appears to
be underexpressed in Cy3, but gene 5 appears to be slightly less than twofold
overexpressed.
Finding significant genes
• After normalizing, filtering and averaging the data, one can identify
genes with expression ratios that are significantly different from 1 or -1
• Some genes fluctuates a great deal more than others (Hughes et al.
2000a, b)
• In general the genes whose expression is most variable are those in
which expression is stress induced, modulated by the immune system
or hormonally regulated (Pritchard et al. 2001)
• There is the Missing Value problem in microarray data set
– By interpolation
References
• Hughes TR, et al. (2000a) Functional discovery via a compendium of
expression profiles. Cell 102(1):109-26
• Hughes TR, et al. (2000b) Widespread aneuploidy revealed by DNA
microarray expression profiling. Nat Genet 25(3):333-7
• Pritchard et al. 2001 Project normal: Defining normal variance in
mouse gene expression. PNAS 98, 13266.
Measure of similarity – definition of distance
A measure of similarity - distance
Euclidean distance between two genes
- for example: p53 and mdm2
(9  10) 2  (3  2) 2  (7  9) 2  6
Measure of similarity – definition of distance
Non-Euclidean metrics
•
Any distance dij be the distance between two vectors, i and j must satisfy a
number of rules:
1. The distance must be positive definite
2. The distance must be symmetric, dij = dji
3. An object is zero distance from itself, dii =0
4. Triangle inequality dik ≦ dij + djk
•
Manhattan distance (or city block) distance is an example of non-Euclidean
distance metric, The Mahattan distance is defined as the sum of the
absolute distances between the components of each expression vector, x
and y,
n
d   | xi  yi |
i 1
It measures the route one might have to travel between two points in a place
such as Manhattan where the streets and avenues are arranged at right
angles to one another. It is known as Hamming distance when applied to data
expressed in binary form, e.g. if the expression levels of the genes have been
discretised into 1s and 0s.
Measure of similarity – definition of distance
• Minkowski distance is a generalization of the
Euclidean distance and is expressed as
n
d Minkowski ( A, B)   (| ai  bi |)1/ p
i 1
The parameter p is called the order. The higher the value
of p, the more significant is the contribution of the largest
components |ai – bi |.
p=1  Manhattan distance
p=2  Euclidean distance
http://library.thinkquest.org/05aug/01273/whoswho.html
Herman Minkowski
(1864-1909)
Measure of similarity – definition of distance
• Euclidean distance is one of the most intuitive ways to measure the
distance between points in space, but it is not always the most
appropriate one for expression profiles.
• We need to define distance measures that score as similar gene
expression profiles that show similar trend, rather than those that
depend on the absolute levels.
• Two simple measures that can be used are the angle and chord
distances.
A
B
chord distance
angular distance
Measure of similarity – definition of distance
•
•
A = (ax, ay), B = (bx, by)
The cosine of the angle between the two vectors A and B is given by their
dot product, and can be used as a similarity measure.
cos  
a x bx  a y by
A B
In n-dimensional space for vectors A = (a1, …. an) and B = (b1, …. bn), the
n
cosine is defined as
cos  
a b
i 1
i i
A B
The chord distance is defined as the length of
the chord between the vectors of unit length
having the same directions as the original
ones.
2
2
d chord ( A, B)  (a x  bx )  (a y  by )
use _(a  b) 2  a 2  2ab  b 2
 d chord ( A, B)  2(1  cos  ) 2
A
B
chord distance
angular distance
Semimetric distance – Pearson correlation coefficient or Covariance
Statistics – standard deviation and variance, var(X)=s2, for 1-dimension data
How about higher dimension data ?
- It is useful to have a similar measure to find out how much the
dimensions vary from the mean with respect to each other.
- Covariance is measured between 2 dimensions,
- suppose one have a 3-dimension data set (X,Y,Z), then one can
calculate Cov(X,Y), Cov(X,Z) and Cov(Y,Z)
- to compare heterogenous pairs of variables, define the correlation
coefficient or Pearson correlation coefficient, -1≦ rXY ≦1
r XY
Cov( X , Y )

(var X )(var Y )
-1  perfect anticorrelation
0  independent
+1 perfect correlation
Semimetric distance – Pearson correlation coefficient or Covariance
- The resulting rXY value will be larger than 0 if a and b tend to increase
together, below 0 if they tend to decrease together, and 0 if they are
independent.
Remark: rXY only test whether there is a linear dependence, Y=aX+b
- if two variables independent  low rXY,
- a low rXY may or may not  independent, it may be a non-linear relation
- a high rXY is a sufficient but not necessary condition for variable dependence
Semimetric distance – Pearson correlation coefficient or
Covariance matrix
A covariance matrix is merely collection of many covariances in the form of a d x d matrix:
Semimetric distance – the squared Pearson correlation coefficient
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•
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Pearson correlation coefficient is useful for examining correlations in the data,
but not useful for identifying genes whose expression levels are
anticorrelated.
One may imagine an instance, for example, in which the same TF can cause
both enhancement and repression of expression.
A better alternative is the squared Pearson correlation coefficient (pcc),
r sq  r
2
XY
Cov( X , Y )

var( X ) var(Y )
The square pcc takes the values in the range 0 ≦ rsq ≦ 1.
0  uncorrelate vector
1  perfectly correlated or anticorrelated
pcc are measures of similarity
Similarity and distance have a reciprocal relationship
similarity↑  distance↓
 d = 1 – r is typically used as a measure of distance
Statistical Analysis of Microarray Data
• Normalize each channel separately  Gn-<G> and Rn-<R>
• Subtraction of the mean log fluorescence intensity for the channel
from each value transforms the measurements such that the
abundance of each transcript is represented as a fold increase or
decrease relative to the sample mean, namely as a relative
fluorescence intensity.
• Log Gn - <log Gn>, Log Rn - <log Rn>, where n=1,2,….
 (Log G
n
-  log G n )  0 and
n
n
Log ZG n 
 (Log R
Log G n -  log G n 
 Log(G)
 LogZ G  0, and _  Log ( Z G )  1
n
-  log R n )  0
Statistical Analysis of Microarray Data