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Microarray data analysis
25 January 2006
David A. McClellan, Ph.D.
Introduction to Bioinformatics
[email protected]
Brigham Young University
Dept. Integrative Biology
Inferential statistics
Inferential statistics are used to make inferences
about a population from a sample.
Hypothesis testing is a common form of inferential
statistics. A null hypothesis is stated, such as:
“There is no difference in signal intensity for the gene
expression measurements in normal and diseased
samples.” The alternative hypothesis is that there
is a difference.
We use a test statistic to decide whether to accept or
reject the null hypothesis. For many applications,
we set the significance level a to p < 0.05.
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Inferential statistics
A t-test is a commonly used test statistic to assess
the difference in mean values between two groups.
t=
x1 – x2
s
difference between mean values
=
variability (noise)
Questions
Is the sample size (n) adequate?
Are the data normally distributed?
Is the variance of the data known?
Is the variance the same in the two groups?
Is it appropriate to set the significance level to p < 0.05?
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Inferential statistics
Paradigm
Parametric test
Nonparametric
Compare two
unpaired groups
Unpaired t-test
Mann-Whitney test
Compare two
paired groups
Paired t-test
Wilcoxon test
Compare 3 or
more groups
ANOVA
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ANOVA
ANalysis Of VAriance
ANOVA calculates the probability that
several conditions all come from the
same distribution
Parametric vs. Nonparametric
Parametric tests are applied to data sets that
are sampled from a normal distribution (ttests & ANOVAs)
Nonparametric tests do not make assumptions
about the population distribution – they rank
the outcome variable from low to high and
analyze the ranks
Mann-Whitney test
(a two-sample rank test)
Actual measurements are not employed; the
ranks of the measurements are used instead
n1 n1  1
U  n1n2 
 R1
2
n1 and n2 are the number of observations in
samples 1 and 2, and R1 is the sum of the
ranks of the observations in sample 1
Mann-Whitney example
Mann-Whitney table
Wilcoxon paired-sample test
A nonparametric analogue to the pairedsample t-test, just as the Mann-Whitney
test is a nonparametric procedure
analogous to the unpaired-sample t-test
Wilcoxon example
Wilcoxon table
Inferential statistics
Is it appropriate to set the significance level to p < 0.05?
If you hypothesize that a specific gene is up-regulated,
you can set the probability value to 0.05.
You might measure the expression of 10,000 genes and
hope that any of them are up- or down-regulated. But
you can expect to see 5% (500 genes) regulated at the
p < 0.05 level by chance alone. To account for the
thousands of repeated measurements you are making,
some researchers apply a Bonferroni correction.
The level for statistical significance is divided by the
number of measurements, e.g. the criterion becomes:
p < (0.05)/10,000 or p < 5 x 10-6
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Significance analysis of microarrays (SAM)
SAM -- an Excel plug-in
-- URL: www-stat.stanford.edu/~tibs/SAM
-- modified t-test
-- adjustable false discovery rate
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observed
upregulated
expected
downregulated
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Descriptive statistics
Microarray data are highly dimensional: there are
many thousands of measurements made from a small
number of samples.
Descriptive (exploratory) statistics help you to find
meaningful patterns in the data.
A first step is to arrange the data in a matrix.
Next, use a distance metric to define the relatedness
of the different data points. Two commonly used
distance metrics are:
-- Euclidean distance
-- Pearson coefficient of correlation
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Euclidean Distance
Pearson Correlation Coefficient
Descriptive statistics: clustering
Clustering algorithms offer useful visual descriptions
of microarray data.
Genes may be clustered, or samples, or both.
We will next describe hierarchical clustering.
This may be agglomerative (building up the branches
of a tree, beginning with the two most closely related
objects) or divisive (building the tree by finding the
most dissimilar objects first).
In each case, we end up with a tree having branches
and nodes.
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Agglomerative clustering
0
1
2
3
4
a
b
a,b
c
d
e
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Agglomerative clustering
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b
a,b
c
d
e
d,e
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Agglomerative clustering
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a
b
a,b
c
d
e
c,d,e
d,e
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Agglomerative clustering
0
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a
b
a,b
a,b,c,d,e
c
d
e
c,d,e
d,e
…tree is constructed
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Divisive clustering
a,b,c,d,e
4
3
2
1
0
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Divisive clustering
a,b,c,d,e
c,d,e
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Divisive clustering
a,b,c,d,e
c,d,e
d,e
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3
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1
0
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Divisive clustering
a,b
a,b,c,d,e
c,d,e
d,e
4
3
2
1
0
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Divisive clustering
a
b
a,b
a,b,c,d,e
c
c,d,e
d
d,e
e
4
3
2
1
0
…tree is constructed
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agglomerative
0
1
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a
b
a,b
a,b,c,d,e
c
c,d,e
d
d,e
e
4
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1
0
divisive
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1
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1
12
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Cluster and TreeView
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Cluster and TreeView
clustering K means SOM PCA
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Cluster and TreeView
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Cluster and TreeView
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Two-way
clustering
of genes (y-axis)
and cell lines
(x-axis)
(Alizadeh et al.,
2000)
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Self-Organizing Maps (SOM)
To download GeneCluster:
http://www.genome.wi.mit.edu/MPR/software.html
SOMs are unsupervised neural net
algorithms that identify coregulated genes
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Two pre-processing steps essential to apply SOMs
1. Variation Filtering:
Data are passed through a variation filter to eliminate
those genes showing no significant change in
expression across the k samples. This step is needed
to prevent nodes from being attracted to large sets
of invariant genes.
2. Normalization:
The expression level of each gene is normalized
across experiments. This focuses attention on the
'shape' of expression patterns rather than absolute
levels of expression.
Principal components analysis (PCA)
An exploratory technique used to reduce the
dimensionality of the data set to 2D or 3D
For a matrix of m genes x n samples, create a new
covariance matrix of size n x n
Thus transform some large number of variables into
a smaller number of uncorrelated variables called
principal components (PCs).
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Principal component axis #2
(10%)
Principal components analysis (PCA),
an exploratory technique that reduces data dimensionality,
distinguishes lead-exposed from control cell lines
P4
N2
Legend
Lead (P)
C2
P1
N3
C3
P2 P3
C4
N4
Sodium (N)
Control (C)
C1
Principal component axis #1
(87%)
Principal components analysis (PCA): objectives
• to reduce dimensionality
• to determine the linear combination of variables
• to choose the most useful variables (features)
• to visualize multidimensional data
• to identify groups of objects (e.g. genes/samples)
• to identify outliers
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http://www.okstate.edu/artsci/botany/ordinate/PCA.htm
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http://www.okstate.edu/artsci/botany/ordinate/PCA.htm
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Use of PCA to demonstrate increased levels of gene
expression from Down syndrome (trisomy 21) brain
Chr 21