PowerPoint-Präsentationfrom Microarray Data
Page 33/3/2008| Achim Tresch
Correlation
close
to 1 or
-1 strong linear dependence Correlation close to 0 no or weak linear dependence
(Pearson) Correlation
Correlation
close
to 1 or
-1 strong linear dependence Correlation close to 0 no or weak linear dependence
(Pearson) Correlation
Correlation
close
to 1 or
-1 strong linear dependence Correlation close to 0 no or weak linear dependence
(Pearson) Correlation
Correlation
close
to 1 or
-1 strong linear dependence Correlation close to 0 no or weak linear dependence
(Pearson) Correlation
Correlation
close
to 1 or
-1 strong linear dependence Correlation close to 0 no or weak linear dependence
(Pearson) Correlation
Correlation
close
to 1 or
-1 strong linear dependence Correlation close to 0 no or weak linear dependence
(Pearson) Correlation
An expression profile is a collection of expression vectors { Xg
= (Xg,s
)s

samples
Genes }
Correlation graph: Genes are the vertices of the graph and an undirected edge (i, j) is drawn
if
(Pearson correlation, Spearman rank correlation, Kendall’s tau) between Xi
and Xj
is sufficiently different from zero.
Advantage: This representation of the marginal dependence structure is easy to interpret and can be estimated accurately even if
p > N the number of features (genes)
> the number of samples
Application: Stuart et al. (Science, 2003) build a graph from coexpression
across multiple organisms.
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It is impossible to distinguish direct from indirect dependence Three reasons why X, Y , and Z may be highly correlated:
Problems of correlation based approaches
A strong correlation is not a strong evidence for regulatory dependence (lots of false positives). But a low correlation is a
strong evidence for no regulatory dependence. Possible remedies:
•search for correlations which cannot be explained by other variables. •measure effects of gene perturbations
Page 163/3/2008| Achim Tresch
In other words: Knowing Z, knowing Y is irrelevant for knowing X (and vice versa). Z “explains”
any observed dependence between X and Y .
Conditional Independence
0 5
10 15
0 5
10 15
Gaussian Graphical Models (GGM)
Page 223/3/2008| Achim Tresch
Gaussian Graphical Models (GGM)
If we assume that the common expression distribution of all genes follows a multivariate Gaussian distribution (which is of course ridiculous), conditional independence can be assessed as follows:
Page 233/3/2008| Achim Tresch
Full conditional relationships can only be accurately estimated if the number of samples N is relatively large compared to the number of variables p.
Graph from Basso et al (Nat Genet, 2005)
Thus, if p »
N, you can . . . •
and
Strimmer, Bioinformatics´05) •
resort to a simpler model that does not rely on full conditional independence
What if p » N?
Page 243/3/2008| Achim Tresch
Correlation
Graphs
GGMs
probability distributions.
pa(N) = {S,R}
spring 1/4
dry wet\),(
Page 263/3/2008| Achim Tresch
spring 1/4
dry wet\),(
{}J
a
a
a
a
=⋅==⋅==⋅=== ===== nNrRsSjJP
summer)P(J summer)J|offP(S summer)J|rainP(R rain)Roff,S|wetP(N
)wet,rain,off,summer(
(instead
for
metric
wíth
of random
acyclic
graphs
can
versa. However
there exist
due to
Page 313/3/2008| Achim Tresch
G2 G3 , G4 , G5
G2 G3 , G4 , G5
necessarily direct
not?
“non est ponenda pluritas sine necessitate” (pluralities ought not to be proposed without necessity)
G2G1 G3
Need
for
algorithm
for which there
•
a sign +1 or –1 according to
the direction of the regulatory effect. Remove a→b
if product of all signs
along the path a→... →b equals the sign of the edge a→b
[Tringe
“Weights“. Let every edge be weighted with a non-
negative number. Edges with low weights are meant to represent edges for which there is strong evidence for a direct regulatory interaction.
Remove a→b
if sum of the weights along the path a→... →b is smaller than the weight of the edge a→b
[Tresch
a x1 b
Finding non-necessary edges
1 1 0
1 1 0
observable
is
linked
to
exactly
one
action
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expressedally differenti IS log R
s P
s P =
0 20
0 40
0 60
graphs, ~ 4seconds
Lo g
Li ke
lih oo
•
•
•
•
Nessy, nem: Implementation and estimation of the Nested Effects Model
Networks in R/Bioconductor
Holger Fröhlich
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