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STATISTICAL TOOLS FOR SYNTHESIZING LISTS OF
DIFFERENTIALLY EXPRESSED FEATURES IN MICROARRAY EXPERIMENTS
Marta Blangiardo and Sylvia Richardson1
1 Centre
for Biostatistics, Imperial College, St Mary’s Campus, Norfolk Place London W2 1PG, UK.
[email protected]
SCOPE OF THE WORK
Consider two different but related experiments, how to assess whether there are more differentially
expressed genes in common than expected by chance?
R(q*)=1.0
RANKED LISTS
q*=0.05
Suppose we have two experiments, each reporting a measure (e.g. p-value,…) of differential
expression on a probability scale:
O11(q*)=18
Small p value: MOST differentially expressed
O1+(q)
Large p value: NOT differentially expressed
Experiment A
Experiment B
pA1
pA2
…
pAn
pB1
pB2
…
pBn
All the computations have
been performed in R and are
available on BGX website
(www.bgx.org.uk)
O+1(q)
SIMULATION
We rank the genes according to the probability measures. For each cut off q we obtain a 2X2 table:
We use three batches of simulations differing by level of association between experiments and
percentage of DE genes. For every batch we simulate two lists of 2000 p-values (Allison et
al.2002) averaging the results over 100 simulations.
Conditional Model
Permutation Test
Exp B
Exp A
DE
DE
DE
O11(q)
DE O+1(q)-O11(q)
O1+(q)-O11(q)
n-O1+(q)- O+1(q)+O11(q)
O1+(q)
n- O1+(q)
n- O+1(q)
n
O+1(q)
The number of genes in common by chance is
O1 (q)  O1 (q)
E (O11 (q) | H 0 ) 
n
The number of genes observed in common is O11(q)
RATIO
T(q*)
q*
Joint Model
Bayesian Analysis
O11(q*) O1+(q*) O+1(q*)
MC p-value
R(q*)
95% CI
q*
O11(q*) O1+(q*)
O+1(q*)
= 0 ,
1.1
DE = 10%
0.040
10
115
120
0.550
1.0
0.050
[0.4-1.5]
18
125
130
= 0.25 ,
DE = 10%
5.7
0.010
6
49
50
0.060
5.0
0.020
[2.2-10.6]
8
59
59
= 0.25 ,
DE = 20%
3.0
0.019
11
82
82
0.030
2.9
0.026
[1.5-4.9]
17
105
106
= 0.25 ,
DE = 30%
2.5
0.023
21
125
126
0.002
2.4
0.030
[1.4-3.6]
28
148
150
NO association is declared when the two lists are not associated (MC p-value not significant, CI include 1)
We propose to calculate the maximum of the observed to expected ratio:
When there is a TRUE association:
*
O
(
q
)
*
11
T (q )  max q T (q) 
E (O11 (q* ) | H 0 )
Conditional Model
Joint Model
•The ratio T(q*) decrease
•q*, O1+(q), O+1(q), and O11(q) increase
It is the maximal deviation from the underneath independence model.
Increasing
% of
DE genes
•MC p-value is more significant
1
•By using the maximum ratio, multiple
testing issues for different list sizes are
avoided
•Returns a single list of O11(q) genes for
further biological investigation
•The ratio R(q*) decreases
•q*, O1+(q), O+1(q), and O11(q) increase
•CI95 are narrower
R(q*) is always smaller than T(q*) and its q* is slightly bigger as it accounts for the additional variability
APPLICATION: analysis of deleterious effect of mechanical ventilation on lung gene
expression
We re-analyse the experiment presented in Ma et al, 2005, investigating the deleterious effect of
mechanical ventilation on lung gene expression through a model of mechanical ventilation-induced
lung injury (VILI) on rodents (mice and rats).
We analyse separately the two dataset using Cyber-T (Baldi and Long, 2001)
0
0.05
0.1
 We use RESOURCERER to reconstruct the list of orthologs for the two species
We apply the methodology described to the lists of 2969 p-values (ortholog genes)
PERMUTATION TEST
Given a threshold q and fixed margins
T (q)  Hyper (O1 (q), O1 (q), n)
•97 genes found in common between mice and rats
But the distribution of T(q*) is not easily obtained since the tables are nested in each other. We
take advantage of the empirical distribution for T(q*) obtained via permutations.
•15 genes in common with the original analysis (which
highlighted 48 genes)
•Two enriched pathways with our methodology:
We perform a Monte Carlo test of T under the
null hypothesis of independence between the
two experiments using permutations. This
returns a Monte Carlo p-value.
Not associated
Associated
P-value 0.8
P-value <0.001
1) MAPK signalling activity. 6 out of the
significant orthologs are involved in this KEGG
pathway (Fgfr1, Gadd45a, Hspa8, Hspa1a, Il1b,
Il1r2) while only 4 were highlighted in the original
one.
T(q*)=1.44
R(q*) = 1.43
•The uncertainty of the margins is not taken into account
q*=0.01
q* =
•The size of the list of genes in common can be vary small (typically when the total number of DE
genes is small) and this can cause an instability in the estimate of T(q)
MC p-value <0.001
CI95 = [1.13-1.75]
LIMITATIONS OF THE TEST
O11(q*) = 97
•We propose a Bayesian model treating also the margins as random variables
0.01
2) Cytokine-Cytokine receptor interaction. 5 out
of the significant orthologs are involved in this
KEGG pathway (IL6, Il1b, Il1r2, CCL2, Kit)
while only 4 were highlighted in the original
one.
O1+(q*) = 393
BAYESIAN MODEL
O+1(q*) = 886
Starting with the 2x2 table we specify a multinomial distribution of dimension 3 for the vector of joint
frequencies:
Multi (O, θ, n)  q
q2
O1 1 ( q )
1
[ O1 ( q ) O1 1 ( q )]
q
[ O1 ( q ) O1 1 ( q )]
3
3
(1   q i )[ n O1 ( q ) O1 ( q ) O1 1 ( q )]
i 1
and the vector of parameters q is modelled as non informative Dirichlet:
q ~ Di(0.05,0.05,0.05,0.05)
The derived quantity of interest is the ratio of the probability that a gene is in common to the
probability that a gene is in common by chance:
R(q) 
q1 (q)
[q1 (q)  q 2 (q)]  [q1 (q)q 3 (q)]
Since the model is conjugated the posterior distribution for q is Dirichlet
θ ~ Di (0.05  O11 (q),0.05  [O1 (q)  O11 (q)],0.05  [O1 (q)  O11 (q)],0.05  [n  O1 (q)  O1 (q)  O11 (q)])
DECISION RULE
We can obtain a sample from the posterior distribution of the derived quantities R(q) and calculate
the credibility interval (CI) at 95% for each threshold q. We define q* as the value of the argument
for which the median of R(q) attains its maximum value, only for the subset of credibility intervals
which do not include 1:
q*  {arg max q Median ( R(q) | O, n) over the set of values of q for which CI95 (q) excludes 1}
Then R(q*) is the ratio associated to q*.
DISCUSSION
•This is a simple procedure to evaluate if two (or more) experiments are associated
•The permutation test gives a first look under the model where the marginal frequencies are fixed
•The Bayesian model permits to enlarge the scenario introducing variability on all the components
•It is very flexible and adaptable for comparisons of several experiments at different levels (gene
level, biological processes level) and for different problems (e.g. comparison between species ,
comparison between platforms )
REFERENCES
Allison et al. (2002), “A mixture model approach for the analysis of microarray gene expression
data”, Computational Statistics And Data Analysis, 39, 1-20.
Baldi and Long, (2001) “A Bayesian framework for the analysis of microarray expression data:
regularized t-test and statistical inferences of gene changes”, Bioinformatics, 17, 509-519.
Ma et al., (2005) “Bioinformatics identification of novel early stress response genes in rodent models
of lung injury”, Am J Physiol Lung Cell Mol Physiol 289(3), 468-477.
ACKNOWLEDGEMENTS
We would like to thank Natalia Bochkina, Alex Lewin and Anne-Mette Hein for helpful discussions.
This work has been supported by a Wellcome Trust Functional Genomics Development Initiative
(FGDI) thematic award ``Biological Atlas of Insulin Resistance (BAIR)", PC2910_DHCT