Diapositive 1

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Combination of Independent Component Analysis
and statistical modeling

for the identification of metabonomic
biomarkers in 1H-NMR spectroscopy Réjane
Rousseau (Institut de Statistique, UCL, Belgium)
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The metabonomics
specific region biomarker
Biofluid (Urine)
1H-NMR spectroscopy

Whithout contact
METABOLITES TISSUES Organs
After contact
Frequency domain
Spectral alterations Altered metabolites
detection

• One molecule several peaks
• ( 1 to 3 ) with specific positions
• in the frequency domain.
• The concentration of a molecule is
• proportional to the area under the
• curve in its peaks.

Identification of biomarkers   which part of
the spectrum to examine? 

• How?
• In an experimental database
• with a methodology based on Principal Component
  Analysis (PCA)

Objective propose a methodology combining ICA
and statistical modeling
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Identification of biomarkers

• Experimental data
• Biologists have a pool of n rats with a
  control pathological state.
• They collect n samples of urine .
• The H-NMR produces n spectra of m values
  Spectral data
• Each sample or spectrum is described by l
  variables in a matrix of design Y.
• One of these variables describes the
  characteristic related to the biomarkers yk
• Biomarker identification statistical methods to
  answer to the question
•  In these multivariate data,
  which are the most altered variables xj

Spectral data X(n x m)
Design data Y(n x l)
y1 y2 yl
13 320
11 200
11 100
12 270
y1 age of the rat yky2 severity
of diabetis
xj
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Example controlled data

• Advantage of controlled data
• we know the spectral regions that should be
  identified as biomarkers.
• The controlled data
• 28 spectra of 600 points X(28 x 600)
• Each spectrum a sample of urine
• a chosen concentration of Citrate
• a
  chosen concentration of Hippurate
• X(28x600) Y (28x2) y1
  concentration of citrate
• 
  y2 concentration of hippurate
• We need a biomarker to detect changes of the
  level of citrate described by y1
•  Which are the spectral regions xj the most
  altered when the y1 changes?
• Spectral regions corresponding to Citrate
  the biomarkers to identify.

hippurate (y2)
citrate (y1)
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spectrum 3000 points
A spectrum of 600 values xj with ? xj 1

Urine Citrate Hippurate

Natural urine
Hippurate y2

Hippurate

Citrate y1
14 mixtures in 2 replications 28 samples
Citrate
The biomarkers to identify. spectral regions
corresponding to Citrate
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NEW
USUAL
I. Reduction of the dimension PCA
I. Reduction of the dimension ICA

• XTC SAT
• Components
• are independent
• with a biological meaning
• Examination of the ALL components
• to visualize unconnected
• molecules in samples

XC TP

• Principal components are
• uncorrelated
• in the direction of maximum of variance
• Examination of the 2 first components

Score plot Loadings
L1
L2
II. Biomarker discovery through Statistical
modelling
ex Citrate plays an important role
Comparison of the intensities of biomarkers
between spectra from ? conditions
Identificationof biomarker
Identificationof biomarker
This is only powerful if the biological
question is related to the highest
variance in the dataset!
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The proposed methodology

• Resulting components
• meaningfull component
• - with some advantages over
• the principal components

• Part I Dimension reduction
• with ICA on the spectral data
• XTC S.AT
  

Part II Biomarker discovery through
statistical modelling
Identification of biomarker
Comparison of the values in biomarker between
spectra from different conditions
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What is Independent component analysis (ICA)?

• The idea
• Each observed vector of data is a linear
  combination of unknown independent (not only
  linearly independent) components
• The ICA provides the independent components
  (sources, sk) which have created a vector of data
  and the corresponding mixing weights aki.
• How do we estimate the sources?
• with linear transformations of observed
  signals that maximize the independence of the
  sources.
• How do we evaluate this property of independence?
• Using the Central Limit Theorem (), the
  independence of sources components can be reflect
  
• by non-gaussianity.
• Solving the ICA problem consists of
  finding a demixing matrix which maximises the
  non-gaussianity of the estimated sources under
  the constraint that their variances are constant.
• Fast-ICA algorithm
• - uses an objective function related to
  negentropy
• - uses fixed-point iteration scheme.
• almost any measured quantity which depends on
  several underlying independent factors has a
  Gaussian PDF

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I. Dimension reduction by ICA
X (nxm) n spectra defined by m variables
ex (28x600)
Transposition
XT (mxn)
Centering
mixture and sj have zero mean
XTC S.AT
XTC (mxn)

• Each spectrum is
• a weighted sum of the
• independent spectral expressions
• which each one can correspond to
• an independent (composite) metabolite contained
  in
• the studied sample.
• (aT , weight ? quantity)

Whitening PCA

1. we obtain uncorrelated scores with unit variance
   ? demixing matrix is orthogonal
2. possibility to discard irrelevant scores chose
   the number of sources to estimate

T (mxq) XTC. P
ICA
S (mxq) XTC. P.W XTC. A
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Example I. Dimension reduction by ICA
XTC S.AT
XTC(600 x 28) S (600 x 6)
AT (6x28)
s1 s2 s3 s4 s5 s6
xTC1 xTC28
s1,1 s1,6

sij


s600,1
at1,1 at1,28




at6,1
at1 at2 at3 at4 at5 at6
........ .... ....
Urine citrate hippurate
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S (600 x 6)
AT
28 spectra

Natural urine
aTi,8
Citrate
Hippurate
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Note Comparison with the usual PCA

• Similarities projection methods linearly
  decomposing multi-dimensional into components.
• Differences
• The number of sources, q, has to be fixed
• Sources are not naturally sorted according to
  their importances
• The independence condition the biggest
  advantage of the ICA
• - independent components are more
  meaningful than uncorrelated components
• - more suitable for our question in which
  the component of interest are not always in the
  direction
• with the maximum variance .

PCA
ICA
1
2
Natural urine
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PCA
ICA
Hippurate Citrate
Natural urine
Loading 1
s1
Citrate
Loading 2
s2
Hippurate Citrate
Hippurate
Loading 3
s3
PC2
aT3
PC1
aT2
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The proposed methodology

• Part I Dimension reduction
• with ICA on the spectral data
• XTC S.AT
  

q sources representing the spectra of
independent (unrelated) composite metabolites
contained in the samples.
Part II Biomarker discovery
through statistical modeling - on
the mixing matrix AT - with
covariates chosen in the design matrix
Identification of biomarkers
Comparison of the intensities in biomarkers
between spectra from different conditions
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PART II Biomarker discovery with statistical
model

• The idea Among the q recovered sj , we
  suppose that some sources
• present biomarker regions for a chosen factor yk
• are interpretable as the spectra of pure or
  composite independent
• metabolite which has a concentration in
  the samples influenced by a chosen factor yk
• have weights influenced by a chosen factor yk

AT (q x n)

• Modelisation of the relation between the weight
  vector and the design variables

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PART II

• Part I ICA on the spectral data XTC S.AT
  

Part II Biomarker discovery through statistical
modelling
Step 1 Fit a linear model on AT

• relation between the weight vector and the
  covariates in design variables
• different models.

Step 2 Biomarker identification
Step 3 comparison of the intensities
in biomarkers between spectra
from different conditions

• apply statistical tests on the parameters
• of the models
• selection of sources with
• significant effects.

• prediction of mixing weights by the model
• reconstruction with biomarkers sources
• comparison between factor levels.

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Step 1 Fit a model on AT

• The design matrix Y is rewritten into 2 separate
  matrixs
• Z1 the (n x p1) incidence matrix for the p1
  covariates with fixed effects
• Z2 the (n x p2) incidence matrix for the p2
  covariates with random effects
• For each of the q recovered sj , we assume a
  linear relation between its vector of weights and
  the design variables
• 
  aj Z1 ßj Z2
  ?j ej
• Models with only fixed effects covariates
  aj Z1 ß j ej
• Case 1 categorical covariates ANOVA
• ? ex1 biomarker to discriminate 2 groups of
  subjects disease sane.
• ex2 biomarker to discriminate 3 groups of
  subjects disease1, disease2 sane
• Case 2 quantitative covariates linear
  regression

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Step 1 Fit a model example

• For each of the q 6 recovered sj, we construct
  a multiple linear regression model
• with 2 quantitative covariates ( p 3) and
  no interaction
• aj ßj0 ßj1 y1 ßj2 y2 ej
• with ßj0 the intercept
• y1 the citrate concentration in
  mg (quantitative)
• y2 the hippurate concentration in
  mg (quantitative)
• ß1 the effect of citrate on the
  mean aj for a fixed value of hippurate
• ß2 the effect of hippurate on the
  mean aj for a fixed value of citrate
• ej the vector of independent
  random error N (0,s2)
• For each of the q recovered sj, the fitted model
  by least square technique is
• âj bj0 bj1 y1 bj2 y2
• In this example, we want to identify biomarkers
  for the concentration of citrate.
• The covariate of interest yk y1

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Step 1 Fit a model example

s3Hippurate
s2 Citrate
a2
a3
Citrate (y1)
Citrate (y1)
hippurate (y2)
hippurate (y2)
(y1)
(y1)
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Step 2 Biomarker identification

• Goal
• Among the q sources , we want to select the ones
  presenting a significant effect of the
• chosen covariate yk on their weights.
• These discriminant sources represent the
  spectrum of an independent metabolite
• with a concentration depending on the chosen
  covariate biomarkers.
• For each source sj, test the significance of the
  parameter ßjk of the covariate of interest yk
• (ex research of biomarkers for the dose
  of citrate y1, we test each of the 6 ßj1)
• H0 ßjk 0 vs H1 ßjk ? 0
• compute the following statistic tj bjk /
  s(bjk) t(n-p)
• take the corresponding p-value pj P( t (n-p)
  ? tj )
• We are in a multiple tests situation
• the selection of a significant set of r
  coefficients ßjk based on q pj obtained from q
  individual tests.
• ? Bonferroni correction select, in a (m
  x r) matrix S, the r sources with pj lt 0.05/q

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Step 2 Biomarker identification example
We research of biomarkers for the dose of
citrate y1yK ? we test each of the 6 ßj1
P-values
a 0.05/6
Sources
9.18 x 10-13
1.84x10-15
2.86 x 10-31
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Step 3 Comparison of the intensities in
biomarkers

• Goal comparison of the effects on the biomarker
  caused by ? changes in yk.
• Choose 3 or more values of yk
• yk1 a first value of reference of yk
• yk2 a new value of interest of yk
• yk3 a second new value of interest of yk
• Compute
• The effect on the biomarker of the change of yk
  from yk1 to yk2
• C1 S ßk (yk2- yk1 )
• The effect on the biomarker of the change of yk
  from yk1 to yk3
• C2 S ßk (yk3- yk1 )

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Step 3 example
yk1 yk2 yk3 yk4
Citrate yk
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Conclusions

• With the presented methodology combining ICA with
  statistical modeling,
• we visualize the independent metabolites
  contained in the studied biofluid (through the
  sources) and their quantity (through the mixing
  weights)
• we identify biomarkers or spectral regions
  changing according to a chosen factor by a
  selection of source.
• we compare the effects on this spectral
  biomarkers caused by different changes of this
  factor.
• In comparison with the PCA, ICA
• gives more biologically meaningful and natural
  representations of this data.

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• Thank you
  for your attention

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Example2 the data

• 18 spectra of 600 values
• 1 characteristic in Y
• X(18x600) Y (18x1)
• y1 disease
  group of the rat (qualitative)
• We want biomarkers for group of disease described
  in y1.
• ? a model with
  qualitative covariates

Group 1 disease 1 Group 2 disease 2 Group 3 no
disease
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Example2 Part I. Dimension reduction by ICA
XTC S.AT
S (600 x 5)
AT (5x18)
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Example 2 Part II biomarkers discovery through
statistical modeling

• Step 1 Fit a model on AT Models with
  only a categorical covariate with fixed effects
  ANOVA I
• 
  aj Z1 ß j ej
• Step 2 Biomarker identification
• For each of the q recovered sj, test the effect
  of y1 ? Fj statistics? pj
• Bonferroni correction select, in a (m x r)
  matrix S, the r sources with pj lt 0.05/q

0.009797431
0.0002412604
0.005710213
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Step 3 Comparison of the intensities in
biomarkers

• Goal comparison of the effects on the biomarker
  caused by ? changes in yk.
• Choose 3 or more values of yk
• yk1 a first value of reference of yk
• yk2 a new value of interest of yk
• yk3 a second new value of interest of yk
• Compute
• The effect on the biomarker of the change of yk
  from yk1 to yk2
• C1 S ßk (yk2- yk1 )
• The effect on the biomarker of the change of yk
  from yk1 to yk3
• C2 S ßk (yk3- yk1 )

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Step 3 Comparison of the intensities in
biomarkers
Goal comparison of the effects on the biomarker
caused by the changes of group.
Citrate
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• Others slides

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Example 1 Step 4 Comparison of the intensities
in biomarkers

• For a first chosen value of interest of yk (not
  necessary observed) yk0
• Choose a value of reference for the other
  factors y0?k
• ? Z01 vector of values (1,
  yk0 , y0?k)
• For each of the r source selected in S, use the
  model to predict weights for the biomarkers
• âj (yk0) bj Z01
• ? â (yk0) a vector
  of r new weights
• Reconstruction of the values to expect in the
  biomarkers
• 
  Sâ(yk0)

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Example 2 the reconstructed spectra
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