The metabonomics

Rousseau Réjane – 1/02/2008

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)


The metabonomics

METABOLITES

TISSUESOrgans

Biofluid

(Urine)

1H-NMR

spectroscopy

Spectral alterations

= Altered metabolites

= detection

Whithout contact

After contact

Frequency domain

Identification of biomarkers

« which part of the spectrum to examine? »

specific region = biomarker

How?

• In an experimental database

• with a methodology based on Principal Component Analysis (“PCA”)

Objective: propose a methodology combining ICA and statistical modeling

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:• 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 when yk changes? »

Spectral data: X(n x m)

xj

y1 y2yl

13 32011 20011 10012 270

Design data Y(n x l)

y1= age of the rat

yk=y2= severity of diabetis


• 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.

Example: controlled data

citrate (y1)

hippurate (y2)


=

spectrum

3000 points

+

+

The biomarkers to identify.= spectral regions corresponding to Citrate

A spectrum of 600 values xj with ∑ xj = 1

Natural urine

Citrate

Hippurate

Citrate y1

Hippurate

y2

14 mixtures in 2 replications

28 samples

UrineCitrate

Hippurate

…


ex: Citrate plays an important role

NEW

II. Biomarker discovery through Statistical modelling

USUAL

Principal components are :

- uncorrelated

- in the direction of maximum of variance

Examination of the 2 first components:

This is only powerful if the biological question

is related to the highest variance in the dataset!

XTC = SAT

Components :

- are independent

- with a biological meaning

Examination of the ALL components:

to visualize unconnected

molecules in samples Score plot Loadings

Identificationof biomarker

Comparison of the intensities of biomarkers

between spectra from ≠ conditions

I. Reduction of the dimension: PCA

I. Reduction of the dimension: ICA

XC = TP

L1

L2

-0.2 -0.1 0.0 0.1 0.2 0.3

-0.3

-0.2

-0.1

0.0

0.1

0.2

PC1

PC

2

Identificationof biomarker


The proposed methodology:

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

Part II: Biomarker discovery through statistical modelling

Resulting components:

- meaningfull component - with some advantages over the principal components

Identification of biomarker

Comparison of the values

in biomarker

between spectra from different conditions


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

1 1 2 21

... l

i k ki i i l lik

x s a s a s a s a


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)

X (nxm) n spectra defined by m variables ex: (28x600)

XT (mxn)

XTC (mxn)

T (mxq) = XTC. P

S (mxq) = XTC. P.W

= XTC. A

XTC= S.AT

ICA

“Whitening”: PCA

Centeringmixture and sj have zero mean

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

I. Dimension reduction by ICA :

Transposition


Example: I. Dimension reduction by ICA XTC= S.AT

s1,1 s1,6

sij

s600,1

04

8

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

39.42 %

39.42 %

04

8

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

29.61 %

29.61 %

05

10

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

27.84 %

27.84 %

-55

15

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.77 %

0.77 %

-10

010

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.28 %

0.28 %

-60

4

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.04 %

at1,1 at1,28

at6,1

at1

at2

at3

at4

at5

at6

0 5 10 15 20 25

0.00

240.

0027

Index

00 00 01 0102 02

04 04

1010 11 11

12 12 20 20 2121

22 22

24 24 40 4042

42 4444

0 5 10 15 20 25

0.00

180.

0026

Index

00 0001 01

02 02

04 04

10 1011 11

12 12

20 2021 21

22 22

24 24

40 40

42 42

44 44

0 5 10 15 20 25

0.00

180.

0026

Index

00 00 01 01 02 02 04 04

10 10 11 11 12 12

20 20 21 21 22 2224 24

40 40 42 42 44 44

0 5 10 15 20 25

0.00

025

Index

00

0001

01

02 0204

04

10

1011

11

12

12

2020

2121

22

2224

24

4040

42 42

44

44

0 5 10 15 20 25

0.00

015

0.00

030

Index

0000

01

01 02

02 04 0410

10

11

1112

1220 20

21

21

22

22

24

24

4040

42

42

44

44

0 5 10 15 20 25

0 e

+00

00

00

01

0102 02 04 04

10

10

1111

1212 20 20

21

21

22 22

24

24

40

4042

42

44

44

s1 s2 s3 s4 s5 s6

XTC(600 x 28) = S (600 x 6) AT

(6x28)

....

....

....

....

xTC1 xTC

28

Urine

+ citrate

+ hippurate


S (600 x 6) AT

Natural urine

Citrate

Hippurate

28 spectra

0 5 10 15 20 25

0.00

240.00

27

Index

00 00 01 0102 02

04 04

1010 11 11

12 12 20 20 2121

22 22

24 24 40 4042

42 4444

0 5 10 15 20 25

0.00

180.00

26

Index

00 0001 01

02 02

04 04

10 1011 11

12 12

20 2021 21

22 22

24 24

40 40

42 42

44 44

0 5 10 15 20 25

0.00

180.00

26

Index

00 00 01 01 02 02 04 04

10 10 11 11 12 12

20 20 21 21 22 22 24 24

40 40 42 42 44 44

0 5 10 15 20 25

0.00

025

Index

00

0001

01

02 02

0404

10

1011

11

12

12

20 2021

21

22

2224

24

4040

42 42

44

44

0 5 10 15 20 25

0.00

015

0.00

030

Index

0000

01

01 02

02 04 0410

10

11

1112

1220 20

21

21

22

22

24

24

4040

42

42

44

44

0 5 10 15 20 25

0 e

+00

00

00

01

0102 02 04 04

10

10

1111

1212 20 20

21

21

22 22

24

24

40

4042

42

44

44

04

8

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

39.42 %

39.42 %

04

8

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

29.61 %

29.61 %

05

10

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

27.84 %

27.84 %

-55

15

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.77 %

0.77 %

-10

010

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.28 %

0.28 %

-60

4

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.04 %

aTi,8


• 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 .

Note: Comparison with the usual PCA

Hippurate

Citrate

Hippurate

Citrate

Hippurate

Citrate

PCA ICA1

2

Natural urine


Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

Index

9 .991700 4 8 .897418 7 .8194544 6 .7249886 5 .647086 4 .5528036 3 .4749012 2 .3804354 1 .302472 0 .2082506

0.0020 0.0022 0.0024 0.0026

0.00

180.

0022

0.00

26

s2

s3

0000

01 010202

1010

11 11 1212

202021

21 2222

2424

4242

4444

Index

9.9917004 7.8194544 5.647086 3.4749012 1.302472

41.14 %

41.14 %

Index

9.9917004 7.8194544 5.647086 3.4749012 1.302472

27.79 %

27.79 %

9.9917004 7.8194544 5.647086 3.4749012 1.302472

25.95 %

-0.3 -0.1 0.0 0.1 0.2 0.3 0.4

-0.3

-0.1

0.1

0.3

0000

0101

0202

1010

1111

1212

2020

2121

2222

2424

4242

4444

PCA ICA

Loading 1

Loading 3

Loading 2

Natural urine

Citrate

Hippurate

Hippurate & Citrate

Hippurate & Citrate

PC1

PC2

s1

s2

s3

aT2

aT3


The proposed methodology:

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

Part II: Biomarker discovery through statistical modeling - on the mixing matrix AT

- with covariates chosen in the design matrix

q sources representing the spectra of independent (unrelated) composite metabolites contained in the samples.

Identificationof biomarkers

Comparison of the intensities

in biomarkers

between spectra from different conditions


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

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

AT (q x n)

0 5 10 15 20 25

0.00

240.

0027

Index

00 00 01 0102 02

04 04

1010 11 11

12 12 20 20 2121

22 2224 24 40 40

4242 44

44

0 5 10 15 20 25

0.00

180.

0026

Index

00 0001 01

02 02

04 04

10 1011 11

12 12

20 2021 21

22 22

24 24

40 40

42 42

44 44

0 5 10 15 20 25

0.00

180.

0026

Index

00 00 01 01 02 02 04 04

10 10 11 11 12 12

20 20 21 21 22 22 24 24

40 40 42 42 44 44

0 5 10 15 20 25

0.00

025

Index

00

0001

01

02 02

0404

10

1011

11

12

12

20 2021

21

22

2224

24

4040

42 42

44

44

0 5 10 15 20 25

0.00

015

0.00

030

Index

0000

01

01 02

02 04 0410

10

11

1112

1220 20

21

21

22

22

24

24

4040

42

42

44

44

0 5 10 15 20 25

0 e

+00

00

00

01

0102 02 04 04

10

10

1111

1212 20 20

21

21

22 22

24

24

40

4042

42

44

44

04

8

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

39.42 %

39.42 %

04

8

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

29.61 %

29.61 %

05

10

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

27.84 %

27.84 %

-55

15

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.77 %

0.77 %

-10

010

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.28 %

0.28 %

-60

4

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.04 %


Part I : ICA on the spectral data: XTC= S.AT

Part II: Biomarker discovery through statistical modelling

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

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.

PART II:

Step 1: Fit a linear model on AT

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


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 βj+ Z2 γj +εj

Models with only fixed effects covariates : aj= Z1 β j + εj

• 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→ ex: biomarker to explore the severity of an illness, the concentration of a drug

Models with only random effects covariates : aj= Z2 γj + εj

→ ex: biomarker to explore variance component (machines, subjects, laboratories)

Models with fixed and random effects covariates : Mixed model: aj= Z1 β j+ Z2 γj +εj


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= βj0 + βj1 y1 + βj2 y2 +εj

with βj0 = the intercept

y1= the citrate concentration in mg (quantitative)

y2= the hippurate concentration in mg (quantitative)

β1= the effect of citrate on the mean aj for a fixed value of hippurate

β2= the effect of hippurate on the mean aj for a fixed value of citrate

εj= the vector of independent random error ~ N (0,σ2)

• 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


0 100 200 300 400 500 600

05

1015

s 3

0 50 100 150 200 250 300

0.00

180.

0024

Citrate doses in mg

a 3

0 100 200 300 400 500 6000

48

12

s 2

0 50 100 150 200 250 300

0.00

180.

0024

Citrate doses in mg

a 2

Step 1: Fit a model: examples3:Hippurate

s2: Citrate

0 100 200 300 400 500 600

05

1015

s 3

0 50 100 150 200 250 300

0.00

180.

0024

Citrate doses in mg

a 3

300200

0.00200

0.00225

0.00250

1000

0.00275

100 0200 300

AC

Citrate

Hippurate

0 100 200 300 400 500 600

04

812

s 2

0 50 100 150 200 250 300

0.00

180.

0024

Citrate doses in mg

a 2

Citrate (y1)

a2

hippurate (y2)

(y1) (y1)

3002000.0018

0.0021

0.0024

1000

0.0027

100 0200 300

AH

Citrate

HippurateCitrate

(y1)hippurate (y2)

a3


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 βjk of the covariate of interest yk : (ex: research of biomarkers for the dose of citrate y1, we test each of the 6 βj1)

• H0: βjk = 0 vs H1: βjk ≠ 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 βjk based on q pj obtained from q individual tests. → Bonferroni correction: select, in a (m x r) matrix S*, the r sources with pj < 0.05/q

Step 2: Biomarker identification


1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

Sources

P-v

alue

Step 2: Biomarker identification: example

9.18 x 10-13 1.84x10-15

s 2 s 3s 1

α = 0.05/6Sources

P-v

alue

s

We research of biomarkers for the dose of citrate y1=yK

→ we test each of the 6 βj1

2.86 x 10-31


Goal: comparison of the effects on the biomarker caused by ≠ changes in yk.

Choose 3 or more values of yk:• yk

1 : 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 yk

2 :

C1= S* βk* (yk

2- yk1 )


3 :

C2= S* βk* (yk

3- yk1 )

Step 3: Comparison of the intensities in biomarkers


Hippurate

Step 3: example

-0.0

020.

002

0.00

6

C1

9.9917004 5.647086 2.3804354

-0.0

020.

002

0.00

6

C2

9.9917004 5.647086 2.3804354

-0.0

020.

002

0.00

6

C3

9.9917004 5.647086 2.3804354

Citrate yk

yk1

yk2

yk3

yk4


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.


Thank you for your attention


Hippurate

Citrate 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

Example2: the data

Group 1= disease 1

Group 2= disease 2

Group 3= no disease


Example2 :Part I. Dimension reduction by ICA XTC= S.AT

S (600 x 5) AT (5x18)

5 10 15

0.00

245

0.00

260

Index

5 10 15

0.00

180.

0022

0.00

26

Index

5 10 15

0.00

180.

0022

0.00

26Index

5 10 15

0.00

048

0.00

054

0.00

060

Index

5 10 15-0.0

0030

-0.0

0020

-0.0

0010

02

46

812

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

40.74 %

40.74 %

02

46

812

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

27.97 %

27.97 %

05

10

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

27.42 %

27.42 %

-50

510

15

Index

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

1.88 %

1.88 %

-50

510

9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506

0.28 %


Example 2: Part II: biomarkers discovery through statistical modelingStep 1: Fit a model on AT

Models with only a categorical covariate with fixed effects: ANOVA I aj= Z1 β j + εj

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 < 0.05/q

0.0

0.1

0.2

0.3

0.4

Sources

P-v

alue

s 1 s 2s 3

0.0002412604 0.005710213

0.009797431


Goal: comparison of the effects on the biomarker caused by ≠ changes in yk.

Choose 3 or more values of yk:• yk

1 : 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:


2 :

C1= S* βk* (yk

2- yk1 )


3 :

C2= S* βk* (yk

3- yk1 )



-0.0

06-0

.002

0.00

20.

006

C1

9.9917004 5.647086 2.3804354

-0.0

06-0

.002

0.00

20.

006

C2

9.9917004 5.647086 2.3804354

-0.0

06-0

.002

0.00

20.

006

C3

9.9917004 5.647086 2.3804354

Hippurate

Citrate


Goal: comparison of the effects on the biomarker caused by the changes of group.


Others slides


For a first chosen value of interest of yk (not necessary observed): yk0

Choose a value of reference for the other factors: y0≠k

→ Z0

1 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: S*â*(yk

0)

Example 1 : Step 4: Comparison of the intensities in biomarkers


0.00

00.

010

0.02

00.

030

0 mg

9.9917004 3.4749012

0.00

00.

010

0.02

00.

030

75 mg

9.9917004 3.4749012

0.00

00.

010

0.02

00.

030

150 mg

9.9917004 3.4749012

0.00

00.

010

0.02

00.

030

300 mg

9.9917004 3.4749012


Example 2: the reconstructed spectra 0.

000.

010.

020.

03

9.9917004 6.7249886 3.4749012 0.2082506

0.00

0.01

0.02

0.03

9.9917004 6.7249886 3.4749012 0.2082506

0.00

0.01

0.02

0.03

9.9917004 6.7249886 3.4749012 0.2082506


Documents

The metabonomics