Medical, Pharma, Engineering, Science, Technology and Business

^{1}Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA), UCL, Belgium

^{2}Machine Learning Group, UCL, Belgium

^{4}Center for Interdisciplinary Research on Medicines (CIRM), Université de Liège (ULg), Belgium

- *Corresponding Author:
- Govaerts B

Institute of Statistics

Biostatistics and Actuarial Sciences (ISBA)

UCL, Belgium

**Tel:**+32 (0) 10 47 43 38

**E-mail:**[email protected]

**Received Date:** March 25, 2017; **Accepted Date:** July 12, 2017; **Published Date:** August 23, 2017

**Citation: **Féraud B, Rousseau R, de Tullio P, Verleysen M, Govaerts B (2017)
Independent Component Analysis and Statistical Modelling for the Identification
of Metabolomics Biomarkers in ^{1}H-NMR Spectroscopy. J Biom Biostat 8: 367. doi:
10.4172/2155-6180.1000367

**Copyright:** © 2017 Féraud B, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Biometrics & Biostatistics

In order to maintain life, living organism’s product and transform small molecules called metabolites. Metabolomics aims at studying the development of biological reactions resulting from a contact with a physio-pathological stimulus, through these metabolites. The 1H-NMR spectroscopy is widely used to graphically describe a metabolite composition via spectra. Biologists can then confirm or invalidate the development of a biological reaction if specific NMR spectral regions are altered from a given physiological situation to another. However, this pro-cess supposes a preliminary identification step which traditionally consists in the study of the two first components of a Principal Component Analysis (PCA). This paper presents a new methodology in two main steps providing knowledge on specific 1H-NMR spectral areas via the identification of biomarkers and via the visualization of the effects caused by some external changes. The first step implies Independent Component Analysis (ICA) in order to decompose the spectral data into statistically independent components or sources of information. The in-dependent (pure or composite) metabolites contained in bio fluids are discovered through the sources, and their quantities through mixing weights. Specific questions related to ICA like the choice of the number of components and their ordering are discussed. The second step consists in a statistical modelling of the ICA mixing weights and introduces statistical hypothesis tests on the parameters of the estimated models, with the objective of selecting sources which present biomarkers (or significantly fluctuating spectral regions). Statistical models are considered here for their adaptability to different possible kinds of data or contexts. A computation of contrasts which can lead to the visualization of changes on spectra caused by changes of the factor of interest is also proposed. This methodology is innovative because multi-factors studies (via the use of mixed models) and statistical confirmations of the factors effects are allowed together.

Metabolomics; Multivariate statistics; Independent
component analysis; Biomarker identification; ^{1}H-NMR spectroscopy;
Linear mixed models

In a metabolomics context, proton nuclear magnetic resonance
(^{1}H-NMR) spectroscopy generates spectral profiles describing the
metabolite composition of collected bio fluid samples. A comparison
of several spectra of metabolites in various specific states permits a
preliminary graphical and qualitative investigation of changes in bio
fluid metabolite composition inherent to the presence of a stressor.
However, the complexity of ^{1}H-NMR spectra and the number of spectra
(of samples) usually available in metabolomics studies require a semiautomated
data analysis. In addition, systematic differences between
samples are often hidden behind biological noise and/or behind
peak shifts. Adequate data pre-processing and multivariate statistical
methodologies are then required to extract spectral regions with stable
differences between the spectra obtained in various conditions [1-5]. These regions, directly linked with biomarkers, are assumed to be
associated with the alteration of an endogenous metabolite in reaction
to the contact with a considered stressor. A biomarker can then be
isolated to detect and follow changes in biological systems. Beside this
goal of biomarker identification, statistical analysis, through predictive
models, also provides a measure of statistical significance of the
identified biomarkers.

The first and the most common chemo metric tool used in
preliminary metabolomics studies is Principal Component Analysis
(PCA). PCA is a starting point for analysing multivariate data and can
rapidly provide an overview of the hidden information. PCA produces
a two-dimensional plot (score plot) where the coordinate axes correspond to the two first principal components [6]. If spectra differ
according to a specific characteristic (presence or absence of a stress, for
example), the score plot reveals the presence of natural clusters in the
datasets. An examination of the loadings leads to identify biomarkers
or key portions of the ^{1}H-NMR spectra giving rise to these clusters.

However, variations within groups are sometimes larger than
variations between groups, resulting in a score plot with clusters that
overlap or do not directly correlate to the studied characteristics. In such
cases, additional information can be extracted by using more advanced
data decomposition methods such as partial least squares (PLS),
discriminant PLS (PLS-DA) or orthogonal PLS (O-PLS). As PCA, these
methods look for systematic variances between samples. In contrast,
they use information about samples such as groups of the characteristic
of interest. Therefore, these methods often allow a better separation of
samples and a clearer identification of significant biomarker variables
[7,8]. Another limitation of PCA is its high sensitivity to noise for the
analysis of ^{1}H-NMR data: very small and random fluctuations within noise of the ^{1}H-NMR spectrum can result in irrelevant clusters in the
score plot formed by the two first principal components.

Despite these limitations, PCA often remains the main statistical
standard for the analysis of ^{1}H-NMR data. In a previous work,
Rousseau et al. extend the standard PCA methodology by selecting
the two most discriminants factors for the score plot (instead of using
systematically the two first ones), and by using statistical methods for
the identification of biomarkers [9-11]. It is suggested to use PLS-DA
or ICA (Independent Component Analysis) for the decomposition of
spectra, resulting in improvements for the identification of biomarkers
(in comparison to PCA).

Motivated by these previous results, and by the good results obtained
with ICA in close domains such as genomics and Mass Spectroscopy
metabolomics, this paper expands the use of ICA to the identification
of specific ^{1}H-NMR spectral regions that are discriminant for two or
more categories of spectra. PCA and ICA share common properties.
Both of them are projections methods which linearly decompose data
into components. As for PCA, the ICA results can then be supported
by visual representations. However, the ICA components have a
more stringent nature than principal components: PCA decomposes
data into uncorrelated components when ICA decomposes them
into independent ones; independence is a stronger statistical concept
than un-correlation for non-Gaussian data. Independence of the
components is also adequate for biological interpretation because
the analysed bio fluid (e.g. plasma, urine) can be seen as a mixture of
unrelated metabolites. ^{1}H-NMR spectra may then be interpreted as
weighted sums of ^{1}H-NMR spectra of these independent metabolites.
The application of ICA should then ideally recover components which
may represent the independent metabolites contained in the media.

In this context, this paper proposes a two-steps methodology for the
identification of ^{1}H-NMR metabolomics biomarkers. Having introduced
a typical experimental dataset, used to illustrate the methodology
throughout this paper, the first step consists in the implementation of
ICA in order to reduce the dimension and decompose the multivariate
spectral dataset into statistically independent components. Solutions
are proposed to select the optimal number of components and to rank
them by importance. The second step of this methodology consists in
a statistical modelling of the ICA resulting mixing weights [12-15]. A
panel of various mixed linear statistical models adapted to the nature
of the domain are considered. The model coefficients and appropriate
statistical tests are used to decide which ICA sources can be considered
as biomarkers of the stressor(s) of interest, including a visualization of
the effect of the latter on the ^{1}H-NMR spectra. In addition, contrasts
are computed from the selected sources to visualize the spectral effects
when one factor of interest changes. Finally, available in the Supporting
Information, the methodology has been used on real medical data to
successfully find biomarkers for Age related Macular Degeneration
(AMD).

A simple set of metabolomics data is used as running example to illustrate the methodology detailed in this paper. This section details this dataset, including the acquisition steps.

**Typical metabolomics data**

A typical experimental metabolomics database is formed by three
sets of data: a design, a set of ^{1}H-NMR spectra and biological and/
or histopathological data. The design describes the experimental
conditions underlying each available spectrum. Typical design factors are: subject ID (animal or human) and its characteristics, treatment,
dose and time of sampling. A ^{1}H-NMR dataset contains the spectral
evaluations of bio fluid samples which were collected according to
the design. A primary data reduction ("binning") is carried out by
digitizing the one-dimensional spectrum into a series of typically
250 to 3000 integrated regions or descriptor variables. However, a
typical metabolomics study involves about 30 to 200 spectra or sample
measurements. The resulting dataset is thus typically characterized by a
larger number m of variables than the number n of observations.

Another important characteristic of ^{1}H-NMR data is the strong
association (dependency) existing between some descriptors, due
to the fact that each molecule can have more than one spectral peak
and hence may contribute to several descriptors. Moreover, as a large
variety of dynamic biological systems and processes are reflected in
spectra, a range of physiological conditions, for example the nutritional
status, can also represent a source of variability into spectra. Noise and
biological fluctuations are thus natural and unavoidable in spectral data.
Finally, each spectrum in the ^{1}H-NMR dataset is also usually linked
with one or more variable(s), which tends to confirm the presence of a
response of the organism to the stressor. This confirmation is obtained
via the current gold-standard examinations (biological measures or
histopathological ones) on the subject for which spectra are measured.

**Experimental data**

Experimental data were produced according to a specific design, in order to provide a database in which one controls the alterations of known descriptors. The next sections detail the design, the acquisition and the pre-processing steps on data. A more detailed description and analysis of these data is available.

**Experimental design:** Homogeneous urine samples were spiked
with two products at different levels of concentration and analysed
through spectroscopy [16-18]. The products are citric acid ("citrate")
and hippuric acid ("hippu-rate"). They were added to urine at four
levels of concentrations, respectively 0, 2, 4 and 8 mM for citric acid
(Qc=8 mM), and 0, 1, 2 and 4 mM for hippuric acid (Qh=4 mM). The
resulting 14 points design is illustrated in **Figure 1**.

As shown in **Figure 2**, the peaks corresponding to each product are
located in distant areas. The hippurate is characterized by three peaks,
with two of them in region containing a low level of noise (around 7
ppm). On the contrary, citrate peaks are located in the noisy region
(around 2 ppm). Note that during the spectral pre-processing these
peaks are aggregated in a single one to avoid alignment problems.

**Sample preparation and acquisition of the ^{1}H-NMR data:** The
two products (citrate and hippurate) were first mixed with phosphate
buffer containing TSP (Trisodium Phosphate). The volume of buffer was adapted in order to obtain a volume of 600 ml. Each urine sample
came from a pool of 344 female Fischer rats and had a volume of
1200 ml. Each mixture of TSP, citrate and hippurate was added to a
urine sample, centrifuged, frozen at-80°C and unfrozen at 40°C the
day before the

**The post-acquisition treatments:** Each acquired spectrum was
processed using Bubble, a MATLAB tool for automatic processing and
reducing NMR spectra. Bubble performs sequentially: suppression of
the water resonance, apodisation (with a line broadening factor of 1
Hz), Fourier transform and phase correction, baseline correction using
a Whithaker smoother, median normalization and warping in order to
align shifted peaks. The last step of the Bubble process reduces, by simple
integration, the part of the spectrum situated between 0.2 and 10 ppm
to 600 descriptors. We manually added several pre-processing tools to
the spectra prepared by Bubble [20]. First, we replaced all the negative
values by zero. Secondly, we set to zero the ppm values corresponding
to the large non-informative urea peak and to the already treated
water peak (4.5-6.0 ppm). Then, the spectral region around the citrate
resonances (2.56-2.72 ppm) was integrated and summarized in just one
peak to suppress large shifts. Finally, we normalized again the dataset.
Indeed, the effect of the first normalization by the median, necessary
to realize an accurate warping, is cancelled due to the reduction. The
second normalization consists in constant sum normalization: each
spectrum is divided by the sum of intensities on all its ppms values.

**Notations**

Let X be the (m n) matrix of spectral data containing n spectra, each of them being described by m descriptors. Y is a (n l) matrix of design data describing each sample or spectrum by l variables. In our experimental data, n=28, m=600 and the l=2 design variables correspond to the citrate and hippurate concentrations. This dataset is used for illustrating the methodology developed in the following sections. Some of the steps are illustrated on a 24-spectra dataset only. The latter results from removing 4 spectra from the original dataset, corresponding to the two replicates (with and without water dilution) of the samples with maximum concentration of hippurate and without citrate, and vice-versa. The 24-spectra dataset has the advantage to correspond to a non-orthogonal design of experiments and will allow to emphasize the differences between PCA and ICA results.

Dimension reduction and signal decomposition by Independent Component Analysis and biomarkers identification by statistical modelling of the ICA weights.

The basic idea of Independent Component Analysis (ICA) is
to recover unobserved multidimensional independent signals from
linearly mixed observed ones [21]. In the metabolomics context, it
is used to extract metabolite profiles of potential biomarkers from
available ^{1}H-NMR spectra.

**The ICA methodology**

ICA was originally developed for signal processing to solve the problem of blind source separation (BSS). In the basic noiseless ICA model, each observed signal is a mixture of unknown statistically independent signals (named sources or components):

X=SA^{T} (1)

Where X denotes the (m n) matrix that contains n original signal
vectors of m observations (x_{i}), S denotes the (m q) matrix that contains
q unknown source vectors s_{j}, and A is a mixing matrix. Both S and A
are unknown. The "unmixing" problem considered by ICA is to find an
unmixing matrix such that the sources can be estimated by , where denotes the matrix formed by q estimations of scaled independent source
vectors s_{j} (as columns). The ICA model introduces an undetermination
in the scale of the recovered sources. Indeed, scaling a source by a
factor l is exactly compensated by dividing the corresponding column
of the mixing matrix by l. A natural way for fixing the magnitudes
of independent components is thus to assume that each component
has unit variance. It should be noted that the ambiguity of the sign
remains as we can multiply any component by -1 without affecting the
model. The key assumption of ICA is that the sources are statistically
independent. Under the ICA model, the observed data tend to be more
Gaussian than the independent components due to the Central Limit
Theorem (the distribution of a sum of independent random variables is
generally more Gaussian than the summands). Thus, the independence
of random variables can be reflected by non-gaussianity. Solving the
ICA problem aims then at finding a matrix W that maximizes the nongaussianity
of the estimated sources, under the constraint that their
variances are constant. The non-gaussianity may be estimated by the
negentropy, as in the FastICA algorithm used in this work. Other ways
of estimating the sources exist. Often, data are pre-processed before
applying ICA. First, mixtures are reduced to zero mean without loss of
generality. The second steps consist in ‘whitening’, i.e. applying PCA.
This reduces roughly by half the number of parameters to be estimated
by ICA, therefore facilitating the task of the latter. In addition using
PCA allows us to reduce the number of mixtures to be used by ICA;
the number q of sources to be computed can be fixed in this step via a
method discussed.

**ICA on metabolomic data and algorithm application:** In the
context of metabolomics ^{1}H-NMR data, the analyzed biofluid (e.g.
plasma, urine) can be seen as a mixture of individual metabolites; NMR
spectra may then be interpreted as weighted sums of NMR spectra of
these single metabolites. If the matrix X of ^{1}H-NMR spectra is rich
enough, the application of ICA to ^{1}H-NMR data should then ideally
recover components included in the mixture, interpretable as spectra of
pure or complex metabolites. The Fast ICA algorithm recovers sources
and linked weights from the spectral matrix through following steps:

• Pre-processing step 1: centre X by columns:

Where is the 1 × n vector of spectral means and 1 m a m 1 unit vector.

• Pre-processing step 2 ("whitening"): reduce by PCA the (m ×
n) matrix X^{c} to a (m × q) matrix of scores T (q £ min(n; m)):

X^{c}=T*P^{*}=T P + E.

Where P^{*} is a (n n) matrix defined on the basis of the eigenvectors
of the covariance matrix (X^{cT} X^{c})/n. Then, P is defined as the q first
lines of P^{*} and E is the error matrix. The column vectors of the full
score matrix T^{*} are centred, uncorrelated and their variances are equal
to one. In other words, the variance-covariance matrix of T^{*} equals
the identity matrix: Var(T^{*})=I_{n}. Note that this PCA differs from usual
PCA for metabolomics biomarkers identification as the resulting
components are linear combinations of observations (spectra) and not
of variables (spectral descriptors), and centring is done by spectra and
not by descriptor. The number of sources q to be estimated must be
fixed to less than min (n, m). This is performed here by selecting the q
first scores vectors (columns) of T* in order to build the (m × q) matrix
T. The choice of q is discussed.

• Extraction of S and A^{T} from T with the fastICA algorithm

The (m × q) matrix S contains q estimated independent components
(IC) s_{j}. Each s_{j} has a zero mean and a unit variance, and at least (q−1)
sources are non-gaussian. The A mixing matrix is a (n q) matrix.
Each column a _{j} is then a (n × 1) vector containing the weights (or
contributions) of the corresponding source s_{j} in the construction of the
n observed spectra. A source s_{j} playing a major role in the contribution
of an observed spectrum x_{i} has then a potentially large absolute value .

Choice of the number of sources to estimate: One important
parameter that may influence the ICA results is the number q of
estimated components. The effective number of independent sources
contributing to the signal is obviously un-known. ICA algorithms make
the fundamental assumption that the number of sources q is less than
or equal to the number of observed mixtures n. Moreover, to make
the implementation of the fast ICA algorithm effective, the maximal
value for q is the smallest dimension of its input matrix T, i.e. q min (n,
m). In ^{1}H-NMR metabolomics datasets, the resolution of a spectrum
m is typically higher than the number of spectra n. The maximal value
for q is then the number n of observed spectra. Anyway, when n is
large, choosing q=n can produce convergence problems or very high
computational costs. Choosing q<n by discarding some score vectors
obtained via the whitening matrix T helps convergence and discards
noise. PCA provides a natural ordering of the columns of T* according
to the eigenvalues l_{j} of X^{cT} X^{c}. The q first score vectors associated with
the largest eigenvalues are then selected to form the reduced matrix T.
Let us define D_{q} the proportion of the variation of X^{c} explained by the first q principal components:

(2)

We propose to choose q on the basis of a scree plot in order to guarantee the preservation of most of the original information.

**Measure of the information contained in ICA sources and
sources ordering:** ICA does not provide a natural ordering of the
computed sources. This section presents a possible solution to this
limitation. Given a set of q estimated sources s _{j}, we can reconstruct the
data as . Let us define the error when X^{c} is reconstructed with
source s j only:

This error is equivalent to the data reconstructed with all the
other sources contained in the (m (q1)) matrix S_{6=j}. For sources with
zero mean and unit variance, it can be shown that a measure of the
proportion of the variation in T explained by s_{j} is:

The proportion of the variance of signals in X^{c} explained by a
source s_{j} is then defined by:

With D_{q} the proportion of variance explained by the q scores in T
(see eqn. (4)). This measure of importance finally allows ordering the
sources s_{j} according to their respective C_{j}.

**Example:** In this section, the ICA procedure is applied on the
n=24 spectra with m=600 ppms dataset described. **Figure 3** shows the
expected improvement of ICA over PCA for this specific dataset: PCA
will produce principal components of maximum variance, while ICA
should provide independent directions, corresponding to the sources
of interest. As the experimental samples are mixtures of three products,
we ideally expected to find three independent sources of variation: the
urine, the citrate and the hippurate.

Of course, in the data analysis, it is supposed that we do not have
the information on the sources and expect to recover them blindly
according to our methodology. Based on the screeplot (**Figure 4**),
we first chose to estimate q=6 sources. The percentage of explained
variance with these first six PCs is D_{6}=0.9796.

A discussion about the three more important ICA sources derived by the ICA algorithm and the first three sources (or loadings vectors) obtained by applying classical PCA to the spectral matrix are detailed in the Supporting Information additional figure file.

**Biomarkers identification by statistical modelling of the ICA
weights**

The second step of the methodology fits a statistical model in order to identify metabolomics biomarkers from ICA results. More precisely, the model will search for a link between the ICA mixing weight matrix A and the design factors of the metabolomics study. This modelling step will provide a list of sources which significantly influence the spectra when the level of a factor of interest changes [22]. The profiles of these sources will then help the biologist to identify corresponding metabolites and designate them as candidate biomarkers.

**Principle:** The fundamental principle underlying this step of
the methodology is the following. A ^{1}H-NMR spectrum reflects the
concentrations of pure or complex metabolites contained in the
analysed sample. The design factors, as for example the dose of a
drug, can influence these concentrations and consequently modify
the spectra in a specific way. The methodology presented in this paper
supposes that the q sources recovered by ICA are the spectral images
of pure or complex metabolites that are influenced by the (observed
or unobserved) variables underlying the study. Under this assumption,
the mixing weights a_{ij} should be proportional to the concentrations of
the identified metabolites in the samples.

This step aims then at finding, through the mixing weights, which sources affect significantly the spectra when the factor of interest of the study changes (e.g. presence/absence of a disease, dose of a drug...) in spectral matrices where several other noise or controlled factors potentially affect the spectra (e.g. subject, nutrition status,... ). Linear mixed statistical models are used in this context to (1) allow to decorrelate the effect of the factor of interest on the mixing weights from the effects of the other covariables and noise factors, (2) allow to take into account the random character of some covariables and (3) provide a measure of statistical significances of the link between the factor of interest and the sources.

**Linear mixed model specification, estimation, testing
and interpretation:** Let a_{j} be the (n 1) vector of mixing weights
corresponding to the j^{th} ICA source and Y the (n l) experimental design
matrix containing the variables of the study (as the factor of interest
for which biomarkers are searched for) and other covariables which
may have affected the spectra. In order to find how these variables
are linked to each vector of weights a_{j}, fitting a linear mixed statistical
model is a flexible solution and a very classical approach in the context
of biomedical studies. For each of the q sources s_{j}, the following linear
mixed model can be written as follows:

(3)

Where,

Z^{1} is a (n × p_{1}) incidence matrix containing the fixed effects of the
model: typically a constant term, coded categorical design variables,
continuous variables and interactions or other high-order terms.

Z^{2}, a (n × p_{2}) incidence matrix containing the random effects of the
model: typically coded random design variables as subject, batch, day
and interactions between fixed and random variables.

β_{j}, a (p_{1} 1) vector of constant parameters to be estimated, γ_{j} is a (p_{2} ×
1) vector of random effects distributed as a multivariate normal N(0,G)
and e j is a (n × 1) vector of residuals distributed as a multivariate
normal N(0 × R).

Different specific cases of this general model are possible according
to the inclusion of Z^{1} or Z^{2} or both. Models using only Z^{1} are also called
GLM models in the statistical literature, and include ANOVA and
regression models depending of the categorical or continuous nature
of the variables included in the model. In most cases, the variable of
interest of the study will be included in this matrix as a dose of a drug or
a treatment versus placebo or the presence/absence of a disease. It can
also include covariables that are not directly of interest but may greatly
affect the spectra as the age or sex of a patient. Models using only Z2
are "variance components" models including only random factors. This
arises when one is interested by the effect of various populations (or
analytical factors) on the spectrum variability (e.g. subject, hospital,
operator, batch…), but this is not yet common in metabolomics.
Complex metabolomics studies will typically include both fixed and
random effects as for example in longitudinal studies where n subjects
belonging to p categories of treatments are followed over time.

Depending of the generality of the specified model, the estimated
parameters and related significance measures will be provided by basic
statistical softwares or by more advanced ones like the PROC MIXED
procedure in SAS or *lme* function in R.

Testing the significance of the factor(s) of interest is a key step in the methodology and is typically neglected in most metabolomics studies. It will allow more generalized and powerful conclusions to the population of interest from which the data are issued. In general mixed models, several common procedures exist to test the significance of model terms. They are different for fixed and random effects, depend on the method applied to estimate the model and may be controversial when complex random effects occur.

Let us suppose that the model contains only fixed continuous and
categorical effects and that the effect of interest is the main effect of a
continuous covariate y_{k}. The significance of y_{k} is derived for each source
s_{j} through the p-value related to a t-statistic:

(4)

(5)

and where is the coefficient of yk in the fitted model on is
the standard error associated with , n is the number of observations
(spectra), p is the number of parameters into the model b and t_{(n-p)} is a
t random variable with (n-p) degrees of freedom.

If one supposes that the effect of interest is a categorical covariate
with q levels, the significance of y_{k} is then derived for each source s_{j} through a F statistic as follows:

(6)

With,

(7)

And where MSR j is the mean square of model residuals for source
s_{j}, MSy^{k}_{j} the mean square related to y_{k} effect and F_{q–1;n p} a F random
variable with (q–1) and (n–p) degrees of freedom.

If such procedure is applied on K variables with more complex
effects of interest (and for each of the q sources), (K × q) tests are
performed and the decision of significance via the p-values must take
into account the multiplicity situation. If (K × q) remains reasonably
small, a simple Bonferroni correction is still applicable and the
significance of the effect of y_{k} for source s_{j} is confirmed if p_{jk}£/(K ×
q), where a is a chosen total error rate (e.g. a=0.05). For larger (K ×
q), procedures like False Discovery Rate (FDR) can be used. Through
these testing procedures, the modelling step can be summarized into a
table containing for each mixing weight vector (and related source) a
measure of significance for each factor of interest (**Table 1**). This result
is the basis of biomarker identification and interpretation.

Sources | Linear Regression p-values | F(j,1) |
ANOVA p-values | |
---|---|---|---|---|

S_{1} |
-6.60e^{-7} |
1.94e^{-15} |
105.46 | 8e^{-13} |

S_{2} |
-5.52e^{-7} |
4.77e^{-16} |
152.71 | 2.04e^{-14} |

S_{3} |
2.65e^{-6} |
8.30e^{-35} |
4468.90 | 1.31e^{-29} |

S_{4} |
-1.07e^{-7} |
0.27 | 0.83 | 0.50 |

S_{5} |
2.21e^{-7} |
0.01 | 2.86 | 0.06 |

S_{6} |
3.70e^{-9} |
0.96 | 0.02 | 0.99 |

**Table 1:** Results of Linear Regression and ANOVA models.

Let us define S^{*} as the (m × r) matrix of the r significant sources
identified for the (or a) factor of interest in the study. A first way to extract biomarkers from these sources consists in examining their
profile and identifying the known pure or complex metabolites with
close profiles. This approach is appropriate when r is quite small and
has "clean fingerprints". Also, sources do not provide quantification or
a direction of the metabolite effect.

An additional and more informative approach is then proposed. It is a generalization of the concept of contrast estimation in classical linear models and gives an answer to the following question: which average change is expected in the spectrum when the covariate of interest changes from one level to another (e.g. if a patient is or is not affected by a disease, or if the dose of a drug is increased)?

Let us introduce y^{1}_{k} and y^{2}_{k}, two levels of interest for a quantitative
covariate yk (e.g. two drug doses). Let us then define as the vector of the differences of predictions for these two covariate
levels and for the r identified sources. For models without interaction,
these differences are only influenced by the terms in y_{k}. For models
with interactions, the values of the other factors should be fixed to
chosen levels.

Consequently, the expected change in spectra can simply be obtained via the following contrast:

(8)

Where C_{2–1} is a (m × 1) vector and can be drawn as a spectrum to
visualize the spectral zones which are affected by the covariate.

In particular, if y_{k} is introduced as a continuous variable in
the model and if is the vector of the coefficients for y_{k} and the r
identified sources, the expected change between the spectra for the two
levels y^{1}_{k} and y^{2}_{k} is given by . If y_{k} is introduced
as a categorical variable in the model and if and are the vectors
of the estimated effects for the two levels of interest for the r sources,
the change in spectra is provided by .

**Example:** This section illustrates the modelling step on the
experimental data presented. All 28 spectra are used in this section and
the two design factors (hippurate and citrate levels) are used as variables.
With 28 spectra, the screeplot suggests to calculate six ICA sources. The
profiles of the three first sources are very similar than those obtained
with the reduced design. Let us define y_{1} as the hippurate dose and take
it as the factor of interest for which biomarkers are investigated, and y_{2} as the citrate level supposed to be an additional covariate in the study.

These variables can be introduced either as continuous or as
categorical variables in the linear model. In the first case, matrix Z will
be a (28 3) matrix with a constant term as first column and y_{1} and y_{2} as second and third columns. For each source s _{j}, the following linear
model is written as:

(9)

The β_{1}’s estimated by linear regression for the six sources and the
corresponding p-values are given in **Table 1** (the β2’s are not provided
since this covariate is not considered of interest). Note that higher
order terms (quadratic or interaction terms) could be included in such
model. If the two covariates are introduced as categorical variables
in the model, Z becomes a (28 7) matrix with a constant term as first
column and two blocks of three columns corresponding to the binary
coding of the 4-levels categorical variables. Such model can then be
estimated by regression but corresponds also to a two ways ANOVA
model which can be fitted through classical ANOVA formulae when
the design is balanced. This model is written in the ANOVA literature as:

(10)

Where indices i and h refer to the levels of the two variables y_{1} and
y_{2}, and and to the corresponding main effects according to
source s_{j}. Note that one could also introduce an interaction term in this
model. **Table 1** provides the F statistic and corresponding p-values for
the effect of the first factor on the six sources. If y_{1} is considered as the
only variable of interest, a p-value will be declared significant if smaller
than a/6=0:00833 with a= 0:05 according to the Bonferroni correction.

Four sources can then be declared as significant in the regression
model and three sources in the ANOVA model. The most important
source seems to be s_{3}. Spectral regions linked with s3 may then represent
biomarkers or spectral expression of metabolites significantly affected
by a change of the factor of interest y_{1}. As expected in this example
with y_{1} being the hippurate dose, s_{3} presents as biomarkers the peaks
in the spectral zone of the hippurate molecule. Logically, the model
recovers that a change of concentration of hippurate in the mixture
introduces a signal corresponding to the third source in the resulting
spectra. However, when other covariables affect the spectra, note that
the methodology presented here is able to extract from the signal
the effect of the variable of interest and keep in other possible nonorthogonal
sources of variability. This is a crucial property in biological
and medical applications, where controlled or noise covariables can
greatly affect the signal and hide the effect of the variable of interest.

When a source is declared as significant, the model parameters
also provide a quantification of the effect of the variable of interest on
the spectra through the mixing weights. **Figure 5** illustrates the linear
effect of the hippurate dose on the mixing weights for the four levels of
citrate. The slope of the line is 2:6410–6, the parameter β_{1} of the linear
model for s_{3}.

Additionally, both linear regression and ANOVA models select s_{1} (spectral profile of pure urine) and s_{2} (spectral profile of pure
citrate) as significant sources. In linear regression models, the sign
of corresponding parameters and is negative, indicating a
negative contribution of these sources to the observed spectra. This
is easily explained by the constant sum normalization pre-processing
applied to the spectra: if the peak heights corresponding to one
metabolite in the spectra increase, the peaks corresponding to all other
products (urine and citrate) decrease accordingly.

When more sources are declared as significant (but with less
interpretable sources), the methodology presented in this paper
allows to reconstruct the effect of the change of one factor level on
the spectra independently to the effect of possible confounding model
factors. In the design matrix Y, the hippurate dose y_{1} is observed at
the following values: 0, 75, 150 and 300 mg. Three contrasts, C_{2–1}, C_{3–1} and C_{4–1}, respectively describe the expected changes in spectra when
the drug dose goes from 0 to 75 mg, 0 to 150 mg and 0 to 300 mg. **Figure 6** presents the three contrasts obtained when y_{1} is introduced
as a continuous variable in the model. This figure shows that, as the
dose goes from 0 to a positive value in each of the three contrasts, the
hippurate peaks increase. On the contrary, negative values appear
everywhere else due to the normalization. The corresponding figure
when y_{1} is introduced as a qualitative variable is very similar.

More discussion concerning this data set and a generalization to more complex mixed models may be found in ref. [18].

The biomarker identification in ^{1}H-NMR based metabolomics
is traditionally realised, with some limitations, via the examination
of the two first components of a PCA, but without any statistical
testing confirmation of factors effects. In this paper, we presented a
new methodology providing three kinds of knowledge on ^{1}H-NMR
metabolomics data: the identification of biomarkers, a statistical
confirmation of the significance of these biomarkers and the
visualization of the effects on the biomarkers caused by factor changes.

The methodology involves a dimension reduction by ICA followed by statistical modelling approaches. This paper first presents a process to decompose by ICA the spectral data into statistically independent components and shows, on experimental data, that ICA allows to visualize, through the resulting sources, the spectral profile of independent metabolites contained in the studied bio fluid and their quantity through the corresponding mixing weights. Then, linear mixed statistical modelling is applied on ICA results to select the sources or spectral regions changing significantly according to the factors of interest. Finally, the selected sources are used to reconstruct the spectra and to compute contrasts presenting the alterations in specific regions caused by different changes of the factor of interest. Beside their discovery, contrasts also allow to visualize the alterations of potential biomarkers for defined changes of covariate conditions or context.

As exposed on experimental data, the ICA solves the weaknesses of the PCA dimension reduction by providing more natural and also more biologically meaningful representations of the data. Additionally, the combination of ICA with statistical models has the advantage to base the component selection on an inferential criterion: biomarkers are identified from components for which the covariate of interest shows a significant effect. In the usual PCA, biomarkers are identified from the component with the largest percentage of variance, without any inferential information.

In this paper, source selection is based on t-statistics computed on the weight vectors without using their significance levels. An accurate source selection is provided, due to its inferential character but also to the fact that models give the possibility to include all the design covariates jointly with the covariate of interest. The large diversity of statistical models accepted by this methodology allows us to apply it to a large variety of complex metabolomics situations: models can include quantitative and qualitative design variables as well as combinations of fixed and random effects (linear mixed models). As a result, additionally to the proposed biomarker search, the methodology provides information on spectral regions affected by other factors of the study.

Furthermore, this methodology has been applied on a real metabolomic AMD dataset (see Supporting Information). The spectral biomarkers linked with this disease correspond to a metabolite supporting biological explanation of the setting of AMD.

The authors are grateful to the Centre Intrafacultaire de Recherche du Médicament, Laboratoire de Pharmacognosie et de Chimie Pharmaceutique, ULg, for providing data. Support from the IAP Re-search Network P7/06 of the Belgian State (Belgian Science Policy) is gratefully acknowledged.

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