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Statistics in
Metabolomics
David Banks
ISDS
Duke University
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1. Background
Metabolomics is the next step after
genomics and proteomics.
There are about 25,000 genes, most
of which have unknown functions.
There are about 1,000,000 proteins,
most of which are unstudied.
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In contrast to the *omics areas:
 There are only about 900 main
metabolites, and we know their
chemical structures
 Also, we know (pretty well) the
biochemical pathways that
determine their production rates
Metabolites are low-weight molecular
compounds produced in the course
of processing raw materials.
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Some common metabolites include:
 cholesterol
 glucose, sucrose, fructose
 amino acids
 lactic acid, uric acid
 ATP, ADP
 drug metabolites, legal and illegal
These are produced in metabolic
pathways, such as the Krebs
(citrate) cycle for oxidation of
glucose.
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These pathways contain important
information about the amount of
each metabolite:
 Stoichiometric equations show how
much material is produced in a
given reaction; i.e., mass balance.
 Rate equations govern the speed
at which reactions take place, and
the location of the Gibbs
equilibrium
This gives metabolomics an edge.
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Biochemical Profile Map to
Metabolic Pathways
Biochemical Profile
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The purposes of metabolomics are:
 Early detection of disease, such as
necrosis, ALS, Alzheimer’s, and
infection or inflammation.
 Assessment of toxicity (especially
liver toxicity) in new drugs.
 Diet strategies, drug testing.
 Elucidating biochemical pathways.
There is less raw information than for
other *omics, but more context.
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2. Measurement Issues
To obtain data, a tissue sample is
taken from a patient. Then:
 The sample is prepped and put
onto wells on a silicon plate.
 Each well’s aliquot is subjected to
gas and/or liquid chromatography.
 After separation, the sample goes
to a mass spectrometer.
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The sample prep involves stabilizing
the sample, adding spiked-in
calibrants, and creating multiple
aliquots (some are frozen) for QC
purposes. This is roboticized.
Sources of error in this step include:
 within-subject variation
 within-tissue variation
 contamination by cleaning solvents
 calibrant uncertainty
 evaporation of volatiles.
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Gas chromatography creates an
ionized aerosol, and each droplet
evaporates to a single ion. This is
separated by mass in the column,
then ejected to the spectrometer.
Sources of error in this step include:
 imperfect evaporation
 adhesion in the column
 ion fragmentation or adductance
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The fourier mass spectrometer
determines the mass to charge
ratio of the ion from the field
strength required to keep the ion
spinning in a circle. This avoids
the entry-time uncertainty in TOF
machines, so the only main error is
uncertainty about the field strength
Some laboratories use MALDI-TOF
equipment, and the error sources
are slightly different.
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The result of this is a set of m/z ratios
and timestamps for each ion,
which can be viewed as a 2-D
histogram in the m/z x time plane.
One now estimates the amount of
each metabolite. This entails
normalization, which also
introduces error.
The caveats pointed out in Baggerley
et al. (Proteomics, 2003) apply.
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3. Statistical Problems
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Understanding the uncertainty
budget in metabolomic data, which
entails both quality control and
cross-platform comparisons.
Identifying the peaks in the m/z x t
plane, and estimating quantity of
specific metabolites.
Finding markers for disease or
toxicity, or measuring change.
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3.1 Uncertainty
The classical NIST approach to this
is to:
 build a model for the error terms
 do a designed experiment with
replicated measurements
 fit a measurement equation to the
data
See Cameron, “Error Analysis,” ESS
Vol. 9, 1982.
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Let z be the vector of raw data, and
let x be the estimates. Then the
measurement equation is:
G(z) = x = µ + ε
where µ is the vector of unknown
true values and ε is
decomposable into separate
components.
For metabolite i, the estimate Xi is:
gi(z) = lnΣ wij ∫∫sm(z) – c(m,t)dm dt.
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The law of propagation of error (this
is essentially the delta method)
says that the variance in X is about
Σni=1 (∂g /∂ zi)2 Var[zi] +
Σi≠k 2 (∂g/∂zi)(∂g/∂zk) Cov[zi, zk]
The weights depend upon the values
of the spiked in calibrants, so this
gets complicated.
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Cross-platform experiments are also
crucial for medical use. This leads
to key comparison designs. Here
the same sample (or aliquots of a
standard solution or sample) are
sent to multiple labs. Each lab
produces its spectrogram.
It is impossible to decide which lab is
best, but one can estimate how to
adjust for interlab differences.
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The Mandel bundle-of-lines model is
what we suggest for interlaboratory
comparisons. This assumes:
Xik = αi + βi θk + εik
where Xik is the estimate at lab i for
metabolite k, θk is the unknown
true quantity of metabolite k, and
εik ~ N(0,σik2).
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To solve the equations given values
from the labs, one must impose
constraints. A Bayesian can put
priors on the laboratory coefficients
and the error variance.
Metabolomics needs a multivariate
version, with models for the rates
at which compounds volatilize.
We plan to use this model to
compare the Metabolon lab in RTP
to Chris Newgard’s lab at Duke.
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3.2 Peak Identification
A classic problem in proteomics is to
locate peaks and estimate their
area or volume.
Unlike proteomics, metabolite peak
location is mostly known. So
Bayesian methods seem good (cf.
Clyde and House). Metabolon uses
proprietary software.
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GC Data
Confidential
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Tissue Differences
Confidential
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Cancer Type - CNS cancer
Cancer Type - breast cancer
Cancer Type - colon cancer
Cancer Type - leukemia
Cancer Type - melanoma
Cancer Type - non small cell lung cancer
Cancer Type - ovarian cancer
Cancer Type - prostate cancer
Cancer Type - renal cancer
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3.3 Data Mining
Different tools are appropriate for
different kinds of metabolomic
studies. The work we have done
focuses on:
 Random Forests
 Support Vector Machines
 Robust Singular Value
Decomposition
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We had abundance data on 317
metabolites from 63 subjects. Of
these, 32 were healthy, 22 had
ALS but were not on medication,
and 9 had ALS and were taking
medication.
The goal was to classify the two ALS
groups and the healthy group.
Here p>n. Also, some abundances
were below detectability.
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Using the Breiman-Cutler code for
Random Forests, the out-of-bag
error rate was 7.94%; 29 of the
ALS patients and 29 of the healthy
patients were correctly classified.
20 of the 317 metabolites were
important in the classification, and
three were dominant.
RF can detect outliers via proximity
scores. There were four such.
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Several support vector machine
approaches were tried on this data:
 Linear SVM
 Polynomial SVM
 Gaussian SVM
 L1 SVM (Bradley and
Mangasarian, 1998)
 SCAD SVM (Fan and Li, 2000)
The SCAD SVM had the best loo
error rate, 14.3%.
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The L1 SVM attempts to mimic the
automatic variable selection in the
LASSO (Tibshirani, 1996) by
solving the programming problem:
Minb,w Σ[1 – yi(b+wTxi)]+ + λΣ | wk |
where the first sum is over n and
the second is over p.
SCAD replaces the L1 penalty with a
nonconvex penalty.
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The SCAD SVM selected 18 of the
metabolites as being important; the
L1 selected 32. This suggests that
the automatic variable selection in
L1 SVM is not very effective.
A further multiple tree analysis with
FIRMPlusTM software from the
GoldenHelix Co. did not achieve
good classification.
So Random Forests wins. And the
selected metabolites make sense.
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Robust SVD (Liu et al., 2003) is used
to simultaneously cluster patients
(rows) and metabolites (columns).
Given the patient by metabolite
matrix X, one writes
Xik = ri ck + εik
where ri and ck are row and column
effects. Then one can sort the
array by the effect magnitudes.
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To do a rSVD use alternating L1
regression, without an intercept, to
estimate the row and column
effects. First fit the row effect as a
function of the column effect, and
then reverse. Robustness stems
from not using OLS.
Doing similar work on the residuals
gives the second singular value
solution.
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3.3.1 Preterm Labor
The NIH wanted to decide whether
amniotic fluid samples from women
in preterm labor could support
classification:
 Term delivery
 Preterm delivery with inflammation
 Preterm delivery without
inflammation.
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The analysis had samples from 113
women in preterm labor. We tried
all of the usual classification
methods.
As before, Random Forests gave the
best results. The various SVMs
were about 5-10% less predictive.
The main information was contained
in amino acids and carbohydrates.
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Predicted
Term
True
Term
Inflamm.
No Inf.
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7
2
Inflamm.
1
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2
No Inf.
0
1
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RF accuracy was 100/113 = 88.49%.
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For those with term delivery, amino
acids were low, carbohydrates
were high.
For those who had preterm delivery
without inflammation, both amino
acids and carbohydrates were low.
For those who had inflammation, the
carbohydrates were very low and
the amino acids were high.
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My collaborators in this research are:
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Chris Beecher, Metabolon, Inc.
Adele Cutler, USU
Leanna House, Duke University
Jackie Hughes-Oliver, NCSU
Xiadong Lin, U. of Cincinnati
Susan Simmons, UNC-Wilmington
Young Truong, UNC-Chapel Hill
Stan Young, NISS
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