Statistics in Metabolomics

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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
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• The purposes of metabolomics are
• Early detection of disease, such as necrosis,
  ALS, Alzheimers, 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 wells 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

• 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 e
• where µ is the vector of unknown true values
  and e is decomposable into separate components.
• For metabolite i, the estimate Xi is
• gi(z) lnS 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
• Sni1 (?g /? zi)2 Varzi
• Si?k 2 (?g/?zi)(?g/?zk) Covzi, 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 ai ßi ?k eik
• where Xik is the estimate at lab i for
  metabolite k, ?k is the unknown true quantity of
  metabolite k, and
• eik N(0,sik2).

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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 Newgards 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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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 pgtn. 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 S1 yi(bwTxi) ?S 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 eik
• 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 Inflamm.
  No Inf.
• Term 39 1
  0
• True Inflamm. 7 32
  1
• No Inf. 2 2
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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
• 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