Kiebel

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Kiebel Friston 2004b( 2cnd article ) A
Framework for discussion
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ERP analysis

• Traditionally
• distinguish between stimulus locked component
  and residual (error)
• Avereged ERP over peristimulus window
• Do ANOVA between trial types/subjects with
  nonsphericity correction

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Hierarchal model

• Yx(1)b(1) e(1)
• b(1)x(2)b(2) e(2)
• 1st level observation model for multiple ERP
• Within ERP- temporal effect fixed effect
  (single subject)
• 2cnd level model 1st level parameters over
  subjects trial types
• Between ERP- experimental effect (group,
  condition) random effect (population inference)
• Need to estimate associated error covariance

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model
contrasts
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Observation model at 1st level Wavelets

• Time-frequency decomposition
• Power in a specific frequency range within
  peristimulus window
• Can characterize induced response
• Reduces the ERP to a few wavelet parameters that
  capture the difference between conditions
• Advantages
• Sparse representation of salient features
• Efficient error covariance estimation at 2cnd
  level
• Can also use Fourier transform

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1st level observation modeltemporal matrix

• Yx(1)b(1) e(1)
• b(1)x(2)b(2) e(2)
• Y ERP for each subject trial
• X(1) I(NsubjectsNtypes) X(t)
• X(t) N bines X N p models temporal components
  of single ERP
• X(t) - Any linear transform of ERP- wavelets
• fig 1, p 501

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1st level

• Yx(1)b(1) e(1)
• If x(1) is square-x(t) wavelet is nontruncated,
  the covariance matrix of e(1)- c(1) cant be
  estimated
• NpltNbins (X(t) N bines X N p )
• E(1) normally distributed

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1st level observation modelERROR

• Yx(1)b(1) e(1)
• E(1) Error covariance matrix c(1)diag (?)
  C(t)
• C(1)-ERP specific components weighted by variance
  ?(k)
• ?(k) between ERP components- observation error
• k 1.. NsubjectsNtypes
• C(t) within ERP components
• Assume homogenous var over peristimulus time
  -gtERP error white noise-gt C(t)I bins.
• Motivation physiology noise (C(t)) is dominated
  by
• measurement noise (?(k))
• 
  Fig 2.
  p505

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2cnd level design matrix

• Yx(1)b(1) e(1)
• b(1)x(2)b(2) e(2)
• X(2)x(d) Ip (p from 1st level time matrix)
• X(d) subject trial type specific treatment
  effect
• Example x(d) 1 subjects X I types-
• averages over
  subjects across trial types
• Fig 3. p506

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2cnd level- ERROR

• Yx(1)b(1) e(1)
• b(1)x(2)b(2) e(2)
• Error covariance c(2) ? (Qd Qp)
• Qd- design specific components (within subj corr)
• Qp- physiological error, random error which can
  represent highly structured variation in ERP.
• Nonspherical
• intersubject var in the way ERP is expressed
• Var changes with peristimulus time (var increases
  around phasic components - N1, due to latency
  var)

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2cnd levelERROR- nonsphercity

• 2cnd level model must capture nonstationary var
  attending correlation structure
• Problem not enough observation to asses highly
  parameterized model
• Solution
• Use prior ERP knowledge to reduce 1st level
  parameters associated covariance parameters at
  2cnd level
• Compute estimation using other voxels
• This paper assumes c(2) is known
• SPM uses 2cnd solution(other voxels)

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Emperical bayes

• In a Bayesian framework, parameters of one level
  can be made priors on distribution of parameters
  at lower level Parametric Empirical Bayes
  (Friston et al, 2002)
• C(2) is equivalent to prior covariance of b(1)
• C(2) places prior constrains on subject specific
  estimate of b(1)

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Estimation

• After specifying the model..
• Estimate the parameters using ML or ordinary
  least squares (OLS)
• Variance parameters by ReML
• Two stage procedure
• 1st level b(1)
• 2cnd level b(2), ? (2)

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Inference-contrasts

• Contrast- linear combination of parameter
  estimates that defines a specific null hypothesis
  about the parameters
• In this model- parameters are wavelet
  coefficients . but not intuitive -gt
• contrast weights in peristimulus time
  projected to 2nd level.

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Contrast vector

• C(2)- contrast vector
• C(2)Cd Cw
• Cd- comparison on the level of the experiment
• 2 trial types -1 1
• Cw- aspect of ERP we are interested in.
• Restricted in time-frequency.
• Time window 0 0 0 0 1 1 0 0 0

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Multidimensional 2cnd level contrast

• W N bins X N con
• Test for effects distributed over multiple
  windows
• Allows a large range of tests in time-frequency
  domain.
• Fig 5,6 p508

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Multidimensional 2cnd level contrast
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Hierarchal vs conventional approach

• Conventional analysis ERP is predetermined by
  the window averaging
• SPM avraging represents 1 of many hypotheses
  that could be tested, by 2cnd level contrast
• Temporally long, freq restricted window stimulus
  locked transients in prespecified freq band
• Short window, multiple freq temporally localized
  shape of unspecified form or frequency

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Hierarchal vs conventional approach

• Conventional no partition of error to
  observation noise, physiological ERP variability.
  e(1)0
• Conventional Single contrast per ERP to 2cnd
  level
• Hierarchal parameters are estimated once
  for a family of hypothesis
• Conventional precludes hypothesis tests for
  treatment effects that span several dimensions
  e.g time-freq
• Fig 8 p 510

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