DCM

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Group analyses of fMRI data
Klaas Enno Stephan Laboratory for Social and
Neural Systems Research Institute for Empirical
Research in Economics University of
Zurich Functional Imaging Laboratory
(FIL) Wellcome Trust Centre for
Neuroimaging University College London
With many thanks for slides images to FIL
Methods group, particularly Will Penny
Methods models for fMRI data analysis26
November 2008
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Overview of SPM
Statistical parametric map (SPM)
Design matrix
Image time-series
Kernel
Realignment
Smoothing
General linear model
Gaussian field theory
Statistical inference
Normalisation
p lt0.05
Template
Parameter estimates
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Why hierachical models?
fMRI, single subject
EEG/MEG, single subject
time
fMRI, multi-subject
ERP/ERF, multi-subject
Hierarchical models for all imaging data!
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Reminder voxel-wise time series analysis!
Time
Time
BOLD signal
single voxel time series
SPM
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The model voxel-wise GLM
X

y

• Model is specified by
• Design matrix X
• Assumptions about e

N number of scans p number of regressors
The design matrix embodies all available
knowledge about experimentally controlled factors
and potential confounds.
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GLM assumes Gaussian spherical (i.i.d.) errors
sphericity iiderror covariance is scalar
multiple of identity matrix Cov(e) ?2I
Examples for non-sphericity
non-identity
non-independence
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Multiple covariance components at 1st level
enhanced noise model
error covariance components Q and
hyperparameters??
V
Q1
Q2
?1
?2

Estimation of hyperparameters ? with ReML
(restricted maximum likelihood).
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t-statistic based on ML estimates
c 1 0 0 0 0 0 0 0 0 0 0
For brevity
ReML-estimates
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Group level inference fixed effects (FFX)

• assumes that parameters are fixed properties of
  the population
• all variability is only intra-subject
  variability, e.g. due to measurement errors
• Laird Ware (1982) the probability distribution
  of the data has the same form for each individual
  and the same parameters
• In SPM simply concatenate the data and the
  design matrices ? lots of power (proportional
  to number of scans), but results are only valid
  for the group studied, cant be generalized to
  the population

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Group level inference random effects (RFX)

• assumes that model parameters are
  probabilistically distributed in the population
• variance is due to inter-subject variability
• Laird Ware (1982) the probability distribution
  of the data has the same form for each
  individual, but the parameters vary across
  individuals
• In SPM hierarchical model? much less power
  (proportional to number of subjects), but
  results can (in principle) be generalized to the
  population

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Recommended reading
Linear hierarchical models
Mixed effect models
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Linear hierarchical model
Multiple variance components at each level
Hierarchical model
At each level, distribution of parameters is
given by level above.
What we dont know distribution of parameters
and variance parameters.
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Example Two-level model




Second level
First level
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Two-level model
random effects
random effects
Friston et al. 2002, NeuroImage
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Mixed effects analysis
Non-hierarchical model
Estimating 2nd level effects
Variance components at 2nd level
between-level non-sphericity
Additionally within-level non-sphericity at both
levels!
Friston et al. 2005, NeuroImage
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Estimation
EM-algorithm
E-step
M-step
Assume, at voxel j
Friston et al. 2002, NeuroImage
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Algorithmic equivalence
Parametric Empirical Bayes (PEB)
Hierarchical model
EM PEB ReML
Single-level model
Restricted Maximum Likelihood (ReML)
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Mixed effects analysis
Summary statistics
Step 1
EM approach
Step 2
Friston et al. 2005, NeuroImage
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Practical problems
Most 2-level models are just too big to compute.
And even if, it takes a long time!
Moreover, sometimes we are only interested in
one specific effect and do not want to model all
the data.
Is there a fast approximation?
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Summary statistics approach
Second level
First level
Data Design Matrix Contrast Images
SPM(t)
One-sample t-test _at_ 2nd level
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Validity of the summary statistics approach
The summary stats approach is exact if for each
session/subject
Within-session covariance the same
First-level design the same
One contrast per session
All other cases Summary stats approach seems to
be fairly robust against typical violations.
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Reminder sphericity
Scans
sphericity means
i.e.
Scans
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2nd level non-sphericity
Error covariance
Errors are independent but not identical e.g.
different groups (patients, controls)
Errors are not independent and not
identical e.g. repeated measures for each
subject (like multiple basis functions)
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Example 1 non-indentical independent errors
Auditory Presentation (SOA 4 secs) of (i) words
and (ii) words spoken backwards
Stimuli
e.g. Book and Koob
(i) 12 control subjects (ii) 11 blind subjects
Subjects
fMRI, 250 scans per subject, block design
Scanning
Noppeney et al.
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Controls
Blinds
1st level
2nd level
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Example 2 non-indentical non-independent errors
Stimuli
Auditory Presentation (SOA 4 secs) of words
1. Motion 2. Sound 3. Visual 4. Action
jump click pink turn
Subjects
(i) 12 control subjects
1. Words referred to body motion. Subjects
decided if the body movement was slow. 2. Words
referred to auditory features. Subjects decided
if the sound was usually loud 3. Words referred
to visual features. Subjects decided if the
visual form was curved. 4. Words referred to
hand actions. Subjects decided if the hand action
involved a tool.
fMRI, 250 scans per subject, block design
Scanning
What regions are affected by the semantic content
of the words?
Question
Noppeney et al.
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Repeated measures ANOVA
1st level
3.Visual
4.Action
1.Motion
2.Sound
?
?
?
2nd level
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Repeated measures ANOVA
1st level
3.Visual
4.Action
1.Motion
2.Sound
?
?
?
2nd level
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Practical conclusions

• Linear hierarchical models are general enough
  for typical multi-subject imaging data (PET,
  fMRI, EEG/MEG).
• Summary statistics are robust approximation to
  mixed-effects analysis.
• Use mixed-effects model only, if seriously in
  doubt about validity of summary statistics
  approach.
• RFX If not using multi-dimensional contrasts at
  2nd level (F-tests), use a series of 1-sample
  t-tests at the 2nd level.
• To minimize number of variance components to be
  estimated at 2nd level, compute relevant
  contrasts at 1st level and use simple test at 2nd
  level.

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Thank you