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Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 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
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Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

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Page 1: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Group analyses of fMRI data

Methods & models for fMRI data analysis28 April 2009

Klaas Enno Stephan

Laboratory for Social and Neural Systems ResearchInstitute for Empirical Research in EconomicsUniversity of Zurich

Functional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London

With many thanks for slides & images to:

FIL Methods group, particularly Will Penny

Page 2: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Overview of SPM

RealignmentRealignment SmoothingSmoothing

NormalisationNormalisation

General linear modelGeneral linear model

Statistical parametric map (SPM)Statistical parametric map (SPM)Image time-seriesImage time-series

Parameter estimatesParameter estimates

Design matrixDesign matrix

TemplateTemplate

KernelKernel

Gaussian Gaussian field theoryfield theory

p <0.05p <0.05

StatisticalStatisticalinferenceinference

Page 3: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Why hierachical models?

fMRI, single subjectfMRI, single subject

fMRI, multi-subjectfMRI, multi-subject ERP/ERF, multi-subjectERP/ERF, multi-subject

EEG/MEG, single subjectEEG/MEG, single subject

Hierarchical models for all imaging data!

Hierarchical models for all imaging data!

time

Page 4: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Time

BOLD signalTim

esingle voxel

time series

single voxel

time series

Reminder: voxel-wise time series analysis!

modelspecificati

on

modelspecificati

onparameterestimationparameterestimation

hypothesishypothesis

statisticstatistic

SPMSPM

Page 5: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

The model: voxel-wise GLM

=

e+yy XX

N

1

N N

1 1p

p

Model is specified by1. Design matrix X2. Assumptions about

e

Model is specified by1. Design matrix X2. Assumptions about

e

N: number of scansp: number of regressors

N: number of scansp: number of regressors

eXy eXy

The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

),0(~ 2INe ),0(~ 2INe

Page 6: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

GLM assumes Gaussian “spherical” (i.i.d.) errors

sphericity = iid:error covariance is scalar multiple of identity matrix:Cov(e) = 2I

sphericity = iid:error covariance is scalar multiple of identity matrix:Cov(e) = 2I

10

01)(eCov

10

04)(eCov

21

12)(eCov

Examples for non-sphericity:

non-identity

non-independence

Page 7: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Multiple covariance components at 1st level

),0(~ 2VNe ),0(~ 2VNe

iiQV

eCovV

)(

iiQV

eCovV

)(

= 1 + 2

Q1 Q2

Estimation of hyperparameters with ReML (restricted maximum likelihood).

V

enhanced noise model error covariance components Qand hyperparameters

Page 8: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

WeWXWy

c = 1 0 0 0 0 0 0 0 0 0 0c = 1 0 0 0 0 0 0 0 0 0 0

)ˆ(ˆ

ˆ

T

T

cdts

ct

cWXWXc

cdtsTT

T

)()(ˆ

)ˆ(ˆ

2

)(

ˆˆ

2

2

Rtr

WXWy

ReML-estimates

ReML-estimates

WyWX )(̂

)(2

2/1

eCovV

VW

)(WXWXIRX

t-statistic based on ML estimates

iiQ

V

TT XWXXWX 1)()( TT XWXXWX 1)()( For brevity:

Page 9: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

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, can’t be generalized to the population

Page 10: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

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 be generalized to the population

Page 11: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Recommended reading

Linear hierarchical models

Mixed effect models

Page 12: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Linear hierarchical model

)()()()1(

)2()2()2()1(

)1()1()1(

nnnn X

X

Xy

)()()( i

k

i

kk

i QC

Hierarchical modelHierarchical model Multiple variance components at each level

Multiple variance components at each level

At each level, distribution of parameters is given by level above.

At each level, distribution of parameters is given by level above.

What we don’t know: distribution of parameters and variance parameters (hyperparameters).

What we don’t know: distribution of parameters and variance parameters (hyperparameters).

Page 13: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Example: Two-level model

=

2221

111

X

Xy

1

1+ 1 = 2X

2

+ 2y

)1(1X

)1(2X

)1(3X

Second levelSecond level

First levelFirst level

Page 14: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Two-level model

(1) (1) (1)

(1) (2) (2) (2)

y X

X

(1) (2) (2) (2) (1)

(1) (2) (2) (1) (2) (1)

y X X

X X X

Friston et al. 2002, NeuroImage

fixed effects random effects

Page 15: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Mixed effects analysis

(1) (2) (2) (1) (2) (1)y X X X

(1) (1)

(2) (2) (2) (1) (1)

(2) (2) (2)

ˆ X y

X X

X

(2) (2) (1) (1) (1) T

Cov C X C X

Non-hierarchical modelNon-hierarchical model

Variance components at 2nd level

Variance components at 2nd level

Estimating 2nd level effectsEstimating 2nd level effects

between-level non-sphericity

( ) ( )( ) i iik k

kQC

Within-level non-sphericity at both levels: multiple

covariance components

Within-level non-sphericity at both levels: multiple

covariance componentsFriston et al. 2005, NeuroImage

within-level non-sphericity

Page 16: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Estimation

EM-algorithmEM-algorithm

gJ

d

LdJ

d

dLg

1

2

2

E-stepE-step

M-stepM-step

kk

kQC

Assume, at voxel j:

Assume, at voxel j: kjjk

maximise L ln ( )p y | λ

111

NppNN

Xy

yCXC

XCXCT

yy

Ty

1||

11| )(

Friston et al. 2002, NeuroImage

GN gradient ascent

Page 17: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Algorithmic equivalence

)()()()1(

)2()2()2()1(

)1()1()1(

nnnn X

X

Xy

Hierarchicalmodel

Hierarchicalmodel

ParametricEmpirical

Bayes (PEB)

ParametricEmpirical

Bayes (PEB)

EM = PEB = ReMLEM = PEB = ReML

RestrictedMaximumLikelihood

(ReML)

RestrictedMaximumLikelihood

(ReML)

Single-levelmodel

Single-levelmodel

)()()1(

)()1()1(

)2()1()1(

...

nn

nn

XXXX

Xy

Page 18: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Mixed effects analysis

Summarystatistics

Summarystatistics

EMapproach

EMapproach

(2)̂

jjj

i

Tii QXQXV

XXY

)2()2()1()1()1()1(

)2(

)1(ˆ

yVXXVX TT 111)2( )(ˆ yVXXVX TT 111)2( )(ˆ

yVXXVX TT 111)1( )(ˆ yVXXVX TT 111)1( )(ˆ

},,{ QXnyyREML T },,{ QXnyyREML T

Step 1

Step 2

},,,{

][)1()2(

1)1()1(

1

)2()1()0(

TXQXQQ

XXXX

IV

XXX

][ )1()0(

datay

Friston et al. 2005, NeuroImage

non-hierarchical model

1st level non-sphericity

2nd level non-sphericity

pooling over voxels

Page 19: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Practical problems

Most 2-level models are just too big to compute.

Most 2-level models are just too big to compute.

And even if, it takes a long time! 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.

Moreover, sometimes we are only interested in one specific effect and do not want to model all the data.

Is there a fast approximation?Is there a fast approximation?

Page 20: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Summary statistics approach

Data Design Matrix Contrast Images )ˆ(ˆ

ˆ

T

T

craV

ct

SPM(t)1̂

11̂

12̂

21̂

22̂

211̂

212̂

Second levelSecond levelFirst levelFirst level

One-samplet-test @ 2nd level

One-samplet-test @ 2nd level

Page 21: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Validity of the summary statistics approach

The summary stats approach is exact if for each session/subject:

The summary stats approach is exact if for each session/subject:

All other cases: Summary stats approach seems to be fairly robust against typical violations.

All other cases: Summary stats approach seems to be fairly robust against typical violations.

Within-session covariance the sameWithin-session covariance the same

First-level design the sameFirst-level design the same

One contrast per sessionOne contrast per session

Page 22: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Reminder: sphericity

„sphericity“ means:„sphericity“ means:

ICov 2)(

Xy )()( TECovC

Scans

Sca

ns

i.e.2)( iVar

10

01)(Cov

Page 23: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

2nd level: non-sphericity

Errors are independent but not identical:

e.g. different groups (patients, controls)

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)

Errors are not independent and not identical:

e.g. repeated measures for each subject (like multiple basis

functions)

Errorcovariance

Errorcovariance

Page 24: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Example 1: non-indentical & independent errors

Stimuli:Stimuli: Auditory Presentation (SOA = 4 secs) of(i) words and (ii) words spoken backwards

Auditory Presentation (SOA = 4 secs) of(i) words and (ii) words spoken backwards

Subjects:Subjects:

e.g. “Book”

and “Koob”

e.g. “Book”

and “Koob”

fMRI, 250 scans per subject, block design

fMRI, 250 scans per subject, block design

Scanning:Scanning:

(i) 12 control subjects(ii) 11 blind subjects

(i) 12 control subjects(ii) 11 blind subjects

Noppeney et al.

Page 25: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

1st level:1st level:

2nd level:2nd level:

ControlsControls BlindsBlinds

X

]11[ TcV

Page 26: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Stimuli:Stimuli: Auditory Presentation (SOA = 4 secs) of words Auditory Presentation (SOA = 4 secs) of words

Subjects:Subjects:

fMRI, 250 scans persubject, block design

fMRI, 250 scans persubject, block designScanning:Scanning:

(i) 12 control subjects(i) 12 control subjects

1. Motion 2. Sound 3. Visual 4. Action

“jump” “click” “pink” “turn”

Question:Question:

What regions are generally affected by the semantic content of the words?Contrast: semantic decisions > auditory decisions on reversed words (gender identification task)

What regions are generally affected by the semantic content of the words?Contrast: semantic decisions > auditory decisions on reversed words (gender identification task)

Example 2: non-indentical & non-independent errors

Noppeney et al. 2003, Brain

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.

Page 27: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Repeated measures ANOVA

1st level:1st level:

2nd level:2nd level:

3.Visual3.Visual 4.Action4.Action

X

?=

?=

?=

1.Motion1.Motion 2.Sound2.Sound

Page 28: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Repeated measures ANOVA

1st level:1st level:

2nd level:2nd level:

3.Visual3.Visual 4.Action4.Action

X

?=

?=

?=

1.Motion1.Motion 2.Sound2.Sound

1100

0110

0011Tc

V

X

Page 29: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Practical conclusions

• Linear hierarchical models are used for group analyses of multi-subject imaging data.

• The main challenge is to model non-sphericity (i.e. non-identity and non-independence of errors) within and between levels of the hierarchy.

• This is done using EM or ReML (which are equivalent for linear models).

• The summary statistics approach is robust approximation to a full mixed-effects analysis.

– Use mixed-effects model only, if seriously in doubt about validity of summary statistics approach.

Page 30: Group analyses of fMRI data Methods & models for fMRI data analysis 28 April 2009 Klaas Enno Stephan Laboratory for Social and Neural Systems Research.

Thank you