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

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

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

Time

BOLD signalTim

esingle voxel

time series

single voxel

time series

Reminder: voxel-wise time series analysis!

modelspecificati

on

modelspecificati

onparameterestimationparameterestimation

hypothesishypothesis

statisticstatistic

SPMSPM

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

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

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

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:

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

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

Recommended reading

Linear hierarchical models

Mixed effect models

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).

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

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

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

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

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

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

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?

Summary statistics approach

Data Design Matrix Contrast Images )ˆ(ˆ

ˆ

T

T

craV

ct

SPM(t)1̂

2̂

11̂

12̂

21̂

22̂

211̂

212̂

Second levelSecond levelFirst levelFirst level

One-samplet-test @ 2nd level

One-samplet-test @ 2nd level

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

Reminder: sphericity

„sphericity“ means:„sphericity“ means:

ICov 2)(

Xy )()( TECovC

Scans

Sca

ns

i.e.2)( iVar

10

01)(Cov

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

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.

1st level:1st level:

2nd level:2nd level:

ControlsControls BlindsBlinds

X

]11[ TcV

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.

Repeated measures ANOVA

1st level:1st level:

2nd level:2nd level:

3.Visual3.Visual 4.Action4.Action

X

?=

?=

?=

1.Motion1.Motion 2.Sound2.Sound

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

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.

Thank you