Group analysesGroup analyses

Wellcome Dept. of Imaging Neuroscience

University College London

Will PennyWill Penny

Data

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 model for all imaging data!

Hierarchical model for all imaging data!

Time

Time

Intensity

Tim

e

single voxeltime series

single voxeltime series

Reminder: voxel by voxel

modelspecification

modelspecification

parameterestimation

parameterestimation

hypothesishypothesis

statisticstatistic

SPMSPM

General Linear Model Xy

=

+X

N

1

N N

1 1p

p

Model is specified by1. Design matrix X2. Assumptions about

Model is specified by1. Design matrix X2. Assumptions about

N: number of scansp: number of regressors

N: number of scansp: number of regressors

y

k kk

QC Error Covariance

Estimation

1. ReML-algorithm1. ReML-algorithm

gJ

d

LdJ

d

dLg

1

2

2

Friston et al. 2002, Neuroimage

Friston et al. 2002, Neuroimage

kk

kQC Maximise L ln ) ln , )p(y | λ p(y | λ d

111

NppNN

Xy

L

g

2. Weighted Least Squares2. Weighted Least Squares

1 1( )T T Te eX C X X C y

)()()()1(

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

)1()1()1(

nnnn X

X

Xy

)()()( i

k

i

kk

i QC

Hierarchical model

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.

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

=

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

)()()()1(

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

)1()1()1(

nnnn X

X

Xy

Estimation

Hierarchicalmodel

Hierarchicalmodel

Single-levelmodel

Single-levelmodel

(1) (1) (2)

(1) ( 1) ( )

(1) ( ) ( )

...n n

n n

y X

X X

X X

X e

Group analysis in practice

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

Many 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!

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

Data Design Matrix Contrast Images )ˆ(ˆ

ˆ

T

T

craV

ct

SPM(t)

Summary Statistics approach

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 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 robust against typical violations.

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

Within-session covariance the sameWithin-session covariance the same

First-level design the sameFirst-level design the same

Friston et al. (2004) Mixed effects and fMRI studies, Neuroimage

Friston et al. (2004) Mixed effects and fMRI studies, Neuroimage

Summarystatistics

Summarystatistics

HierarchicalModel

HierarchicalModel

Auditory Data

Multiple contrasts per subject

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:

U. Noppeney et al.U. Noppeney et al.

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

Motion Sound Visual Action

“jump” “click” “pink” “turn”

Question:Question:What regions are affectedby the semantic content ofthe words?

What regions are affectedby the semantic content ofthe words?

ANOVA

1st level:1st level:

2nd level:2nd level:

3.Visual3.Visual 4.Action4.Action

X

?=

?=

?=

1.Motion1.Motion 2.Sound2.Sound

1st level:1st level:

2nd level:2nd level:

VisualVisual ActionAction

X

1100

0110

0011Tc

?=

?=

?=

MotionMotion SoundSound

V

X

ANOVA

Summary

Linear hierarchical models are general enough for typical multi-subject imaging data (PET, fMRI, EEG/MEG).

Linear hierarchical models are general enough for typical multi-subject imaging data (PET, fMRI, EEG/MEG).

Summary statistics are robust approximation for group analysis.

Summary statistics are robust approximation for group analysis.

Also accomodates multiple contrasts per subject.Also accomodates multiple contrasts per subject.