Group analyses of fMRI data

Methods & models for fMRI data analysisNovember 2012

With many thanks for slides & images to:

FIL Methods group, particularly Will Penny & Tom Nichols

Klaas Enno Stephan

Translational Neuromodeling Unit (TNU)Institute for Biomedical Engineering, University of Zurich & ETH Zurich

Laboratory for Social & Neural Systems Research (SNS), University of Zurich

Wellcome Trust Centre for Neuroimaging, University College London

Overview of SPM

Realignment Smoothing

Normalisation

General linear model

Statistical parametric map (SPM)Image time-series

Parameter estimates

Design matrix

Template

Kernel

Gaussian field theory

p <0.05

Statisticalinference

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

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

),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

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)()( For brevity:

Distribution of population effect

8

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

Subj. 6

0

Fixed vs.random effectsanalysis

• Fixed Effects– Intra-subject variation

suggests most subjects different from zero

• Random Effects– Inter-subject variation

suggests population is not very different from zero

Distribution of each subject’s estimated effect 2

FFX

2RFX

Fixed Effects

• Assumption: variation (over subjects) is only due to measurement error

• parameters are fixed properties of the population (i.e., they are the same in each subject)

• two sources of variation (over subjects)– measurement error– Response magnitude: parameters are probabilistically

distributed in the population

• effect (response magnitude) in each subject is randomly distributed

Random/Mixed Effects

Random/Mixed Effects

• two sources of variation (over subjects)– measurement error– Response magnitude: parameters are probabilistically

distributed in the population

• effect (response magnitude) in each subject is randomly distributed• variation around population mean

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 and cannot 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

• hierarchical model much less power (proportional to number of subjects), but results can be generalized to the population

FFX vs. RFX

• FFX is not "wrong", it makes different assumptions and addresses a different question than RFX

• For some questions, FFX may be appropriate (e.g., low-level physiological processes).

• For other questions, RFX is much more plausible (e.g., cognitive tasks, disease processes in heterogeneous populations).

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

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

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

Estimation by Expectation Maximisation (EM)

EM-algorithmEM-algorithm

gJ

d

LdJ

d

dLg

1

2

2

E-stepE-step

M-stepM-step

kk

kQC

maximise L ln ( )p y | λ

111

NppNN

Xy

Friston et al. 2002, NeuroImage

Gauss-Newtongradient ascent

• E-step: finds the (conditional) expectation of the parameters, holding the hyperparameters fixed

• M-step: updates the maximum likelihood estimate of the hyperparameters, keeping the parameters fixed

1 1 1

|

1| |

Ty

Ty y

C X C X C

η C X C y C η

Practical problems

• Full MFX inference using REML or EM for a whole-brain 2-level model has enormous computational costs

• for many subjects and scans, covariance matrices become extremely large

• nonlinear optimisation problem for each voxel

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

• Is there a fast approximation?

Summary statistics approach: Holmes & Friston 1998

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:

But: Summary stats approach is fairly robust against violations of these

conditions.

But: Summary stats approach is fairly robust against violations of these

conditions.

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

Mixed effects analysis: spm_mfx

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)1( )(ˆ

},,{ 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

2nd level non-sphericity modeling in SPM8

• 1 effect per subject→use summary statistics approach

• >1 effect per subject→model sphericity at 2nd level using variance basis

functions

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 (multiple basis functions,

multiple conditions etc.)

Errors are not independent and not identical:

e.g. repeated measures for each subject (multiple basis functions,

multiple conditions etc.)

Errorcovariance

Errorcovariance

Example of 2nd level non-sphericity

Qk:

Error Covariance

...

...

Example of 2nd level non-sphericity

30

y = X + eN 1 N p p 1 N 1

N

N

error covariance

• 12 subjects, 4 conditions

• Measurements between subjects uncorrelated

• Measurements within subjects correlated

• Errors can have different variances across subjects

Cor(ε) =Σk λkQk

2nd level non-sphericity modeling in SPM8:assumptions and limitations

• Cor() assumed to be globally homogeneous

• lk’s only estimated from voxels with large F

• intrasubject variance assumed homogeneous

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 by estimating hyperparameters 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.

Recommended reading

Linear hierarchical models

Mixed effect models

Thank you