Variance componentsVariance componentsVariance componentsVariance components

Wellcome Dept. of Imaging Neuroscience

Institute of Neurology, UCL, London

Stefan KiebeStefan Kiebell

Modelling in SPM

pre-processingpre-processinggenerallinearmodel

generallinearmodel

SPMsSPMs

functional datafunctional data

templatestemplates

smoothednormalised

data

smoothednormalised

data

design matrixdesign matrix

variance componentsvariance components

hypotheseshypotheses

adjustedP-values

adjustedP-values

parameterestimation

parameterestimation

general linear model Xy

=

+X

N

1

N N

1 1p

p

model specified by1. design matrix X2. assumptions about

model specified by1. design matrix X2. assumptions about

N: number of observations p: number of regressors

N: number of observations p: number of regressors

error normally

distributed

error normally

distributed

y

Summary

Sphericity/non-sphericitySphericity/non-sphericity

Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)

Estimation in SPM2Estimation in SPM2

Summary

Sphericity/non-sphericitySphericity/non-sphericity

Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)

Estimation in SPM2Estimation in SPM2

Sphericity/non-sphericitySphericity/non-sphericity

‚sphericity‘

‚sphericity‘ means:‚sphericity‘ means:

ICov 2)(

Xy )()( TECovC

Scans

Sca

nsi.e.

2)( iVar12

‚non-sphericity‘

non-sphericity means that the error covariance doesn‘t

look like this*:

non-sphericity means that the error covariance doesn‘t

look like this*:

*: or can be brought through a linear transform to this form

*: or can be brought through a linear transform to this form

ICov 2)(

10

01)(Cov

10

04)(Cov

21

12)(Cov

Example: serial correlations

withwithttt a 1 ),0(~ 2 Nt

autoregressive process of order 1 (AR(1))

)(Covautocovariance-

function

N

N

Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)

Summary

Sphericity/non-sphericitySphericity/non-sphericity

Estimation in SPM2Estimation in SPM2

Restricted Maximum Likelihood

Xy ?)(Cov observed

ReMLReMLestimated

2211ˆˆ QQ

j

Tjj yy

voxel

1Q

2Q

t-statistic (OLS estimator)

Xy

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

cVXXccdtSTTT 2ˆ)ˆ(ˆ

)(

ˆˆ

2

2

RVtr

Xy

approximate degrees of freedom following

Satterthwaite

approximate degrees of freedom following

SatterthwaiteReML-estimate

ReML-estimate

yX ̂

)(2 CovV

XXIR

VX

Variance components

Variance components Q model the error

Variance components Q model the error

KKQQQCovV 2211)(

Xy

model for sphericitymodel for sphericity

IQ 12

1 and model for inhomogeneous

variances (2 groups)

model for inhomogeneousvariances (2 groups)

1Q1Q 2Q

The variance parameters are estimated by ReML.

The variance parameters are estimated by ReML.

Example I

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:

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

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

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

Population differences

1st level:1st level:

2nd level:2nd level:

ControlsControls BlindsBlinds

X

]11[ TcV

Estimation in SPM2Estimation in SPM2

Summary

Sphericity/non-sphericitySphericity/non-sphericity

Restricted Maximum Likelihood (ReML)Restricted Maximum Likelihood (ReML)

Estimating variances

111

NppNN

Xy EM-algorithmEM-algorithm

yCXC

XCXCT

yy

Ty

1||

11| )(

gJ

d

LdJ

d

dLg

1

2

2

E-stepE-step

M-stepM-step

K. Friston et al. 2002, Neuroimage

K. Friston et al. 2002, Neuroimage

kk

kQC

Assume, at voxel j:

Assume, at voxel j: kjjk

)lnL maximise p(y|λ

Time

Intensity

Tim

e

Time series inone voxel

Time series inone voxel

voxelwise

model specification

model specification

parameterestimation

parameterestimation

hypothesishypothesis

statisticstatistic

SPMSPM

Spatial ‚Pooling‘

Assumptions in SPM2:• global correlation matrix V • local variance

Assumptions in SPM2:• global correlation matrix V • local variance

observed

ReMLReML

estimated

2211ˆˆˆ QQC

jvoxel

Tjj yy

Matrix is where

, )ˆ(

ˆ

NNV

Ctrace

nCV

global

)( ,

)(ˆ

2/12/121

2

XVXVIRyRVr

Rtr

rr

j/

j

jTj

j

local in voxel j: VC jj2ˆˆ

Estimation in SPM2

jjj Xy

jOLSj yX ,̂

),,ReML()(ˆˆ

QXyyvoCCjvoxel

Tjj

jTT

MLj yVXXVX 111, )(ˆ

‚quasi‘-Maximum Likelihood‚quasi‘-Maximum LikelihoodOrdinary least-squaresOrdinary least-squares

ReML (pooled estimate)ReML (pooled estimate)

•optional in SPM2•one pass through data•statistic using (approximated) effective degrees of freedom

•optional in SPM2•one pass through data•statistic using (approximated) effective degrees of freedom

•2 passes (first pass for selection of voxels)

•more precise estimate of V

•2 passes (first pass for selection of voxels)

•more precise estimate of V

t-statistic (ML-estimate)

Xy

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

cWXWXccdtSTTT )()(ˆ)ˆ(ˆ 2

)(

ˆˆ

2

2

Rtr

WXWy

ReML-estimate

ReML-estimate

WyWX )(̂)(2

2/1

CovV

VW

)(WXWXIR

VX

Example II

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?

Repeated measures Anova

1st level:1st level:

2nd level:2nd level:

VisualVisual ActionAction

X

1100

0110

0011Tc

?=

?=

?=

MotionMotion SoundSound

V

X