The general linear model and Statistical Parametric Mapping The ...

The The general general linear linear model model and and Statistical Statistical Parametric Parametric Mapping Mapping I: I: Introduction Introduction to to the the GLM GLM Alexa Morcom

and Stefan Kiebel, Rik Henson, Andrew Holmes & J-B Poline

Overview • Introduction • Essential concepts – – – –

Modelling Design matrix Parameter estimates Simple contrasts

• Summary

Some terminology • SPM is based on a mass univariate approach that fits a model at each voxel – Is there an effect at location X? Investigate localisation of function or functional specialisation – How does region X interact with Y and Z? Investigate behaviour of networks or functional integration

• A General(ised) Linear Model – Effects are linear and additive – If errors are normal (Gaussian), General (SPM99) – If errors are not normal, Generalised (SPM2)

Classical statistics... • Parametric – – – – – – – – – –

one sample t-test two sample t-test paired t-test ANOVA ANCOVA correlation linear regression multiple regression F-tests etc…

all cases of the (univariate) General Linear Model Or, with non-normal errors, the Generalised Linear Model

Classical statistics... • Parametric – – – – – – – – – –

one sample t-test two sample t-test paired t-test ANOVA ANCOVA correlation linear regression multiple regression F-tests etc…

all cases of the (univariate) General Linear Model Or, with non-normal errors, the Generalised Linear Model

Classical statistics... • Parametric – – – – – – – – – –

one sample t-test two sample t-test paired t-test ANOVA ANCOVA correlation linear regression multiple regression F-tests etc…

• Multivariate?

all cases of the (univariate) General Linear Model Or, with non-normal errors, the Generalised Linear Model

→ PCA/ SVD, MLM

Classical statistics... • Parametric – – – – – – – – – –

one sample t-test two sample t-test paired t-test ANOVA ANCOVA correlation linear regression multiple regression F-tests etc…

• Multivariate? • Non-parametric?

all cases of the (univariate) General Linear Model Or, with non-normal errors, the Generalised Linear Model

→ PCA/ SVD, MLM → SnPM

Why modelling? Why?

Make Makeinferences inferencesabout abouteffects effectsof ofinterest interest

How?

1.1. Decompose Decomposedata datainto intoeffects effectsand anderror error 2.2. Form Formstatistic statisticusing usingestimates estimatesof ofeffects effectsand anderror error

Model?

Use Useany anyavailable availableknowledge knowledge

effects effectsestimate estimate data data

model model

statistic statistic error errorestimate estimate

Choose your model “All models are wrong, but some are useful” George E.P.Box

AAvery verygeneral general model model

Variance Variance

Bias Bias

No Nonormalisation normalisation

Lots Lotsofofnormalisation normalisation

No Nosmoothing smoothing Massive Massivemodel model

default default SPM SPM

Lots Lotsof ofsmoothing smoothing Sparse Sparsemodel model

Captures signal

High sensitivity

but but... ...not notmuch much sensitivity sensitivity

but but......may maymiss miss signal signal

Modelling with SPM Functional Functionaldata data

Design Designmatrix matrix

Smoothed Smoothed normalised normalised data data

Preprocessing Preprocessing

Templates Templates

Generalised Generalised linear linear model model

Contrasts Contrasts

Parameter Parameter estimates estimates

Variance Variancecomponents components

SPMs SPMs

Thresholding Thresholding

Modelling with SPM Design Designmatrix matrix

Contrasts Contrasts

Preprocessed Preprocessed data: data:single single voxel voxel

Generalised Generalised linear linear model model

Parameter Parameter estimates estimates

Variance Variancecomponents components

SPMs SPMs

fMRI example One Onesession session

Time Timeseries seriesofofBOLD BOLD responses responsesininone onevoxel voxel

Passive Passiveword wordlistening listening versus versusrest rest 77cycles cyclesof of rest restand andlistening listening Each Eachepoch epoch66scans scans with with77sec secTR TR

Question: Question:IsIsthere thereaachange changeininthe theBOLD BOLD response between listening and rest? response between listening and rest?

Stimulus Stimulusfunction function

GLM essentials • The model – Design matrix: Effects of interest – Design matrix: Confounds (aka effects of no interest) – Residuals (error measures of the whole model)

• Estimate effects and error for data – Parameter estimates (aka betas) – Quantify specific effects using contrasts of parameter estimates

• Statistic – Compare estimated effects – the contrasts – with appropriate error measures – Are the effects surprisingly large?

GLM essentials • The model – Design matrix: Effects of interest – Design matrix: Confounds (aka effects of no interest) – Residuals (error measures of the whole model)

• Estimate effects and error for data – Parameter estimates (aka betas) – Quantify specific effects using contrasts of parameter estimates

• Statistic – Compare estimated effects – the contrasts – with appropriate error measures – Are the effects surprisingly large?

Regression model

Intensity

+ β2

x1

Question: Question:IsIsthere thereaachange changeininthe theBOLD BOLD response responsebetween betweenlistening listeningand andrest? rest?

+

x2

error

Time

= β1

General case

ε∼Ν(0, σ2Ι) (error is normal and independently and identically distributed)

ε Hypothesis test: β1 = 0? (using t-statistic)

Regression model

= βˆ1

Y

=

βˆ1 x1

General case

+ βˆ2

+

ˆ β + 2 x2

+

εˆ

Model Modelisisspecified specifiedby by 1.1. Design Designmatrix matrixXX 2.2. Assumptions Assumptionsabout aboutεε

Matrix formulation Yi = β1 xi + β2 + εi

i = 1,2,3

Y1 = β1 x1 + β2 × 1 + ε1 Y2 = β1 x2 + β2 × 1 + ε2 Y3 = β1 x3 + β2 × 1 + ε3 dummy variables

Y1 x1 1 Y2 = x2 1 Y3 x3 1

β1

Y =

β+ ε

X

ε1 + ε2 β1 ε 3

Matrix formulation

^ Hats Hats== estimates estimates

Linear regression Yi = β1 xi + β2 + εi

i = 1,2,3

Y1 = β1 x1 + β2 × 1 + ε1 Y2 = β1 x2 + β2 × 1 + ε2

Yi

y

× ^ β

2

×

1

×

Y3 = β1 x3 + β2 × 1 + ε3

^ β

1

^ Yi x

dummy variables

Y1 x1 1 Y2 = x2 1 Y3 x3 1

β1

Y =

β+ ε

X

ε1 + ε2 β1 ε 3

Fitted values

^ ^ β1 & β2 ^ ^ &Y Y

Residuals

^ ε1 & ^ε2

Parameter estimates

1

2

Geometrical perspective Y1 x1 1 Y2 = x2 1 Y3 x3 1

β1

ε1 + ε2 β2 ε3

DATA (Y1, Y2, Y3) Y

Y = β1 × X1 + β2 × X2 + ε

X1 (x1, x2, x3) (1,1,1) O

X2

design space

GLM essentials • The model – Design matrix: Effects of interest – Design matrix: Confounds (aka effects of no interest) – Residuals (error measures of the whole model)

• Estimate effects and error for data – Parameter estimates (aka betas) – Quantify specific effects using contrasts of parameter estimates

• Statistic – Compare estimated effects – the contrasts – with appropriate error measures – Are the effects surprisingly large?

Parameter estimation

Ordinary least squares

Y = Xβ + ε T −1 T ˆ β = (X X ) X Y

εˆ = Y − Xβˆ residuals residuals

Parameter Parameter estimates estimates

Estimate Estimateparameters parameters such suchthat that

N

∑ εˆ

2 t

t =1

minimal minimal

Least Leastsquares squares parameter parameterestimate estimate

Parameter estimation

Ordinary least squares

Y = Xβ + ε T −1 T ˆ β = (X X ) X Y

Parameter Parameter estimates estimates

εˆ = Y − Xβˆ residuals residuals==rr

2 Error Errorvariance varianceσσ2== (sum (sumof) of)squared squaredresiduals residuals standardised standardisedfor fordf df ==rrTTrr//df df (sum (sumof ofsquares) squares) …where …wheredegrees degreesof offreedom freedomdf df(assuming (assumingiid): iid): ==NN--rank(X) rank(X) (=N-P (=N-PififXXfull fullrank) rank)

Estimation (geometrical) Y1 x1 1 Y2 = x2 1 Y3 x3 1

β1

ε1 + ε2 β2 ε3

DATA (Y1, Y2, Y3) Y RESIDUALS ^ ε = (ε^1,ε^2,ε^3)T

Y = β1 × X1 + β2 × X2 + ε

X1 (x1, x2, x3)

^ Y

(1,1,1) O

X2

design space

^ ^ ,Y^ ,Y (Y 1 2 3)

GLM essentials • The model – Design matrix: Effects of interest – Design matrix: Confounds (aka effects of no interest) – Residuals (error measures of the whole model)

• Estimate effects and error for data – Parameter estimates (aka betas) – Quantify specific effects using contrasts of parameter estimates

• Statistic – Compare estimated effects – the contrasts – with appropriate error measures – Are the effects surprisingly large?

Inference - contrasts Y = Xβ + ε

A contrast = a linear combination of parameters: c´ x β à spm_con*img

t-test: one-dimensional contrast, difference in means/ difference from zero boxcar boxcarparameter parameter>>00?? Null Nullhypothesis: hypothesis: β1

=0

à spmT_000*.img SPM{t} map

F-test: tests multiple linear hypotheses – does subset of model account for significant variance Does Doesboxcar boxcarparameter parameter model modelanything? anything? Null Nullhypothesis: hypothesis:variance variance of oftested testedeffects effects==error error variance variance

à ess_000*.img SPM{F} map

t-statistic - example Y = Xβ + ε

c βˆ t= Stˆd (cT βˆ ) T

cc==1100000000000000000000

X

Contrast Contrastof ofparameter parameter estimates estimates Variance Varianceestimate estimate

Standard Standarderror errorof ofcontrast contrastdepends dependson onthe thedesign, design, and andisislarger largerwith withgreater greaterresidual residualerror errorand and ‚greater‘ ‚greater‘covariance/ covariance/autocorrelation autocorrelation Degrees Degreesof offreedom freedomd.f. d.f.then then==n-p, n-p,where wherenn observations, observations,ppparameters parameters Tests Testsfor foraadirectional directional difference differenceininmeans means

F-statistic - example Do Domovement movementparameters parameters(or (orother otherconfounds) confounds)account accountfor foranything? anything?

H0: True model is X0 H0: β3-9 = (0 0 0 0 ...) X0

X1 (β3−9)

This model ?

test H0 : c´ x β = 0 ?

X0

Or this one ?

Null Nullhypothesis hypothesisHH00:: That are Thatall allthese thesebetas betasββ3-9 3-9 are zero, zero,i.e. i.e.that thatno nolinear linear combination combinationof ofthe theeffects effects accounts accountsfor forsignificant significant variance variance This Thisisisaanon-directional non-directionaltest test

F-statistic - example Do Domovement movementparameters parameters(or (orother otherconfounds) confounds)account accountfor foranything? anything?

H0: True model is X0 H0: β3-9 = (0 0 0 0 ...) X0

X1 (β3−9)

X0 c’ =

This model ?

Or this one ?

test H0 : c´ x β = 0 ?

00100000 00010000 00001000 00000100 00000010 00000001

SPM{F}

Summary so far • The essential model contains – Effects of interest

• A better model? – A better model (within reason) means smaller residual variance and more significant statistics – Capturing the signal – later – Add confounds/ effects of no interest – Example of movement parameters in fMRI – A further example (mainly relevant to PET)…

Example PET experiment β2

β3

β4

rank(X)=3

β1

12 12scans, scans,33conditions conditions(1-way (1-wayANOVA) ANOVA) yyj ==xx1j bb1 ++xx2j bb2 ++xx3j bb3 ++xx4j bb4 ++eej j 1j 1 2j 2 3j 3 4j 4 j

where where(dummy) (dummy)variables: variables: xx1j ==[0,1] [0,1]==condition conditionAA(first (first44scans) scans) 1j xx2j ==[0,1] [0,1]==condition conditionBB(second (second44scans) scans) 2j xx3j ==[0,1] [0,1]==condition conditionCC(third (third44scans) scans) 3j xx4j ==[1] [1]== grand grandmean mean(session (sessionconstant) constant) 4j

xx x x oo xx xx x x oo o xx o o xx x oo o o x oo o o o global o gCBF o

rCBF

x x

xx

x xo o x o xo o

rCBF

o

AnCova

xx

xx

α k2

xx

xx

α k1

xx 1 oo

ζ k

oo

xx oo

oo

oo oo

global gCBF

x x x x x x o o o o o o

rCBF

g.. rCBF (adj)

•• May Maybe bevariation variationin inPET PETtracer tracerdose dose from fromscan scanto toscan scan •• Such Such“global” “global”changes changesin inimage image intensity intensity(gCBF) (gCBF)confound confoundlocal local// regional regional(rCBF) (rCBF)changes changesof of experiment experiment •• Adjust Adjustfor forglobal globaleffects effectsby: by: --AnCova AnCova(Additive (AdditiveModel) Model)--PET? PET? --Proportional --fMRI? ProportionalScaling Scaling fMRI? •• Can Canimprove improvestatistics statisticswhen when orthogonal orthogonalto toeffects effectsof ofinterest… interest… •• …but …butcan canalso alsoworsen worsenwhen wheneffects effects of ofinterest interestcorrelated correlatedwith withglobal global

rCBF

Global effects

x

Scaling

xx o x o x x xx o oo oo o o o o x

x xo

x

global 0

0

50

gCBF

Global effects (AnCova) β2

β3

β4

β5

rank(X)=4

β1

12 12scans, scans,33conditions, conditions,11confounding confoundingcovariate covariate yyj ==xx1j bb1 ++xx2j bb2 ++xx3j bb3 ++xx4j bb4 ++xx5j bb5 ++eej j 1j 1 2j 2 3j 3 4j 4 5j 5 j

where where(dummy) (dummy)variables: variables: xx1j ==[0,1] [0,1]==condition conditionAA(first (first44scans) scans) 1j xx2j ==[0,1] [0,1]==condition conditionBB(second (second44scans) scans) 2j xx3j ==[0,1] [0,1]==condition conditionCC(third (third44scans) scans) 3j xx4j ==[1] [1]== grand grandmean mean(session (sessionconstant) constant) 4j

xx5j ==global globalsignal signal(mean (meanover overall allvoxels) voxels) 5j

Global effects (AnCova) β3

β4

β1

•• Global Globaleffects effectsnot not accounted accountedfor for •• Maximum Maximumdegrees degrees of offreedom freedom(global (global uses usesone) one)

β2

β3

β4

β5

Correlated Correlatedglobal global β1

rank(X)=4

β2

rank(X)=3

β1

Orthogonal Orthogonalglobal global

•• Global Globaleffects effects independent independentof of effects effectsof ofinterest interest •• Smaller Smallerresidual residual variance variance •• Larger LargerTTstatistic statistic •• More Moresignificant significant

β2

β3

β4

β5

rank(X)=4

No NoGlobal Global

•• Global Globaleffects effects correlated correlatedwith with effects effectsof ofinterest interest •• Smaller Smallereffect effect&/or &/or larger largerresiduals residuals •• Smaller SmallerTTstatistic statistic •• Less Lesssignificant significant

Global effects (scaling) •• Two Twotypes typesof ofscaling: scaling:Grand GrandMean Meanscaling scalingand andGlobal Globalscaling scaling -- Grand GrandMean Meanscaling scalingisisautomatic, automatic,global globalscaling scalingisisoptional optional -- Grand GrandMean Meanscales scalesby by100/mean 100/meanover overall allvoxels voxelsand andALL ALLscans scans (i.e, (i.e,single singlenumber numberper persession) session) -- Global Globalscaling scalingscales scalesby by100/mean 100/meanover overall allvoxels voxelsfor forEACH EACHscan scan (i.e, (i.e,aadifferent differentscaling scalingfactor factorevery everyscan) scan) •• Problem Problemwith withglobal globalscaling scalingisisthat thatTRUE TRUEglobal globalisisnot not(normally) (normally) known… known…… …only onlyestimated estimatedby bythe themean meanover overvoxels voxels -- So Soififthere thereisisaalarge largesignal signalchange changeover overmany manyvoxels, voxels,the theglobal global estimate estimatewill willbe beconfounded confoundedby bylocal localchanges changes -- This Thiscan canproduce produceartifactual artifactualdeactivations deactivationsininother otherregions regionsafter after global globalscaling scaling •• Since Sincemost mostsources sourcesof ofglobal globalvariability variabilityin infMRI fMRIare arelow lowfrequency frequency (drift), (drift),high-pass high-passfiltering filteringmay maybe besufficient, sufficient,and andmany manypeople peopledo do not notuse useglobal globalscaling scaling

Summary • General(ised) linear model partitions data into – Effects of interest & confounds/ effects of no interest – Error

• Least squares estimation – Minimises difference between model & data – To do this, assumptions made about errors – more later

• Inference at every voxel – Test hypothesis using contrast – more later – Inference can be Bayesian as well as classical

• Next: Applying the GLM to fMRI