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

Transcription

1 The general linear mdel and Statistical Parametric Mapping I: Intrductin t the GLM Alexa Mrcm and Stefan Kiebel, Rik Hensn, Andrew Hlmes & J-B J Pline

2 Overview Intrductin Essential cncepts Mdelling Design matrix Parameter estimates Simple cntrasts Summary

3 Sme terminlgy SPM is based n a mass univariate apprach that fits a mdel at each vxel Is there an effect at lcatin X? Investigate lcalisatin f functin r functinal specialisatin Hw des regin X interact with Y and Z? Investigate behaviur f netwrks r functinal integratin A General(ised) Linear Mdel Effects are linear and additive If errrs are nrmal (Gaussian), General (SPM99) If errrs are nt nrmal, Generalised (SPM2)

4 Classical statistics... Parametric ne sample t-test tw sample t-test paired t-test ANOVA ANCOVA crrelatin linear regressin multiple regressin F-tests etc all cases f the (univariate) General Linear Mdel Or, with nn-nrmal errrs, the Generalised Linear Mdel

5 Classical statistics... Parametric ne sample t-test tw sample t-test paired t-test ANOVA ANCOVA crrelatin linear regressin multiple regressin F-tests etc all cases f the (univariate) General Linear Mdel Or, with nn-nrmal errrs, the Generalised Linear Mdel

6 Classical statistics... Parametric ne sample t-test tw sample t-test paired t-test ANOVA ANCOVA crrelatin linear regressin multiple regressin F-tests etc Multivariate? all cases f the (univariate) General Linear Mdel Or, with nn-nrmal errrs, the Generalised Linear Mdel fi PCA/ SVD, MLM

7 Classical statistics... Parametric ne sample t-test tw sample t-test paired t-test ANOVA ANCOVA crrelatin linear regressin multiple regressin F-tests etc Multivariate? Nn-parametric? all cases f the (univariate) General Linear Mdel Or, with nn-nrmal errrs, the Generalised Linear Mdel fi PCA/ SVD, MLM fi SnPM

8 Why mdelling? Why? Make inferences abut effects f f interest Hw? Decmpse data data int int effects and and errr errr Frm statistic using estimates f f effects and and errr errr Mdel? Use Use any any available knwledge data data mdel mdel effects effects estimate errr errr estimate statistic statistic

9 Chse yur mdel A very very general general mdel mdel All mdels are wrng, but sme are useful Gerge E.P.Bx Variance N N nrmalisatin N N smthing Massive Massive mdel mdel Captures signal but nt much sensitivity default default SPM SPM Bias Lts Lts f f nrmalisatin Lts Lts f f smthing Sparse Sparse mdel mdel High sensitivity but may miss signal

10 Mdelling with SPM Functinal data data Design Design matrix matrix Cntrasts Preprcessing Smthed nrmalised data data Generalised linear mdel Parameter estimates SPMs Templates Variance cmpnents Threshlding

11 Mdelling with SPM Design Design matrix matrix Cntrasts Preprcessed data: data: single single vxel vxel Generalised linear mdel Parameter estimates SPMs Variance cmpnents

12 fmri example One One sessin Passive wrd listening versus rest rest Time Time series series f f BOLD BOLD respnses in in ne ne vxel vxel 7 cycles f f rest rest and and listening Each Each epch 6 scans with with 7 sec sec TR TR Questin: Is Is there there a a change change in in the the BOLD BOLD respnse between listening and and rest? rest? Stimulus functin functin

13 GLM essentials The mdel Design matrix: Effects f interest Design matrix: Cnfunds (aka effects f n interest) Residuals (errr measures f the whle mdel) Estimate effects and errr fr data Parameter estimates (aka betas) Quantify specific effects using cntrasts f parameter estimates Statistic Cmpare estimated effects the cntrasts with apprpriate errr measures Are the effects surprisingly large?

14 GLM essentials The mdel Design matrix: Effects f interest Design matrix: Cnfunds (aka effects f n interest) Residuals (errr measures f the whle mdel) Estimate effects and errr fr data Parameter estimates (aka betas) Quantify specific effects using cntrasts f parameter estimates Statistic Cmpare estimated effects the cntrasts with apprpriate errr measures Are the effects surprisingly large?

15 Regressin mdel General case Time = b + 1 b + 2 errr e~n(0, s 2 I) (errr is nrmal and independently and identically distributed) Intensity x 1 x 2 e Questin: Is Is there there a change in in the the BOLD respnse between listening and and rest? rest? Hypthesis test: β 1 = 0? (using t-statistic)

16 Regressin mdel General case ˆβ ˆβ 1 2 = + + ˆ Y x + β ˆ x + εˆ = β Mdel is is specified by by Design matrix X Assumptins abut ε

17 Matrix frmulatin Y i = b 1 x i + b 2 + e i i = 1,2,3 Y 1 = b 1 x 1 + b e 1 Y 2 = b 1 x 2 + b e 2 Y 3 = b 1 x 3 + b e 3 dummy variables Y 1 x 1 1 b 1 e 1 Y 2 = x e 2 Y 3 x 3 1 b 1 e 3 Y = X b + e

18 Linear regressin Matrix frmulatin ^ Hats Hats = estimates Y i = b 1 x i + b 2 + e i i = 1,2,3 Y 1 = b 1 x 1 + b e 1 Y 2 = b 1 x 2 + b e 2 y ^ b 2 Y i 1 ^ Y i ^b1 Y 3 = b 1 x 3 + b e 3 Y 1 x 1 1 b 1 e 1 Y 2 = x e 2 Y 3 x 3 1 b 1 e 3 Y = X b + e dummy variables Parameter estimates ^ ^ b 1 & b 2 Fitted values Y ^ 1 & Y ^ 2 Residuals e^ 1 & ^e 2 x

19 Gemetrical perspective Y = Y 1 x 1 1 b 1 e 1 Y 2 = x e 2 Y 3 x 3 1 b 2 e 3 DATA (Y 1, Y 2, Y 3 ) b 1 X 1 +b 2 X 2 + e Y X 1 (x1, x2, x3) O X 2 (1,1,1) design space

20 GLM essentials The mdel Design matrix: Effects f interest Design matrix: Cnfunds (aka effects f n interest) Residuals (errr measures f the whle mdel) Estimate effects and errr fr data Parameter estimates (aka betas) Quantify specific effects using cntrasts f parameter estimates Statistic Cmpare estimated effects the cntrasts with apprpriate errr measures Are the effects surprisingly large?

21 Parameter estimatin Ordinary least squares Y = Xβ + ε βˆ = ( X X ) T 1 X T Y εˆ = Y Xβˆ Parameter estimates residuals Estimate parameters N such such that that t= 1 ˆε 2 t minimal Least squares parameter estimate

22 Parameter estimatin Ordinary least squares Y = Xβ + ε βˆ = ( X X ) T 1 X T Y εˆ = Y Xβˆ Parameter estimates residuals = rr Errr variance s 2 2 = (sum (sum f) f) squared residuals standardised fr fr df df = rr T T r r // df df (sum (sum f f squares) where degrees f f freedm df df (assuming iid): iid): = N --rank(x) (=N-P if if X full full rank)

23 Estimatin (gemetrical) Y = b Y 1 x 1 1 b 1 e 1 Y 2 = x e 2 Y 3 x 3 1 e 3 2 Y b 1 X 1 +b 2 X 2 + e DATA (Y 1, Y 2, Y 3 ) RESIDUALS ^e = (e^ 1,e ^ 2,e^ 3 ) T X 1 (x1, x2, x3) Y^ (Y^ 1,Y ^ 2,Y ^ 3 ) O X 2 (1,1,1) design space

24 GLM essentials The mdel Design matrix: Effects f interest Design matrix: Cnfunds (aka effects f n interest) Residuals (errr measures f the whle mdel) Estimate effects and errr fr data Parameter estimates (aka betas) Quantify specific effects using cntrasts f parameter estimates Statistic Cmpare estimated effects the cntrasts with apprpriate errr measures Are the effects surprisingly large?

25 Inference - cntrasts Y = Xβ + ε A cntrast = a linear cmbinatin f parameters: c x b spm_cn*img t-test: ne-dimensinal cntrast, difference in means/ difference frm zer bxcar parameter > 0? Null Null hypthesis: β 1 = 0 spmt_000*.img SPM{t} map F-test: tests multiple linear hyptheses des subset f mdel accunt fr significant variance Des bxcar parameter mdel anything? Null Null hypthesis: variance f f tested effects = errr errr variance ess_000*.img SPM{F} map

26 t-statistic - example Y = Xβ + ε c c = = t = T c βˆ ˆ T Std( c βˆ) Cntrast f f parameter estimates Variance estimate X Standard errr errr f f cntrast depends n n the the design, and and is is larger with with greater residual errr and and greater cvariance/ autcrrelatin Degrees f f freedm d.f. d.f. then then = n-p, n-p, where n bservatins, p parameters Tests fr fr a directinal difference in in means

27 F-statistic - example D D mvement parameters (r (r ther cnfunds) accunt fr fr anything? H 0 : True mdel is X 0 H 0 : b 3-9 = ( ) test H 0 : c x b = 0? X 0 X 1 (b 3-9 ) X 0 This mdel? Or this ne? Null Null hypthesis H 0 : 0 : That That all all these betas b 3-9 are 3-9 are zer, zer, i.e. i.e. that that n n linear cmbinatin f f the the effects accunts fr fr significant variance This This is is a nn-directinal test test

28 F-statistic - example D D mvement parameters (r (r ther cnfunds) accunt fr fr anything? H 0 : True mdel is X 0 H 0 : b 3-9 = ( ) test H 0 : c x b = 0? X 0 X 1 (b 3-9 ) X 0 c = This mdel? Or this ne? SPM{F}

29 Summary s far The essential mdel cntains Effects f interest A better mdel? A better mdel (within reasn) means smaller residual variance and mre significant statistics Capturing the signal later Add cnfunds/ effects f n interest Example f mvement parameters in fmri A further example (mainly relevant t PET)

30 Example PET experiment b 1 b 2 b 3 b 4 rank(x)= scans, 3 cnditins (1-way ANOVA) y j = j x 1j b 1j x 2j b 2j x 3j b 3j x 4j b 4j e j j where (dummy) variables: x 1j = 1j [0,1] = cnditin A (first (first 4 scans) x 2j = 2j [0,1] = cnditin B (secnd 4 scans) x 3j = 3j [0,1] [0,1] = cnditin C (third 4 scans) x 4j = 4j [1] [1] = grand mean (sessin cnstant)

31 Glbal effects May May be be variatin in in PET PET tracer dse dse frm scan scan t t scan scan Such glbal changes in in image intensity (gcbf) cnfund lcal lcal // reginal (rcbf) changes f f experiment Adjust fr fr glbal effects by: by: --AnCva (Additive Mdel) --PET? --Prprtinal Scaling --fmri? Can Can imprve statistics when rthgnal t t effects f f interest but can can als als wrsen when effects f f interest crrelated with with glbal rcbf (adj) rcbf x x x x x x x rcbf rcbf a k 2 a k 1 rcbf 0 0 AnCva Scaling x x x x x x x x x x x x x x x x 1 x x x x g.. z k x x x x x x 50 glbal gcbf glbal gcbf glbal gcbf

32 Glbal effects (AnCva) b 1 b 2 b 3 b 4 b 5 rank(x)= scans, 3 cnditins, 1 cnfunding cvariate y j = j x 1j b 1j x 2j b 2j x 3j b 3j x 4j b 4j x 5j b 5j e j j where (dummy) variables: x 1j = 1j [0,1] = cnditin A (first (first 4 scans) x 2j = 2j [0,1] = cnditin B (secnd 4 scans) x 3j = 3j [0,1] [0,1] = cnditin C (third 4 scans) x 4j = 4j [1] [1] = grand mean (sessin cnstant) x 5j = 5j glbal signal (mean ver ver all all vxels)

33 Glbal effects (AnCva) N N Glbal Orthgnal glbal Crrelated glbal b 1 b 2 b 3 b 4 b 1 b 2 b 3 b 4 b 5 b 1 b 2 b 3 b 4 b 5 rank(x)=3 rank(x)=4 rank(x)=4 Glbal effects nt nt accunted fr fr Maximum degrees f f freedm (glbal uses uses ne) ne) Glbal effects independent f f effects f f interest Smaller residual variance Larger T statistic Mre significant Glbal effects crrelated with with effects f f interest Smaller effect &/r &/r larger residuals Smaller T statistic Less Less significant

34 Glbal effects (scaling) Tw Tw types f f scaling: Grand Mean scaling and and Glbal scaling -- Grand Mean scaling is is autmatic, glbal scaling is is ptinal -- Grand Mean scales by by 100/mean ver ver all all vxels and and ALL ALL scans (i.e, (i.e, single number per per sessin) -- Glbal scaling scales by by 100/mean ver ver all all vxels fr fr EACH scan scan (i.e, (i.e, a different scaling factr every scan) Prblem with with glbal scaling is is that that TRUE glbal is is nt nt (nrmally) knwn nly nly estimated by by the the mean ver ver vxels -- S S if if there there is is a large large signal change ver ver many vxels, the the glbal estimate will will be be cnfunded by by lcal lcal changes -- This This can can prduce artifactual deactivatins in in ther regins after after glbal scaling Since mst surces f f glbal variability in in fmri are are lw lw frequency (drift), high-pass filtering may may be be sufficient, and and many peple d d nt nt use use glbal scaling

35 Summary General(ised) linear mdel partitins data int Effects f interest & cnfunds/ effects f n interest Errr Least squares estimatin Minimises difference between mdel & data T d this, assumptins made abut errrs mre later Inference at every vxel Test hypthesis using cntrast mre later Inference can be Bayesian as well as classical Next: Applying the GLM t fmri