The general linear model for fmri

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h gnral linar modl for fmri Mthods and Modls in fmri, 7.0.07 Jakob Hinzl hinzl@biomd..thz.

1 h gnral linar modl for fmri Mthods and Modls in fmri, Jakob Hinzl ranslational Nuromodling Unit (NU) Institut for Biomdical Enginring (IB) Univrsit and EH Zürich Man thanks to K. E. Stphan and F. Ptzschnr for matrial ranslational Nuromodling Unit

2 Ovrviw of SPM Imag tim-sris Krnl Dsign matrix Statistical paramtric map (SPM) Ralignmnt Smoothing Gnral linar modl Normalisation Statistical infrnc Gaussian fild thor mplat Paramtr stimats p <0.05 GLM for fmri

3 What is th problm w want to solv? W hav an xprimntal paradigm and want to tst whthr brain activit is (linarl) rlatd to th paradigm. W will tr to solv th problm b modling th data. GLM for fmri 3

4 Modlling th masurd data Wh? How? Mak infrncs about ffcts of intrst. Dcompos data into ffcts and rror. Form statistic using stimats of ffcts and rror stimulus function data linar modl ffcts stimat rror stimat statistic GLM for fmri 4

5 A vr simpl xprimnt tim On sssion 7 ccls of rst and listning Blocks of 6 scans with 7 sc R What is th brain s rspons to such a stimulation? GLM for fmri 5

6 How is brain data rlatd to th input? singl voxl tim sris What w masur. What w know. tim Qustion: Is thr a chang in th BOLD rspons btwn listning and rst? GLM for fmri 6

7 Explain our data A linar modl of th data as a combination of xprimntal manipulation,confounds and rrors im = + + rror BOLD signal x x Singl voxl rgrssion modl: x x rgrssors GLM for fmri 7

8 Writing vrthing in matrix notation im = + rror BOLD signal x x Singl voxl rgrssion modl: GLM for fmri 8

9 h wa it looks in SPM data = + + n n n p n: numbr of scans p: numbr of rgrssors dsign matrix p rror GLM for fmri 9

10 W nd to spcif th dsign matrix. spcif a nois modl,.g. and thn, stimat th paramtrs b N that minimiz th rror t Minimization of th rror dpnds on assumptions about th nois. t I ~ N (0, ) GLM for fmri 0

11 Summar: Mass-univariat GLM p p I ~ N (0, ) = + Modl is spcifid b. Dsign matrix. Assumptions about N N N N: numbr of scans p: numbr of rgrssors h dsign matrix mbodis all availabl knowldg about xprimntall controlld factors and potntial confounds. GLM for fmri

12 How to fit th modl paramtrs. = + rror OLS (Ordinar Last Squars) ŷ ˆ ŷ ˆ Data prdictd b our modl min( ) min(( ˆ) ( ˆ)) = rror btwn prdictd and actual data Goal is to dtrmin th btas that minimiz th quadratic rror GLM for fmri

13 OLS Ordinar last squars ( ˆ) ( ˆ) ( ˆ )( ˆ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ W want to minimiz th quadratic rror btwn data and modl ˆ ˆ 0 ˆ ˆ ( ) GLM for fmri 3

14 OLS Ordinar last squars GLM for fmriv 4 ) ( ˆ ˆ 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ) )( ˆ ( ˆ) ( ˆ) ( OLS stimat for

15 Summar: OLS solution = + Objctiv: stimat paramtrs to minimiz N t t Ordinar last squars stimation (OLS) (assuming i.i.d. rror): ˆ ( ) GLM for fmri 5

16 Gomtric prspctiv ˆ OLS stimats ( ) Rsidual forming matrix R R R I P x ˆ ˆ Dsign spac dfind b x Projction matrix P ˆ P P ( ) GLM for fmri 6

17 Corrlatd and orthogonalizd rgrssors GLM for fmri 7 x x x * Whn x is orthogonalizd with rgard to x, onl th paramtr stimat for x changs, not that for x! Corrlatd rgrssors = xplaind varianc is shard btwn rgrssors x x x x ; * * * x x ; * * * x x Dsign spac dfind b

18 W ar narl thr linar modl ffcts stimat rror stimat statistic GLM for fmri 8

19 Problms of this modl. BOLD rsponss hav a dlad and disprsd form (cf. Lctur ). HRF. h BOLD signal includs substantial amounts of lowfrqunc nois. 3. h data ar sriall corrlatd (tmporall autocorrlatd) this violats th assumptions of th nois modl in th GLM GLM for fmri 9

20 Summar: Mass-univariat GLM p p I ~ N (0, ) = + Modl is spcifid b. Dsign matrix. Assumptions about N N N N: numbr of scans p: numbr of rgrssors h dsign matrix mbodis all availabl knowldg about xprimntall controlld factors and potntial confounds. GLM for fmri 0

21 Problm : h BOLD rspons f t g( t) f ( ) g( t ) d h rspons of a linar tim-invariant (LI) sstm is th convolution of th input with th sstm's rspons to an impuls (dlta function). 0 GLM for fmri

22 Basic math: What is a convolution? f t g( t) f ( ) g( t ) d 0 GLM for fmri

23 Solution: Convolution with th HRF xpctd BOLD rspons = input function impuls rspons function (HRF) f t g( t) f ( ) g( t ) d 0 HRF blu = grn = rd = data prdictd rspons, taking convolvd with HRF prdictd rspons, NO taking into account th HRF GLM for fmri 3

24 Problm : Low frqunc nois blu = black = grn = rd = data man + low-frqunc drift prdictd rspons, taking into account low-frqunc drift prdictd rspons, NO taking into account low-frqunc drift GLM for fmri 4

25 Solution : High-pass filtring discrt cosin transform (DC) st GLM for fmri 5

26 Solution : High-pass filtring Linar modl blu = black = grn = rd = data man + low-frqunc drift prdictd rspons, taking into account low-frqunc drift prdictd rspons, NO taking into account low-frqunc drift GLM for fmri 6

27 Problm 3: Srial corrlations sphricit = i.i.d. rror covarianc is a scalar multipl of th idntit matrix: Cov() = I Exampls for non-sphricit: 4 0 Cov() 0 non-idntit Cov() 0 0 Cov() non-indpndnc GLM for fmri 7

28 Problm 3: Srial corrlations with ~ N (0, ) t a t t st ordr autorgrssiv procss: AR() t autocovarianc function n Cov() n n: numbr of scans GLM for fmri 8

29 Solution 3: Pr-whitning Pr-whitning:. Us an nhancd nois modl with multipl rror covarianc componnts, i.. ~ N(0, V) instad of ~ N(0, I).. Us stimatd srial corrlation to spcif filtr matrix W for whitning th data. his is i.i.d W W W GLM for fmri 9

30 How to dfin W? Enhancd nois modl Rmmbr linar transform for Gaussians V ~ N (0, ) x ~ N (, ), ~ N ( a, a ) ax Choos W such that rror covarianc bcoms sphrical W ~ N (0, W V ) Conclusion: W is a simpl function of V so how do w stimat V? W W V V I / W W W GLM for fmri 30

maximisation) or RML (rstrictd maximum liklihood).

31 Find W multipl covarianc componnts. V ~ N (0, ) V V Cov ( ) Q i i nhancd nois modl V rror covarianc componnts Q and hprparamtrs Q Q = + Estimation of hprparamtrs with EM (xpctation maximisation) or RML (rstrictd maximum liklihood). For mor dtails s (Friston t al, Nuroimag, 6:465; 00) GLM for fmri 3

32 linar modl c = ffcts stimat rror stimat Null hpothsis: 0 t statistic c ˆ Std ( c ˆ) Lctur: Classical (frquntist) infrnc GLM for fmri 3

33 Outlook: Contrasts and statistical maps c = Q: activation during listning? Null hpothsis: 0 t c ˆ Std ( c ˆ) GLM for fmri 33

34 Summar of GLM W W W ˆ ( W ) W c = t W V c ˆ ˆ std( c ˆ) V RMLstimats / Cov( ) V i Q i std ˆ ( c ˆ c ˆ ( W ) ˆ) ( W ) ( W ) tr( R) W W ˆ R I W (W ) For brvit: ( W ) c GLM for fmri 34

35 Phsiological confounds had movmnts artrial pulsations (particularl bad in brain stm) brathing blinks (visual cortx) adaptation ffcts, fatigu, fluctuations in concntration, tc. Lctur: Nois modls in fmri and nois corrction GLM for fmri 35

36 Outlook furthr challngs corrction for multipl comparisons variabilit in th HRF across voxls slic timing limitations of frquntist statistics Basian analss GLM ignors intractions among voxls modls of ffctiv connctivit hs issus ar discussd in futur lcturs. GLM for fmri 36

37 Corrction for multipl comparison Mass-univariat approach: W appl th GLM to ach of a hug numbr of voxls (usuall > 00,000). hrshold of p<0.05 mor than 5000 voxls significant b chanc! Massiv problm with multipl comparisons! Solution: Gaussian random fild thor Lctur: Multipl comparison corrction GLM for fmri 37

38 Variabilit in th BOLD rspons HRF varis substantiall across voxls and subjcts For xampl, latnc can diffr b ± scond Solution: us multipl basis functions S talk on vnt-rlatd fmri GLM for fmri 38

39 Summar Mass-univariat approach: sam GLM for ach voxl GLM includs all known xprimntal ffcts and confounds Convolution with a canonical HRF High-pass filtring to account for low-frqunc drifts, implmntd b a st of cosin functions. Estimation of multipl varianc componnts (.g. to account for srial corrlations) GLM for fmri 39

40 Bibliograph Friston, Ashburnr, Kibl, Nichols, Pnn (007) Statistical Paramtric Mapping: h Analsis of Functional Brain Imags. Elsvir. Christnsn R (996) Plan Answrs to Complx Qustions: h hor of Linar Modls. Springr. Friston KJ t al. (995) Statistical paramtric maps in functional imaging: a gnral linar approach. Human Brain Mapping : GLM for fmri 40

41 Supplmntar slids

42 Convolution stp-b-stp (from Wikipdia):. Exprss ach function in trms of a dumm variabl τ.. Rflct on of th functions: g(τ) g( τ). 3. Add a tim-offst, t, which allows g(t τ) to slid along th τ-axis. 4.Start t at - and slid it all th wa to +. Whrvr th two functions intrsct, find th intgral of thir product. In othr words, comput a sliding, wightd-avrag of function f(τ), whr th wighting function is g( τ). h rsulting wavform (not shown hr) is th convolution of functions f and g. If f(t) is a unit impuls, th rsult of this procss is simpl g(t), which is thrfor calld th impuls rspons.

The General Linear Model (GLM)

The General Linear Model (GLM) he General Linear Model (GLM) Klaas Enno Stephan ranslational Neuromodeling Unit (NU) Institute for Biomedical Engineering University of Zurich & EH Zurich Wellcome rust Centre for Neuroimaging Institute