Experimental Design Initial GLM Intro. This Time

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1 Eperimental Design Initial GLM Intr This Time

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4 GLM General Linear Mdel Single subject fmri mdeling Single Subject fmri Data Data at ne vel Rest vs. passive wrd listening Is there an effect? Linear in parameters & Time A Linear Mdel = errr Intensity

5 Linear mdel, in image frm in image matri frm = = Y Y X in matri frm. Linear Mdel Predictrs Y X Y = p X p Signal Predictrs Blck designs Event-related respnses Nuisance Predictrs Drift Regressin parameters Y X N N N N: Number f scans, p: Number f regressrs Signal Predictrs Y X Cnvlutin Eamples Blck Design Event-Related Y X Linear Time-Invariant system LTI specified slely by Stimulus functin f eperiment Blcks Eperimental Stimulus Functin Hemdynamic Respnse Functin (HRF) Respnse t instantaneus impulse Events Hemdynamic Respnse Functin Predicted Respnse

6 LTI Pet Peeve LTI/cnvlutin apprach implies antisymmetry Shape f rise must match inverted shape f fall Bump here must match... Y X...bump here Cannical HRF Mst sensitive if it is crrect If wrng, leads t bias and/r pr fit E.g. True respnse may be faster/slwer E.g. True respnse may have smaller/ bigger undersht HRF Mdels SPM s HRF Y X HRF Mdels Y X HRF Mdels Y X Smth Basis HRFs Mre fleible Less interpretable N ne parameter eplains the respnse Less sensitive relative t cannical (nly if cannical is crrect) Gamma Basis Decnvlutin Mst fleible Allws any shape Even bizarre, nn-sensical nes Least sensitive relative t cannical (again, if cannical is crrect) Decnvlutin Basis Furier Basis Drift Mdels Drift Slwly varying Nuisance variability Even seen in cadavers! A. Smith et al, NI, 999, 9: Mdels Linear, quadratic Discrete Csine Transfrm Y X Sme Terminlgy SPM ( Statistical Parametric Mapping ) is a massively univariate apprach - meaning that a statistic (e.g., T-value) is calculated fr every vel - using the General Linear Mdel Eperimental manipulatins are specified in a mdel ( design matri ) which is fit t each vel t estimate the size f the eperimental effects ( parameter estimates ) in that vel n which ne r mre hyptheses ( cntrasts ) are tested t make statistical inferences ( p-values ), crrecting fr multiple cmparisns acrss vels (using Randm Field Thery ) Discrete Csine Transfrm Basis The parametric statistics assume cntinuus-valued data and additive nise that cnfrms t a Gaussian distributin ( nnparametric versin SNPM eschews such assumptins)

7 Sme Terminlgy Overview SPM usually fcused n functinal specializatin - i.e. lcalizing different functins t different regins in the brain One might als be interested in functinal integratin - hw different regins (vels) interact Multivariate appraches wrk n whle images and can identify spatial/tempral patterns ver vels, withut necessarily specifying a design matri (PCA, ICA)... r with an eperimental design matri (PLS, CVA), r with an eplicit anatmical mdel f cnnectivity between regins - effective cnnectivity - eg using Dynamic Causal Mdeling. General Linear Mdel Design Matri Estimatin/Cntrasts Cvariates (eg glbal) Estimability/Crrelatin. fmri timeseries Highpass filtering HRF cnvlutin Autcrrelatin (nnsphericity) Overview. General Linear Mdel Design Matri Estimatin/Cntrasts Cvariates (eg glbal) Estimability/Crrelatin. fmri timeseries Highpass filtering HRF cnvlutin Autcrrelatin (nnsphericity) General Linear Mdel Parametric statistics ne sample t-test tw sample t-test paired t-test Anva AnCva crrelatin linear regressin multiple regressin F-tests etc all cases f the General Linear Mdel General Linear Mdel Equatin fr single (and all) vels: Matri Frmulatin Equatin fr scan j y j = j jl L j j ~ N(0, ) y j jl l j : data fr scan, j = J : eplanatry variables / cvariates / regressrs, l = L : parameters / regressin slpes / fied effects : residual errrs, independent & identically distributed ( iid ) (Gaussian, mean f zer and standard deviatin f σ) Simultaneus equatins fr scans.. J Regressrs that can be slved fr parameters.. L Equivalent matri frm: y = X Scans X : design matri / mdel

8 Overview General Linear Mdel (Estimatin) Estimate parameters frm least squares fit t data, y:. General Linear Mdel Design Matri Estimatin/Cntrasts Cvariates (eg glbal) Estimability/Crrelatin. fmri timeseries Highpass filtering HRF cnvlutin Autcrrelatin (nnsphericity) ^ = (X T X) - X T y = X y (OLS estimates) Fitted respnse is: ^ Y = X Residual errrs and estimated errr variance are: ^ = y - Y ^ = ^ T ^/ df where df are the degrees f freedm (assuming iid): df = J - rank(x) (=J-L if X full rank) ( R = I - XX = Ry df = trace(r) ) Simple LS Derivatin GLM Estimatin Gemetric Perspective Y = y y = y X X y DATA (y, y, y 3 ) ^RESIDUALS = ( ^ ^ ^ ) T X (,, 3) Y (Y,Y,Y 3 ) O X (,,) design space General Linear Mdel (Inference) Z-scre and T-statistic Specify cntrast (hypthesis), c, a linear cmbinatin ^ f parameter estimates, c T Calculate T-statistic fr that cntrast: ^ ^ ^ T = c T / std(c T ) = c T / sqrt(^ c T (X T X) - c) (c is a vectr), r an F-statistic: F = [( 0T ε 0 ε T ε) / (L-L 0 )] / [ε T ε / (J-L)] where ε 0 and L 0 are residuals and rank resp. frm the reduced mdel specified by c (which is a matri) Prb. f falsely rejecting Null hypthesis, H 0 : c T =0 ( p-value ) T F T-distributin p( t 0) c = [ - 0 0] c = [ ] u t f ( y) Z T s s s N The z-scre describes the relative lcatin f a particular scre () when the mean (μ) and standard deviatin (σ) are knwn The t-scre describes the relative lcatin f the sample mean () when the ppulatin mean is knwn and the ppulatin standard deviatin is estimated with the sample standard deviatin (s)

9 Simple ANOVA-like Eample Overview scans, 3 cnditins (-way ANOVA) y j = j j 3j 3 4j 4 j where (dummy) variables: j = [0,] = cnditin A (first 4 scans) j = [0,] = cnditin B (secnd 4 scans) 3j = [0,] = cnditin C (third 4 scans) 4j = [] = grand mean T-cntrast : [ - 0 0] tests whether A>B [- 0 0] tests whether B>A F-cntrast: [ ] tests main effect f A,B,C = rank(x)= c=[- 0 0], T=0/sqrt(3.3*8) df=-3=9, T(9)=.94, p<.05. General Linear Mdel Design Matri Estimatin/Cntrasts Cvariates (eg glbal) Estimability/Crrelatin. fmri timeseries Highpass filtering HRF cnvlutin Autcrrelatin (nnsphericity) Glbal Effects Simple ANCOVA Eample May be variatin in verall image intensity frm scan t scan rcbf rcbf scans, 3 cnditins, cnfunding cvariate Such glbal changes may cnfund lcal / reginal induced by eperiment Adjust fr glbal effects by: - AnCva (Additive Mdel) - PET? - Prprtinal Scaling - fmri? Can imprve statistics when rthgnal t effects f interest (as here) but can als wrsen when effects f interest crrelated with glbal (as net) rcbf (adj) k rcbf k rcbf 0 0 AnCva Scaling g.. glbal gcbf k glbal gcbf glbal gcbf 50 y j = j j 3j 3 4j 4 5j 5 j where (dummy) variables: j = [0,] = cnditin A (first 4 scans) j = [0,] = cnditin B (secnd 4 scans) 3j = [0,] = cnditin C (third 4 scans) 4j = grand mean 5j = glbal signal (mean ver all vels) (further mean-crrected ver all scans) Glbal crrelated here with cnditins (and time) = c=[- 0 0], T=3.3/sqrt(3.8*8) df=-4=8, T(8)=0.6, p>.05 Glbal Effects (fmri) Overview Tw types f scaling: Grand Mean scaling and Glbal scaling Grand Mean scaling is autmatic, glbal scaling is ptinal Grand Mean scales by 00/mean ver all vels and ALL scans (i.e, single number per sessin) Glbal scaling scales by 00/mean ver all vels fr EACH scan (i.e, a different scaling factr every scan) Prblem with glbal scaling is that TRUE glbal is nt (nrmally) knwn we nly estimate it by the mean ver vels S if there is a large signal change ver many vels, the glbal estimate will be cnfunded by lcal changes This can prduce artifactual deactivatins in ther regins after glbal scaling Since mst surces f glbal variability in fmri are lw frequency (drift), high-pass filtering may be sufficient, and many peple t nt use glbal scaling. General Linear Mdel Design Matri Estimatin/Cntrasts Cvariates (eg glbal) Estimability/Crrelatin. fmri timeseries Highpass filtering HRF cnvlutin Autcrrelatin (nnsphericity)

10 A wrd n crrelatin/estimability A wrd n crrelatin/estimability If any clumn f X is a linear cmbinatin f any thers (X is rank deficient), sme parameters cannt be estimated uniquely (inestimable) which means sme cntrasts cannt be tested (eg, nly if sum t zer) This has implicatins fr whether baseline (cnstant term) is eplicitly r implicitly mdelled Rank deficiency can be thught f as perfect crrelatin rank(x)= = 0.9 = 0.7 c d * = [ 0]*= 0.9 c m = [ 0 0] c d = [ - 0] A B AB implicit =.6 c m = [ = 0.7 0] c d = [ - ] A B c d * = [ -]*= 0.9 eplicit A AB c d = [ 0] When there is high (but nt perfect) crrelatin between regressrs, parameters can be estimated but the estimates will be inefficiently estimated (ie highly variable) meaning sme cntrasts will nt lead t very pwerful tests SPM shws pairwise crrelatin between regressrs but this will NOT tell yu that, eg, X X is highly crrelated with X 3 s sme cntrasts can still be inefficient/efficient, even thugh pairwise crrelatins are lw/high A B AB cnvlved with HRF! A B AB c m = [ 0 0] c d = [ - 0] c m = [ 0 0] () c d = [ - 0] A wrd n rthgnalizatin A wrd n rthgnalizatin T remve crrelatin between tw regressrs, yu can eplicitly rthgnalize ne (X ) with respect t the ther (X ): X = X (X X )X (Gram-Schmidt) Y X X = 0.9 = 0.7 Paradically, this will NOT change the parameter estimate fr X, but will fr X In ther wrds, the parameter estimate fr the rthgnalized regressr is unchanged! This reflects fact that parameter estimates autmatically reflect rthgnal cmpnent f each regressr s n need t rthgnalize, UNLESS yu have a priri reasn fr assigning cmmn variance t the ther regressr X X X Orthgnalize X (Mdel M) X X (M) =.6 (M) = 0.7 Orthgnalize X (Mdel M) X X (M) = 0.9 =. = = ( Overview fmri Analysis. General Linear Mdel Design Matri Estimatin/Cntrasts Cvariates (eg glbal) Estimability/Crrelatin. fmri timeseries Highpass filtering HRF cnvlutin Autcrrelatin (nnsphericity) Scans are treated as a timeseries and can be filtered t remve lw-frequency (/f) nise Effects f interest are cnvlved with haemdynamic respnse functin (HRF), t capture sluggish nature f (BOLD) respnse Scans can n lnger be treated as independent bservatins they are typically temprally autcrrelated (fr TRs<8s)

11 fmri Analysis (Epch) fmri eample Scans are treated as a timeseries and can be filtered t remve lw-frequency (/f) nise Effects f interest are cnvlved with haemdynamic respnse functin (HRF), t capture sluggish nature f (BOLD) respnse Scans can n lnger be treated as independent bservatins they are typically temprally autcrrelated (fr TRs<8s) = (t) (b-car uncnvlved) vel timeseries b-car functin baseline (mean) (Epch) fmri eample Lw frequency nise Lw frequency nise: Physical (scanner drifts) Physilgical (aliased) cardiac (~ Hz) respiratry (~0.5 Hz) aliasing = pwer spectrum nise signal (eg infinite 30s n-ff) pwer spectrum highpass filter y = X (Epch) fmri eample...with highpass filter (Epch) fmri eample fitted and adjusted data Raw fmri timeseries Adjusted data fitted b-car = highpass filtered (and scaled) Residuals fitted high-pass filter y = X

12 fmri Analysis Cnvlutin with HRF Uncnvlved fit Residuals Scans are treated as a timeseries and can be filtered t remve lw-frequency (/f) nise Effects f interest are cnvlved with haemdynamic respnse functin (HRF), t capture sluggish nature f (BOLD) respnse = Scans can n lnger be treated as independent bservatins they are typically temprally autcrrelated (fr TRs<8s) Bcar functin Cnvlved fit hæmdynamic respnse cnvlved with HRF Residuals (less structure) fmri Analysis Tempral autcrrelatin Scans are treated as a timeseries and can be filtered t remve lw-frequency (/f) nise Effects f interest are cnvlved with haemdynamic respnse functin (HRF), t capture sluggish nature f (BOLD) respnse Scans can n lnger be treated as independent bservatins they are typically temprally autcrrelated (fr TRs<8s) Because the data are typically crrelated frm ne scan t the net, ne cannt assume the degrees f freedm (dfs) are simply the number f scans minus the dfs used in the mdel need effective degrees f freedm In ther wrds, the residual errrs are nt independent: Y = X ~ N(0, V) V I, V=AA' where A is the intrinsic autcrrelatin Generalized least squares: KY = KX K K ~ N(0, V) V = KAA'K' (autcrrelatin is a special case f nnsphericity ) Linear Mdel Errrs Tw sample cvariance matrices... Cv( ) Cv( ) N Y X Cnclusin GLM Encmpasses almst any statistical mdel yu need Imprtant t accunt fr fmri autcrrelatin Cntrasts Way f interrgating GLM T cntrasts (is this effect psitive [negative]?) F cntrasts (are ne r mre f these effects different frm zer?) N independence autcrrelatin