Presentation on theme: "The General Linear Model (GLM)"— Presentation transcript:
1 The General Linear Model (GLM) Klaas Enno StephanWellcome Trust Centre for NeuroimagingUniversity College LondonWith many thanks to my colleagues in the FIL methods group, particularly Stefan Kiebel, for useful slidesSPM Course, ICN May 2008
2 Overview of SPM p <0.05 Statistical parametric map (SPM) Image time-seriesKernelDesign matrixRealignmentSmoothingGeneral linear modelStatisticalinferenceGaussianfield theoryNormalisationp <0.05TemplateParameter estimates
3 A very simple fMRI experiment One sessionPassive word listeningversus rest7 cycles ofrest and listeningBlocks of 6 scanswith 7 sec TRStimulus functionQuestion: Is there a change in the BOLD response between listening and rest?
4 Modelling the measured data Why?Make inferences about effects of interestDecompose data into effects and errorForm statistic using estimates of effects and errorHow?stimulus functioneffects estimatelinearmodelstatisticdataerror estimate
5 Voxel-wise time series analysis modelspecificationparameterestimationhypothesisstatisticTimeTimeBOLD signalsingle voxeltime seriesSPM
6 Single voxel regression model error=+12+Timex1x2eBOLD signal
7 Mass-univariate analysis: voxel-wise GLM =+yModel is specified byDesign matrix XAssumptions about eN: number of scansp: number of regressorsThe design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.
8 GLM assumes Gaussian “spherical” (i.i.d.) errors sphericity = iid: error covariance is scalar multiple of identity matrix:Cov(e) = 2IExamples for non-sphericity:non-identitynon-identitynon-independence
9 = + Parameter estimation y X Least squares parameter estimate Estimate parameters such thatminimalLeast squares parameter estimate(assuming iid error)
10 A geometric perspective yeDesign space defined by Xx2x1
11 What are the problems of this model? BOLD responses have a delayed and dispersed form.HRFThe BOLD signal includes substantial amounts of low- frequency noise.The data are serially correlated (temporally autocorrelated) this violates the assumptions of the noise model in the GLM
12 Problem 1: Shape of BOLD response Solution: Convolution model hemodynamic response function (HRF)The response of a linear time-invariant (LTI) system is the convolution of the input with the system's response to an impulse (delta function).expected BOLD response= input function impulse response function (HRF)
13 Convolution model of the BOLD response Convolve stimulus function with a canonical hemodynamic response function (HRF): HRF
14 Problem 2: Low-frequency noise Solution: High pass filtering S = residual forming matrix of DCT setdiscrete cosine transform (DCT) set
15 High pass filtering: example blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account low-frequency driftred = predicted response, NOT taking into account low-frequency drift
16 Problem 3: Serial correlations with1st order autoregressive process: AR(1)autocovariancefunction
17 Dealing with serial correlations Pre-colouring: impose some known autocorrelation structure on the data (filtering with matrix W) and use Satterthwaite correction for df’s.Pre-whitening: Use an enhanced noise model with hyperparameters for multiple error covariance components Use estimated autocorrelation to specify filter matrix W for whitening the data.
18 How do we define W? Enhanced noise model Remember how Gaussians are transformed linearlyChoose W such that error covariance becomes sphericalConclusion: W is a function of V so how do we estimate V?
19 Multiple covariance components enhanced noise modelVQ1Q2=1+ 2Estimation of hyperparameters with ReML (restricted maximum likelihood).
21 t-statistic based on ML estimates ReML-estimate
22 Physiological confounds head movementsarterial pulsationsbreathingeye blinksadaptation affects, fatigue, fluctuations in concentration, etc.
23 Correlated and orthogonal regressors yx2x2*x1Correlated regressors =explained variance is shared between regressorsWhen x2 is orthogonalized with regard to x1, only the parameter estimate for x1 changes, not that for x2!
24 Outlook: further challenges correction for multiple comparisonsvariability in the HRF across voxelsslice timinglimitations of frequentist statistics Bayesian analysesGLM ignores interactions among voxels models of effective connectivity
25 Correction for multiple comparisons Mass-univariate approach: We apply the GLM to each of a huge number of voxels (usually > 100,000).Threshold of p<0.05 more than 5000 voxels significant by chance!Massive problem with multiple comparisons!Solution: Gaussian random field theory
26 Variability in the HRFHRF varies substantially across voxels and subjectsFor example, latency can differ by ± 1 secondSolution: use multiple basis functionsSee talk on event-related fMRI
27 Summary Mass-univariate approach: same GLM for each voxel GLM includes all known experimental effects and confoundsConvolution with a canonical HRFHigh-pass filtering to account for low-frequency driftsEstimation of multiple variance components (e.g. to account for serial correlations)Parametric statistics