The general linear model and Statistical Parametric Mapping Stefan Kiebel Andrew Holmes Wellcome Dept. of Imaging Neuroscience Institute of Neurology, UCL, London

Passive word listening fMRI example One session Time series of BOLD responses in one voxel Passive word listening versus rest 7 cycles of rest and listening Each epoch 6 scans with 7 sec TR Stimulus function Question: Is there a change in the BOLD response between listening and rest?

Why modelling? Why? Make inferences about effects of interest Decompose data into effects and error Form statistic using estimates of effects and error How? Model? Use any available knowledge Stimulus function effects estimate statistic data model error estimate

Modelling with SPM General linear model Preprocessing SPMs functional data Design matrix Contrasts Smoothed normalised data Parameter estimates General linear model Preprocessing SPMs Random field theory templates Variance components

but ... not much sensitivity Choose your model Eyeballing Variance Bias No normalisation Lots of normalisation default SPM No smoothing Lots of smoothing Massive model Sparse model Captures signal High sensitivity but ... not much sensitivity but ... may miss signal

Voxel by voxel Time Time model specification parameter estimation hypothesis statistic Time Intensity single voxel time series SPM

Classical statistics General linear model etc... etc... one sample t-test correlation ANOVA multiple regression General linear model Fourier transform wavelet transform etc... etc...

identically distributed) Regression model error e~N(0, s2I) (error is normal and independently and identically distributed) = + b1 b2 + Time x1 x2 e Intensity Question: Is there a change in the BOLD response between listening and rest? Hypothesis test: b1 = 0? (using t-statistic)

Regression model = + + error e~N(0, s2I) b1 b2 x1 x2 e Time Intensity (error is normal and independently and identically distributed) = + b1 b2 + Time x1 x2 e Intensity Stimulus function is not expected BOLD response Data is serially correlated What‘s wrong with this model?

Regression model = + +

Design matrix = + +

General Linear Model = + Y Model is specified by Design matrix X Assumptions about e N: number of scans p: number of regressors

Least squares parameter estimate Parameter estimation Assume iid error = + residuals Estimate parameters Least squares parameter estimate such that minimal

Estimation, example Assume iid error residuals and Least squares estimate Another estimate

Convolve stimulus function with model of BOLD response Improved model Convolve stimulus function with model of BOLD response Haemodynamic response function fitted data

High pass filter high pass filter implemented by residuals of DCT set discrete cosine transform set

data and three different models High pass filter data and three different models

Mass univariate approach = Y X +

autoregressive process of order 1 (AR(1)) Serial correlation with autoregressive process of order 1 (AR(1)) autocovariance function COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

Error covariance matrix i.i.d. AR(1) sampled error covariance matrices (103 voxels) COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!! Serial correlation

Restricted Maximum Likelihood observed Q1 ReML estimated correlation matrix Q2 COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

Inference -- t-statistic boxcar parameter > 0 ? Null hypothesis: COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!

t-statistic -- Computations least squares estimates c = +1 0 0 0 0 0 0 0 0 0 0 X V COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!! compute df using Satterthwaite approximation ReML

Summary General linear model: Unified and comprehensive approach Partition data into effects (of interest and no interest) and error Model serial correlation using variance components Maximum Likelihood and Restricted Maximum Likelihood estimators t-contrast: tests for single dimension in parameter space F-contrast: tests for multiple dimensions inference at first or second level (fixed or random effects) over conditions or groups: main effect, difference, interaction: average over time window parametric modulation with extrinsic variable power comparison in time-frequency domain Inference Use classical statistics Hypothesis test using contrast