1. Idiot's guide to... General Linear Model & fMRI Elliot Freeman, ICN. fMRI model, Linear Time Series, Design Matrices, Parameter estimation, *&%@!

2. General Linear Model & fMRI How does GLM apply to fMRI experiments? Y = X. β + ε Observed = Predictors * Parameters + Error BOLD= Design Matrix * Betas + Error

3. Observed data Preprocessing... Intensity Time Y Y = X. β + ε Y is a matrix of BOLD signals: Each column represents a single voxel sampled at successive time points.

4. Univariate analysis Each voxel considered as independent observation Analysis of individual voxels over time, not groups over space SPM would still work on an Amoeba! Y = X. β + ε

5. Continuous predictors X can contain values quantifying experimental variable YXYX Y = X. β + ε

6. Binary predictors X can contain values distinguishing experimental conditions YXYX Y = X. β + ε

7. Parameters & error this line is a 'model' of the data slope β = 0.23 intercept = 54.5 β: slope of line relating X to Y ‘how much of X is needed to approximate Y?’ the best estimate of β minimises ε: deviations from line Y = X. β + ε

8. Design Matrix Matrix represents values of X Different columns = different predictors Y X 1 X 2 X 1 X 2 Y = X. β + ε

9. Matrix formulation (t) Y X 1 X 2 Y1Y2YNY1Y2YN β1β2βLβ1β2βL ε (t1) ε (t2) ε (tN) + = X 1 (t1) X 2 (t1)... X L (t1) X 1 (t2) X 2 (t2)... X L (tS) X 1 (tN) X 2 (tN)... X L (tN) Y 1 = (5 * β 1 ) + (1 * β 2 ) ^ Y 2 = (4 * β 1 ) + (1 * β 2 )... ^^ Y N = ( X 1 (tN) * β 1 ) + ( X 2 (tN) * β 2 ) ^ X 1 X 2 Y = X. β + ε

10. Parameter estimation and stats Find betas (by least squares estimation) Y= βX -> “B = Y / X” (B= estimated β) Matlab magic: >> B = inv(X) * Y Now find error term: e = Y – (X * B )...and use these results for statistics: t = betas / standard error

11. Covariates vs. conditions Covariates: parametric modulation of independent variable e.g. task-difficulty 1 to 6 -> regression: beta = slope Conditions: 'dummy' codes identify different levels of experimental factor specify time of onset and duration e.g. integers 0 or 1: 'off' or 'on' -> ANOVA: beta = effect mean on off off on

12. Modelling haemodynamics Brain does not just switch on and off! -> Reshape (convolve) regressors to resemble HRF HRF basic function Original HRF Convolved

13. Anatomy of a design matrix Example: 5 subjects 2 conditions per subject 6 replications per condition 1 covariate covariates subjects conditions

14. Interesting vs. uninteresting Important to model all known variables, even if not experimentally interesting: e.g. head movement, block and subject effects  minimise residual error variance for better stats  effects-of-interest means adjusted to eliminate effects- of-no-interest subjects global activity or movement conditions: effects of interest

15. Selecting and comparing betas A beta value is estimated for each column in design matrix A contrast variable is used to select (groups of) conditions and compare with others e.g. mean β (2 4 6...) - mean β (1 3 5...) t statistic = ( β 1 β 2 β 3... ). / SE t-test: t > critical value ? 1...

16. fMRI characteristics which may increase error Variable gain & scanner drift Variations of signal amplitude with every volume and between scanning sessions -> Proportional & Grand-mean scaling of data High-pass filtering in design matrix Serial temporal correlations breathing, heartbeat: activity at one time point correlates with other times -> adjust error term Temporal uncertainty & slice timing delays: model (and eliminate) first derivative of HRF

17. Summary: Reverse Cookery You start with the finished product and want to know how it was made You specify which ingredients to add (design matrix variables) For each ingredient, GLM finds the quantities (betas) that produce the best reproduction (model) Now you can compare your recipe with others (null hypothesis) to see if they differ! (statistical tests)

18. How dumb was that? Sources: http://www.fil.ion.ucl.ac.uk/spm/doc/papers/SPM_3/welcome.html http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch7.pdf http://www.mrc-cbu.cam.ac.uk/Imaging/Common/spmstats.shtml