The general linear model and Statistical Parametric Mapping

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

2. 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?

3. 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

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

5. 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

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

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

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

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

10. Regression model = + +

11. Design matrix = + +

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

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

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

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

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

17. High pass filter data and three different models

18. Mass univariate approach = Y X +

19. Serial correlation with autoregressive process of order 1 (AR(1)) autocovariance function

20. Error covariance matrix i.i.d. AR(1) sampled error covariance matrices (103 voxels) Serial correlation

21. Restricted Maximum Likelihood observed Q1 ReML estimated correlation matrix Q2

22. Inference -- t-statistic c = +1 0 0 0 0 0 0 0 0 0 0 boxcar parameter > 0 ? Null hypothesis:

23. t-statistic -- Computations least squares estimates c = +1 0 0 0 0 0 0 0 0 0 0 X V compute df using Satterthwaite approximation ReML

24. 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 Inference Use classical statistics Hypothesis test using contrast