The General Linear Model (GLM)

Transcription

1 he General Linear Model (GLM) Klaas Enno Stephan ranslational Neuromodeling Unit (NU) Institute for Biomedical Engineering University of Zurich & EH Zurich Wellcome rust Centre for Neuroimaging Institute of Neurology, University College London With many thanks to the FIL Methods group Zurich SPM Course February

2 Overview of SPM Image time-series Kernel Design matrix Statistical parametric map (SPM) Realignment Smoothing General linear model Normalisation Statistical inference Gaussian field theory emplate Parameter estimates p <0.05

3 One session Passive word listening versus rest 7 cycles of rest and listening A very simple fmri experiment Blocks of 6 scans with 7 sec R Stimulus function Question: Is there a change in the BOLD response between listening and rest?

4 Modelling the measured data Why? How? Make inferences about effects of interest 1. Decompose data into effects and error. Form statistic using estimates of effects and error stimulus function data linear model effects estimate error estimate statistic

5 Voxel-wise time series analysis ime model specification parameter estimation hypothesis statistic single voxel time series BOLD signal SPM

6 Single voxel regression model ime = β + 1 β + error BOLD signal x 1 x e β β 1 y x + x + = 1 e

7 Mass-univariate analysis: voxel-wise GLM 1 p 1 1 y = Xβ + e β y = X p + e e ~ N (0, σ I Model is specified by 1. Design matrix X. Assumptions about e ) N N N N: number of scans p: number of regressors he design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

8 GLM assumes Gaussian spherical (i.i.d.) errors sphericity = i.i.d. error covariance is scalar multiple of identity matrix: Cov(e) = σ I Examples for non-sphericity: 4 0 Cov(e) = 0 1 non-identity Cov(e) = Cov(e) = 1 non-independence

9 Parameter estimation β1 = + β Objective: estimate parameters to minimize N t= 1 e t y X e Ordinary least squares estimation (OLS) (assuming i.i.d. error): y = Xβ + e ˆ β = ( X X ) 1 X y

10 A geometric perspective on the GLM ˆ β = OLS estimates ( X X ) 1 X y y e Residual forming matrix R e R = = Ry I P x ˆ Xβˆ y = Design space defined by X x 1 yˆ P Projection matrix P = Py = X ( X X ) 1 X

11 Deriving the OLS equation X e= 0 ( ) X y Xβˆ = 0 X y X Xβˆ = 0 X X β ˆ = X y ( ) X X 1 ˆ = X y OLS estimate β

12 Correlated and orthogonal regressors y x * x x 1 y = 1 1 β = β x β + x 1 = 1 β + e y = x β + x 1 1 * β 1 ; * 1 > β = 1 β * + e Correlated regressors = explained variance is shared between regressors When x is orthogonalized with regard to x 1, only the parameter estimate for x 1 changes, not that for x!

13 What are the problems of this model? 1. BOLD responses have a delayed and dispersed form. HRF. he BOLD signal includes substantial amounts of lowfrequency noise. 3. he data are serially correlated (temporally autocorrelated) this violates the assumptions of the noise model in the GLM

14 Problem 1: Shape of BOLD response Solution: Convolution model hemodynamic response function (HRF) f t g( t) = f ( τ ) g( t τ ) dτ he response of a linear time-invariant (LI) 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) 0

15 Convolution model of the BOLD response Convolve stimulus function with a canonical hemodynamic response function (HRF): f t g( t) = f ( τ ) g( t τ ) dτ 0 HRF

16 Problem : Low-frequency noise Solution: High pass filtering Sy = SXβ + Se S = residual forming matrix of DC set discrete cosine transform (DC) set

17 High pass filtering: example blue = black = data mean + low-frequency drift green = predicted response, taking into account low-frequency drift red = predicted response, NO taking into account low-frequency drift

18 e = 1 + ε with ε ~ N (0, σ ) t ae t Problem 3: Serial correlations t t 1 st order autoregressive process: AR(1) autocovariance function Cov(e) N N

19 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: 1. Use an enhanced noise model with multiple error covariance components, i.e. e ~ N(0, σ V) instead of e ~ N(0, σ I).. Use estimated serial correlation to specify filter matrix W for whitening the data. Wy = WXβ + We

20 How do we define W? Enhanced noise model e ~ N (0, σ V ) Remember linear transform for Gaussians x ~ y ~ N( µ, σ N( aµ, a ), y = ax σ ) Choose W such that error covariance becomes spherical We ~ N(0, σ W V ) Conclusion: W is a simple function of V so how do we estimate V? W W V = V = I 1/ Wy = WXβ + We

21 Estimating V: Multiple covariance components V e ~ N(0, σ ) V = enhanced noise model V Cov( e) λiq i error covariance components Q and hyperparameters λ V Q 1 Q = λ 1 + λ Estimation of hyperparameters λ with ReML (restricted maximum likelihood).

22 Contrasts & statistical parametric maps c = X Q: activation during listening? Null hypothesis: β 1 = 0 t = c ˆ β Std( c ˆ) β

23 t-statistic based on ML estimates Wy = WXβ + We ˆβ = ( WX ) + Wy c = X c ˆ β t = ˆ std( c ˆ) β σ W V 1/ = V = Cov( e) V = λ i Q i std ˆ ( c ˆ σ c ˆ σ = ˆ) β = ( WX ) + ( WX ) + ( Wy WX ˆ β ) tr( R) R = I WX (WX ) + c For brevity: ReMLestimates ( WX ) + = ( X WX ) 1 X

24 Physiological confounds head movements arterial pulsations (particularly bad in brain stem) breathing eye blinks (visual cortex) adaptation effects, fatigue, fluctuations in concentration, etc.

25 Outlook: further challenges correction for multiple comparisons variability in the HRF across voxels slice timing limitations of frequentist statistics Bayesian analyses GLM ignores interactions among voxels models of effective connectivity hese issues are discussed in future lectures.

26 Correction for multiple comparisons Mass-univariate approach: We apply the GLM to each of a huge number of voxels (usually > 100,000). hreshold of p<0.05 more than 5000 voxels significant by chance! Massive problem with multiple comparisons! Solution: Gaussian random field theory

27 Variability in the HRF HRF varies substantially across voxels and subjects For example, latency can differ by ± 1 second Solution: use multiple basis functions See talk on event-related fmri

28 Summary Mass-univariate approach: same GLM for each voxel GLM includes all known experimental effects and confounds Convolution with a canonical HRF High-pass filtering to account for low-frequency drifts Estimation of multiple variance components (e.g. to account for serial correlations)

29 Bibliography Friston, Ashburner, Kiebel, Nichols, Penny (007) Statistical Parametric Mapping: he Analysis of Functional Brain Images. Elsevier. Christensen R (1996) Plane Answers to Complex Questions: he heory of Linear Models. Springer. Friston KJ et al. (1995) Statistical parametric maps in functional imaging: a general linear approach. Human Brain Mapping :

30 Supplementary slides

31 Convolution step-by-step (from Wikipedia): 1. Express each function in terms of a dummy variable τ.. Reflect one of the functions: g(τ) g( τ). 3. Add a time-offset, t, which allows g(t τ) to slide along the τ-axis. 4.Start t at - and slide it all the way to +. Wherever the two functions intersect, find the integral of their product. In other words, compute a sliding, weighted-average of function f(τ), where the weighting function is g( τ). he resulting waveform (not shown here) is the convolution of functions f and g. If f(t) is a unit impulse, the result of this process is simply g(t), which is therefore called the impulse response.