Ged Ridgway Wellcome Trust Centre for Neuroimaging University College London. [slides from the FIL Methods group] SPM Course Vancouver, August 2010
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1 Ged Ridgway Wellcome Trust Centre for Neuroimaging University College London [slides from the FIL Methods group] SPM Course Vancouver, August 2010
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9 one sample t-test two sample t-test paired t-test Analysis of Variance (ANOVA) Factorial designs correlation linear regression multiple regression F-tests fmri time series models etc
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12 sphericity = i.i.d. error covariance is scalar multiple of identity matrix: Cov(e) = σ 2 I
13 y X
14 y e Smallest errors (shortest error vector) when e is orthogonal to X x 2 x 1 X Ordinary Least Squares (OLS)
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16 Impulses HRF Expected BOLD =
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18 HRF Sliding the reversed HRF past a unitintegral pulse and integrating the product recovers the HRF (hence the name impulse response) The step response shows a delay and a slight overshoot, before reaching a steady-state. This gives some intuition for a square wave
19 Slow square wave. Main (fundamental) frequency of output matches input, but with delay (phase-shift) and change in amplitude (and shape; sinusoids keep their shape) Fast square wave. Amplitude is reduced, phase shift is more dramatic (output almost in anti-phase). Fourier transform of HRF yields magnitude and phase responses.
20 Convolve stimulus function with a canonical hemodynamic response function (HRF):
21 blue = black = data mean + low-frequency drift green = predicted response, taking into account low-frequency drift red = predicted response, NOT taking into account low-frequency drift
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23 with 1 st order autoregressive process: AR(1) autocovariance function
24 enhanced noise model at voxel i error covariance components Q and hyperparameters λ V Q 1 Q 2 = λ 1 + λ 2 Estimation of hyperparameters λ with ReML (Restricted Maximum Likelihood).
25 Parameters can then be estimated using Weighted Least Squares (WLS) Let Then where WLS equivalent to OLS on whitened data and design
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27 Mass univariate approach. Fit GLMs with design matrix, X, to data at different points in space to estimate local effect sizes, GLM is a very general approach Hemodynamic Response Function High pass filtering Temporal autocorrelation
28 Correlated regressors = explained variance is shared between regressors When x 2 is orthogonalized with regard to x 1, only the parameter estimate for x 1 changes, not that for x 2!
29 c = ReMLestimates For brevity:
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