1st level analysis Basis functions, parametric modulation and correlated regressors
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1 1st level analysis Basis functions, parametric modulation and correlated regressors 1
2 First Level Analysis Bold impulse response Temporal Basis Functions Parametric modulation Correlated regressors
3 Blocked design vs. event-related design
4 Hemodynamic Response Function (HRF) Function of blood oygenation, flow, volume Peak (ma. oygenation) 4-6s poststimulus; baseline after 20-30s Initial undershoot can be observed Similar across V1, A1, S1 but possible differences across: other regions individuals Initial Undershoot Brief Stimulus Peak Undershoot
5 Hemodynamic Response Function (HRF) Long SOA BOLD response returns to baseline, no overlap Overlap can be accommodated if the BOLD response is eplicitly modelled (linear superposition) Short SOAs are more sensitive Initial Undershoot Brief Stimulus Peak Undershoot
6 General Linear (convolution) model
7 General linear model
8 Temporal basis functions
9 Temporal basis functions Fourier Set Windowed sines & cosines Any shape (up to frequency limit) Inference via F-test Finite Impulse Response Mini timebins (selective averaging) Any shape (up to frequency limit) Inference via F-test
10 Temporal basis functions Gamma Functions Bounded, asymmetrical (like BOLD) Set of different lags Inference via F-test Two Gamma functions added
11 Temporal basis functions Gamma Functions Bounded, asymmetrical (like BOLD) Set of different lags Inference via F-test Informed Basis Set Best guess of canonical BOLD response Variability captured by Taylor epansion Magnitude inferences via t-test?
12 Temporal basis functions Canonical Informed Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions)
13 Temporal basis functions Canonical Temporal Informed Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) plus Multivariate Taylor epansion in: time (Temporal Derivative)
14 Temporal basis functions Canonical Temporal Dispersion Informed Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) plus Multivariate Taylor epansion in: time (Temporal Derivative) width (Dispersion Derivative)
15 Design Matri Left Right Mean 3 regressors used to model each condition The three basis functions are: 1. Canonical HRF 2. Derivatives with respect to time 3. Derivatives with respect to dispersion
16 Temporal basis functions Informed Basis Set Magnitude inferences via t-test on canonical parameters (providing canonical is a reasonable fit) Latency inferences via tests on ratio of derivative : canonical parameters
17 Which temporal basis set? Eample: rapid motor response to faces, Henson et al, 2001 Canonical + Temporal + Dispersion + FIR canonical + temporal + dispersion derivatives appear sufficient may not be for more comple trials (eg stimulus-delay-response) but then such trials better modelled with separate neural components (ie activity no longer delta function) + constrained HRF (Zarahn, 1999)
18 Comparison of the fitted response Haemodynamic response in a single voel. Left: Estimation using the simple model Right: More fleible model with basis functions 18
19 Summary SPM uses basis functions to model the hemodynamic response using a single basis function or a set of functions. The most common choice is the `Canonical HRF' (Default in SPM) By adding the time and dispersion derivatives one can account for variability in the signal change over voels
20 Part II: Correlated regressors parametric/non-parametric design
21 Multicollinearity y i = ß 0 + ß 1 i1 + ß 2 i2 + + ß N in + e Coefficients reflect an estimated change in y with every unit change in i while controlling for all other regressors
22 { Multicollinearity y i = ß 0 + ß 1 i1 + ß 2 i2 + + ß N in + e i1 = l 0 + l i2 + v X i1 (e.g. age) low variance of v X i1 high variance of v X i2 (e.g. chronic disease duration) X i2
23 Multicollinearity and estimability y e OLS minimizes e by Xe = 0 2 yˆ X ˆ with e = Y (X estim ) -1 1 (SPM course Oct. 2010, Guillaume Flandin) which gives estim = (X T X) -1 X T Y high multicollinearity (i.e. variance of v small) inaccuracy of individual estim, high standard error perfect multicollinearity (i.e. variance of v = 0) det(x) = 0 (X T X) not invertible estim not unique cf covariance matri
24 Multicollinearity (t- and [unidimensional] F-) testing of a single regressor (e.g. R 1 ) = testing for the component that is not eplained by (is orthogonal to) the other/the reduced model (e.g. R 2 ) R 1 R 1 R 2 X i estim multicollinearity is contrast specific conflating correlated regressors by means of (multidimensional) F-contrasts permits assessing common contribution to variance (X i estim = projection of Y i onto X space)
25 regressor 2 hrf Multicollinearity (relatively) little spread after projection onto -ais, y-ais or f() = reflecting reduced efficiency for detecting dependencies of the observed data on the respective (combination of) regressors regressor 1 hrf (MRC CBU Cambridge,
26 Orthogonality matri reflects the cosine of the angles between respective pairs of columns (SPM course Oct. 2010, Guillaume Flandin)
27 Orthogonalizing leaves the parameter estimate of R 1 unchanged but alters the estimate of the R 2 parameter assumes unambiguous causality between the orthogonalized predictor and the dependent variable by attributing the common variance to this one predictor only R 1 orth R 1 R 2 X estim R 2 new hence rarely justified
28 Dealing with multicollinearity Avoid. (avoid dummy variables; when sequential scheme of predictors (stimulus response) is inevitable: inject jittered delay (see B) or use a probabilistic R 1 -R 2 sequence (see C)) Obtain more data to decrease standard error of parameter estimates Use F-contrasts to assess common contribution to data variance Orthogonalizing might lead to selffulfilling prophecies (MRC CBU Cambridge,
29 parametric factorial Parametric vs. factorial design Widely-used eample (Statistical Parametric Mapping, Friston et al. 2007) Four button press forces
30 parametric factorial Parametric vs. factorial design Which when? Limited prior knowledge, fleibility in contrasting beneficial ( screening ): Large number of levels/continuous range:
31 Resources Slides from Methods for Dummies 2011 Rik Henson Short SPM Course slides SPM 2012 Course SPM Manual and Data Set Special thanks to Guillaume Flandin
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