regressors 1 st of February 2012 Sylvia Kreutzer MaxPhilipp Stenner Methods for Dummies 20112012 1 First Level Analysis Data analysis with SPM Preprocessing of the data Alignment smoothing etc ID: 533164
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Slide1
1st level analysis: basis functions, parametric modulation and correlated regressors.
1st of February 2012Sylvia KreutzerMax-Philipp Stenner
Methods for Dummies 2011/2012
1Slide2
First Level AnalysisData analysis with SPM
Pre-processing of the data (Alignment, smoothing etc.)First Level Analysis
Basis Functions (Sylvia)Experimental design and correlated
regressors
(Max)Random Field theory (next talk)Second Level Analysis
Methods for Dummies 2011/2012
2Slide3
Normalisation
Statistical Parametric Map
Image time-series
Parameter estimates
General Linear Model
Realignment
Smoothing
Design matrix
Anatomical
reference
Spatial filter
Statistical
Inference
RFT
p <
0.05Slide4
Basis FunctionsTemporal basis functions are used to model a more complex function
Function of interest in fMRI Percent signal change over timeHow to approximate the signal?We have to find the combination of functions that give the best representation of the measured BOLD response
Default in SPM: Canonical hemodynamic response function (HDRF)
Methods for Dummies 2011/2012
4Slide5
Basis FunctionsMany different possible functions can be used
Methods for Dummies 2011/2012
5
Fourier analysis
The complex wave at the top can be decomposed into the sum of the three simpler waves shown below.
f(t)=h1(t)+h2(t)+h3(t)
f(t)
h1(t)
h2(t)
h3(t)
Finite Impulse Response
(FIR)Slide6
Hemodynamic Response Function (HRF)Methods for Dummies 2011/2012
6
Since we know the shape of the hemodynamic response, we should use this knowledge and find a similar function to model the percentage signal change over time.
This is our best prediction
of the signal.
Brief
Stimulus
Undershoot
Initial
Undershoot
Peak
Hemodynamic response functionSlide7
Hemodynamic Response Function(HRF)
Methods for Dummies 2011/20127
Gamma functions Two Gamma functions added
Two gamma functions added together form a good representation of the
haemodynamic
response, although they lack the initial undershoot!Slide8
Fits of a boxcar epoch model with (red) and without (black) convolution by a canonical HRF, together with the data (blue).
HRF versus boxcarSlide9
Limits of HRF
General shape of the BOLD impulse response similar across early sensory regions, such as V1 and S1.
Variability across higher cortical regions.
Considerable variability across people.
These types of variability can be accommodated by expanding the HRF...Slide10
Informed Basis Set
Canonical HRF
Temporal derivative
Dispersion derivative
Methods for Dummies 2011/2012
10
Canonical HRF (2 gamma
functions) plus two expansions in:
Time:
The temporal derivative can model (small) differences in the latency of the peak
response
Width: The
dispersion derivative can model (small) differences in the duration of the peak response.Slide11
Design MatrixMethods for Dummies 2011/2012
11
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 dispersionSlide12
General (convoluted) Linear Model
Ex: Auditory words
every 20s
SPM{F}
0 time {secs} 30
Sampled every
TR = 1.7s
Design matrix,
X
[x(t)
ƒ
1
(
) | x(t)
ƒ
2
(
) |...]
…
Gamma functions
ƒ
i
(
) of
peristimulus
time
REVIEW DESIGNSlide13
Comparison of the fitted response
Methods for Dummies 2011/201213
Left: Estimation using
the simple
model
Right: More
flexible model with basis
functions
Haemodynamic
response in a single
voxel
. Slide14
SummaryMethods for Dummies 2011/2012
14
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
voxelsSlide15
SourcesMethods for Dummies 2011/2012
15
www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt
http://www.fil.ion.ucl.ac.uk/spm/doc/manual.pdf
And thanks to Guillaume!Slide16
Part II: Correlated regressors
parametric/non-parametric designMethods for Dummies 2011/2012
16Slide17
MulticollinearityMethods for Dummies 2011/2012
17
yi
= ß
0 + ß1xi1 +
ß2xi2
+… +
ß
N
x
iN
+ eCoefficients reflect
an estimated change in y with every unit change in xi while controlling for
all other regressorsSlide18
Multicollinearity
yi = ß0 + ß
1xi1 + ß
2
x
i2 +… + ßNx
iN + e
x
i1
=
l
0 + lxi2 + v
Methods for Dummies 2011/2012
18{
X
i1
(e.g. age)
Xi2
(e.g. chronic disease duration)x
x x
x
x x
x
x
x
x
x
low variance of v
high variance of v
x
x
x
x
x
x
x
x x x
x
Xi1
Xi2Slide19
Multicollinearity and estimability
Methods for Dummies 2011/201219
y
e
x
1
x
2
(SPM course Oct. 2010, Guillaume
Flandin
)
OLS minimizes
e
by
Xe
= 0
with
e =
Y – (
X
b
estim
)
-1
which gives
b
estim
= (X
T
X)
-1
X
T
Y
cf
covariance matrix
perfect
multicollinearity
(i.e. variance of
v
= 0)
det
(X) = 0
(
X
T
X
)
not invertible
b
estim
not
unique
high
multicollinearity
(i.e. variance of
v
small)
inaccuracy of individual
b
estim
, high standard error Slide20
MulticollinearityMethods for Dummies 2011/2012
20
X
i
b
estim
R
1
R
2
R
1
’
(t- and [
unidimensional
] F-) testing of a single
regressor
(e.g. R
1
) =
̂
testing for the component that is
not explained by (is orthogonal to) the other/the reduced model (e.g. R2)
multicollinearity is contrast specific“conflating” correlated regressors by means of (multidimensional) F-contrasts permits assessing common contribution to variance(X
ibestim
= projection of Yi onto X space)Slide21
MulticollinearityMethods for Dummies 2011/2012
21
(relatively) little spread after projection onto
x-axis,
y-axis or f(x) = xreflecting reduced efficiency for detecting dependencies of the observed data on the respective (combination of) regressors
regressor
1 x
hrf
regressor
2 x
hrf
(MRC
CBU Cambridge,
http
://imaging.mrc-cbu.cam.ac.uk/imaging/DesignEfficiency)Slide22
Orthogonality matrix
Methods for Dummies 2011/201222
reflects the cosine of the angles
between respective pairs of columns
(SPM course Oct. 2010, Guillaume
Flandin
)Slide23
OrthogonalizingMethods for Dummies 2011/2012
23
X
b
estim
R
1
R
2
new
R
1
orth
R
2
leaves the parameter estimate of R
1
unchanged but alters the estimate of the R
2
parameter
a
ssumes unambiguous causality between the
orthogonalized
predictor and the dependent variable by attributing the common variance to this one predictor
only
hence rarely justifiedSlide24
Dealing with multicollinearity
Methods for Dummies 2011/201224
Avoid.
(avoid dummy variables; when
sequential scheme of predictors (stimulus – response) is inevitable: inject jittered delay (see B) or use a
probabilistic R1-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 self-fulfilling prophecies
(MRC CBU Cambridge, http
://imaging.mrc-cbu.cam.ac.uk/imaging/DesignEfficiency)Slide25
Parametric vs. factorial design
Methods for Dummies 2011/201225
factorial
parametric
Widely-used example
(Statistical Parametric Mapping,
Friston
et al. 2007)
Four button press forcesSlide26
Parametric vs. factorial design
Methods for Dummies 2011/201226
factorial
parametric
W
hich
–
when?
Limited prior knowledge, flexibility in contrasting beneficial (“screening”):
Large number of levels/continuous range:Slide27
Methods for Dummies 2011/2012
27Happy mapping!