Post on 19-Jan-2016
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SPM 2002SPM 2002SPM 2002SPM 2002
C1C2C3X =X =
C1C1C2C2
XbXb
LC1
LC2
C1C1C2C2
XbXb
LC1
LC2
YY
XbXb
ee
Space of XSpace of X
C1C1
C2C2
XbXb Sp
ace
XSp
ace
X
C1C2
C1
C3 PC1C2
Xb
XbXbSpace of XSpace of X
C1
C2C1
C3
P C1C3 Xb
C2
C3
XbXb Space of XSpace of XC1C2C1
C3P C1C3
XbC2
C3
= (X’X)= (X’X)-- X’Y = X’Y = ^̂ ( )C1Y/S1
2
C2Y/S22
Jean-Baptiste PolineOrsay SHFJ-CEA
A priori complicated … a posteriori very simple !
SPM course - 2002SPM course - 2002LINEARLINEAR MODELS and CONTRASTSMODELS and CONTRASTS
The RFT
Hammering a Linear Model
Use forNormalisation
T and F tests : (orthogonal projections)
Jean-Baptiste PolineOrsay SHFJ-CEAwww.madic.org
realignement &coregistration smoothing
normalisation
Corrected p-values
images Adjusted signalDesignmatrix
Anatomical Reference
Spatial filter
Random Field Theory
Your question:a contrast
Statistical MapUncorrected p-values
General Linear Model Linear fit
statistical image
Make sure we understand the testing procedures : t and F testsMake sure we understand the testing procedures : t and F tests
Correlation in our model : do we mind ? Correlation in our model : do we mind ?
PlanPlan
A bad model ... And a better oneA bad model ... And a better one
Make sure we know all about the estimation (fitting) part ...Make sure we know all about the estimation (fitting) part .... .
A (nearly) real exampleA (nearly) real example
Temporal series fMRI
Statistical image(SPM)
voxel time course
One voxel = One test (t, F, ...)One voxel = One test (t, F, ...)amplitude
time
General Linear Modelfittingstatistical image
Regression example…Regression example…Regression example…Regression example…
= + +
voxel time series
90 100 110
box-car reference function
-10 0 10
90 100 110
Mean value
Fit the GLM
-2 0 2
Regression example…Regression example…Regression example…Regression example…
= + +
voxel time series
90 100 110
90 100 110
Mean value
box-car reference function
-2 0 2
error
-2 0 2
……revisited : matrix formrevisited : matrix form……revisited : matrix formrevisited : matrix form
= + +
s= + +f(ts)1Ys
error
Box car regression: design matrix…Box car regression: design matrix…Box car regression: design matrix…Box car regression: design matrix…
= +
= +Y X
data v
ecto
r
(v
oxel
time s
eries
)
design
mat
rix
param
eters
erro
r vec
tor
Add more reference functions ...Add more reference functions ...Add more reference functions ...Add more reference functions ...
Discrete cosine transform basis functionsDiscrete cosine transform basis functions
……design matrixdesign matrix……design matrixdesign matrix
=
+
= +Y X
data v
ecto
r
design
mat
rix
param
eters
erro
r vec
tor
= the b
etas (
here :
1 to
9)
Fitting the model = finding some Fitting the model = finding some estimateestimate of the betas of the betas= = minimising the sum of square of the error Sminimising the sum of square of the error S22
Fitting the model = finding some Fitting the model = finding some estimateestimate of the betas of the betas= = minimising the sum of square of the error Sminimising the sum of square of the error S22
raw fMRI time series adjusted for low Hz effects
residuals
fitted “high-pass filter”
fitted box-car
the squared values of the residuals s2number of time points minus the number of estimated betas
We put in our model regressors (or covariates) that represent We put in our model regressors (or covariates) that represent how we think the signal is varying (of interest and of no interest how we think the signal is varying (of interest and of no interest alike)alike)
Summary ...Summary ...
Coefficients (=estimated parameters or betas) are found such Coefficients (=estimated parameters or betas) are found such that the sum of the weighted regressors is as close as possible to that the sum of the weighted regressors is as close as possible to our dataour data As close as possible = such that the sum of square of the As close as possible = such that the sum of square of the residuals is minimalresiduals is minimal These estimated parameters (the “betas”) These estimated parameters (the “betas”) dependdepend on the scaling on the scaling of the regressors : keep in mind when comparing !of the regressors : keep in mind when comparing ! The residuals, their sum of square, the tests (t,F), The residuals, their sum of square, the tests (t,F), do notdo not depend depend on the scaling of the regressors on the scaling of the regressors
Make sure we understand t and F testsMake sure we understand t and F tests
Correlation in our model : do we mind ? Correlation in our model : do we mind ?
PlanPlan
A bad model ... And a better oneA bad model ... And a better one
Make sure we all know about the estimation (fitting) part ...Make sure we all know about the estimation (fitting) part .... .
A (nearly) real exampleA (nearly) real example
SPM{t}
A contrast = a linear combination of parameters: c´
c’ = 1 0 0 0 0 0 0 0
test H0 : c´ b > 0 ?
T test - one dimensional contrasts - SPM{T test - one dimensional contrasts - SPM{tt}}
T =
contrast ofestimated
parameters
varianceestimate
T =
ss22c’(X’X)c’(X’X)++cc
c’bc’b
box-car amplitude > 0 ?=
b > 0 ? (b : estimation of ) =
1xb + 0xb + 0xb + 0xb + 0xb + . . . > 0 ?=
b b b b b ....
How is this computed ? (t-test)How is this computed ? (t-test)How is this computed ? (t-test)How is this computed ? (t-test)^̂̂̂contrast ofestimated
parameters
varianceestimate
YY = = X X + + ~ ~ N(0,I) N(0,I) (Y : at one position)(Y : at one position)
b = (X’X)b = (X’X)+ + X’Y X’Y (b = estimation of (b = estimation of ) -> ) -> beta??? images beta??? images
e = Y - Xbe = Y - Xb (e = estimation of (e = estimation of ))
ss22 = (e’e/(n - p)) = (e’e/(n - p)) (s = estimation of (s = estimation of n: temps, p: paramètres)n: temps, p: paramètres) -> -> 1 image ResMS1 image ResMS
Estimation [Y, X] [b, s]
Test [b, s2, c] [c’b, t]
Var(c’b) Var(c’b) = s= s22c’(X’X)c’(X’X)++c c (compute for each contrast c)(compute for each contrast c)
t = c’b / sqrt(st = c’b / sqrt(s22c’(X’X)c’(X’X)++c) c) (c’b -> (c’b -> images spm_con???images spm_con???
compute the t images -> compute the t images -> images spm_t??? images spm_t??? ))
under the null hypothesis Hunder the null hypothesis H00 : t ~ Student( df ) df = n-p : t ~ Student( df ) df = n-p
Tests multiple linear hypotheses : Does X1 model anything ?
F test (SPM{F test (SPM{FF}) : a reduced model or ...}) : a reduced model or ...
X1 X0
This model ?
H0: True model is X0
S2
Or this one ?
X0
S02 > S2 F =
errorvarianceestimate
additionalvariance
accounted forby tested
effects
F ~ ( S02 - S2 ) / S2
H0: 3-9 = (0 0 0 0 ...)
0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1
c’ =c’ = +1 0 0 0 0 0 0 0
SPM{F}
tests multiple linear hypotheses. Ex : does HPF model anything?
F test (SPM{F test (SPM{FF}) : a reduced model or ...}) : a reduced model or ...multi-dimensional contrasts ? multi-dimensional contrasts ?
test H0 : c´ b = 0 ?
X1 (3-9)X0
This model ? Or this one ?
H0: True model is X0
X0
How is this computed ? (F-test)How is this computed ? (F-test)How is this computed ? (F-test)How is this computed ? (F-test)^̂̂̂Error
varianceestimate
additionalvariance accounted for
by tested effects
Test [b, s, c] [ess, F]
F = (eF = (e00’e’e0 0 - e’e)/(p - p- e’e)/(p - p00) / s) / s2 2 -> image (e-> image (e00’e’e0 0 - e’e)/(p - p- e’e)/(p - p00) : ) : spm_ess???spm_ess???
-> image of F : -> image of F : spm_F???spm_F???
under the null hypothesis : F ~ F(under the null hypothesis : F ~ F(df1df1,,df2df2) ) p - pp - p0 0 n-p n-p
bb00 = (X = (X’X’X))+ + XX’Y’Y
ee00 = Y - X = Y - X0 0 bb00 (e(e = estimation of = estimation of ))
ss2200 = (e = (e00’e’e00/(n - p/(n - p00)) )) (s(s = estimation of = estimation of n: time, pn: time, p: parameters): parameters)
Estimation [Y, X0] [b0, s0] (not really like that )
Estimation [Y, X] [b, s]
YY == X X + + ~ N(0, ~ N(0, I) I)
YY == XX + + ~ N(0, ~ N(0, I) I) XX : X Reduced : X Reduced
Make sure we understand t and F testsMake sure we understand t and F tests
Correlation in our model : do we mind ? Correlation in our model : do we mind ?
PlanPlan
A bad model ... And a better oneA bad model ... And a better one
Make sure we all know about the estimation (fitting) part ...Make sure we all know about the estimation (fitting) part .... .
A (nearly) real exampleA (nearly) real example
A A badbad model ... model ...A A badbad model ... model ...
True signal and observed signal (---)
Model (green, pic at 6sec) et TRUE signal (blue, pic at 3sec)
Fitting (b1 = 0.2, mean = .11)
=> Test for the green regressor non significant
Noise (still contains some signal)
= +
Y X
b1= 0.22 b2= 0.11
A A badbad model ... model ...A A badbad model ... model ...
Residual Variance = 0.3
P( b1 > 0 ) = 0.1 (t-test)
P( b1 = 0 ) = 0.2 (F-test)
A « A « betterbetter » model ... » model ...A « A « betterbetter » model ... » model ...
True signal + observed signal
Global fit (blue)and partial fit (green & red)Adjusted and fitted signal
=> Test of the green regressor significant=> Test F very significant=> Test of the red regressor very significant
Noise (a smaller variance)
Model (green and red)and true signal (blue ---)Red regressor : temporal derivative of the green regressor
A A betterbetter model ... model ...A A betterbetter model ... model ...
= +
Y X
b1= 0.22 b2= 2.15 b3= 0.11
Residual Var = 0.2
P( b1 > 0 ) = 0.07 (t test)
P( [b1 b2] [0 0] ) = 0.000001 (F test)
=
Flexible models :Flexible models :Fourier BasisFourier Basis
Flexible models :Flexible models :Fourier BasisFourier Basis
Flexible models : Flexible models : Gamma BasisGamma BasisFlexible models : Flexible models : Gamma BasisGamma Basis
We rather test flexible models if there is little a priori information, and precise ones with a lot a priori information
The residuals should be looked at ...(non random structure ?)
In general, use the F-tests to look for an overall effect, then look at the betas or the adjusted signal to characterise the origin of the signal
Interpreting the test on a single parameter (one function) can be very confusing: cf the delay or magnitude situation
Summary ... (2)Summary ... (2)
Make sure we understand t and F testsMake sure we understand t and F tests
Correlation in our model : do we mind ? Correlation in our model : do we mind ?
PlanPlan
A bad model ... And a better oneA bad model ... And a better one
Make sure we all know about the estimation (fitting) part ...Make sure we all know about the estimation (fitting) part .....
A (nearly) real exampleA (nearly) real example
?
True signal
CorrelationCorrelation between regressors between regressorsCorrelationCorrelation between regressors between regressors
Fitting (blue : global fit)
Noise
Model (green and red)
= +
Y X
b1= 0.79 b2= 0.85 b3 = 0.06
CorrelationCorrelation between regressors between regressorsCorrelationCorrelation between regressors between regressors
Residual var. = 0.3
P( b1 > 0 ) = 0.08 (t test)
P( b2 > 0 ) = 0.07 (t test)
P( [b1 b2] 0 ) = 0.002 (F test)
=
true signal
CorrelationCorrelation between regressors - 2 between regressors - 2CorrelationCorrelation between regressors - 2 between regressors - 2
Noise
Fit
Model (green and red)red regressor has been
orthogonolised with respect to the green one= remove every thing that can correlate with
the green regressor
= +
Y X
b1= 1.47 b2= 0.85 b3 = 0.06
Residual var. = 0.3
P( b1 > 0 ) = 0.0003 (t test)
P( b2 > 0 ) = 0.07 (t test)
P( [b1 b2] <> 0 ) = 0.002 (F test)
See « explore design »
CorrelationCorrelation between regressors -2 between regressors -2CorrelationCorrelation between regressors -2 between regressors -2
0.79 0.85 0.06
Design orthogonality :Design orthogonality : « explore design » « explore design » Design orthogonality :Design orthogonality : « explore design » « explore design »
BewareBeware : when there is more than 2 regressors (C1,C2,C3...), you : when there is more than 2 regressors (C1,C2,C3...), you may think that there is little correlation (light grey) between them, may think that there is little correlation (light grey) between them, but C1 + C2 + C3 may be correlated with C4 + C5 but C1 + C2 + C3 may be correlated with C4 + C5
Black = completely correlated White = completely orthogonal
Corr(1,1) Corr(1,2)1 2
1 2
1
2
1 2
1 2
1
2
Implicit or explicit Implicit or explicit ((decorrelation (or decorrelation (or orthogonalisation)orthogonalisation)
Implicit or explicit Implicit or explicit ((decorrelation (or decorrelation (or orthogonalisation)orthogonalisation)
C1C1C2C2
XbXb
YY
XbXb
ee
Space of XSpace of X
C1C1
C2C2
LC2 :
LC1:
test of C2 in the implicit model
test of C1 in the explicit model
C1C1C2C2
XbXb
LC1
LC2
C2C2
See Andrade et al., NeuroImage, 1999
This GENERALISES when testing several regressors (F tests)
1 0 11 0 10 1 1 0 1 1 1 0 11 0 10 1 10 1 1
X =X =
MeanMeanCond 1Cond 1 Cond 2Cond 2
Y = Xb + e Y = Xb + e
C1C1
C2C2
Mean = C1+C2Mean = C1+C2
^̂̂̂ ““completely” correlated ... completely” correlated ... ““completely” correlated ... completely” correlated ...
Parameters are not unique in general ! Some contrasts have no meaning: NON ESTIMABLE
Example here : c’ = [1 0 0] is not estimable ( = no specific information in the first regressor);
c’ = [1 -1 0] is estimable;
We are implicitly testing additional effect only, so we may miss We are implicitly testing additional effect only, so we may miss the signal if there is some correlation in the model using t teststhe signal if there is some correlation in the model using t tests
Summary ... (3)Summary ... (3)
Orthogonalisation is not generally needed - parameters and test Orthogonalisation is not generally needed - parameters and test on the changed regressor don’t change on the changed regressor don’t change
It is always simpler (when possible !) to have orthogonal It is always simpler (when possible !) to have orthogonal (uncorrelated) regressors (uncorrelated) regressors
In case of correlation, use F-tests to see the overall significance. In case of correlation, use F-tests to see the overall significance. There is generally no way to decide where the « common » part There is generally no way to decide where the « common » part shared by two regressors should be attributed toshared by two regressors should be attributed to
In case of correlation and you need to orthogonolise a part of In case of correlation and you need to orthogonolise a part of the design matrix, there is no need to re-fit a new model : the the design matrix, there is no need to re-fit a new model : the contrast only should change. contrast only should change.
PlanPlan
Make sure we understand t and F testsMake sure we understand t and F tests
Correlation in our model : do we mind ? Correlation in our model : do we mind ?
A bad model ... And a better oneA bad model ... And a better one
Make sure we all know about the estimation (fitting) part ...Make sure we all know about the estimation (fitting) part .....
A (nearly) real exampleA (nearly) real example
A real example A real example (almost !)(almost !) A real example A real example (almost !)(almost !)
Factorial design with 2 factors : modality and category 2 levels for modality (eg Visual/Auditory)3 levels for category (eg 3 categories of words)
Experimental Design Design Matrix
V
A
C1
C2
C3C1
C2
C3
V A C1 C2 C3
Asking ouselves some questions ...Asking ouselves some questions ...Asking ouselves some questions ...Asking ouselves some questions ...V A C1 C2 C3
• Design Matrix not orthogonal • Many contrasts are non estimable• Interactions MxC are not modeled
Test C1 > C2 : c = [ 0 0 1 -1 0 0 ]Test V > A : c = [ 1 -1 0 0 0 0 ]Test the modality factor : c = ? Use 2-Test the category factor : c = ? Use 2-Test the interaction MxC ?
2 ways :1- write a contrast c and test c’b = 02- select colons of X for themodel under the null hypothesis.
The contrast manager allows that
Modelling the interactionsModelling the interactionsModelling the interactionsModelling the interactions
Asking ouselves some questions ...Asking ouselves some questions ...Asking ouselves some questions ...Asking ouselves some questions ...
V A V A V A
• Design Matrix orthogonal• All contrasts are estimable• Interactions MxC modelled• If no interaction ... ? Model too “big”
Test C1 > C2 : c = [ 1 1 -1 -1 0 0 0]Test V > A : c = [ 1 -1 1 -1 1 -1 0]
Test the differences between categories :[ 1 1 -1 -1 0 0 0]
c = [ 0 0 1 1 -1 -1 0]
C1 C1 C2 C2 C3 C3
Test the interaction MxC ? :[ 1 -1 -1 1 0 0 0]
c = [ 0 0 1 -1 -1 1 0][ 1 -1 0 0 -1 1 0]
Test everything in the category factor , leaves out modality :[ 1 1 0 0 0 0 0]
c = [ 0 0 1 1 0 0 0][ 0 0 0 0 1 1 0]
Asking ouselves some questions ... With a Asking ouselves some questions ... With a more flexible modelmore flexible model
Asking ouselves some questions ... With a Asking ouselves some questions ... With a more flexible modelmore flexible model
V A V A V ATest C1 > C2 ?Test C1 different from C2 ?from c = [ 1 1 -1 -1 0 0 0]to c = [ 1 0 1 0 -1 0 -1 0 0 0 0 0 0]
[ 0 1 0 1 0 -1 0 -1 0 0 0 0 0]becomes an F test!
C1 C1 C2 C2 C3 C3
Test V > A ? c = [ 1 0 -1 0 1 0 -1 0 1 0 -1 0 0]
is possible, but is OK only if the regressors coding for the delay are all equal
The RFT
Hammering a Linear Model
Use forNormalisation
T and F tests : (orthogonal projections)
We don’t use that one
Jean-Baptiste PolineOrsay SHFJ-CEAwww.madic.org
SPM course - 2002SPM course - 2002LINEARLINEAR MODELS and CONTRASTSMODELS and CONTRASTS