Download presentation

Presentation is loading. Please wait.

Published byEduardo Ditton Modified over 2 years ago

2
SPM 2002 C1C2C3 X = C1 C2 Xb L C1 L C2 C1 C2 Xb L C1 L C2 Y Xb e Space of X C1 C2 Xb Space X C1 C2 C1 C3 P C1C2 Xb Xb Space of X C1 C2 C1 C3 P C1C3 Xb C2 C3 Xb Space of X C1 C2 C1 C3 P C1C3 Xb C2 C3 = (X’X) - X’Y = = (X’X) - X’Y = ^ ( ) C 1Y /S 1 2 C 2Y /S 2 2 Jean-Baptiste Poline Orsay SHFJ-CEA A priori complicated … a posteriori very simple !

3
SPM course - 2002 LINEAR MODELS and CONTRASTS The RFT Hammering a Linear Model Use for Normalisation T and F tests : (orthogonal projections) Jean-Baptiste Poline Orsay SHFJ-CEA www.madic.org

4
realignement & coregistration smoothing normalisation Corrected p-values images Adjusted signal Design matrix Anatomical Reference Spatial filter Random Field Theory Your question: a contrast Statistical Map Uncorrected p-values General Linear Model Linear fit Ü statistical image

5
w Make sure we understand the testing procedures : t and F tests w Correlation in our model : do we mind ? Plan w A bad model... And a better one w Make sure we know all about the estimation (fitting) part.... w A (nearly) real example

6
Temporal series fMRI Statistical image (SPM) voxel time course One voxel = One test (t, F,...) amplitude time General Linear Model Üfitting Üstatistical image

7
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

8
Regression example… = ++ voxel time series 90 100 110 Mean value box-car reference function -2 0 2 error -2 0 2

9
…revisited : matrix form = + + ss = ++ f(t s )1YsYs error

10
Box car regression: design matrix… =+ = +YX data vector (voxel time series) design matrix parameters error vector

11
Add more reference functions... Discrete cosine transform basis functions

12
…design matrix = + = + YX data vector design matrix parameters error vector = the betas (here : 1 to 9)

13
Fitting the model = finding some estimate of the betas = minimising the sum of square of the error S 2 raw fMRI time series adjusted for low Hz effects residuals fitted “high-pass filter” fitted box-car the squared values of the residuals s 2 number of time points minus the number of estimated betas

14
w We put in our model regressors (or covariates) that represent how we think the signal is varying (of interest and of no interest alike) Summary... w Coefficients (=estimated parameters or betas) are found such that the sum of the weighted regressors is as close as possible to our data w As close as possible = such that the sum of square of the residuals is minimal w These estimated parameters (the “betas”) depend on the scaling of the regressors : keep in mind when comparing ! w The residuals, their sum of square, the tests (t,F), do not depend on the scaling of the regressors

15
w Make sure we understand t and F tests w Correlation in our model : do we mind ? Plan w A bad model... And a better one w Make sure we all know about the estimation (fitting) part.... w A (nearly) real example

16
SPM{t} A contrast = a linear combination of parameters: c´ c’ = 1 0 0 0 0 0 0 0 test H 0 : c´ b > 0 ? T test - one dimensional contrasts - SPM{t} T = contrast of estimated parameters variance estimate T = s 2 c’(X’X) + c c’b box-car amplitude > 0 ? = b > 0 ? (b : estimation of ) = 1 x b + 0 x b + 0 x b + 0 x b + 0 x b +... > 0 ? = b b b b b ....

17
How is this computed ? (t-test) ^^ contrast of estimated parameters variance estimate Y = X + ~ N(0,I) (Y : at one position) b = (X’X) + X’Y (b = estimation of ) -> beta??? images e = Y - Xb (e = estimation of ) s 2 = (e’e/(n - p)) (s = estimation of n: temps, p: paramètres) -> 1 image ResMS -> 1 image ResMS Estimation [Y, X] [b, s] Test [b, s 2, c] [c’b, t] Var(c’b) = s 2 c’(X’X) + c (compute for each contrast c) t = c’b / sqrt(s 2 c’(X’X) + c) (c’b -> images spm_con??? compute the t images -> images spm_t??? ) compute the t images -> images spm_t??? ) under the null hypothesis H 0 : t ~ Student( df ) df = n-p

18
Tests multiple linear hypotheses : Does X1 model anything ? F test (SPM{F}) : a reduced model or... X1X1 X0X0 This model ? H 0 : True model is X 0 S2S2 Or this one ? X0X0 S 0 2 > S 2 F = error variance estimate additional variance accounted for by tested effects F ~ ( S 0 2 - S 2 ) / S 2

19
H 0 : 3-9 = (0 0 0 0...) 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 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}) : a reduced model or... multi-dimensional contrasts ? test H 0 : c´ b = 0 ? X 1 ( 3-9 ) X0X0 This model ?Or this one ? H 0 : True model is X 0 X0X0

20
How is this computed ? (F-test) ^^ Error variance estimate additional variance accounted for by tested effects Test [b, s, c] [ess, F] F = (e 0 ’e 0 - e’e)/(p - p 0 ) / s 2 -> image (e 0 ’e 0 - e’e)/(p - p 0 ) : spm_ess??? -> image of F : spm_F??? -> image of F : spm_F??? under the null hypothesis : F ~ F(df1,df2) p - p 0 n-p p - p 0 n-p b 0 = (X ’X ) + X ’Y e 0 = Y - X 0 b 0 (e = estimation of ) s 2 0 = (e 0 ’e 0 /(n - p 0 )) (s = estimation of n: time, p : parameters) Estimation [Y, X 0 ] [b 0, s 0 ] (not really like that ) Estimation [Y, X] [b, s] Y = X + ~ N(0, I) Y = X + ~ N(0, I) X : X Reduced

21
w Make sure we understand t and F tests w Correlation in our model : do we mind ? Plan w A bad model... And a better one w Make sure we all know about the estimation (fitting) part.... w A (nearly) real example

22
A bad 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)

23
=+ Y X b 1 = 0.22 b 2 = 0.11 A bad model... Residual Variance = 0.3 P( b1 > 0 ) = 0.1 (t-test) P( b1 = 0 ) = 0.2 (F-test)

24
A « better » 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

25
A better model... =+ Y X b 1 = 0.22 b 2 = 2.15 b 3 = 0.11 Residual Var = 0.2 P( b1 > 0 ) = 0.07 (t test) P( [b1 b2] [0 0] ) = 0.000001 (F test) =

26
Flexible models : Fourier Basis

27
Flexible models : Gamma Basis

28
w We rather test flexible models if there is little a priori information, and precise ones with a lot a priori information w The residuals should be looked at...(non random structure ?) w 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 w Interpreting the test on a single parameter (one function) can be very confusing: cf the delay or magnitude situation Summary... (2)

29
w Make sure we understand t and F tests w Correlation in our model : do we mind ? Plan w A bad model... And a better one w Make sure we all know about the estimation (fitting) part.... w A (nearly) real example ?

30
True signal Correlation between regressors Fitting (blue : global fit) Noise Model (green and red)

31
=+ Y X b 1 = 0.79 b 2 = 0.85 b3 = 0.06 Correlation 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) =

32
true signal Correlation 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

33
=+ Y X b 1 = 1.47 b 2 = 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 » Correlation between regressors -2 0.79 0.85 0.06

34
Design orthogonality : « explore design » Beware : when there is more than 2 regressors (C1,C2,C3...), you may think that there is little correlation (light grey) between them, but C1 + C2 + C3 may be correlated with C4 + C5 Black = completely correlated White = completely orthogonal Corr(1,1)Corr(1,2) 12 12 1 2 12 12 1 2

35
Implicit or explicit (decorrelation (or orthogonalisation) Implicit or explicit ( decorrelation (or orthogonalisation) C1 C2 Xb Y Xb e Space of X C1 C2 L C2 : L C1 : test of C2 in the implicit model test of C1 in the explicit model C1 C2 Xb L C1 L C2 C2 C2 See Andrade et al., NeuroImage, 1999 This GENERALISES when testing several regressors (F tests)

36
1 0 1 0 1 1 1 0 1 0 1 1 X = Mean Cond 1 Cond 2 Y = Xb + e C1 C2 Mean = C1+C2 ^^ “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;

37
w We are implicitly testing additional effect only, so we may miss the signal if there is some correlation in the model using t tests Summary... (3) w Orthogonalisation is not generally needed - parameters and test on the changed regressor don’t change w It is always simpler (when possible !) to have orthogonal (uncorrelated) regressors w In case of correlation, use F-tests to see the overall significance. There is generally no way to decide where the « common » part shared by two regressors should be attributed to w 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 contrast only should change.

38
Plan w Make sure we understand t and F tests w Correlation in our model : do we mind ? w A bad model... And a better one w Make sure we all know about the estimation (fitting) part.... w A (nearly) real example

39
A real example (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 DesignDesign Matrix V A C1 C2 C3 C1 C2 C3 V A C 1 C 2 C 3

40
Asking ouselves some questions... V A C 1 C 2 C 3 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 = 0 2- select colons of X for the model under the null hypothesis. The contrast manager allows that

41
Modelling the interactions

42
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] C 1 C 1 C 2 C 2 C 3 C 3 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]

43
Asking ouselves some questions... With a more flexible model V A V A V A Test 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! C 1 C 1 C 2 C 2 C 3 C 3 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

44
The RFT Hammering a Linear Model Use for Normalisation T and F tests : (orthogonal projections) We don’t use that one Jean-Baptiste Poline Orsay SHFJ-CEA www.madic.org SPM course - 2002 LINEAR MODELS and CONTRASTS

Similar presentations

OK

SPM short course – May 2003 Linear Models and Contrasts The random field theory Hammering a Linear Model Use for Normalisation T and F tests : (orthogonal.

SPM short course – May 2003 Linear Models and Contrasts The random field theory Hammering a Linear Model Use for Normalisation T and F tests : (orthogonal.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on nuclear family and joint family in india Ppt on satellite orbiting A ppt on computer networking Download ppt on forms of business organisation Ppt on environment and development Sources of water for kids ppt on batteries Ppt on ac to dc rectifier circuit Ppt on index numbers in excel Ppt on critical thinking skills Ppt on anti parkinson drugs