1. SPM short course – Mai 2007 Linear Models and Contrasts
Jean-Baptiste Poline
Neurospin,I2BM, CEA
Saclay, France
Among the various tools that SPM uses to anlyse brain images, I will concentrate on two aspects : the linear model that is fitted to the data and the test.
In particular, I wish to give you enough information such that you may be able to undezrstand some difficulties that may arise when using linear model.

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

3. Plan Make sure we know all about the estimation (fitting) part ....
Make sure we understand the testing procedures : t- and F-tests
But what do we test exactly ?
An example – almost real

4. One voxel = One test (t, F, ...)
amplitude
General Linear Model
fitting
statistical image
time
Statistical image
(SPM)
Temporal series fMRI
voxel time course

5. Regression example… + + = b2 b1 b1 = 1 voxel time series 90 100 110
box-car reference function b1 b2
Mean value + + =
Fit the GLM
b1 = 1
b2 = 100

6. Regression example… + + = -2 0 2 b2 b1 b1 = 5 voxel time series
-2 0 2 b1 b2
Mean value + + =
Fit the GLM
b1 = 5
b2 = 100
box-car reference function

7. …revisited : matrix form= b1 b2 + +
error Y = b1
f(t) + b2 1 + es

8. Box car regression: design matrix…
(voxel time series)
data vector
parameters
design matrix
error vector b1 b2 a = + m Y = X b + e

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

10. …design matrix = the betas (here : 1 to 9) = + Y = X ´ b + e
parameters
error vector
design matrix
data vector b1 a m b3 b4 b5 b6 b7 b8 b9 b2 = + Y = X b + e

11. Fitting the model = finding some estimate of the betas = minimising the sum of square of the residuals S2
raw fMRI time series
adjusted for low Hz effects
fitted box-car
fitted “high-pass filter”
residuals
the squared values of the residuals S
= s2
number of time points minus the number of estimated betas

12. Take home ...
We put in our model regressors (or covariates) that represent how we think the signal is varying (of interest and of no interest alike) QUESTION IS : WHICH ONE TO INCLUDE ?
Coefficients (= parameters) are estimated using the Ordinary Least Squares (OLS) by minimizing the fluctuations, - variability – variance – of the noise – the residuals
Because the parameters depend on the scaling of the regressors included in the model, one should be careful in comparing manually entered regressors
The residuals, their sum of squares and the resulting tests (t,F), do not depend on the scaling of the regressors.

13. Plan Make sure we all know about the estimation (fitting) part ....
Make sure we understand t and F tests
But what do we test exactly ?
An example – almost real

14. T test - one dimensional contrasts - SPM{t}
A contrast = a linear combination of parameters: c´ ´ b
c’ =
box-car amplitude > 0 ? =
b1 > 0 ?
=>
b1 b2 b3 b4 b5 ....
Compute 1xb1 + 0xb2 + 0xb3 + 0xb4 + 0xb
and
divide by estimated standard deviation
SPM{t}
T =
contrast of estimated parameters
variance estimate
s2c’(X’X)+c
c’b

15. How is this computed ? (t-test)
contrast of estimated parameters
variance estimate
How is this computed ? (t-test)
Y = X b + e e ~ s2 N(0,I) (Y : at one position)
b = (X’X)+ X’Y (b: estimate of b) -> beta??? images
e = Y - Xb (e: estimate of e)
s2 = (e’e/(n - p)) (s: estimate of s, n: time points, p: parameters)
-> 1 image ResMS
Estimation [Y, X] [b, s]
Test [b, s2, c] [c’b, t]
Var(c’b) = s2c’(X’X)+c (compute for each contrast c, proportional to s2)
t = c’b / sqrt(s2c’(X’X)+c) (c’b -> images spm_con???
compute the t images -> images spm_t??? )
under the null hypothesis H0 : t ~ Student-t( df ) df = n-p

16. F-test : a reduced model or ...
Tests multiple linear hypotheses : Does X1 model anything ?
H0: True (reduced) model is X0 X1 X0 S2
Or this one? X0
S02
F =
error variance estimate
additional variance accounted for by tested effects
F ~ ( S S2 ) / S2
This (full) model ?

17. F-test : a reduced model or ... multi-dimensional contrasts ?
tests multiple linear hypotheses. Ex : does DCT set model anything?
H0: b3-9 = ( )
X1 (b3-9) X0
This model ?
Or this one ?
H0: True model is X0
test H0 : c´ ´ b = 0 ?
c’ =
SPM{F}

18. How is this computed ? (F-test)
additional variance accounted for by tested effects
How is this computed ? (F-test)
Error variance estimate
Estimation [Y, X] [b, s]
Y = X b + e e ~ N(0, s2 I)
Y = X0 b0 + e e0 ~ N(0, s02 I) X0 : X Reduced
b0 = (X0’X0)+ X0’Y
e0 = Y - X0 b (eà: estimate of eà)
s20 = (e0’e0/(n - p0)) (sà: estimate of sà, n: time, pà: parameters)
Estimation [Y, X0] [b0, s0]
Test [b, s, c] [ess, F]
F ~ (s0 - s) / s > image spm_ess???
-> image of F : spm_F???
under the null hypothesis : F ~ F(p - p0, n-p)

19. T and F test: take home ...
T tests are simple combinations of the betas; they are either positive or negative (b1 – b2 is different from b2 – b1)
F tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler model, or
F test the sum of the squares of one or several combinations of the betas
in testing “single contrast” with an F test, for ex. b1 – b2, the result will be the same as testing b2 – b1. It will be exactly the square of the t-test, testing for both positive and negative effects.

20. Plan Make sure we all know about the estimation (fitting) part ....
Make sure we understand t and F tests
But what do we test exactly ? Correlation between regressors
An example – almost real

21. « Additional variance » : Again
Independent contrasts

22. « Additional variance » : Again
Testing for the green
correlated regressors, for example
green: subject age
yellow: subject score

23. « Additional variance » : Again
Testing for the red
correlated contrasts

24. « Additional variance » : Again
Testing for the green
Entirely correlated contrasts ?
Non estimable !

25. « Additional variance » : Again
Testing for the green and yellow
If significant ? Could be G or Y !
Entirely correlated contrasts ?
Non estimable !

26. SAME PRINCIPLE APPLIES:
Less simple contrasts
SAME PRINCIPLE APPLIES:
Think of it as constructing 3 regressors from the 3 differences and complement this new design matrix such that data can be fitted in the exact same way (same error, same fitted data)

27. Plan Make sure we all know about the estimation (fitting) part ....
Make sure we understand t and F tests
But what do we test exactly ? Correlation between regressors
An example – almost real

28. A real example (almost !)
Experimental Design
Design Matrix
Factorial design with 2 factors : modality and category
2 levels for modality (eg Visual/Auditory)
3 levels for category (eg 3 categories of words)
V A C1 C2 C3 V A C1 C2 C3

29. Asking ourselves some questions ...
V A C1 C2 C3
Test C1 > C : c = [ ]
Test V > A : c = [ ]
[ ]
Test C1,C2,C3 ? (F) c = [ ]
[ ]
Test the interaction MxC ?
Design Matrix not orthogonal
Many contrasts are non estimable
Interactions MxC are not modelled

30. Modelling the interactions

31. Asking ourselves some questions ...
C1 C1 C2 C2 C3 C3
Test C1 > C : c = [ ]
V A V A V A
Test V > A : c = [ ]
Test the category effect :
[ ]
c = [ ]
[ ]
Test the interaction MxC :
[ ]
c = [ ]
[ ]
Design Matrix orthogonal
All contrasts are estimable
Interactions MxC modelled
If no interaction ... ? Model is too “big” !

32. Asking ourselves some questions ... With a more flexible model
C1 C1 C2 C2 C3 C3
V A V A V A
Test C1 > C2 ?
Test C1 different from C2 ?
from
c = [ ] to
c = [ ]
[ ]
becomes an F test!
Test V > A ?
c = [ ]
is possible, but is OK only if the regressors coding
for the delay are all equal

33. Convolution
model
Design and
contrast
SPM(t) or
SPM(F)
Fitted and
adjusted data

34. Toy example: take home ... F tests have to be used when
Testing for >0 and <0 effects
Testing for more than 2 levels
Conditions are modelled with more than one regressor
F tests can be viewed as testing for
the additional variance explained by a larger model wrt a simpler model, or
the sum of the squares of one or several combinations of the betas (here the F test b1 – b2 is the same as b2 – b1, but two tailed compared to a t-test).

35. Plan Make sure we all know about the estimation (fitting) part ....
Make sure we understand t and F tests
But what do we test exactly ? Correlation between regressors
A (nearly) real example
A bad model ... And a better one

36. A bad model ... True signal and observed signal (---)
Model (green, pic at 6sec)
TRUE signal (blue, pic at 3sec)
Fitting (b1 = 0.2, mean = 0.11)
=> Test for the green regressor not significant
Residual (still contains some signal)

37. A bad model ... Residual Variance = 0.3 P(Y| b1 = 0) =>
p-value = 0.1
(t-test)
p-value = 0.2
(F-test) = + Y
X b e

38. A « better » model ... True signal + observed signal
Model (green and red)
and true signal (blue ---)
Red regressor : temporal derivative of the green regressor
Global fit (blue)
and partial fit (green & red)
Adjusted and fitted signal
=> t-test of the green regressor significant
=> F-test very significant
=> t-test of the red regressor very significant
Residual (a smaller variance)

39. A better model ... Residual Var = 0.2 P(Y| b1 = 0) p-value = 0.07
(t-test)
P(Y| b1 = 0, b2 = 0)
p-value =
(F-test) = + Y
X b e

40. Flexible models : Gamma Basis

41. Summary ... (2) The residuals should be looked at ...!
Test flexible models if there is little a priori information
In general, use the F-tests to look for an overall effect, then look at the response shape
Interpreting the test on a single parameter (one regressor) can be difficult: cf the delay or magnitude situation
BRING ALL PARAMETERS AT THE 2nd LEVEL

42. Lunch ?

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

44. Correlation between regressors
True signal
Model (green and red)
Fit (blue : global fit)
Residual

45. Correlation between regressors
Residual var. = 0.3
P(Y| b1 = 0)
p-value = 0.08
(t-test)
P(Y| b2 = 0)
p-value = 0.07
P(Y| b1 = 0, b2 = 0)
p-value =
(F-test) = + Y
X b e

46. Correlation between regressors - 2
true signal
Model (green and red)
red regressor has been
orthogonalised with respect to the green one
 remove everything that correlates with
the green regressor
Fit
Residual

47. Correlation between regressors -2
0.79***
0.85
0.06
Residual var. = 0.3
P(Y| b1 = 0)
p-value =
(t-test)
P(Y| b2 = 0)
p-value = 0.07
P(Y| b1 = 0, b2 = 0)
p-value =
(F-test) = + Y
X b e
See « explore design »

48. Design orthogonality : « explore design »
Black = completely correlated White = completely orthogonal 1 2 1 2
Corr(1,1)
Corr(1,2)
Beware: when there are 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

49. Summary
We implicitly test for an additional effect only, be careful if there is correlation
Orthogonalisation = decorrelation : not generally needed
Parameters and test on the non modified regressor change
It is always simpler to have orthogonal regressors and therefore designs !
In case of correlation, use F-tests to see the overall significance. There is generally no way to decide to which regressor the « common » part should be attributed to
Original regressors may not matter: it’s the contrast you are testing which should be as decorrelated as possible from the rest of the design matrix

50. Conclusion : check your models
Check your residuals/model
- multivariate toolbox
Check group homogeneity
- Distance toolbox
Check your HRF form
- HRF toolbox !

52. Implicit or explicit (^) decorrelation (or orthogonalisation)Xb Y Xb e
Space of X C1 C2 C1 C2 Xb
LC1^
LC2
C2^
This generalises when testing several regressors (F tests)
LC2 :
LC1^ :
test of C2 in the
implicit ^ model
test of C1 in the
explicit ^ model
cf Andrade et al., NeuroImage, 1999

53. “completely” correlated ...^
b ?
“completely” correlated ... Y
1 0 1
0 1 1
X =
Mean
Cond 1
Cond 2 e
Y = Xb+e;
Space of X C2 Xb
Mean = C1+C2 C1
Parameters are not unique in general ! Some contrasts have no meaning: NON ESTIMABLE
c = [1 0 0] is not estimable (no specific information in the first regressor);
c = [1 -1 0] is estimable;