SPM short course – Mai 2008 Linear Models and Contrasts Jean-Baptiste Poline Neurospin, I2BM, CEA Saclay, France

realignment & coregistration smoothing normalisation Corrected p-values images Adjusted data 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

w Make sure we understand the testing procedures : t- and F-tests Plan w Examples – almost real w REPEAT: model and fitting the data with a Linear Model w But what do we test exactly ?

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

Regression example… = ++ voxel time series       box-car reference function   Mean value Fit the GLM

Regression example… = ++ voxel time series       box-car reference function   Mean value Fit the GLM

…revisited : matrix form =  + +  =++ Y   11 22  f(t)

Box car regression: design matrix…   =+  =  +YX data vector (voxel time series) design matrix parameters error vector    

What if we believe that there are drifts?

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

…design matrix =  =  + YX error vector      … + data vector

…design matrix =  +  =  + YX data vector design matrix parameters error vector  = the betas (here : 1 to 9)  

Fitting the model = finding some estimate of the betas raw fMRI time series adjusted for low Hz effects residuals fitted “high-pass filter” fitted signal the squared values of the residuals   s 2 number of time points minus the number of estimated betas fitted drift Raw data Noise Variance

Fitting the model = finding some estimate of the betas =            + Y = X  ∥ Y−X ∥ 2 = Σ i [ y i − X i ] 2 finding the betas = minimising the sum of square of the residuals finding the betas = minimising the sum of square of the residuals when are estimated: let’s call them b when  are estimated: let’s call them b when is estimated : let’s call it e when  is estimated : let’s call it e estimated SD of : let’s call it s estimated SD of  : let’s call it s

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) BUT WHICH ONE TO INCLUDE ? Take home... w Coefficients (= parameters) are estimated by minimizing the fluctuations, - variability – variance – of the noise – the residuals. w Because the parameters depend on the scaling of the regressors included in the model, one should be careful in comparing manually entered regressors

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

A contrast = a weighted sum of parameters: c´   c’ = divide by estimated standard deviation of b  T test - one dimensional contrasts - SPM{t} SPM{t} T = contrast of estimated parameters variance estimate T = s 2 c’(X’X) - c c’b   > 0 ? Compute 1 x b  + 0 x b  + 0 x b  + 0 x b  + 0 x b  +... b  b  b  b  b ....

How is this computed ? (t-test) Y = X  +   ~   N(0,I)  (Y : at one position) b = (X’X) + X’Y (b: estimate of  ) -> beta??? images e = Y - Xb (e: estimate of  ) s 2 = (e’e/(n - p)) (s: estimate of  n: time points, p: parameters) -> 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, proportional to s 2 ) 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-t( df ) df = n-p

F-test : a reduced model H 0 :  1 = 0 X1X1 X0X0 H 0 : True model is X 0 X0X0 c’ = T values become F values. F = T 2 Both “activation” and “deactivations” are tested. Voxel wise p-values are halved. This (full) model ? Or this one? S2S2 S02S02 F ~ ( S S 2 ) / S 2

Tests multiple linear hypotheses : Does X1 model anything ? F-test : a reduced model or... This (full) model ? H 0 : True (reduced) model is X 0 X1X1 X0X0 S2S2 Or this one? X0X0 S02S02 F = error variance estimate additional variance accounted for by tested effects F ~ ( S S 2 ) / S 2

c’ = tests multiple linear hypotheses. Ex : does drift functions model anything? F-test : a reduced model or... multi-dimensional contrasts ? H 0 :  3-9 = ( ) X1X1 X0X0 H 0 : True model is X 0 X0X0 Or this one? This (full) model ?

How is this computed ? (F-test) Error variance estimate additional variance accounted for by tested effects Test [b, s, c] [ess, F] F ~ (s 0 - s) / s 2 -> image spm_ess??? -> image of F : spm_F??? -> image of F : spm_F??? under the null hypothesis : F ~ F( p - p0, n-p) Estimation [Y, X] [b, s] Y = X  +   ~ N(0,   I) Y = X     +     ~ N(0,    I) X  : X Reduced

w T tests are simple combinations of the betas; they are either positive or negative (b1 – b2 is different from b2 – b1) T and F test: take home... w F tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler model, or w F tests the sum of the squares of one or several combinations of the betas w 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.

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

« Additional variance » : Again Independent contrasts

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

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

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

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

An example: real Testing for first regressor: T max = 9.8

Including the movement parameters in the model Testing for first regressor: activation is gone !

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

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

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

Asking ourselves some questions... V A C 1 C 2 C 3 Design Matrix not orthogonal Many contrasts are non estimable Interactions MxC are not modelled Test C1 > C2 : c = [ ] Test V > A : c = [ ] [ ] Test C1,C2,C3 ? (F) c = [ ] [ ] Test the interaction MxC ?

Modelling the interactions

Asking ourselves some questions... V A V A V A Test C1 > C2 : c = [ ] Test the interaction MxC : [ ] c =[ ] [ ] Design Matrix orthogonal All contrasts are estimable Interactions MxC modelled If no interaction... ? Model is too “big” ! C 1 C 1 C 2 C 2 C 3 C 3 Test V > A : c = [ ] Test the category effect : [ ] c =[ ] [ ]

Asking ourselves some questions... With a more flexible model V A V A V A Test C1 > C2 ? Test C1 different from C2 ? from c = [ ] to c = [ ] [ ] becomes an F test! C 1 C 1 C 2 C 2 C 3 C 3 Test V > A ? c = [ ] is possible, but is OK only if the regressors coding for the delay are all equal

w F tests have to be used when - Testing for >0 and 0 and <0 effects - Testing for more than 2 levels in factorial designs - Conditions are modelled with more than one regressor Toy example: take home... w 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).