Jean-Baptiste Poline

Transcription

1 Edinburgh course Avril 2010 Linear Models Contrasts Variance components Jean-Baptiste Poline Neurospin, I2BM, CEA Saclay, France Credits: Will Penny, G. Flandin, SPM course authors

2 Outline Part I: Linear model and contrast: going through it again and going further Part II: Variance component and group analyses* * (shamelessly( stolen from Will Penny SPM course)

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

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

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

6 Regression example = β 1 + β 2 + β 1 = 1 β 2 = 1 Fit the GLM voxel time series box-car reference function Mean value

7 Regression example = + + β 1 β 2 β 1 = 5 β 2 = 100 Fit the GLM voxel time series box-car reference function Mean value

8 revisited : matrix form = β β 2 Y = f(t) + + β 1 β 2 1 ε

9 Box car regression: design matrix data vector (voxel time series) design matrix parameters error vector α β 1 = + µ β 2 Y = X β + ε

10 Fact: model parameters depend on regressors scaling Q: When do I care? A: ONLY when comparing manually entered regressors (say you would like to compare two scores) What if two conditions A and B are not of the same duration before convolution HRF?

11 What if we believe that there are drifts?

12 Add more reference functions / covariates... Discrete cosine transform basis functions

13 design matrix data vector β 1 β 2 β 3 β 4 error vector = + Y = X β + ε

14 error vector design matrix β α 1 β µ 2 β 3 = β 4 β 5 + β 6 β 7 β 8 β 9 Y = X β + ε data vector design matrix parameters = the betas (here : 1 to 9)

15 Fitting the model = finding some estimate of the betas raw fmri time series adjusted for low Hz effects Raw data fitted signal fitted low frequencies fitted drift residuals How do we find the betas estimates? By minimizing the residual variance

16 Fitting the model = finding some estimate of the betas β 1 = β 2 β 5 β 6 + Y = X β + ε β 7... finding the betas = minimising the sum of square of the residuals Y X 2 = Σ i [ y i X i ] 2 when β are estimated: let s s call them b when ε is estimated : let s s call it e estimated SD of ε : let s s call it s

17 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) WHICH ONE TO INCLUDE? What if we have too many? Coefficients (= parameters) are estimated by minimizing the fluctuations, - variability variance of estimated 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, or conditions of different durations

18 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

19 T test - one dimensional contrasts - SPM{t} c = b 1 b 2 b 3 b 4 b 5... A contrast = a weighted sum of parameters: c b b 1 > 0? Compute 1xb 1 + 0xb 2 + 0xb 3 + 0xb 4 + 0xb divide by estimated standard deviation of b 1 T = contrast of estimated parameters variance estimate T = c b s 2 c (X X) X) - c SPM{t}

20 From one time series to an image voxels Y: data = X * B beta??? images + E scans Var(E) = s 2 spm_resms c = T = c b s 2 c (X X) X) - c = spm_con??? images spm_t??? images

21 F-test : a reduced model H 0 : True model is X 0 H 0 : β 1 = 0 X 1 X 0 X 0 c = F ~ ( S 0 2 -S 2 ) / S 2 T values become F values. F = T 2 S 2 S 0 2 This (full) model? Or this one? Both activation and deactivations are tested. Voxel wise p-values are halved.

22 F-test : a reduced model or... Tests multiple linear hypotheses : Does X1 model anything? H 0 : True (reduced) model is X 0 X 0 X 1 X 0 additional variance accounted for by tested effects S 2 S 0 2 F = error variance estimate This (full) model? Or this one? F ~ ( S 0 2 -S 2 ) / S 2

23 F-test : a reduced model or... multi-dimensional contrasts? tests multiple linear hypotheses. Ex : does drift functions model anything? H 0 : True model is X 0 H 0 : β 3-9 = ( ) X 0 X 1 X c = This (full) model? Or this one?

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

25 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 tests the sum of the squares of one or several combinations of o 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, t test, testing for both positive and negative effects.

26 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

27 «Additional variance» : Again No correlation between green red and yellow

28 Testing for the green correlated regressors, for example green: subject age yellow: subject score

29 Testing for the red correlated contrasts

30 Testing for the green Very correlated regressors? Dangerous!

31 Testing for the green and yellow If significant? Could be G or Y!

32 Testing for the green Completely correlated regressors? Impossible to test! (not estimable)

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

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

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

36 Correlation between regressors: take home... Do we care about correlation in the design? Yes, always Start with the experimental design : conditions should be as uncorrelated as possible use F tests to test for the overall variance explained by several (correlated) regressors

37 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

38 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 C1 C2 C3 V A C 1 C 2 C 3

39 Asking ourselves some questions... V A C 1 C 2 C 3 Test C1 > C2 : 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

40 Modelling the interactions

41 Test C1 > C2 : c = [ ] C 1 C 1 C 2 C 2 C 3 C 3 Test V > A : c = [ ] V A V A V A 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!

42 With a more flexible model C 1 C 1 C 2 C 2 C 3 C 3 V A V A V A Test C1 > C2? Test C1 different from C2? from c = [ ] to c = [ ] [ ] becomes an F test! What if we use only: c = [ ] OK only if the regressors coding for the delay are all equal

43 Toy example: take home... use F tests when - Test for >0 and <0 effects - Test for more than 2 levels in factorial designs - Conditions are modelled with more than one regressor Check post hoc

44 1 General Linear Model p 1 1 y = Xθ + ε θ y = X p + ε Error Covariance N N N C ε = λ k k Q k N: N: number number of of scans scans p: p: number number of of regressors Model is is specified by by Design matrix X Assumptions about εε

45 Estimation y = X θ + ε N 1 N p p 1 N ReML-algorithm C ε = λkq k k Maximise L = ln p(y λ) = ln p(y θ, λ) dθ L g λ dl g = dλ 2 d L J = 2 dλ λ = λ + J 1 g Weighted Least Squares θ = ( X C X ) X C y T 1 T T 1 e e Friston Fristonet et al. al. 2002, 2002, Neuroimage

46 Hierarchical model θ Hierarchical model θ y (1) ( n 1) = = = X X X (1) (2) ( n) θ θ M θ (1) (2) ( n) + ε + ε + ε (1) (2) ( n) Multiple variance components at at each level C ( i ) = ( i ) λ Q ( i ) ε k k k At At each level, distribution of of parameters is is given by by level above. What we we don t know: distribution of of parameters and variance parameters.

47 θ y Example: Two level model = () 1 () 1 ( 1) θ ( 1) ( 2 ) ( 2 ) ( 2 ) = X X θ + + ε ε y = (1) X 1 (1) X 2 ( 1) θ + ( 1 ) ( 1) ε θ = X ( 2) ( 2 ) θ + ε ( 2) (1) X 3 First level Second level

48 Estimation Hierarchical model θ y (1) = = X X (1) (2) θ θ M (1) (2) + ε + ε (1) (2) θ ( n 1) = X ( n) θ ( n) + ε ( n) Single-level model y (1) (1) (2) = ε + X ε + X... + X (1) ( n 1) ( n) X K K X (1) ( n) ( n) = Xθ + e ε θ +

49 Group analysis in practice Many 2-level models are just too big to to compute. And even if, if, it it takes a long time! Is Is there a fast approximation?

50 Summary Statistics approach First level Second level T Data Design Matrix Contrast Images Var( c α ) t = c ˆ T ˆ α ˆ αˆ1 2 σˆ1 SPM(t) αˆ2 2 σˆ 2 αˆ11 2 σˆ11 αˆ12 2 σˆ12 One-sample 2 nd nd level

51 Validity of approach The summary stats approach is is exact if if for for each session/subject: Within-session covariance the the same First-level design the the same All other cases: Summary stats approach seems to to be be robust against typical violations.

52 Auditory Data Summary statistics Hierarchical Model Friston Fristonet et al. al. (2004) (2004) Mixed Mixed effects effects and and fmri fmri studies, studies, Neuroimage Neuroimage

53 Multiple contrasts per subject Stimuli: Auditory Presentation (SOA = 4 secs) secs) of of words Motion Sound Visual Action jump click pink turn Subjects: Scanning: Question: (i) (i) control subjects fmri, scans per per subject, block design What regions are are affected by by the the semantic content of of the the words? U. U. Noppeney et et al. al.

54 ANOVA 1 st st level: 1.Motion 2.Sound 3.Visual 4.Action? =? = X? = 2 nd nd level: 2,1 3,1 4,1 3,2 4,2 4,3

55 ANOVA 1 st st level: Motion Sound Visual Action? =? = X? = 2 nd nd level: T c = V X

56 Summary Linear hierarchical models are are general enough for for typical multisubject imaging data (PET, fmri, EEG/MEG). Summary statistics are are robust approximation for for group analysis. Also accomodates multiple contrasts per subject.

57 Thank you for your attention!