Signal Processing for Functional Brain Imaging: General Linear Model (2)

Transcription

1 Signal Processing for Functional Brain Imaging: General Linear Model (2) Maria Giulia Preti, Dimitri Van De Ville Medical Image Processing Lab, EPFL/UniGE March 5, 2015

2 Overview GLM method (part 1, last week ) intuition of fitting linear models matrix algebra explanation model generation parameter estimation hypothesis testing GLM method (part 2, today ) hypothesis testing continued t-test and F-test multiple comparisons enriching the model accounting for imaging artifacts, physiological noise from single-subject to group-level analysis 2

3 From GLM fitting to hypothesis testing time model get effect size 0.83 ˆ = contrast c T = [1 0 0] get data Null hypothesis H 0 : c T =0 then c T ˆ is asymptotically normal N (0, 2 c T (X T X) 1 c) and t = ˆ c T ˆ pc T (X T X) 1 c follows Student t-distribution with N degrees of freedom L get error effect size t = = 6.42 uncertainty of effect size 3

4 Hypothesis testing: t-test Null hypothesis expresses no effect (i.e., true c T is 0) H 0 : c T =0 q t = c T ˆ/ ˆ 2 c T (X T X) 1 c follows Student t-distribution assuming H 0 reject H 0 if t T, where the -level is the acceptable false positive rate: = P (t 0 T ) (one-sided t-test) p-value indicates the assessment of t assuming H 0 : p = P (t 0 t) specificity: risk of false positives (type I errors) sensitivity: risk of false negatives (type II errors) H0 true H0 false Reject H0 Null hypothesis acceptation/rejection controls specificity only useful as evidence of presence, not evidence of absence (neurosurgeon!) 4

5 F-test - putting the same question... fit model ˆ = = fit reduced model ˆ 0.25 = F = = = 5

6 Hypothesis testing: F-test Partitioning into two blocks of regressors Consider reduced model by design matrix X 0 y = X + e y 0 = X e 0 Null hypothesis H 0 expresses no improvement of X over X 0 F = ê T ê ê 0T ê 0 L L 0 ê T ê N L follows F-distribution (L L 0,N L) assuming H 0 reject H 0 if F T, where the -level is the acceptable false positive rate two-sided test if reduced model removes one regressor 6

7 ... or putting more general questions Use of F-test interested in activation for faces or objects ; some arrogant voxel is activating during faces, deactivating during objects t-contrast: c T = [1 1 0] " F -contrast: C T = # any difference between three conditions (like ANOVA) " # F -contrast: C T =

8 ... or putting more general questions (2) fit model ˆ = F-contrast c T = fit reduced model ˆ = 2.87 F 2,57 =

9 Hypothesis testing: F-test (2) More flexible by contrast matrix reduced model can be made up by linear combinations of regressors avoid reparametrization of model F = ê T ê ê 0T ê 0 L L 0 ê T ê N L = yt My y T Ry N L L = ˆ T X T MXˆ L 0 y T Ry model to remove (specified by C): X c = XC reduced model: X 0 = XC 0 where C 0 = I L contrast matrix) R = I N XX + R 0 = I N X 0 X + 0 M = R 0 R X 0 R 0 y can be computed efficiently N L F (L L 0,N L) L L 0 only needs to be computed once My CC + (residual forming of y Ry X c X 9

10 Multiple comparisons Mass univariate testing (V =10K-100K intracranial voxels) E[FP] = V, so false positives should be controlled adequately! Family-wise error rate: FWE = P ([ V k=1 t0 k T ) Bonferroni correction: assuming independent observations FWE =1 (1 ) V V to obtain FWE, use FWE /V at the individual tests high specificity, low sensitivity since neglecting spatial correlation therefore, too conversative can be applied locally (if ROI is chosen a priori) 10

11 Glance at Gaussian random field theory Consider contrast as lattice representation of continuous Gaussian random field spatially smooth data with 3D Gaussian kernel, typical full-width-at-half-maximum (FWHM) about 6 12 mm Euler characteristic: T =topological measure #blobs #holes Assuming H 0 and high T, we have 1.8mm x 1.8mm FWHM=6mm P ( V k=1 t k >T) = P (max k (t k ) >T) = P (one or more blobs) P ( T 1) (no holes) E[ T ] (one blob) can be further approximated assuming (sufficient) spatial smoothness 11

12 Glance at Gaussian random field theory (2) Advantages: increased sensitivity decreased inter-subject variability (group studies!) Limitations: requires sufficient smoothness: typically FWHM like 3 4 voxel size smoothness needs to be estimated bias if not sufficiently smooth several approximations in cascade (high T ) 12

13 Prof. Keith Worsley

14 5% corrected Bonferroni SPM FWE [VDV et al., IEEE JSTSP, 2008] 14

15 5% corrected Bonferroni SPM FWE [VDV et al., IEEE JSTSP, 2008] 15

16 To correct or not to correct? 16

17 To correct or not to correct? 17

18 To correct or not to correct? 18

19 Remember model specification From stimuli to modeled BOLD response blocks (epochs) h(t; w) = events = Convolution is performed in microtime See exercise 1! 19

20 Enriching the model Hemodynamic variations subject-dependent, regional changes, habituation and anticipation effects approximate h(t + t; w + w), where w is dispersion h(t + t; w + w) h(t; w) + t h t + w h w = More involved techniques using Volterra kernels 20

21 Enriching the model (2) Low-frequency components truncated DCT-basis act like a high-pass filter scanner drifts physiological fluctuations (aliased) intrinsic brain activity Add nuissance regressors realignment parameters spikeregressors to cancel bad scans 21

22 Parameter estimation (revisited) GLM with correlated noise, introduce filter S y = X + e, where e : N (0, 2 V ) Sy = SX + Se {z} e 0, where e 0 : N (0, 2 SV S T ) normal equations: (SX) T Sy =(SX) T (SX) estimate: ˆ =(SX) + Sy assume known covariance matrix V = KK T e.g., parametric form of noise model (1/f-noise, autoregressive model) then BLUE is obtained for S = K 1 Hint: adapt exercise 1 as to incorporate temporal correlation of the noise in the simulation & fitting [Bullmore et al., 1996] 22

23 From single-subject analysis... Analysis of (many) timecourses Time 23

24 ... to group-level analysis Fixed effects analysis concatenate data and design matrices of subjects inference on the observed group Random effects analysis estimate contrast of interest for individual subjects (1st level) enter contrast in basic model and re-estimate (2nd level) inference on the population from which group is sample 24

25 one-sample t-test two-sample t-test paired t-test... to group-level analysis (RFX) Analysis of (many) subjects voxel s intensity One-sample t-test! Subjects... significant? subject See exercise 2! 25

26 Conclusion General linear model many ways of testing the fitted parameters t-test, F-test conceptually simple, yet powerful and flexible many tricks to enrich the model generalizes basic models Multiple comparisons problem Gaussian random field theory is state-of-the-art degrades spatial resolution, improves sensitivity, reduces inter-subject variability Alternatives FP rates (e.g., false discovery rate) Spatial modeling (e.g., wavelets,... ) Bayesian inferences (alternative hypothesis made explicit)... 26

27 Questions you should be able to answer Do you need to refit the data to the reduced model when performing an F-test? Does the GLM require regressors to be orthogonal? What happens in the extreme case when regressors are linearly dependent? Can a standard statistical test (e.g., paired t-test) be expressed with a GLM? What is the design matrix? Explain the difference between a significant group-level effect when performing fixed-effect versus randomeffects analysis? Which one would you prefer in general? Which one would you prefer with very few subjects? (e.g., 3) 27

28 Testing without (m)any assumptions... Parametric testing Assume distribution of statistic under null hypothesis Estimate its parameters Apply threshold according to confidence level Non-parametric testing Use data to find distribution of statistic under null Assumption: Null hypothesis exchangeability 5% Parametric Null Distribution 5% Nonparametric Null Distribution [Adapted from Tom Nichols, 2006] 28

29 Permutation testing Compare contrasts in a 2nd level analysis between 2 groups, 3 subjects in each group group 1: AAA group 2: BBB H0: no difference between groups labels can be exchanged! permutations: 2 6 equivalent labelings compute test statistic (e.g., t-value of GLM contrast) find 95%-percentile of test-statistic distribution or, exceed max of 19 permutations (p-value = 1 / (1+19) = 5%) AAABBB 9.82 ABABAB BAAABB 0.70 BABBAA AABABB 3.24 ABABBA BBBAAA

30 Permutation testing (2) Exchangeability Labels of subjects can be flipped under H0 FMRI scans not due to temporal correlation Becomes within reach today Computational and storage needs! E.g., permutation testing for each voxel Can be extended to deal with multiple comparisons Works usually better than parametric testing More robust against outliers When normal null distribution is less suitable Small number of samples Skewed distribution 30