The General Linear Model. Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London

Transcription

1 The General Linear Model Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course Lausanne, April 2012

2 Image time-series Spatial filter Design matrix Statistical Parametric Map Realignment Smoothing General Linear Model Normalisation Statistical Inference RFT Anatomical reference Parameter estimates p <0.05

3 A very simple fmri experiment One session Passive word listening versus rest 7 cycles of rest and listening Blocks of 6 scans with 7 sec TR Stimulus function Question: Is there a change in the BOLD response between listening and rest?

4 Modelling the measured data Why? How? Make inferences about effects of interest 1. Decompose data into effects and error 2. Form statistic using estimates of effects and error stimulus function data linear model effects estimate error estimate statistic

5 Voxel-wise time series analysis Time Model specification Parameter estimation Hypothesis Statistic single voxel time series BOLD signal SPM

6 error Single voxel regression model Time = BOLD signal x 1 x 2 e y x x e

7 Mass-univariate analysis: voxel-wise GLM 1 p 1 1 y X e y = X p + e e ~ N(0, 2 I Model is specified by 1. Design matrix X 2. Assumptions about e ) N N N N: number of scans p: number of regressors The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

8 GLM: mass-univariate parametric analysis one sample t-test two sample t-test paired t-test Analysis of Variance (ANOVA) Factorial designs correlation linear regression multiple regression F-tests fmri time series models Etc..

9 Parameter estimation 1 = + 2 Objective: estimate parameters to minimize N t1 e 2 t y y X X e e Ordinary least squares estimation (OLS) (assuming i.i.d. error): ˆ ( X X ) T 1 X T y

10 A geometric perspective on the GLM y Smallest errors (shortest error vector) when e is orthogonal to X e x 2 Design space defined by X x 1 yˆ Xˆ X T e 0 X T ( y X ˆ) T X y ˆ ( X X T X ) Xˆ T 1 X 0 T y Ordinary Least Squares (OLS)

11 Example: Two sample T-test B B B B A A A A Mean group A Compare the average difference between two groups Mean group B

12 Example: Two sample T-test ˆ ˆ B B B B A A A A Y X X X T T Compare the average difference between two groups

13 What are the problems of this model? 1. BOLD responses have a delayed and dispersed form. HRF 2. The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift). 3. Due to breathing, heartbeat & unmodeled neuronal activity, the errors are serially correlated. This violates the assumptions of the noise model in the GLM

14 Problem 1: Shape of BOLD response Solution: Convolution model Impulses HRF Expected BOLD = f t g( t) f ( ) g( t ) d 0 expected BOLD response = input function impulse response function (HRF)

15 Convolution model of the BOLD response Convolve stimulus function with a canonical hemodynamic response function (HRF): f t g( t) f ( ) g( t ) d 0 HRF

16 Problem 2: Low-frequency noise Solution: High pass filtering discrete cosine transform (DCT) set blue = data black = mean + low-frequency drift green = predicted response, taking into account low-frequency drift red = predicted response, NOT taking into account low-frequency drift

17 High pass filtering discrete cosine transform (DCT) set

18 Problem 3: Serial correlations e 2 1 with ~ N(0, ) t ae t t 1 st order autoregressive process: AR(1) autocovariance function t Cov(e) N N

19 Multiple covariance components e i ~ N(0, Ci enhanced noise model at voxel i V ) C V i 2 i V Q j error covariance components Q and hyperparameters Q 1 Q 2 j = Estimation of hyperparameters with ReML (Restricted Maximum Likelihood).

20 Summary Mass univariate approach. Fit GLMs with design matrix, X, to data at different points in space to estimate local effect sizes, GLM is a very general approach Hemodynamic Response Function High pass filtering Temporal autocorrelation

21 Statistical Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course Lausanne, April 2012

22 Image time-series Spatial filter Design matrix Statistical Parametric Map Realignment Smoothing General Linear Model Normalisation Statistical Inference RFT Anatomical reference Parameter estimates p <0.05

23 A mass-univariate approach

24 Estimation of the parameters i.i.d. assumptions: OLS estimates:

25 Contrasts [ ] [ ]

26 Hypothesis Testing To test an hypothesis, we construct test statistics. Null Hypothesis H 0 Typically what we want to disprove (no effect). The Alternative Hypothesis H A expresses outcome of interest. Test Statistic T The test statistic summarises evidence about H 0. Typically, test statistic is small in magnitude when the hypothesis H 0 is true and large when false. We need to know the distribution of T under the null hypothesis. Null Distribution of T

27 Hypothesis Testing Significance level α: Acceptable false positive rate α. threshold u α Threshold u α controls the false positive rate Conclusion about the hypothesis: We reject the null hypothesis in favour of the alternative hypothesis if t > u α p-value: p( T u H0) A p-value summarises evidence against H 0. This is the chance of observing value more extreme than t under the null hypothesis. Null Distribution of T t u p-value Null Distribution of T

28 T-test - one dimensional contrasts SPM{t} c T = Question: box-car amplitude > 0? = 1 = c T > 0? Null hypothesis: H 0 : c T =0 Test statistic: T = contrast of estimated parameters variance estimate T T T c ˆ c ˆ ~ T 2 T T 1 var( c ˆ) ˆ c X X c t N p

29 T-contrast in SPM For a given contrast c: ˆ beta_???? images ( X X ) T 1 X T y ResMS image ˆ 2 T ˆ ˆ N p con_???? image c T ˆ spmt_???? image SPM{t}

30 1 T-test: a simple example Passive word listening versus rest c T = [ ] Q: activation during listening? Null hypothesis: 1 0 SPMresults: Height threshold T = {p<0.001} voxel-level T ( Z ) p uncorrected mm mm mm Inf Inf Inf Inf Inf

31 T-test: summary T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate). Alternative hypothesis: T T H 0 : c 0 vs H A : c 0 T-contrasts are simple combinations of the betas; the T- statistic does not depend on the scaling of the regressors or the scaling of the contrast.

32 F-test - the extra-sum-of-squares principle Model comparison: Null Hypothesis H 0 : True model is X 0 (reduced model) X 0 X 1 X 0 Test statistic: ratio of explained variability and unexplained variability (error) RSS ˆ 2 full RSS 0 ˆreduced 2 Full model? or Reduced model? 1 = rank(x) rank(x 0 ) 2 = N rank(x)

33 F-test - multidimensional contrasts SPM{F} Tests multiple linear hypotheses: H 0 : 4 = 5 =... = 9 = 0 X 0 H 0 : True model is X 0 X0 test H 0 : c T = 0? X 1 ( 4-9 ) c T = SPM{F 6,322 } Full model? Reduced model?

34 F-contrast in SPM ˆ beta_???? images ( X X ) T 1 X T y ResMS image ˆ 2 T ˆ ˆ N p ess_???? images ( RSS 0 - RSS ) spmf_???? images SPM{F}

35 F-test example: movement related effects contrast(s) contrast(s) Design matrix Design matrix

36 F-test: summary F-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nested) model model comparison. F tests a weighted sum of squares of one or several combinations of the regression coefficients. In practice, we don t have to explicitly separate X into [X 1 X 2 ] thanks to multidimensional contrasts. Hypotheses: Null Hypothesis H0 : Alternativ e Hypothesis H : at least one In testing uni-dimensional contrast with an F-test, for example 1 2, the result will be the same as testing 2 1. It will be exactly the square of the t-test, testing for both positive and negative effects. A k 0

37 Orthogonal regressors Variability in Y

38 Correlated regressors Shared variance Variability in Y

39 Correlated regressors Variability in Y

40 Correlated regressors Variability in Y

41 Correlated regressors Variability in Y

42 Correlated regressors Variability in Y

43 Correlated regressors Variability in Y

44 Correlated regressors Variability in Y

45 Design orthogonality For each pair of columns of the design matrix, the orthogonality matrix depicts the magnitude of the cosine of the angle between them, with the range 0 to 1 mapped from white to black. If both vectors have zero mean then the cosine of the angle between the vectors is the same as the correlation between the two variates.

46 Topological Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course Lausanne, April 2012

47 Contrast c Random Field Theory Preprocessings General Linear Model Statistical Inference

48 Inference at a single voxel u Null Hypothesis H 0 : zero activation Decision rule (threshold) u: determines false positive rate α Choose u to give acceptable α under H 0 Null distribution of test statistic T

49 Multiple tests u u u u u u If we have 100,000 voxels, α=0.05 5,000 false positive voxels. t t t t t t This is clearly undesirable; to correct for this we can define a null hypothesis for a collection of tests. Noise Signal

50 Multiple tests u u u u u u If we have 100,000 voxels, α=0.05 5,000 false positive voxels. t t t t t t This is clearly undesirable; to correct for this we can define a null hypothesis for a collection of tests. Use of uncorrected p-value, α = % 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% Percentage of Null Pixels that are False Positives

51 Family-Wise Null Hypothesis Family-Wise Null Hypothesis: Activation is zero everywhere If we reject a voxel null hypothesis at any voxel, we reject the family-wise Null hypothesis A FP anywhere in the image gives a Family Wise Error (FWE) Family-Wise Error rate (FWER) = corrected p-value Use of uncorrected p-value, α =0.1 Use of corrected p-value, α =0.1 FWE

52 Bonferroni correction The Family-Wise Error rate (FWER), α FWE, for a family of N tests follows the inequality: where α is the test-wise error rate. Therefore, to ensure a particular FWER choose: This correction does not require the tests to be independent but becomes very stringent if dependence.

53 Spatial correlations 100 x 100 independent tests Spatially correlated tests (FWHM=10) Discrete data Spatially extended data Bonferroni is too conservative for spatial correlated data.

54 Topological inference Topological feature: Peak height u space significant local maxima non significant local maxima

55 Topological inference Topological feature: Cluster extent u clus significant cluster space non significant clusters

56 Topological inference Topological feature: Number of clusters u clus space Here, c=1, only one cluster larger than k.

57 RFT and Euler Characteristic Search volume Roughness (1/smoothness) Threshold

58 Expected Euler Characteristic

59 Random Field Theory The statistic image is assumed to be a good lattice representation of an underlying continuous stationary random field. Typically, FWHM > 3 voxels (combination of intrinsic and extrinsic smoothing) Smoothness of the data is unknown and estimated: very precise estimate by pooling over voxels stationarity assumptions (esp. relevant for cluster size results). RFT conservative for low degrees of freedom (always compare with Bonferroni correction). Afford littles power for group studies with small sample size. A priori hypothesis about where an activation should be, reduce search volume Small Volume Correction: mask defined by (probabilistic) anatomical atlases mask defined by separate "functional localisers" mask defined by orthogonal contrasts (spherical) search volume around previously reported coordinates

60 Conclusion There is a multiple testing problem and corrections have to be applied on p-values (for the volume of interest only (see SVC)). Inference is made about topological features (peak height, spatial extent, number of clusters). Use results from the Random Field Theory. Control of FWER (probability of a false positive anywhere in the image): very specific, not so sensitive. Control of FDR (expected proportion of false positives amongst those features declared positive (the discoveries)): less specific, more sensitive.

61 References Friston KJ, Frith CD, Liddle PF, Frackowiak RS. Comparing functional (PET) images: the assessment of significant change. J Cereb Blood Flow Metab Jul;11(4): Worsley KJ, Marrett S, Neelin P, Vandal AC, Friston KJ, Evans AC. A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping 1996;4: Chumbley J, Worsley KJ, Flandin G, and Friston KJ. Topological FDR for neuroimaging. NeuroImage, 49(4): , Chumbley J and Friston KJ. False Discovery Rate Revisited: FDR and Topological Inference Using Gaussian Random Fields. NeuroImage, Kilner J and Friston KJ. Topological inference for EEG and MEG data. Annals of Applied Statistics, in press.