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Statistical Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course Zurich, February 2013 Image time-series Realignment Spatial filter Design matrix Smoothing General Linear Model Statistical Parametric Map Statistical Inference Normalisation Anatomical reference Parameter estimates RFT p <0.05 A mass-univariate approach Estimation of the parameters i.i.d. assumptions: OLS estimates: ~(0, 2 ) = ( )−1 1 = 3.9831 2−7 = {0.6871, 1.9598, 1.3902, 166.1007, 76.4770, −64.8189} = + 8 = 131.0040 = ~ , 2 ( )−1 2 = − Contrasts A contrast selects a specific effect of interest. A contrast is a vector of length . [1 10 -1 0 00 00 00 00 00 00 00 00 00 00 0] [0 0] is a linear combination of regression coefficients . = [1 0 0 0 … ] = × 1 + × 2 + × 3 + × 4 + ⋯ = = [0 1 − 1 0 … ] = × 1 + × 2 + − × 3 + × 4 + ⋯ = − ~ , 2 ( )−1 Hypothesis Testing To test an hypothesis, we construct “test statistics”. Null Hypothesis H0 Typically what we want to disprove (no effect). The Alternative Hypothesis HA expresses outcome of interest. Test Statistic T The test statistic summarises evidence about H0. Typically, test statistic is small in magnitude when the hypothesis H0 is true and large when false. We need to know the distribution of T under the null hypothesis. Null Distribution of T Hypothesis Testing u Significance level α: Acceptable false positive rate α. threshold uα Threshold uα controls the false positive rate p(T u | H0 ) Null Distribution of T Conclusion about the hypothesis: We reject the null hypothesis in favour of the alternative hypothesis if t > uα p-value: A p-value summarises evidence against H0. This is the chance of observing value more extreme than t under the null hypothesis. > |0 t p-value Null Distribution of T T-test - one dimensional contrasts – SPM{t} cT =10000000 b1 b2 b3 b4 b5 ... Question: box-car amplitude > 0 ? = b1 = c T b> 0 ? H0: cTb=0 Null hypothesis: contrast of estimated parameters T= Test statistic: T cT bˆ var(cT bˆ ) variance estimate cT bˆ ˆ 2cT X T X c 1 ~ tN p T-contrast in SPM For a given contrast c: ResMS image beta_???? images bˆ ( X T X ) 1 X T y con_???? image c T bˆ T ˆ ˆ 2 ˆ Np spmT_???? image SPM{t} T-test: a simple example Passive word listening versus rest Q: activation during listening ? 1 cT = [ 1 0 0 0 0 0 0 0] Null hypothesis: b1 = var 0 SPMresults: Height threshold T = 3.2057 {p<0.001} voxel-level mm mm mm ( Z) T p uncorrected 13.94 12.04 11.82 13.72 12.29 9.89 7.39 6.84 6.36 6.19 5.96 5.84 5.44 5.32 Inf Inf Inf Inf Inf 7.83 6.36 5.99 5.65 5.53 5.36 5.27 4.97 4.87 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -63 -48 -66 57 63 57 36 51 -63 -30 36 -45 48 36 -27 15 -33 12 -21 6 -21 12 -12 -3 -39 6 -30 -15 0 48 -54 -3 -33 -18 -27 9 42 9 27 24 -27 42 T-test: summary T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate). Alternative hypothesis: H0: cT b 0 vs HA: cT b 0 T-contrasts are simple combinations of the betas; the Tstatistic does not depend on the scaling of the regressors or the scaling of the contrast. Scaling issue [1 1 1 1 ]/ 4 T cT bˆ var(c bˆ ) Subject 1 T cT bˆ ˆ c X X c 2 T T 1 The T-statistic does not depend on the scaling of the regressors. The T-statistic does not depend on the scaling of the contrast. Subject 5 [1 1 1 ]/ 3 Contrast cT bˆ depends on scaling. Be careful of the interpretation of the contrasts cT bˆ themselves (eg, for a second level analysis): sum ≠ average F-test - the extra-sum-of-squares principle Model comparison: Null Hypothesis H0: True model is X0 (reduced model) X0 X0 X1 RSS 2 ˆ full Full model ? Test statistic: ratio of explained variability and unexplained variability (error) RSS0 2 ˆ reduced or Reduced model? 1 = rank(X) – rank(X0) 2 = N – rank(X) F-test - multidimensional contrasts – SPM{F} Tests multiple linear hypotheses: H0: True model is X0 X0 X1 (b4-9) H0: b4 = b5 = ... = b9 = 0 X0 cT = test H0 : cTb = 0 ? 000100000 000010000 000001000 000000100 000000010 000000001 SPM{F6,322} Full model? Reduced model? F-contrast in SPM ResMS image beta_???? images bˆ ( X T X ) 1 X T y T ˆ ˆ 2 ˆ Np ess_???? images spmF_???? images ( RSS0 - RSS ) SPM{F} F-test example: movement related effects contrast(s) contrast(s) 10 20 30 40 50 60 70 80 10 20 2 30 40 50 60 70 80 2 4 Design matrix 6 8 4 6 8 Design matrix 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 b. In practice, we don’t have to explicitly separate X into [X1X2] thanks to multidimensional contrasts. Hypotheses: 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 Null HypothesisH0 : b1 b2 b3 0 Alternative HypothesisH A : at least one bk 0 In testing uni-dimensional contrast with an F-test, for example 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. Orthogonal regressors Variability described by 1 Testing for 1 Variability described by 2 Testing for 2 Variability in Y Correlated regressors Variability described by Variability described by Shared variance Variability in Y Correlated regressors Variability described by Variability described by Testing for 1 Variability in Y Correlated regressors Variability described by Variability described by Testing for 2 Variability in Y Variability described by Variability described by Correlated regressors Variability in Y Correlated regressors Variability described by Variability described by Testing for 1 Variability in Y Correlated regressors Variability described by Variability described by Testing for 2 Variability in Y Correlated regressors Variability described by Variability described by Testing for 1 and/or 2 Variability in Y 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. Correlated regressors: summary We implicitly test for an additional effect only. When testing for the first regressor, we are effectively removing the part of the signal that can be accounted for by the second regressor: implicit orthogonalisation. x^2 x2 x2 x1 x1 x^ 1 x2 x^2 = x2 – x1.x2 x1 x1 Orthogonalisation = decorrelation. Parameters and test on the non modified regressor change. Rarely solves the problem as it requires assumptions about which regressor to uniquely attribute the common variance. change regressors (i.e. design) instead, e.g. factorial designs. use F-tests to assess overall significance. 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 Design efficiency The aim is to minimize the standard error of a t-contrast T (i.e. the denominator of a t-statistic). var(cT bˆ ) ˆ 2cT ( X T X )1 c cT bˆ var(cT bˆ ) This is equivalent to maximizing the efficiency e: e(ˆ 2 , c, X ) (ˆ 2cT ( X T X )1 c)1 Noise variance Design variance If we assume that the noise variance is independent of the specific design: 1 e(c, X ) (c ( X X ) c) T T 1 This is a relative measure: all we can really say is that one design is more efficient than another (for a given contrast). Design efficiency A B = 1 −0.9 −0.9 1 = [1 0] : , = 18.1 = [0.5 0.5] : , = 19.0 = [1 − 1] : , = 95.2 A+B A-B High correlation between regressors leads to low sensitivity to each regressor alone. We can still estimate efficiently the difference between them. Bibliography: Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 2007. Plane Answers to Complex Questions: The Theory of Linear Models. R. Christensen, Springer, 1996. Statistical parametric maps in functional imaging: a general linear approach. K.J. Friston et al, Human Brain Mapping, 1995. Ambiguous results in functional neuroimaging data analysis due to covariate correlation. A. Andrade et al., NeuroImage, 1999. Estimating efficiency a priori: a comparison of blocked and randomized designs. A. Mechelli et al., NeuroImage, 2003. Example: Mean Factor 2 One-way ANOVA (unpaired two-sample t-test) images If X is not of full rank then we can have Xb1 = Xb2 with b1≠ b2 (different parameters). The parameters are not therefore ‘unique’, ‘identifiable’ or ‘estimable’. For such models, XTX is not invertible so we must resort to generalised inverses (SPM uses the pseudo-inverse). Factor 1 Estimability of a contrast 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1parameters Rank(X)=2 parameter estimability (gray b not uniquely specified) [1 0 0], [0 1 0], [0 0 1] are not estimable. [1 0 1], [0 1 1], [1 -1 0], [0.5 0.5 1] are estimable. Three models for the two-samples t-test 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 β1=y1 β2=y2 1 1 1 1 1 1 1 1 [1 1].β = y1 β1+β2=y1 [0 1].β = y2 β2=y2 [1 0].β = y1-y2 [.5 1].β = mean(y1,y2) [1 0].β = y1 [0 1].β = y2 [0 -1].β = y1-y2 [.5 .5].β = mean(y1,y2) 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 β1+β3=y1 β2+β3=y2 [1 0 1].β = y1 [0 1 1].β = y2 [1 -1 0].β = y1-y2 [.5 0.5 1].β = mean(y1,y2) Multidimensional contrasts 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 same exact way (same error, same fitted data). Example: working memory A Stimulus B Stimulus Response Stimulus Response Response Time (s) Time (s) Time (s) Correlation = -.65 Efficiency ([1 0]) = 29 C Correlation = +.33 Efficiency ([1 0]) = 40 Correlation = -.24 Efficiency ([1 0]) = 47 B: Jittering time between stimuli and response. C: Requiring a response on a randomly half of trials.