Report

Statistical Inference Christophe Phillips SPM Course London, May 2012 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 Voxel-wise time series analysis Model specification Time Parameter estimation Hypothesis Statistic BOLD signal single voxel time series SPM Overview Model specification and parameters estimation Hypothesis testing Contrasts T-tests F-tests Correlation between regressors Contrast estimability Model Specification: The General Linear Model 1 p 1 1 y X y N = X N p + N Sphericity assumption: Independent and identically distributed (i.i.d.) error terms ~ N (0, I ) 2 N: number of scans, p: number of regressors The General Linear Model is an equation that expresses the observed response variable in terms of a linear combination of explanatory variables X plus a well behaved error term. Each column of the design matrix corresponds to an effect one has built into the experiment or that may confound the results. Parameter Estimation: Ordinary Least Squares Find ˆ that minimises y X 2 T The Ordinary Least Estimates are: ˆ ( X T X ) 1 X T y Under i.i.d. assumptions, the Ordinary Least Squares estimates are Maximum Likelihood. ~ N (0, 2 I ) Y ~ N ( X , 2 I ) T ˆ ˆ 2 ˆ Np 2 T 1 ˆ ~ N ( , ( X X ) ) Hypothesis Testing To test an hypothesis, we construct “test statistics”. The Null Hypothesis H0 Typically what we want to disprove (no effect). The Alternative Hypothesis HA expresses outcome of interest. The 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 ) Observation of test statistic t, a realisation of T The 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. p(T t | H0 ) Null Distribution of T t P-val Null Distribution of T Contrasts We are usually not interested in the whole β vector. A contrast selects a specific effect of interest: a contrast c is a vector of length p. cTβ is a linear combination of regression coefficients β. cT = [1 0 0 0 0 …] cTβ = 1x1 + 0x2 + 0x3 + 0x4 + 0x5 + . . . cT = [0 -1 1 0 0 …] cTβ = 0x1 + -1x2 + 1x3 + 0x4 + 0x5 + . . . Under i.i.d assumptions: T 2 T T 1 ˆ c ~ N (c , c ( X X ) c) T T-test - one dimensional contrasts – SPM{t} cT =10000000 1 2 3 4 5 ... Question: box-car amplitude > 0 ? = 1 = c T > 0 ? H0: cT=0 Null hypothesis: contrast of estimated parameters T= Test statistic: T cT ˆ var(cT ˆ ) variance estimate cT ˆ ˆ 2cT X T X c 1 ~ tN p T-test: a simple example Passive word listening versus rest cT = [ 1 Q: activation during listening ? 1 0 ] 10 Null hypothesis: 1 X 20 30 0 Statistics: 40 c ˆ t T ˆ Std (c ) T 50 60 70 80 0.5 1 1.5 Design matrix 2 2.5 set-level p c 0.000 10 SPMresults: Height threshold T = 3.2057 {p<0.001} voxel-level mm mm mm ( Z) T p uncorrected p-values adjusted for search volume 13.94 Inf 0.000 voxel-level 12.04 Infp 0.000 FWE-corr p FDR-corr T 11.82 Inf 0.000 0.000 520 0.000 0.000 13.94 13.72 Inf0.000 0.000 0.000 0.000 12.04 0.0000.000 0.000 11.82 12.29 Inf 0.000 426 0.000 0.000 13.72 9.89 7.830.000 0.000 0.000 0.000 12.29 7.39 6.360.0000.000 0.000 9.89 0.000 35 0.000 0.000 0.000 7.39 6.84 5.99 0.000 0.000 9 0.000 0.000 0.000 6.84 0.002 6.36 3 0.024 0.001 0.000 5.65 0.000 6.36 0.000 8 0.001 0.001 0.000 6.19 5.530.0030.000 0.000 6.19 9 0.000 0.000 5.96 0.005 5.96 2 0.058 0.000 5.84 5.360.0040.000 0.015 1 0.166 0.000 5.44 5.270.022 0.015 5.84 1 0.166 0.0360.000 0.000 5.32 5.44 4.97 0.000 5.32 4.87 0.000 cluster-level p corrected k E p uncorrected -63 -27 15 mm mm -48p -33 mm12 (Z ) uncorrected -66 -21 6 Inf 57 0.000 -21 -63 12-27 15 Inf 0.000 -48 -33 12 Inf 63 0.000 -12 -66 -3-21 6 Inf 0.000 57 -21 12 57 -39 Inf 0.000 63 6 -12 -3 36 0.000 -30 -15 7.83 57 -39 6 6.36 0.000 36 -30 -15 51 0 48 5.99 0.000 51 0 48 5.65 0.000 -63 -54 -63-3-54 -3 5.53 0.000 -30 -33 -18 -30 -33 -18 5.36 0.000 36 -27 9 5.27 36 0.000 -27 -45 942 9 4.97 0.000 27 24 -45 42 48 4.87 0.000 36 9 -27 42 48 27 24 36 -27 42 T-contrast in SPM For a given contrast c: ResMS image beta_???? images ˆ ( X T X ) 1 X T y con_???? image cT ˆ T ˆ ˆ 2 ˆ Np spmT_???? image SPM{t} T-test: a few remarks T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate). 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. Unilateral test: H0: cT 0 vs HA: cT 0 Scaling issue [1 1 1 1 ]/ 4 T cT ˆ var(c ˆ ) Subject 1 T cT ˆ 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 ˆ depends on scaling. Be careful of the interpretation of the contrasts cT ˆ 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 (4-9) H0: 4 = 5 = ... = 9 = 0 X0 cT = test H0 : cT = 0 ? 000100000 000010000 000001000 000000100 000000010 000000001 SPM{F6,322} Full model? Reduced model? F-contrast in SPM ResMS image beta_???? images ˆ ( 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 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). F-test: a few remarks 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 [X1X2] thanks to multidimensional contrasts. Hypotheses: 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 Null HypothesisH0 : 1 2 3 0 Alternative HypothesisH A : at least one k 0 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. Shared variance Orthogonal regressors. Shared variance Testing for the green: Correlated regressors, for example: green: subject age yellow: subject score Shared variance Testing for the red: Correlated regressors. Shared variance Testing for the green: Highly correlated green & yellow regressors. NOTE: Entirely correlated non estimable Shared variance Testing for the green and yellow If significant, can be G and/or 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. The cosine of the angle between two vectors a and b is obtained by: cos a b a b 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 Orthogonal regressors (=uncorrelated): By varying each separately, one can predict the combined effect of varying them jointly. x 2 x1 Non-orthogonal regressors (=correlated): 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 x1^ x2 x^2 = x2 – x1.x2 x1 x1 It does not reduce the predictive power or reliability of the model as a whole. Example: Mean Factor 2 One-way ANOVA (unpaired two-sample t-test) images If X is not of full rank then we can have X(1) = X(2) with (1)≠ (2) (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 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 1 1 1 1 1 1 1 1 β1=y1 β2=y2 [1 0].β = y1 [0 1].β = y2 [1 -1].β = y1-y2 [.5 .5].β = mean(y1,y2) [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 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 .5 1].β = mean(y1,y2) A few remarks We implicitly test for an additional effect only, be careful if there is correlation - Orthogonalisation = decorrelation : not generally needed - Parameters and test on the non modified regressor change It is always simpler to have orthogonal regressors and therefore designs. In case of correlation, use F-tests to see the overall significance. There is generally no way to decide to which regressor the « common » part should be attributed to. 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 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. With many thanks to J.-B. Poline, G. Flandin, T. Nichols, S. Kiebel, R. Henson for slides. 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 ˆ ) ˆ 2cT ( X T X )1 c cT ˆ var(cT ˆ ) 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 The efficiency of an estimator is a measure of how reliable it is and depends on error variance (the variance not modeled by explanatory variables in the design matrix) and the design variance (a function of the explanatory variables and the contrast tested). XTX represents covariance of regressors in design matrix; high covariance increases elements of (XTX)-1. High correlation between regressors leads to low sensitivity to each regressor alone. T T 1 c (X X ) c 1 0.9 0.9 1 cT=[1 0]: 5.26 cT=[1 1]: 20 cT=[1 -1]: 1.05