Statistical Inference

Transcription

1 Statistical Inference J. Daunizeau Institute of Empirical Research in Economics, Zurich, Switzerland Brain and Spine Institute, Paris, France SPM Course Edinburgh, April 2011 Image time-series Spatial filter Design matrix Statistical Parametric Map Realignment Smoothing General Linear Model Normalisation Statistical Inference RF Anatomical reference Parameter estimates p <0.05 Voxel-wise time series analysis ime ime Model specification Parameter estimation Hypothesis Statistics single voxel time series BOLD signal SPM

2 Overview -tests Overview -tests he GLM 1 p 1 1 β y = Xβ + ε y = X p + ε Sphericity assumption: Independent and identically distributed (i.i.d.) error terms N N N N: number of scans, p: number of regressors ε ~ N(0, σ 2 I ) he 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.

3 Parameter estimation: OLS 2 Find βˆ that minimises y Xβ = ε ε he Ordinary Least Estimates are: ˆ β = ( X X ) 1 Under i.i.d. assumptions, the Ordinary Least Squares estimates are Maximum Likelihood. ε ~ N(0, σ 2 I) Y ~ N( Xβ, σ 2 I) X y ˆ ε ˆ ε ˆ σ 2 = N p ˆ 2 1 β ~ N( β, σ ( X X) ) Overview -tests Hypothesis testing o test an hypothesis, we construct test statistics. he Null Hypothesis H 0 ypically what we want to disprove (no effect). he Alternative Hypothesis H A expresses outcome of interest. he est Statistic he test statistic summarises evidence about H 0. ypically, test statistic is small in magnitude when the hypothesis H 0 is true and large when false. We need to know the distribution of under the null hypothesis. Null Distribution of

4 Significance level and p-value Significance level α: Acceptable false positive rate α. threshold u α hreshold u α controls the false positive rate α = p > u α H ) ( 0 Observation of test statistic t, a realisation of he conclusion about the hypothesis: We reject the null hypothesis in favour of the alternative hypothesis if t > u α P-value: A p-value summarises evidence against H 0. his is the chance of observing value more extreme than t under the null hypothesis. p ( > t H 0) u α α Null Distribution of t P-val Null Distribution of ype I and II errors Neyman-Pearson lemma: the likelihood ratio... ( ) p Y H1 Λ = u p ( Y H0 )...is the most powerful test of size (FPR) ( Λ ) α = p u H 0 Increasing the FPR decreases power ype I error is more serious than type II error We choose to keep the type I error low (5%) Overview -tests

5 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. c β is a linear combination of regression coefficients β. c = [ ] c β = 1xβ 1 + 0xβ 2 + 0xβ 3 + 0xβ 4 + 0xβ c = [ ] c β = 0xβ xβ 2 + 1xβ 3 + 0xβ 4 + 0xβ Under i.i.d assumptions: ˆ 1 2 c β ~ N( c β, σ c ( X X) c) -test: one dimensional contrasts c = Question: box-car amplitude > 0? = β 1 = c β> 0? β 1 β 2 β 3 β 4 β 5... Null hypothesis: H 0 : c β=0 est statistic: = contrast of estimated parameters variance estimate c ˆ β c ˆ β = = ~ t 2 var( c ˆ) β ˆ σ c c 1 ( X X ) N p -contrast in SPM For a given contrast c: beta_???? images ˆ 1 β = ( X X ) X y ResMS image ˆ ε ˆ ε ˆ σ 2 = N p con_???? image c βˆ spm_???? image SPM{t}

6 1 -test: a simple example Passive word listening versus rest Q: activation during listening? H0: β 1 = X Design matrix c = [ 1 0 ] SPMresults: Height threshold = {p<0.001} voxel-level ( Z ) p uncorrected mm mm mm Inf cluster-level voxel-level Inf p (Z ) -66 corrected k E p uncorrected p FWE-corr p Inf FDR-corr p uncorrected Inf Inf Statistics: p-values adjusted for search volume set-level p c mm mm mm Inf Inf Inf Inf Inf test: a few remarks -test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate). -contrasts are simple combinations of the betas; the - statistic does not depend on the scaling of the regressors or the scaling of the contrast. Unilateral test: H 0 : c β = 0 vs H A : c β > 0 Scaling issue Subject 1 Subject 5 [ ] / 4 [1 1 1 ] / 3 = c ˆ β = var( c ˆ) β 2 ˆ σ c c ˆ β 1 ( X X ) c he -statistic does not depend on the scaling of the regressors. he -statistic does not depend on the scaling of the contrast. Contrast c βˆ depends on scaling. Be careful of the interpretation of the contrasts c βˆ themselves (eg, for a second level analysis): sum average

7 F-test: the extra sum-of-squares principle Model comparison: Full vs. Reduced model? Null Hypothesis H 0 : rue model is X 0 (reduced model) X 0 X 1 X 0 est statistic: ratio of explained variability and unexplained variability (error) Full model? RSS 2 εˆ full RSS 0 ˆreduced ε 2 Or Reduced model? F RSS RSS RSS 0 ESS F RSS ~ F ν 1, ν 2 ν 1 = rank(x) rank(x 0 ) ν 2 = N rank(x) F-test: multidimensional contrats ests multiple linear hypotheses: H 0 : rue model is X 0 H 0 : β 3 = β 4 =... = β 9 = 0 test H 0 : c β = 0? X 0 X 1 (β 3-9 ) X c = SPM{F 6,322 } Full model? Reduced model? F-contrast in SPM beta_???? images ˆ 1 β = ( X X ) X y ResMS image ˆ ε ˆ ε ˆ σ 2 = N p ess_???? images ( RSS 0 - RSS ) spmf_???? images SPM{F}

8 F-test example: movement related effects Multidimensional contrasts hink 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 β. In practice, we don t have to explicitly separate X into [X 1 X 2 ] thanks to multidimensional contrasts. Hypotheses: Null Hypothesis H 0 : β1 = β 2 = β 3 = 0 Alternative Hypothesis H : at least one β 0 In testing uni-dimensional contrast with an F-test, for example β 1 β 2, the result will be the same as testing β 2 β 1. It will be exactly the square of the t-test, testing for both positive and negative effects. A k

9 Overview -tests Estimability of a contrast If X is not of full rank then we can have Xβ 1 = Xβ 2 with β 1 β 2 (different parameters). he parameters are not therefore unique, identifiable or estimable. For such models, X X is not invertible so we must resort to generalised inverses (SPM uses the pseudo-inverse). Example: One-way ANOVA (unpaired two-sample t-test) parameters Rank(X)=2 parameter estimability (gray β not uniquely specified) [1 0 0], [0 1 0], [0 0 1] are not estimable. [1 0 1], [0 1 1], [1-1 0], [ ] are estimable. images Factor 1 Factor 2 Mean Overview -tests

10 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. he cosine of the angle between two vectors a and b is obtained by: cos α = 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. Multicollinearity Contrast covariance matrix: 2 1 Var( c ˆ) β =σ c ( X X) c Orthogonal regressors (=uncorrelated): By varying each separately, one can predict the combined effect of varying them jointly. x 2 1 ( X X) is diagonal x 1 x 2 x 1 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 x 2 x 2 x 2 x 1 x = x 2 2 x 1.x 2 x 1 x 1 x 1 x 1 It does not reduce the predictive power or reliability of the model as a whole. Shared variance Orthogonal regressors.

11 Shared variance esting for the green: Correlated regressors, for example: green: subject age yellow: subject score Shared variance esting for the red: Correlated regressors. Shared variance esting for the green: Highly correlated. Entirely correlated non estimable

12 Shared variance esting for the green and yellow If significant, can be G and/or Y Examples 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. here 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 Overview -tests

13 Design eficiency he aim is to minimize the standard error of a t-contrast (i.e. the denominator of a t-statistic). 2 1 var( c ˆ) β = ˆ σ c ( X X ) c = c ˆ β var( c ˆ) β his is equivalent to maximizing the efficiency e: 2 e( ˆ σ, c, X ) ˆ 2 1 = ( σ c ( X X ) c ) 1 Noise variance Design variance If we assume that the noise variance is independent of the specific design: 1 1 e( c, X ) = ( c ( X X ) c) his is a relative measure: all we can really say is that one design is more efficient than another (for a given contrast). Design efficiency he 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). X X represents covariance of regressors in design matrix; high covariance increases elements of (X X) -1. High correlation between regressors leads to low sensitivity to each regressor alone. c ( X X ) 1 c c =[1 0]: 5.26 c =[1 1]: 20 c =[1-1]: 1.05 Bibliography Statistical Parametric Mapping: he Analysis of Functional Brain Images. Elsevier, Plane Answers to Complex Questions: he heory of Linear Models. R. Christensen, Springer, Statistical parametric maps in functional imaging: a general linear approach. K.J. Friston et al, Human Brain Mapping, Ambiguous results in functional neuroimaging data analysis due to covariate correlation. A. Andrade et al., NeuroImage, Estimating efficiency a priori: a comparison of blocked and randomized designs. A. Mechelli et al., NeuroImage, With many thanks to G. Flandin, J.-B. Poline and om Nichols for slides.