1. Statistical Inference Christophe Phillips SPM Course London, May 2012

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

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

4. Overview  Model specification and parameters estimation  Hypothesis testing  Contrasts  T-tests  F-tests  Correlation between regressors  Contrast estimability

5. Model Specification: The General Linear Model = + y y Sphericity assumption: Independent and identically distributed (i.i.d.) error terms 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.

6. Parameter Estimation: Ordinary Least Squares  Find that minimises  The Ordinary Least Estimates are:  Under i.i.d. assumptions, the Ordinary Least Squares estimates are Maximum Likelihood.

7. Hypothesis Testing  The Null Hypothesis H 0 Typically what we want to disprove (no effect).  The Alternative Hypothesis H A expresses outcome of interest. To test an hypothesis, we construct “test statistics”.  The 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

8. Hypothesis Testing  P-value: A p-value summarises evidence against H 0. This is the chance of observing value more extreme than t under the null hypothesis. Observation of test statistic t, a realisation of T Null Distribution of T  Significance level α: Acceptable false positive rate α.  threshold u α Threshold u α controls the false positive rate t P-val Null Distribution of T  uu  The conclusion about the hypothesis: We reject the null hypothesis in favour of the alternative hypothesis if t > u α

9. Contrasts  A contrast selects a specific effect of interest:  a contrast c is a vector of length p.  c T β is a linear combination of regression coefficients β.  Under i.i.d assumptions:  We are usually not interested in the whole β vector. c T β = 1x  1 + 0x  2 + 0x  3 + 0x  4 + 0x  5 +... c T = [1 0 0 0 0 …] c T β = 0x  1 + -1x  2 + 1x  3 + 0x  4 + 0x  5 +... c T = [0 -1 1 0 0 …]

10. c T = 1 0 0 0 0 0 0 0 T = contrast of estimated parameters variance estimate box-car amplitude > 0 ? =  1 = c T  > 0 ?  1  2  3  4  5... T-test - one dimensional contrasts – SPM{t} Question: Null hypothesis: H 0 : c T  =0 Test statistic:

11. T-test: a simple example Q: activation during listening ? c T = [ 1 0 ] Null hypothesis:  Passive word listening versus rest SPMresults: Height threshold T = 3.2057 {p<0.001} Design matrix 0.511.522.5 10 20 30 40 50 60 70 80 voxel-level p uncorrected T ( Z  ) mm mm mm 13.94 Inf0.000-63 -27 15 12.04 Inf0.000-48 -33 12 11.82 Inf0.000-66 -21 6 13.72 Inf0.000 57 -21 12 12.29 Inf0.000 63 -12 -3 9.89 7.830.000 57 -39 6 7.39 6.360.000 36 -30 -15 6.84 5.990.000 51 0 48 6.36 5.650.000-63 -54 -3 6.19 5.530.000-30 -33 -18 5.96 5.360.000 36 -27 9 5.84 5.270.000-45 42 9 5.44 4.970.000 48 27 24 5.32 4.870.000 36 -27 42 1

12. T-contrast in SPM ResMS image con_???? image beta_???? images spmT_???? image SPM{t}  For a given contrast c:

13. 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. H0:H0:vs H A :  Unilateral test:

14. Scaling issue  The T-statistic does not depend on the scaling of the regressors. [1 1 1 1 ] [1 1 1 ]  Be careful of the interpretation of the contrasts themselves (eg, for a second level analysis): sum ≠ average  The T-statistic does not depend on the scaling of the contrast. / 4 / 3 Subject 1 Subject 5  Contrast depends on scaling.

15. F-test - the extra-sum-of-squares principle  Model comparison: Null Hypothesis H 0 : True model is X 0 (reduced model) Full model ? X1X1 X0X0 or Reduced model? X0X0 Test statistic: ratio of explained variability and unexplained variability (error) 1 = rank(X) – rank(X 0 ) 2 = N – rank(X) RSS RSS 0

16. F-test - multidimensional contrasts – SPM{F}  Tests multiple linear hypotheses: 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 c T = H 0 :  4 =  5 =  =  9 = 0 X 1 (  4-9 ) X0X0 Full model?Reduced model? H 0 : True model is X 0 X0X0 test H 0 : c T  = 0 ? SPM{F 6,322 }

17. F-contrast in SPM ResMS image beta_???? images spmF_???? images SPM{F} ess_???? images ( RSS 0 - RSS )

18. F-test example: movement related effects

19. 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).

20. F-test: a few remarks nested Model comparison.  F-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nested) model  Model comparison.  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. sum of squares  F tests a weighted sum of squares of one or several combinations of the regression coefficients . multidimensional contrasts  In practice, we don’t have to explicitly separate X into [X 1 X 2 ] thanks to multidimensional contrasts.  Hypotheses:

21. Shared variance Orthogonal regressors.

22. Shared variance Correlated regressors, for example: green: subject age yellow: subject score Testing for the green:

23. Shared variance Correlated regressors. Testing for the red:

24. Shared variance Highly correlated green & yellow regressors. Testing for the green: NOTE: Entirely correlated  non estimable

25. Shared variance GY If significant, can be G and/or Y Testing for the green and yellow

26. 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:  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.

27. Correlated regressors  Orthogonal regressors (=uncorrelated): By varying each separately, one can predict the combined effect of varying them jointly.  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. x1x1 x2x2 It does not reduce the predictive power or reliability of the model as a whole. x1x1 x2x2 x1x1 x2x2 x1x1 x2x2 xx xx 2 1 2 x  = x 2 – x 1.x 2 x 1

28. Estimability of a contrast  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, X T X is not invertible so we must resort to generalised inverses (SPM uses the pseudo-inverse). 1 0 1 0 1 1 One-way ANOVA (unpaired two-sample t-test) Rank(X)=2 [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.  Example: parameters images Factor 1 2 Mean parameter estimability (gray   not uniquely specified)

29. Three models for the two-samples t-test 1 0 1 1 0 0 1 1 0 1 0 1 1 β 1 =y 1 β 2 =y 2 β 1 +β 2 =y 1 β 2 =y 2 [1 0].β = y 1 [0 1].β = y 2 [1 -1].β = y 1 -y 2 [.5.5].β = mean(y 1,y 2 ) [1 1].β = y 1 [0 1].β = y 2 [1 0].β = y 1 -y 2 [.5 1].β = mean(y 1,y 2 ) β 1 +β 3 =y 1 β 2 +β 3 =y 2 [1 0 1].β = y 1 [0 1 1].β = y 2 [1 -1 0].β = y 1 -y 2 [.5.5 1].β = mean(y 1,y 2 )

30. 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

31. Bibliography:  Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 2007. With many thanks to J.-B. Poline, G. Flandin, T. Nichols, S. Kiebel, R. Henson for slides.  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.

33. Design efficiency  The aim is to minimize the standard error of a t-contrast (i.e. the denominator of a t-statistic).  This is equivalent to maximizing the efficiency e: Noise variance Design variance  If we assume that the noise variance is independent of the specific design:  This is a relative measure: all we can really say is that one design is more efficient than another (for a given contrast).

34. 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).  X T X represents covariance of regressors in design matrix; high covariance increases elements of (X T X) -1.  High correlation between regressors leads to low sensitivity to each regressor alone. c T =[1 0]: 5.26 c T =[1 1]: 20 c T =[1 -1]: 1.05