1. Statistical Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course Zurich, February 2009

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  Contrast estimability  Correlation between regressors  Example(s)  Design efficiency

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 minimize  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  Type I Error α : Acceptable false positive rate α. Level  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. One cannot accept the null hypothesis (one can just fail to reject it)  Absence of evidence is not evidence of absence. If we do not reject H 0, then all that we are saying is that there is no strong evidence in the data to reject that H 0 might be a reasonable approximation to the truth. This certainly does not mean that there is a strong evidence to accept H 0.

10. 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 …]

11. 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:

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

13. 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 1 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

14. T-test: a few remarks  T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate). -5-4-3-2012345 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 =1 =2 =5 =10 =  Probability Density Function of Student’s t distribution  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:

15. Scaling issue  The T-statistic does not depend on the scaling of the regressors. [1 1 1 1 ] [1 1 1 ]  However 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

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

17. F-test - multidimensional contrasts – SPM{F}  Tests multiple linear hypotheses: 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 c T = H 0 :  3 =  4 =  =  9 = 0 X 1 (  3-9 ) X0X0 Full model?Reduced model? H 0 : True model is X 0 X0X0 test H 0 : c T  = 0 ? SPM{F 6,322 }

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

19. F-test example: movement related effects  To assess movement- related activation:  There is a lot of residual movement-related artifact in the data (despite spatial realignment), which tends to be concentrated near the boundaries of tissue types.  Even though we are not interested in such artifact, by including the realignment parameters in our design matrix, we “covary out” linear components of subject movement, reducing the residual error, and hence improve our statistics for the effects of interest.

20. 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).

21. 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:

22. 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)

23. 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.

24. Multicollinearity  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 by the second regressor  implicit orthogonalisation. x1x1 x2x2 x1x1 x2x2 is diagonal 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

25. Shared variance Orthogonal regressors.

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

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

28. Shared variance Highly correlated. Entirely correlated  non estimable Testing for the green:

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

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. 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).

32. 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

33. Bibliography:  Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 2007. With many thanks to J.-B. Poline, Tom 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.  Estimating efficiency a priori: a comparison of blocked and randomized designs. A. Mechelli et al., NeuroImage, 2003.

34. True signal and observed signal (--) Model (green, pic at 6sec) TRUE signal (blue, pic at 3sec) Fitting (  1 = 0.2, mean = 0.11)  Test for the green regressor not significant Residual (still contains some signal) Example I: a first model

35. e =+ Y X   1 = 0.22  2 = 0.11 Residual Var.= 0.3 p(Y|  1 = 0)  p-value = 0.1 (t-test) p(Y|  1 = 0)  p-value = 0.2 (F-test) Example I: a first model

36.  t-test of the green regressor significant  F-test very significant  t-test of the red regressor very significant Example I: an “improved” model True signal + observed signal Model (green and red) and true signal (blue ---) Red regressor : temporal derivative of the green regressor Global fit (blue) and partial fit (green & red) Adjusted and fitted signal Residual (a smaller variance)

37. e =+ Y X   1 = 0.22  2 = 2.15  3 = 0.11 Residual Var. = 0.2 p(Y|  1 = 0)  p-value = 0.07 (t-test) p(Y|  1 = 0,  2 = 0)  p-value = 0.000001 (F-test) Example I: an “improved” model Conclusion

38. True signal Fit (blue : global fit) Residual Model (green and red) Example II: correlation between regressors

39.  =+ Y X   1 = 0.79  2 = 0.85  3 = 0.06 Residual var. = 0.3 p(Y|  1 = 0)  p-value = 0.08 (t-test) P(Y|  2 = 0)  p-value = 0.07 (t-test) p(Y|  1 = 0,  2 = 0)  p-value = 0.002 (F-test) Example II: correlation between regressors 12 1 2

40. True signal Residual Fit Model (green and red) red regressor has been orthogonalised with respect to the green one  remove everything that correlates with the green regressor Example II: correlation between regressors

41.  =+ Y X  Residual var. = 0.3 p(Y|  1 = 0) p-value = 0.0003 (t-test) p(Y|  2 = 0) p-value = 0.07 (t-test) p(Y|  1 = 0,  2 = 0) p-value = 0.002 (F-test) (0.79) (0.85) (0.06) Example II: correlation between regressors  1 = 1.47  2 = 0.85  3 = 0.06 12 1 2 Conclusion