Contrasts & Statistical Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course London, October 2008

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

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

Overview  A recapitulation of model specification and parameters estimation  Hypothesis testing  Contrasts and estimability  T-tests  F-tests  Design orthogonality  A first example  A second example

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.

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

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

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

One cannot accept the null hypothesis (one can just fail to reject it)  Absence of evidence is not evidence of absence.

Contrasts  A contrast select 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  =  1 c T = [ …] c T β = 0x  x  2 + 1x  3 + 0x  4 + 0x  =  3 -  2 c T = [ …]

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) 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], [ ] are estimable.  Example: parameters images Factor 1 2 Mean parameter estimability (gray   not uniquely specified)

c T = 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:

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

T-test: a simple example Q: activation during listening ? c T = [ 1 0 ] Null hypothesis:  Passive word listening versus rest SPMresults: Height threshold T = {p<0.001} Design matrix voxel-level p uncorrected T ( Z  ) mm mm mm Inf Inf Inf Inf Inf

T-test: a few remarks  T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate) =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:

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)

F-test - multidimensional contrasts – SPM{F}  Tests multiple linear hypotheses: 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 }

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

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.

Slightly more complicated… 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 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:

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.

Shared variance Independent contrasts

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

Shared variance correlated contrasts Testing for the red:

Shared variance Entirely correlated contrasts ?  Non estimable ! Testing for the green:

Shared variance If significant ? Could be G or Y ! Entirely correlated contrasts ? Non estimable ! Testing for the green and yellow

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) A “not so good” model

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) A “not so good” model

 t-test of the green regressor significant  F-test very significant  t-test of the red regressor very significant A “better” 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)

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 = (F-test) A “better” model

True signal Fit (blue : global fit) Residual Model (green and red) Correlation between regressors

 =+ 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 = (F-test) Correlation between regressors

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 Correlation between regressors

 =+ Y X  Residual var. = 0.3 p(Y|  1 = 0) p-value = (t-test) p(Y|  2 = 0) p-value = 0.07 (t-test) p(Y|  1 = 0,  2 = 0) p-value = (F-test) (0.79) (0.85) (0.06) Correlation between regressors  1 = 1.47  2 = 0.85  3 =

Bibliography:  Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 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,  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, 2003.