SPM Course Zurich, February 2012 Statistical Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London

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

A mass-univariate approach Time

Estimation of the parameters i.i.d. assumptions: OLS estimates:

Contrasts [ ] [ ]

Hypothesis Testing 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. 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. 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-value Null Distribution of T u Conclusion about the hypothesis: We reject the null hypothesis in favour of the alternative hypothesis if t > u α

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

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

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

T-test: summary 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 : Alternative hypothesis:

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

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

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

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

F-test example: movement related effects Design matrix contrast(s) Design matrix contrast(s)

F-test: summary. 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. F tests a weighted sum of squares of one or several combinations of the regression coefficients. In practice, we dont have to explicitly separate X into [X 1 X 2 ] thanks to multidimensional contrasts. Hypotheses:

Orthogonal regressors Variability in Y

Correlated regressors Shared variance Variability in Y

Correlated regressors Variability in Y

Correlated regressors Variability in Y

Correlated regressors Variability in Y

Correlated regressors Variability in Y

Correlated regressors Variability in Y

Correlated regressors Variability in 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. 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: summary We implicitly test for an additional effect only. 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. Orthogonalisation = decorrelation. Parameters and test on the non modified regressor change. Rarely solves the problem as it requires assumptions about which regressor to uniquely attribute the common variance. change regressors (i.e. design) instead, e.g. factorial designs. use F-tests to assess overall significance. Original regressors may not matter: its the contrast you are testing which should be as decorrelated as possible from the rest of the design matrix x1x1 x2x2 x1x1 x2x2 x1x1 x2x2 x x x = x 2 – x 1.x 2 x 1

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).

Design efficiency AB A+B A-B High correlation between regressors leads to low sensitivity to each regressor alone. We can still estimate efficiently the difference between them.

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

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)

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

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).

Example: working memory B: Jittering time between stimuli and response. Stimulus Response Stimulus Response Stimulus Response ABC Time (s) Correlation = -.65 Efficiency ([1 0]) = 29 Correlation = +.33 Efficiency ([1 0]) = 40 Correlation = -.24 Efficiency ([1 0]) = 47 C: Requiring a response on a randomly half of trials.