Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Parametric and non-parametric multivariate test statistics for high-dimensional fMRI data Daniela Adolf, Johannes Bernarding, Siegfried Kropf Department for Biometry and Medical Informatics Otto-von-Guericke University Magdeburg, Germany COMPSTAT August 24, 2010 Multivariate test statistics for fMRI data Daniela Adolf 1 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Characteristics of fMRI data fMRI = functional magnetic resonance imaging → detection of activated voxels in the brain number of variables (voxels) exceeds the number of measurements extremely spatial dependence temporal dependence in each voxel Hollmann et al. (2010) (assuming a first-order autoregressive model) first-level analyses mostly done by using a univariate general linear model for each voxel including an adjustment for temporal correlation Yielding higher power via multivariate statistics in fMRI data?! Multivariate test statistics for fMRI data Daniela Adolf 2 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary General linear model signal is measured as time series in n time points over p voxels ( n < p ) → presentable in a GLM Y = X B + E N( 0 , I ⊗ Σ ) E ∼ y 11 y 1 p x 11 x 1 s β 11 β 1 p ǫ 11 ǫ 1 p · · · · · · · · · · · · . . . . . . . . ... ... ... ... . . . . . . . . = + . . . . . . . . y n 1 · · · y np x n 1 · · · x ns β s 1 · · · β sp ǫ n 1 · · · ǫ np hypothesis: H 0 : C ′ B = 0 ⇒ multivariate analysis is possible by means of so-called stabilized multivariate test statistics (L¨ auter et al., 1996 and 1998) Multivariate test statistics for fMRI data Daniela Adolf 3 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary General linear model signal is measured as time series in n time points over p voxels ( n < p ) → presentable in a GLM Y = X B + E N( 0 , I ⊗ Σ ) E ∼ y 11 y 1 p x 11 x 1 s β 11 β 1 p ǫ 11 ǫ 1 p · · · · · · · · · · · · . . . . . . . . ... ... ... ... . . . . . . . . = + . . . . . . . . y n 1 · · · y np x n 1 · · · x ns β s 1 · · · β sp ǫ n 1 · · · ǫ np hypothesis: H 0 : C ′ B = 0 ⇒ multivariate analysis is possible by means of so-called stabilized multivariate test statistics (L¨ auter et al., 1996 and 1998) Multivariate test statistics for fMRI data Daniela Adolf 3 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary General linear model signal is measured as time series in n time points over p voxels ( n < p ) → presentable in a GLM Y = X B + E N( 0 , V ⊗ Σ ) E ∼ y 11 y 1 p x 11 x 1 s β 11 β 1 p ǫ 11 ǫ 1 p · · · · · · · · · · · · . . . . . . . . ... ... ... ... . . . . . . . . = + . . . . . . . . y n 1 · · · y np x n 1 · · · x ns β s 1 · · · β sp ǫ n 1 · · · ǫ np hypothesis: H 0 : C ′ B = 0 ⇒ multivariate analysis is possible by means of so-called stabilized multivariate test statistics (L¨ auter et al., 1996 and 1998) but adjustment for temporal correlation necessary: aim of our research Multivariate test statistics for fMRI data Daniela Adolf 3 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Standardized Sum and Principal Component Test creating q (1 ≤ q < min ( p , n − s )) summary variables (scores) by means of a p × q -dimensional weight matrix D , that is any function of the total sums of squares and cross products matrix W ( W = SSQ hypothesis + SSQ residuals ) Z ( n × q ) = Y ( n × p ) D ( p × q ) ⇒ using these low-dimensional scores in classical analyses then Standardized Sum Test: d = Diag ( W ) − 1 2 1 p Principal Component Test: D : computed by means of the eigenvalue problem of W scale dependent: WD = DΛ , D ′ D = I q scale invariant: WD = Diag ( W ) DΛ , D ′ Diag ( W ) D = I q Multivariate test statistics for fMRI data Daniela Adolf 4 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Standardized Sum and Principal Component Test creating q (1 ≤ q < min ( p , n − s )) summary variables (scores) by means of a p × q -dimensional weight matrix D , that is any function of the total sums of squares and cross products matrix W ( W = SSQ hypothesis + SSQ residuals ) Z ( n × q ) = Y ( n × p ) D ( p × q ) ⇒ using these low-dimensional scores in classical analyses then Standardized Sum Test: d = Diag ( W ) − 1 2 1 p Principal Component Test: D : computed by means of the eigenvalue problem of W scale dependent: WD = DΛ , D ′ D = I q scale invariant: WD = Diag ( W ) DΛ , D ′ Diag ( W ) D = I q Multivariate test statistics for fMRI data Daniela Adolf 4 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Parametric adjustment for temporal correlation Satterthwaite approximation temporal correlation is taken into account within the test statistic → adjusting the variance estimation and the degrees of freedom Prewhitening Y = X B + E , E N( 0 , V ⊗ Σ ) ∼ → classical model: Y ⋆ = X ⋆ B + E ⋆ , E ⋆ ∼ N( 0 , I n ⊗ Σ ) via Y ⋆ = V − 1 X ⋆ = V − 1 E ⋆ = V − 1 2 Y , 2 X , 2 E → yields an exact test when V is known ⇒ problem : estimation of the correlation coefficient – assuming AR(1) Multivariate test statistics for fMRI data Daniela Adolf 5 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Parametric adjustment for temporal correlation Satterthwaite approximation temporal correlation is taken into account within the test statistic → adjusting the variance estimation and the degrees of freedom Prewhitening Y = X B + E , E N( 0 , V ⊗ Σ ) ∼ → classical model: Y ⋆ = X ⋆ B + E ⋆ , E ⋆ ∼ N( 0 , I n ⊗ Σ ) via Y ⋆ = V − 1 X ⋆ = V − 1 E ⋆ = V − 1 2 Y , 2 X , 2 E → yields an exact test when V is known ⇒ problem : estimation of the correlation coefficient – assuming AR(1) Multivariate test statistics for fMRI data Daniela Adolf 5 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Non-parametric adjustment for temporal correlation classical permutation original permutations 1. 2. 3. 4. 5. 6. ... Multivariate test statistics for fMRI data Daniela Adolf 6 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Non-parametric adjustment for temporal correlation blockwise permutation of blockwise permutation adjacent elements to account for original permutations temporal correlation 1. 2. 3. 4. 5. 6. ... 1 2 3 1 1 1 4 2 1 2 3 3 4 2 3 3 1 2 4 3 3 4 4 4 4 1 2 2 Multivariate test statistics for fMRI data Daniela Adolf 6 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Non-parametric adjustment for temporal correlation blockwise permutation of blockwise permutation including a random shift adjacent elements to account for original random shift and permutation temporal correlation 1. 2. 3. ... = = = 1 3 10 a a a including a random shift in order 2 3 2 to increase the number of possible blockwise permutations 1 2 4 3 1 3 4 4 1 Multivariate test statistics for fMRI data Daniela Adolf 6 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Non-parametric adjustment for temporal correlation blockwise permutation of blockwise permutation including a random shift adjacent elements to account for original random shift and permutation temporal correlation 1. 2. 3. ... = = = 1 3 10 a a a including a random shift in order 2 3 2 to increase the number of possible blockwise permutations → in each permutation step: 1 2 4 random removal of a (0 ≤ a < n ) elements on top, adding them at 3 1 the end 3 block arrangement and permutation 4 4 calculation of the permuted test 1 statistic Multivariate test statistics for fMRI data Daniela Adolf 6 of 11
Background Stabilized multivariate tests Dealing with correlated sample elements Comparison Summary Simulation studies for multivariate adjustments . . . to control the empirical type I error prewhitening holds the nominal test level (for at least a few hundreds of measurements, which is a common sample size in fMRI studies) Satterthwaite approximation partly exceeds the test level blockwise permutation including a random shift holds the nominal test level (for a block length of at least 40 even when there are just two blocks left) Multivariate test statistics for fMRI data Daniela Adolf 7 of 11
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