Detecting multivariate effective connectivity Keith Worsley, Department of Mathematics and Statistics, McGill University, Montreal Jason Lerch, Montreal Neurological Institute Jonathan Taylor, Department of Statistics, Stanford Francesco Tomaiuolo, IRCCS Fondazione ‘Santa Lucia’, Rome

Examples n 1 =17 subjects with non-missile brain trauma (coma days), and n 2 =19 age and sex matched controls –Y = WM density and vector deformations –Covariates = group –Find regions of damage Between trauma group and control group Between a single trauma case and control group (clinical) n=321 subjects aged years –Y = cortical thickness –Covariates = age, gender –Find cortical thickness differences between males and females All data smoothed 10mm

How do we measure anatomy? Structure density: –Segment image GM/WM/CSF or hippocampus/thalamus/amygdala … –Smooth to produce structure density Deformations: –Find non-linear warps needed to warp structure to atlas (data is 3D deformation vectors) Cortical thickness: –Find inner and outer cortical surface –Find cortical thickness

Atlas

How do we model anatomy? Y = structure density or structure thickness: –linear model: Y = covariate × coef + … + error –T = coef / sd Y 1×3 = vector deformations (x,y,z components): –multivariate linear model: Y 1×3 = covariate × coef 1×3 + … + error 1×3 –Take a linear combination a 3×1 of components to give a univariate linear model: Y = Y 1×3 × a 3×1 = covariate × coef + … + error –Hotelling’s T 2 = max a T 2 = coef 1×3 × var 3×3 -1 × coef 3×1 t

Which method is better? Assess methods / measures by the SD of the difference between cases and controls: –Group use: n 1 =100 cases and n 2 =100 controls sd decreases as sqrt(n) –Clinical use: n 1 =1 case and n 2 =100 controls sd not much affected by n ~ 6 times this sd is 95% detectable (at P=0.05, searching over the whole brain).

Sd for group comparison (n 1 =n 2 =100) WM Density Deformations GM density, %mm ~1.5 × 6 = 9% density difference can be detected ~0.15 × 6 = 0.9 mm deformation difference can be detected For clinical use, multiply everything by

Sd for group comparison (n 1 =n 2 =100) Cortical thickness mm ~0.1 × 6 = 0.6 mm thickness difference can be detected, slightly better than deformations

“Anatomical connectivity” Measured by the correlation between residuals at a pair of voxels Choose one voxel as reference, correlate its values with those at every other voxel –Y ~ refvoxval × coef + error Correlation is equivalent to usual T statistic –for univariate data e.g. WM, cortical thickness … Voxel 2 Voxel Activation only Voxel 2 Voxel Correlation only

“Deformation vector connectivity” Something new for multivariate data, such as vector deformations: There are now three reference voxel values (x,y,z components) –Y 1×3 = refvoxval 1×3 × coef 3×3 + error 1×3 There are several choices of test statistic: –Wilk’s Lambda (likelihood ratio), Pillai trace, Lawley-Hotelling trace, –but the most convenient is Roy’s maximum root Again take a linear combination a 3×1 of components to give a univariate linear model: –Y = Y 1×3 × a 3×1 = refvoxval 1×3 × coef 3×1 + error Roy’s maximum root R = max a F statistic –R = maximum eigenvalue of coef 3×3 × var 3×3 -1 × coef 3×3 t –Equivalent to maximum canonical correlation –P-value random field theory is now available in FMRISTAT

6D connectivity Measured by the correlation between residuals at every pair of voxels (6D data!) Local maxima are larger than all 12 neighbours P-value random field theory now available, even for multivariate data (using maximum canonical correlation) Good at detecting focal connectivity, but PCA of subjects × voxels is better at detecting large regions of co-correlated voxels