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• n1=17 subjects with non-missile brain trauma (coma 3-14

days), and n2=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 20-70 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

Deformations

0

0.2

0.4

0.6

0.8

1Segmented

Structure

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0.2

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0.6

0.8

1Density

Atlas

How do we model anatomy?• Y = structure density or structure thickness:

– linear model:• Y = covariate × coef + … + error

– T = coef / sd

• Y1×3 = vector deformations (x,y,z components):– multivariate linear model:

• Y1×3 = covariate × coef1×3 + … + error1×3

– Take a linear combination a3×1 of components to give a univariate linear model:

• Y = Y1×3 × a3×1 = covariate × coef + … + error

– Hotelling’s T2 = maxa T2 = coef1×3 × var3×3

-1 × coef3×1t

Which method is better?

• Assess methods / measures by the SD of the difference between cases and controls:– Group use: n1=100 cases and n2=100 controls

• sd decreases as sqrt(n)

– Clinical use: n1=1 case and n2=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 (n1=n2=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

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0.5

1

1.5

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110

70

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140

100

60

For clinical use, multiply everything by 7

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0.05

0.1

0.15

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0.25

56

36

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61

41

21

66

46

26

71

51

31 0.3

Sd for group comparison (n1=n2=100) Cortical thickness

mm

~0.1 × 6 = 0.6 mm thickness difference can be detected, slightly better than deformations

0

0.03

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0.15

0.12

“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 1

++ +++ +

Activation onlyVoxel 2

Voxel 1++

+

+

+

+

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)

– Y1×3= refvoxval1×3 × coef3×3 + error1×3

• 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 a3×1 of components to give a univariate linear model:– Y = Y1×3 × a3×1 = refvoxval1×3 × coef3×1 + error

• Roy’s maximum root R = maxa F statistic– R = maximum eigenvalue of coef3×3

× var3×3-1 × coef3×3

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