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 314 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
Structure
Deformations
Atlas
Segmented
1
Density
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
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
30
40
50
3
60
16
21
26
31
0.25
2.5
70
80
90
100
2
36
41
46
51
120
130
140
1
56
61
66
71
0.1
0.5
0.05
0
0
GM density, %
~1.5 × 6 = 9% density
difference can be detected
0.2
0.15
1.5
110
0.3
mm
~0.15 × 6 = 0.9 mm deformation
difference can be detected
For clinical use, multiply everything by 7
Sd for group comparison (n1=n2=100)
Cortical thickness
0.15
0.12
0.09
0.06
0.03
0
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
Activation only
Voxel 2
++
+ +++
Correlation only
Voxel 2
Voxel 1
+
+
++
+
Voxel 1
+
• 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 …
“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 × coef3×3t
– 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