Some problems at the subject level become even more pressing at the group level:

• Independent components have no natural interpretation
• Independent components have no meaningful order (permutation ambiguity)
• The magnitude of independent component maps and time courses is undetermined (scale ambiguity)
• The sign of independent component maps and time courses is arbitrary

For group level analyses, some form of ‘matching’ is required:

• Equal coefficients across subjects?
• Equal distributions across subjects?
• Equal dynamics across subjects?

Principle

• Perform ICA on individual subjects and match similar individual components one-on-one across subjects
• Subjective or (semi)automatic matching on basis of similarity between distribution maps and/or time courses
• Self-organized clustering, spatial correlation of components + hierarchical clustering
• E.g. Calhoun et al., 2001, Esposito et al., 2005
• Also used to test the reproducibility of some stochastic ICA algorithms

Components may differ in strength and dynamics

• Can be applied to inhomogeneous population
• Unequal spatial patterns: brain plasticity
• Unequal dynamics: flexible paradigm

Statistical assessment at group level:

• Voxel-by-voxel SPMs (careful with scaling and bias!)

Principle

• Concatenate subjects’ data matrices along the time dimension
• Perform ICA on aggregate data matrix
• Partition resulting components into individual time courses
• E.g. Calhoun et al., 2001

Components may differ in strength and dynamics

• Can be applied to inhomogeneous population
• Equal spatial patterns: comparable brain organization
• Unequal temporal dynamics: flexible paradigm such as resting state fMRI

Statistical assessment at group level:

• Voxel-by-voxel SPMs, from back-projection (careful with statistics!) or dual-regression.

Principle

• Concatenate subjects’ data matrices along the spatial dimension
• Perform ICA on aggregate data matrix
• Partition resulting components into individual maps
• E.g. Svensén et al., 2002

Components may differ in strength and distribution

• Can be applied to inhomogeneous population
• Unequal spatal patterns: brain plasticity
• Equal temporal dynamics: fixed paradigm;

Statistical assessment at group level:

-Voxel-by-voxel SPMs

Principle

- Average the data sets of all subjects before ICA
- Perform ICA on the mean data
- E.g. Schmithorst et al., 2004
 

All subjects are assumed to have identical components

• Only applied to homogeneous population
• Equal spatial patterns: comparable brain organization
• Equal temporal dynamics: fixed paradigm; resting state impossible

Statistical assessment at group level:

• Enter ICA time courses into linear regression model (back-projection)

Principle

• Stack subjects’ data matrices along a third dimension
• Decompose data tensor into product of (acquisition-dependent) time course, (voxel-dependent) maps, and (subject-dependent) loadings
• E.g.: Beckmann et al., 2005

Components may differ only in strength

• Can be applied to inhomogeneous population
• Equal spatial distributions: comparable brain organization
• Equal temporal dynamics: fixed paradigm; resting state impossible

Statistical assessment at group level:

• Components as a whole

Partition

• Particion individual ICs into the group components \(S\) and subject-specific components \(\epsilon_i\): \(S_i = [S; \epsilon_i]\)

• the results can be very unstable since ICA performance really depends on the specification of the number of ICs.

• By developing regularization methods, we can identify which components do not appear in each individual (or using post-hoc thresholding..)

The group ICs can be expressed differently in subjects:

• \(S_i = S + \epsilon_i\)

• back-reconstruction & dual regression

• More integrated hierarchical modeling (Ying Guo in Emory university) approaches have been developed.

Longitudinal ICA

• Hierarchical modeling for ICA

Multimodal Fusion

• Joint ICA, mICA, Linked ICA etc.