Functional Brain Signal Processing: EEG & fMRI

Lesson 15

Kaushik Majumdar

Indian Statistical Institute Bangalore Center

kmajumdar@isibang.ac.in

M.Tech. (CS), Semester III, Course B50

Group Statistics: An Example

Poldrack et al., 2011, Section 6.1

Fixed Effect vs. Mixed Effect Model

2 210.25

4FFX w is fixed effect variance in average hair length in a gender group. is within group variance, which is assumed to be 1 here.

FFXw

2 22 1 49

12.54 4 4 4w B

MFX

is the average mixed effect variance, where is the between the groups variance.

MFXB

Mixed Effect Model in fMRI

Poldrack et al., 2011

Here β = a.

Linear Regression Analysis

http://en.wikipedia.org/wiki/File:Linear_regression.svg

Multilevel GLM for Group Analysis

k k k k Y M a e Yk is a vector of T time points. k є {1,…..,N},

where N is the number of subjects in the group.Mk is the design matrix of model functions andek is the error vector (for the kth subject with one

element in the vector for each time point. ( ) 0,cov( )k k kE V e e

1 1 1 1

2 2

0 0 . . .

0 0 . . . . .

. 0 0 . . . . . ., , ,

. . . . . . . . .

. . . . . . . . .

0 0 . . .N N N N

Y M a e

Y M

Y M a e

Y M a e

Two Level Model

Level 1: individual analysis

Level 2: group analysis

Y Ma e

G G a M a dMG is the group level design matrix, aG is a vector of group level parameters d is the residual vector of group level parameters.

( ) 0

cov( )

cov( )G

E

d

d V

e V

Beckmann et al., 2003

(1)

Two Level Model as A Single Level Model

, ( ) 0, cov( )

G G

TGE

Y MM a f

f Md e f f W MV M V

This is equivalent to the two level model described in the previous slide.

Parameter Estimation at Two Levels

Linear spaces generated by the columns of M and MTM are the same (Rao, 1974, p. 222).

Proof: Let λ be an eigenvalue of M. Then there is an eigenvector v such that Mv = λv or M = λI MT = λI. So, MTMv = MTλv = λ2v. In other words M and MTM have same eigenvectors and therefore generate the same eigen space.

Parameter Estimation (cont)

In general Y = Ma may be inconsistent (may not have unique solution), but MTMa = MTY always has a unique solution in a, because MTY is in the space generated by columns of MTM. Let â be a solution of MTMa = MTY, then (Y – Ma)T(Y – Ma) = [Y – Mâ + M(â – a)]T[Y – Mâ + M(â – a)] = (Y – Mâ)T(Y – Mâ) + (â – a)TMTM(â – a) ≥ (Y – Mâ)T(Y – Mâ).

This shows that the minimum of (Y – Ma)T(Y – Ma) is (Y – Mâ)T(Y – Mâ) and is attained for

Parameter Estimation (cont)

a = â, which is unique for all solutions â of MTMa = MTY.

Solution for Two Level GL Model

1 1 1

2 1 1

ˆ ( )

ˆ ˆ ( )

T T

T T

a M V M M V Y

aa M V M For individual.

1 1 1

2 1 1

ˆ ˆ( )

ˆ ˆ ( )

T TG G G G G G

T TG G G G G G

a M V M M M a

a a M V M For group.

1ˆ ˆ

G G a M a f This together with (1) gives the second level estimation of the parameters.

Inference of BOLD Activation

Poldrack et al., 2011

Nature of BOLD Signal

Buxton, 2009

CBF = Cerebral blood flow.CMRO2 = Cerebral metabolic rate of O2.CBV = Cerebral blood volume.

BOLD Components

BOLD response is primarily driven by CBF, but also strongly modulated by two other factors:

, and

M, which reflects level of deoxyhemoglobin at the baseline.

2

Fractional CBF changen

Fractional CMRO change

BOLD is Best Captured in Gradient Recall Echo (GRE) Imaging

Buxton, 2009

References

R. A. Poldrack, J. A. Mumford and T. E. Nichols, Handbook of Functional MRI Data Analysis, Cambridge University Press, Cambridge, New York, 2011. Chapter 6.

C. F. Beckmann, M. Jenkinson and S. M. Smith, General multilevel linear modeling for group analysis in fMRI, NeuroImage, 20: 1052 – 1063, 2003.

References (cont)

C. R. Rao, Linear Statistical Inference and Its Applications, 2e, Wiley Eastern Ltd., New Delhi, 1974, Chapter 4 (Theory of least squares and analysis of variance).

THANK YOU

This lecture is available at http://www.isibang.ac.in/~kaushik