Transcript Lesson 15

M.Tech. (CS), Semester III, Course B50
Functional Brain Signal
Processing: EEG & fMRI
Lesson 15
Kaushik Majumdar
Indian Statistical Institute
Bangalore Center
[email protected]
Poldrack et al., 2011, Section 6.1
Group Statistics: An Example
Fixed Effect vs. Mixed Effect
Model


2
FFX
2
MFX
1 2
  w  0.25
4
 w2
 FFX is fixed effect variance in average hair
length in a gender group. w is within group
variance, which is assumed to be 1 here.
 B2
1 49


 
 12.5  MFX is the average mixed effect
variance, where B is the between
4
4 4 4
the groups variance.
Poldrack et al., 2011
Mixed Effect Model in fMRI
Here β = a.
http://en.wikipedia.org/wiki/File:Linear_regression.svg
Linear Regression Analysis
Multilevel GLM for Group Analysis
Yk  Mk ak  ek
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 and
ek is the error vector (for the kth subject with one
k element in the vector for each time point.
E (ek )  0,cov(ek )  V
 Y1 
M1 0
Y 
 0 M
2
 2

 . 
 0
0
Y   , M  
.
 . 
 .
 . 
 .
.
 

0
 YN 
 0
0 . .
0 . .
. . .
. . .
. . .
. . .
. 
 a1 
 e1 
 . 
 . 
. 
 
 
 . 
 . 
. 
, a   , e   
. 
 . 
 . 
 . 
 . 
. 

 
 
M N 
a N 
e N 
Beckmann et al., 2003
Two Level Model

Level 1: individual analysis
Y  Ma  e

Level 2: group analysis
a  M G aG  d
(1)
E (d )  0
cov(d )  VG
cov(e)  V
MG is the group level design matrix, aG is a vector of group level parameters
d is the residual vector of group level parameters.
Two Level Model as A Single Level
Model
Y  MMG aG  f
f  Md  e, E (f )  0, cov(f )  W  MVG MT  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
aˆ  (MT V 1M ) 1 MT V 1Y
ˆ ˆ T   2 (MT V 1M ) 1
aa
For individual.
aˆ  (MTG VG1M G )1 MTG M G1aˆ G
aˆ G aˆ TG   G2 (MTG VG1M G )1
aˆ  MGaˆ G  f1
For group.
This together with (1) gives the second level
estimation of the parameters.
Poldrack et al., 2011
Inference of BOLD Activation
Buxton, 2009
Nature of BOLD Signal
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:
Fractional CBF change
n
, and
Fractional CMRO2 change
M, which reflects level of deoxyhemoglobin
at the baseline.
Buxton, 2009
BOLD is Best Captured in Gradient
Recall Echo (GRE) Imaging
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