Download Building new Statistical Models for fMRI data.

HST 583
fMRI DATA ANALYSIS AND ACQUISITION
A Review of Statistics for
fMRI Data Analysis
Emery N. Brown
Massachusetts General Hospital
Harvard Medical School/MIT Division
of Health, Sciences and Technology
December 2, 2002
Outline
• What Makes Up an fMRI Signal?
• Statistical Modeling of an fMRI Signal
• Maxmimum Likelihoood Estimation for
fMRI
• Data Analysis
• Conclusions
THE STATISTICAL PARADIGM (Box, Tukey)
Question
Preliminary Data (Exploration Data Analysis)
Models
Experiment
(Confirmatory
Analysis)
Model Fit
Goodness-of-Fit
not satisfactory
Assessment
Satisfactory
Make an Inference
Make a Decision
Case 3: fMRI Data Analysis
Question: Can we construct an accurate statistical model
to describe the spatial temporal patterns of activation in fMRI
images from visual and motor cortices during combined motor
and visual tasks? (Purdon et al., 2001; Solo et al., 2001)
A STIMULUS-RESPONSE EXPERIMENT
Acknowledgements: Chris Long and Brenda Marshall
What Makes Up An fMRI Signal?
Hemodynamic Response/MR Physics
i) stimulus paradigm
a) event-related
b) block
ii) blood flow
iii) blood volume
iv) hemoglobin and deoxy hemoglobin content
Noise
Stochastic
i) physiologic
ii) scanner noise
Systematic
i) motion artifact
ii) drift
iii) [distortion]
iv) [registration], [susceptibility]
Physiologic Response
Model: Block Design
Gamma Hemodynamic Response Model
Physiologic Model:
Event-Related Design
Physiologic Model: Flow, Volume and Interaction
Terms
Volume Term
Flow Term
1
1
0.5
0.5
0
0
0
20
40
60
80
100
120
0
20
40
fa=1
Interaction Term
60
80
100
120
fb=-0.5
1
Modeled BOLD Signal
fc=0.2
0.5
0.6
0.4
0
0
20
40
60
80
100
120
0.2
0
-0.2
0
20
40
60
80
100
120
Scanner and Physiologic Noise Models
DATA:
y1 , …, yT
The sequence of image intensity measurements on a single
pixel.
fMRI Signal and Noise Model
yt =  h( t ) + v t
Measurement on a single pixel at time t
yt
Physiologic response
h( t )
Activation coefficient

Physiologic and Scanner Noise
vt for
t = 1,…, T
We assume the vt are independent, identically distributed
Gaussian random variables.
fMRI Signal Model
Physiologic Response
h( t ) =  g( u)c( t - u)du
g( t ) hemodynamic response

c( t ) input stimulus
Gamma model of the hemodynamic response
g ( t ) = t  - 1 e - t
Assume we know the parameters of g(t).
MAXIMUM LIKELIHOOD
Define the likelihood function L( | y ) = f ( y |  ) , the joint
probability density viewed as a function of the parameter 
with the data y fixed. The maximum likelihood estimate
of  is ˆML
ˆML ( y ) = arg max L( | y ) = arg max logL( | y ).


That is, ˆML ( y ) is a parameter value for which L( | y )
attains a maximum as a function of  for fixed y.
ESTIMATION
Joint Distribution
T
 1  2
 1 T ( yt -  ht )2 
f ( y | ) = 
exp -  t =1

2
 2 2 
2

 




Log Likelihood
logf ( y |  ) =
T
1
log( 2 2 ) - Tt = 1 ( yt -  ht ) 2 /  2
2
2
 = (  2 )
Maximum Likelihood

ˆ = Tt = 1 ht2

-1
Tt =1 ht yt
ˆ ε2 = T -1 Tt = 1 ( yt - ˆ ht )2
GOODNESS-OF-FIT/MODEL SELECTION
An essential step, if not the most essential step in a data analysis,
is to measures how well the model describes the data. This
should be assessed before the model is used to make inferences
about that data.
Akaike’s Information Criterion
-2logf (y | ˆ
) + 2p
ML
For maximum likelihood estimates it measures the trade-off
between maximizing the likelihood (minimizing -2logf (y | ˆML ) )
and the numbers of parameters p, the model requires.
GOODNESS-OF-FIT
• Residual Plots:
ˆt = yt - ht ˆ
• KS Plots:
ˆt  Ν (0, 2 )
We can check the Gaussian assumption with our K-S plots.
Measure correlation in the residuals to assess independence.
EVALUATION OF ESTIMATORS
Given w( y ), an estimator of based on y = ( y1 , …, yn )
Mean-Squared Error: E[ w ( y ) -  ]2 = Variance + bias 2
Bias= E[w( y)]-  ;unbiasedness E[w( y )] = 
Consistency: w( y )   as n   (sample size)
Efficiency: Achieves a minimum variance (Cramer-Rao
Lower Bound)
FACTOIDS ABOUT MAXIMUM LIKELIHOOD ESTIMATES
•Generally biased.
•Consistent, hence asymptotically unbiased.
•Asymptotically efficient.
•Variance can be approximated by minus the inverse of the Fisher
information matrix.
•If ˆ is the ML estimate of  , then
estimate of h( ).
h(ˆ )
is the ML
Cramer-Rao Lower Bound
2
 dE[ w ( y )] 


d


Var[ w ( y )] 
[  logf ( y |  ]
-E


CRLB gives the lowest bound on the variance of an estimate.
CONFIDENCE INTERVALS
The approximate probability density of the maximum
likelihood estimates is the Gaussian probability density with
mean  and variance - I ( )-1 where I ( ) is the Fisher
information matrix
  2 logf (y |  ) 
I ( ) = - E 

2



An approximate confidence interval for a component of 
is
ˆi,ML ± z |z  I ( )-ii1 


1
2
THE INFORMATION MATRIX
( 2 )-1 T ht2

0
t =1
I ( ) = - 

 0
( 2 )-2 T 2 -1 
CONFIDENCE INTERVAL
ˆ


1
2 - 2
t
± 2ˆ  T =1 hT
2 -12
ˆ  ± 2 2ˆ  T
2
Kolmogorov-Smirnov Test White Noise Model
White Noise Model
Pixelwise Confidence Intervals for the Slice
  2 

  2 
fMRI Signal and Noise Model 2
yt =  h( t ) + v t
Measurement on a single pixel at time t
yt
Physiologic response
h( t )
Activation coefficient

Physiologic and Scanner Noise
vt   vt -1  t
for
t = 1,…, T
We assume the vt are correlated noise AR(1)
Gaussian random variables.
Simple Convolution
Plus Correlated Noise
Kolmogorov-Smirnov Test Correlated Noise Model
Correlated Noise Model
Pixelwise Confidence Intervals for
the Slice
  2 

  2 
AIC Difference = AIC Colored Noise-AIC White Noise
fMRI Signal and Noise Model 3
yt = s( t ) + v t
Measurement on a single pixel at time t
yt
Physiologic response
q
s( t ) =  Ar cos( rt )  Br sin( rt )
r 1
Physiologic and Scanner Noise
vt for
t = 1,…, T
We assume the vt are independent, identically distributed
Gaussian random variables.
Harmonic Regression Plus White Noise Model
AIC Difference Map= AIC Correlated Noise-AIC Harmonic
Regression
Conclusions
• The white noise model gives a good description
of the hemodynamic response
• The correlated noise model incorporates known
physiologic and biophysical properties and
hence yields a better fit
• The likelihood approach offers a unified way to
formulate a model, compute confidence
intervals, measure goodness of fit and most
importantly make inferences.