1. 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

2. Outline
What Makes Up an fMRI Signal?
Statistical Modeling of an fMRI Signal
Maxmimum Likelihoood Estimation for fMRI
Data Analysis
Conclusions

3. 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

4. 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

6. 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]

7. Physiologic Response Model: Block Design

8. Gamma Hemodynamic Response Model

9. Physiologic Model:
Event-Related Design

10. Physiologic Model: Flow, Volume and Interaction
Terms

11. Scanner and Physiologic Noise Models

12. DATA:
The sequence of image intensity measurements on a single
pixel.

13. fMRI Signal and Noise Model
Measurement on a single pixel at time
Physiologic response
Activation coefficient
Physiologic and Scanner Noise
for
We assume the
are independent, identically distributed
Gaussian random variables.

14. fMRI Signal Model
Physiologic Response
hemodynamic response
input stimulus
Gamma model of the hemodynamic response
Assume we know the parameters of g(t).

15. MAXIMUM LIKELIHOOD
Define the likelihood function
, the joint
probability density viewed as a function of the parameter
with the data
fixed. The maximum likelihood estimate of is
That is,
is a parameter value for which
attains a maximum as a function of
for fixed

16. ESTIMATION
Joint Distribution
Log Likelihood
Maximum Likelihood

17. 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
For maximum likelihood estimates it measures the trade-off
between maximizing the likelihood (minimizing )
and the numbers of parameters
the model requires.

18. GOODNESS-OF-FIT
Residual Plots:
KS Plots:
We can check the Gaussian assumption with our K-S plots.
Measure correlation in the residuals to assess independence.

19. EVALUATION OF ESTIMATORS
Given
an estimator of
based on
Mean-Squared Error:
Bias=
Consistency:
Efficiency: Achieves a minimum variance (Cramer-Rao
Lower Bound)

20. 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
is the
estimate of
then
is the
estimate of

21. Cramer-Rao Lower Bound
CRLB gives the lowest bound on the variance of an estimate.

22. CONFIDENCE INTERVALS
The approximate probability density of the maximum
likelihood estimates is the Gaussian probability density with
mean
and variance
where
is the Fisher
information matrix
An approximate confidence interval for a component of is

23. THE INFORMATION MATRIX

24. CONFIDENCE INTERVAL

26. Kolmogorov-Smirnov Test White Noise Model

27. Pixelwise Confidence Intervals for the Slice
White Noise Model
Pixelwise Confidence Intervals for the Slice

28. fMRI Signal and Noise Model 2
Measurement on a single pixel at time
Physiologic response
Activation coefficient
Physiologic and Scanner Noise
for
We assume the
are correlated noise AR(1)
Gaussian random variables.

29. Simple Convolution Plus Correlated Noise

30. Kolmogorov-Smirnov Test Correlated Noise Model

31. Pixelwise Confidence Intervals for the Slice
Correlated Noise Model
Pixelwise Confidence Intervals for the Slice

32. AIC Difference = AIC Colored Noise-AIC White Noise

33. fMRI Signal and Noise Model 3
Measurement on a single pixel at time
Physiologic response
Physiologic and Scanner Noise
for
We assume the
are independent, identically distributed
Gaussian random variables.

34. Harmonic Regression Plus White Noise Model

35. AIC Difference Map= AIC Correlated Noise-AIC Harmonic
Regression

36. 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.