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 SatisfactoryMake 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-relatedb) block

ii) blood flow iii) blood volume iv) hemoglobin and deoxy hemoglobin contentNoise Stochastic i) physiologic ii) scanner noiseSystematic i) motion artifact ii) drift iii) [distortion] iv) [registration], [susceptibility]

Physiologic Response Model: Block Design

Gamma Hemodynamic Response Model

Physiologic Model:

Event-Related Design

0 20 40 60 80 100 1200

0.5

1

Flow Term

0 20 40 60 80 100 1200

0.5

1

Volume Term

0 20 40 60 80 100 1200

0.5

1

Interaction Term

0 20 40 60 80 100 120-0.2

0

0.2

0.4

0.6

Modeled BOLD Signal

fa=1 fb=-0.5

fc=0.2

Physiologic Model: Flow, Volume and Interaction Terms

Scanner and Physiologic Noise Models

DATA:

, …,1 Ty y

The sequence of image intensity measurements on a singlepixel.

fMRI Signal and Noise Model

= ( ) + vt ty h t

Measurement on a single pixel at time

t

t y

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

-( ) = 1 - tg t t e

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

ˆ ( ) = arg max ( ) = arg max ( ).

ML y L | y logL | y

That is, ˆ ( )ML y is a parameter value for which ( )L | y

attains a maximum as a function of

for fixed y.

ESTIMATION

Joint DistributionT

2 2T t tt 12 2

y h1 1f y

22

=( - )

( | ) = exp -

Log Likelihood

=log ( | ) = log( ) - ( - )

= ( )

2 2 2Tt tt 1

2

T 1f y 2 y h /

2 2

Maximum Likelihood

ˆ

ˆˆ

-

= =

-ε =

=

= ( - )

12T Tt t tt 1 t 1

2 1 2Tt tt 1

h h y

T y h

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 inferencesabout that data. Akaike’s Information Criterion

ˆML2 f | 2p- log (y ) +

For maximum likelihood estimates it measures the trade-off between maximizing the likelihood (minimizing

ˆML2 f |- log (y )

and the numbers of parameters p, the model requires.)

GOODNESS-OF-FIT

• Residual Plots:

ˆˆ = -t t ty h

• KS Plots:

ˆ ( )2t Ν 0,

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 = ( , …, )1 ny y y

Mean-Squared Error: [ ( ) - ] = Variance + bias2 2E w y

Bias= [ ( )]- ;unbiasedness [ ( )] =E w y E w y

Consistency: ( ) as (sample size) w y n

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 ̂ is the ML estimate of , then ˆ( )h is the ML

estimate of

( ).h

Cramer-Rao Lower Bound

2dE w y

dw y

f y-E

[ ( )]

Var[ ( )][ log ( | ]

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 withmean and variance -- ( ) 1I where ( )I is the Fisherinformation matrix

log (y )( ) = -

2

2

f |I E

An approximate confidence interval for a component of is

ˆ -± ( )

121

i,ML |z iiz I

THE INFORMATION MATRIX

-=

- -

( )( ) = -

(

2 1 2Ttt 1

2 2 1

h 0I

0 ) T 2

CONFIDENCE INTERVAL

ˆ ˆ

ˆ ˆ

-

=

-

±

±

12

12

2tTT 1

2 2

2 h

2 2 T

Kolmogorov-Smirnov Test White Noise Model

2 2

White Noise ModelPixelwise Confidence Intervals for the Slice

fMRI Signal and Noise Model 2

= ( ) + vt ty h t

Measurement on a single pixel at time

t

t y

Physiologic response( )h t

Activation coefficient

Physiologic and Scanner Noise

tv v t t -1 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

2 2

Correlated Noise ModelPixelwise Confidence Intervals for

the Slice

AIC Difference = AIC Colored Noise-AIC White Noise

fMRI Signal and Noise Model 3

t ty t= s( ) + v

Measurement on a single pixel at time

t

t y

Physiologic response

1cos( ) sin( )

q

rs t rt B rt

r r( ) = A

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 HarmonicRegression

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.