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Data Modeling General Linear Model & Statistical Inference

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Thomas Nichols, Ph.D.

Assistant Professor

Department of Biostatistics

http://www.sph.umich.edu/~nichols

Brain Function and fMRI

ISMRM Educational Course

July 11, 2002

- Data Modeling
- Characterize Signal
- Characterize Noise

- Statistical Inference
- Detect signal
- Localization (Where’s the blob?)

- Data Modeling
- General Linear Model
- Linear Model Predictors
- Temporal Autocorrelation
- Random Effects Models

- Statistical Inference
- Statistic Images & Hypothesis Testing
- Multiple Testing Problem

- Data at one voxel
- Rest vs.passive word listening

- Is there an effect?

- “Linear” in parameters 1&2

error

=

+

+

b1

b2

Time

e

x1

x2

Intensity

=

+

+

Estimated

=

+

+

=

+

=

+

Y

N: Number of scans, p: Number of regressors

- Signal Predictors
- Block designs
- Event-related responses

- Nuisance Predictors
- Drift
- Regression parameters

- Linear Time-Invariant system
- LTI specified solely by
- Stimulus function ofexperiment
- Hemodynamic ResponseFunction (HRF)
- Response to instantaneousimpulse

Blocks

Events

Block Design

Event-Related

Experimental Stimulus Function

Hemodynamic Response Function

Predicted Response

SPM’s HRF

- Canonical HRF
- Most sensitive if it is correct
- If wrong, leads to bias and/or poor fit
- E.g. True responsemay be faster/slower
- E.g. True response may have smaller/bigger undershoot

- Smooth Basis HRFs
- More flexible
- Less interpretable
- No one parameter explains the response

- Less sensitive relativeto canonical (only if canonical is correct)

Gamma Basis

Fourier Basis

- Deconvolution
- Most flexible
- Allows any shape
- Even bizarre, non-sensical ones

- Least sensitive relativeto canonical (again, ifcanonical is correct)

- Most flexible

Deconvolution Basis

- Drift
- Slowly varying
- Nuisance variability

- Models
- Linear, quadratic
- Discrete Cosine Transform

Discrete Cosine Transform Basis

- Fits data Y as linear combination of predictor columns of X
- Very “General”
- Correlation, ANOVA, ANCOVA, …

- Only as good as your X matrix

- Standard statistical methods assume independent errors
- Error i tells you nothing about j i j

- fMRI errors not independent
- Autocorrelation due to
- Physiological effects
- Scanner instability

- Independence
- Precoloring
- Prewhitening

- Ignore autocorrelation
- Leads to
- Under-estimation of variance
- Over-estimation of significance
- Too many false positives

- Temporally blur, smooth your data
- This induces more dependence!
- But we exactly know the form of the dependence induced
- Assume that intrinsic autocorrelation is negligible relative to smoothing

- Then we know autocorrelation exactly
- Correct GLM inferences based on “known” autocorrelation

[Friston, et al., “To smooth or not to smooth…” NI 12:196-208 2000]

- Statistically optimal solution
- If know true autocorrelation exactly, canundo the dependence
- De-correlate your data, your model
- Then proceed as with independent data

- Problem is obtaining accurate estimates of autocorrelation
- Some sort of regularization is required
- Spatial smoothing of some sort

- Some sort of regularization is required

- Autoregressive
- Error is fraction of previous error plus “new” error
- AR(1): i = i-1 + I
- Software: fmristat, SPM99

- AR + White Noise or ARMA(1,1)
- AR plus an independent WN series
- Software: SPM2

- AR plus an independent WN series
- Arbitrary autocorrelation function
- k = corr( i, i-k )
- Software: FSL’s FEAT

- k = corr( i, i-k )

- For each voxel
- Fit GLM, estimate betas
- Write b for estimate of

- But usually not interested in all betas
- Recall is a length-p vector

- Fit GLM, estimate betas

Predictor of interest

b1

b2

b3

b4

b5

b6

b7

b8

b9

=

+

´

=

+

Y

X

b

e

c’ = 1 0 0 0 0 0 0 0

b1b2b3b4b5....

contrast ofestimatedparameters

c’b

T =

T =

varianceestimate

s2c’(X’X)+c

- Contrast
- A linear combination of parameters
- c’

- So now have a value T for our statistic
- How big is big
- Is T=2 big? T=20?

P-val

- Assume Null Hypothesis of no signal
- Given that there is nosignal, how likely is our measured T?
- P-value measures this
- Probability of obtaining Tas large or larger

- level
- Acceptable false positive rate

T

- GLM has only one source of randomness
- Residual error

- But people are another source of error
- Everyone activates somewhat differently…

Distribution of each subject’s effect

Subj. 1

Subj. 2

- Fixed Effects
- Intra-subject variation suggests all these subjects different from zero

- Random Effects
- Intersubject variation suggests population not very different from zero

Subj. 3

Subj. 4

Subj. 5

Subj. 6

0

- Summary Statistic Approach
- Easy
- Create contrast images for each subject
- Analyze contrast images with one-sample t

- Limited
- Only allows one scan per subject
- Assumes balanced designs and homogeneous meas. error.

- Easy
- Full Mixed Effects Analysis
- Hard
- Requires iterative fitting
- REML to estimate inter- and intra subject variance
- SPM2 & FSL implement this, very differently

- Very flexible

- Hard

- Fixed isn’t “wrong”, just usually isn’t of interest
- If it is sufficient to say“I can see this effect in this cohort”then fixed effects are OK
- If need to say“If I were to sample a new cohort from the population I would get the same result”then random effects are needed

t > 2.5

t > 4.5

t > 0.5

t > 1.5

t > 3.5

t > 5.5

t > 6.5

- Inference on statistic images
- Fit GLM at each voxel
- Create statistic images of effect

- Which of 100,000 voxels are significant?
- =0.05 5,000 false positives!

- Familywise Error Rate (FWER)
- Familywise Error
- Existence of one or more false positives

- FWER is probability of familywise error

- Familywise Error
- False Discovery Rate (FDR)
- R voxels declared active, V falsely so
- Observed false discovery rate: V/R

- FDR = E(V/R)

- R voxels declared active, V falsely so

- Bonferroni
- Maximum Distribution Methods
- Random Field Theory
- Permutation

- Bonferroni
- Maximum Distribution Methods
- Random Field Theory
- Permutation

- FWER & distribution of maximum
FWER= P(FWE)= P(One or more voxels u | Ho)= P(Max voxel u | Ho)

- 100(1-)%ile of max distn controls FWER
FWER = P(Max voxel u | Ho)

u

- Euler Characteristic u
- Topological Measure
- #blobs - #holes

- At high thresholds,just counts blobs
- FWER= P(Max voxel u | Ho)= P(One or more blobs | Ho) P(u 1 | Ho) E(u| Ho)

- Topological Measure

Threshold

Random Field

Suprathreshold Sets

5%

Parametric Null Max Distribution

5%

Nonparametric Null Max Distribution

- Parametric methods
- Assume distribution ofmax statistic under nullhypothesis

- Nonparametric methods
- Use data to find distribution of max statisticunder null hypothesis
- Any max statistic!

- Familywise Error Rate (FWER)
- Familywise Error
- Existence of one or more false positives

- FWER is probability of familywise error

- Familywise Error
- False Discovery Rate (FDR)
- R voxels declared active, V falsely so
- Observed false discovery rate: V/R

- FDR = E(V/R)

- R voxels declared active, V falsely so

Signal

Noise

Signal+Noise

11.3%

11.3%

12.5%

10.8%

11.5%

10.0%

10.7%

11.2%

10.2%

9.5%

6.7%

10.5%

12.2%

8.7%

10.4%

14.9%

9.3%

16.2%

13.8%

14.0%

Control of Per Comparison Rate at 10%

Percentage of Null Pixels that are False Positives

Control of Familywise Error Rate at 10%

FWE

Occurrence of Familywise Error

Control of False Discovery Rate at 10%

Percentage of Activated Pixels that are False Positives

p(i) i/V q

- Select desired limit q on E(FDR)
- Order p-values, p(1)p(2) ... p(V)
- Let r be largest i such that
- Reject all hypotheses corresponding top(1), ... , p(r).

1

p(i)

p-value

i/V q

0

0

1

i/V

- Analyzing fMRI Data
- Need linear regression basics
- Lots of disk space, and time
- Watch for MTP (no fishing!)

- Slide help
- Stefan Keibel, Rik Henson, JB Poline, Andrew Holmes