Data Modeling General Linear Model & Statistical Inference

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Data Modeling General Linear Model & Statistical Inference. 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. Motivations. Data Modeling Characterize Signal

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Data ModelingGeneral Linear Model &Statistical Inference

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

Motivations
• Data Modeling
• Characterize Signal
• Characterize Noise
• Statistical Inference
• Detect signal
• Localization (Where’s the blob?)
Outline
• Data Modeling
• General Linear Model
• Linear Model Predictors
• Temporal Autocorrelation
• Random Effects Models
• Statistical Inference
• Statistic Images & Hypothesis Testing
• Multiple Testing Problem
Basic fMRI Example
• Data at one voxel
• Rest vs.passive word listening
• Is there an effect?
A Linear Model
• “Linear” in parameters 1&2

error

=

+

+

b1

b2

Time

e

x1

x2

Intensity

=

+

Y

… in matrix form.

N: Number of scans, p: Number of regressors

Linear Model Predictors
• Signal Predictors
• Block designs
• Event-related responses
• Nuisance Predictors
• Drift
• Regression parameters
Signal Predictors
• Linear Time-Invariant system
• LTI specified solely by
• Stimulus function ofexperiment
• Hemodynamic ResponseFunction (HRF)
• Response to instantaneousimpulse

Blocks

Events

Block Design

Event-Related

Convolution Examples

Experimental Stimulus Function

Hemodynamic Response Function

Predicted Response

SPM’s HRFHRF Models
• 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
HRF Models
• 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

HRF Models
• Deconvolution
• Most flexible
• Allows any shape
• Even bizarre, non-sensical ones
• Least sensitive relativeto canonical (again, ifcanonical is correct)

Deconvolution Basis

Drift Models
• Drift
• Slowly varying
• Nuisance variability
• Models
• Discrete Cosine Transform

Discrete Cosine Transform Basis

General Linear ModelRecap
• Fits data Y as linear combination of predictor columns of X
• Very “General”
• Correlation, ANOVA, ANCOVA, …
• Only as good as your X matrix
Temporal Autocorrelation
• 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
Temporal AutocorrelationIn Brief
• Independence
• Precoloring
• Prewhitening
Autocorrelation: Independence Model
• Ignore autocorrelation
• Under-estimation of variance
• Over-estimation of significance
• Too many false positives
Autocorrelation:Precoloring
• 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]

Autocorrelation:Prewhitening
• Statistically optimal solution
• If know true autocorrelation exactly, canundo the dependence
• Then proceed as with independent data
• Problem is obtaining accurate estimates of autocorrelation
• Some sort of regularization is required
• Spatial smoothing of some sort
Autocorrelation: Models
• 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
• Arbitrary autocorrelation function
• k = corr( i, i-k )
• Software: FSL’s FEAT
Statistic Images &Hypothesis Testing
• 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
Building Statistic Images

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

Building Statistic Images
• Contrast
• A linear combination of parameters
• c’
Hypothesis Test
• So now have a value T for our statistic
• How big is big
• Is T=2 big? T=20?
P-val Hypothesis Testing
• 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

Random Effects Models
• 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 effectFixed vs.RandomEffects

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

Random Effects for fMRI
• 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.
• Full Mixed Effects Analysis
• Hard
• Requires iterative fitting
• REML to estimate inter- and intra subject variance
• SPM2 & FSL implement this, very differently
• Very flexible
Random Effects for fMRIRandom vs. Fixed
• 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

Multiple Testing Problem
• 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!
MCP Solutions:Measuring False Positives
• Familywise Error Rate (FWER)
• Familywise Error
• Existence of one or more false positives
• FWER is probability of familywise error
• False Discovery Rate (FDR)
• R voxels declared active, V falsely so
• Observed false discovery rate: V/R
• FDR = E(V/R)
FWER MCP Solutions
• Bonferroni
• Maximum Distribution Methods
• Random Field Theory
• Permutation
FWER MCP Solutions
• Bonferroni
• Maximum Distribution Methods
• Random Field Theory
• Permutation
FWER MCP Solutions: Controlling FWER w/ Max
• 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

FWER MCP Solutions:Random Field Theory
• 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)

Threshold

Random Field

Suprathreshold Sets

5%

Parametric Null Max Distribution

5%

Nonparametric Null Max Distribution

Controlling FWER: Permutation Test
• Parametric methods
• Assume distribution ofmax statistic under nullhypothesis
• Nonparametric methods
• Use data to find distribution of max statisticunder null hypothesis
• Any max statistic!
Measuring False Positives
• Familywise Error Rate (FWER)
• Familywise Error
• Existence of one or more false positives
• FWER is probability of familywise error
• False Discovery Rate (FDR)
• R voxels declared active, V falsely so
• Observed false discovery rate: V/R
• FDR = E(V/R)
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 qControlling FDR:Benjamini & Hochberg
• 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

Conclusions
• Analyzing fMRI Data
• Need linear regression basics
• Lots of disk space, and time
• Watch for MTP (no fishing!)
Thanks
• Slide help
• Stefan Keibel, Rik Henson, JB Poline, Andrew Holmes